diff --git a/_freeze/materials/glm-practical-logistic-proportion/execute-results/html.json b/_freeze/materials/glm-practical-logistic-proportion/execute-results/html.json index bda0306..e3b7c21 100644 --- a/_freeze/materials/glm-practical-logistic-proportion/execute-results/html.json +++ b/_freeze/materials/glm-practical-logistic-proportion/execute-results/html.json @@ -1,8 +1,8 @@ { - "hash": "a05f16cdde599e9aa77982adf6dd4c1c", + "hash": "54eb4ea078864673aad5ca1389ed9046", "result": { "engine": "knitr", - "markdown": "---\ntitle: \"Proportional response\"\n---\n\n::: {.cell}\n\n:::\n\n\n::: {.callout-tip}\n## Learning outcomes\n\n- How do I analyse proportion responses?\n- Be able to create a logistic model to test proportion response variables\n- Be able to plot the data and fitted curve\n- Assess the significance of the fit\n:::\n\n## Libraries and functions\n\n::: {.callout-note collapse=\"true\"}\n## Click to expand\n\n::: {.panel-tabset group=\"language\"}\n## R\n\n### Libraries\n### Functions\n\n## Python\n\n### Libraries\n\n\n::: {.cell}\n\n```{.python .cell-code}\n# A maths library\nimport math\n# A Python data analysis and manipulation tool\nimport pandas as pd\n\n# Python equivalent of `ggplot2`\nfrom plotnine import *\n\n# Statistical models, conducting tests and statistical data exploration\nimport statsmodels.api as sm\n\n# Convenience interface for specifying models using formula strings and DataFrames\nimport statsmodels.formula.api as smf\n```\n:::\n\n\n### Functions\n:::\n:::\n\nThe example in this section uses the following data set:\n\n`data/challenger.csv`\n\nThese data, obtained from the [faraway package](https://www.rdocumentation.org/packages/faraway/versions/1.0.7), contain information related to the explosion of the USA Space Shuttle Challenger on 28 January, 1986. An investigation after the disaster traced back to certain joints on one of the two solid booster rockets, each containing O-rings that ensured no exhaust gases could escape from the booster.\n\nThe night before the launch was unusually cold, with temperatures below freezing. The final report suggested that the cold snap during the night made the o-rings stiff, and unable to adjust to changes in pressure. As a result, exhaust gases leaked away from the solid booster rockets, causing one of them to break loose and rupture the main fuel tank, leading to the final explosion.\n\nThe question we're trying to answer in this session is: based on the data from the previous flights, would it have been possible to predict the failure of most o-rings on the Challenger flight?\n\n## Load and visualise the data\n\nFirst we load the data, then we visualise it.\n\n::: {.panel-tabset group=\"language\"}\n## R\n\n\n::: {.cell}\n\n```{.r .cell-code}\nchallenger <- read_csv(\"data/challenger.csv\")\n```\n\n::: {.cell-output .cell-output-stderr}\n\n```\nRows: 23 Columns: 2\n── Column specification ────────────────────────────────────────────────────────\nDelimiter: \",\"\ndbl (2): temp, damage\n\nℹ Use `spec()` to retrieve the full column specification for this data.\nℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.\n```\n\n\n:::\n:::\n\n:::\n\nThe data set contains several columns:\n\n1. `temp`, the launch temperature in degrees Fahrenheit\n2. `damage`, the number of o-rings that showed erosion\n\nBefore we have a further look at the data, let's calculate the proportion of damaged o-rings (`prop_damaged`) and the total number of o-rings (`total`) and update our data set.\n\n::: {.panel-tabset group=\"language\"}\n## R\n\n\n::: {.cell}\n\n```{.r .cell-code}\nchallenger <-\nchallenger %>%\n mutate(total = 6, # total number of o-rings\n intact = 6 - damage, # number of undamaged o-rings\n prop_damaged = damage / total) # proportion damaged o-rings\n\nchallenger\n```\n\n::: {.cell-output .cell-output-stdout}\n\n```\n# A tibble: 23 × 5\n temp damage total intact prop_damaged\n \n 1 53 5 6 1 0.833\n 2 57 1 6 5 0.167\n 3 58 1 6 5 0.167\n 4 63 1 6 5 0.167\n 5 66 0 6 6 0 \n 6 67 0 6 6 0 \n 7 67 0 6 6 0 \n 8 67 0 6 6 0 \n 9 68 0 6 6 0 \n10 69 0 6 6 0 \n# ℹ 13 more rows\n```\n\n\n:::\n:::\n\n:::\n\nPlotting the proportion of damaged o-rings against the launch temperature shows the following picture:\n\n::: {.panel-tabset group=\"language\"}\n## R\n\n\n::: {.cell}\n\n```{.r .cell-code}\nggplot(challenger, aes(x = temp, y = prop_damaged)) +\n geom_point()\n```\n\n::: {.cell-output-display}\n![](glm-practical-logistic-proportion_files/figure-html/unnamed-chunk-5-1.png){width=672}\n:::\n:::\n\n:::\n\nThe point on the left is the data point corresponding to the coldest flight experienced before the disaster, where five damaged o-rings were found. Fortunately, this did not result in a disaster.\n\nHere we'll explore if we could have predicted the failure of both o-rings on the Challenger flight, where the launch temperature was 31 degrees Fahrenheit.\n\n## Creating a suitable model\n\nWe only have 23 data points in total. So we're building a model on not that much data - we should keep this in mind when we draw our conclusions!\n\n::: {.panel-tabset group=\"language\"}\n## R\n\nWe are using a logistic regression for a proportion response in this case, since we're interested in the proportion of o-rings that are damaged.\n\nWe can define this as follows:\n\n\n::: {.cell}\n\n```{.r .cell-code}\nglm_chl <- glm(cbind(damage, intact) ~ temp,\n family = binomial,\n data = challenger)\n```\n:::\n\n\nDefining the relationship for proportion responses is a bit annoying, where you have to give the `glm` model a two-column matrix to specify the response variable.\n\nHere, the first column corresponds to the number of damaged o-rings, whereas the second column refers to the number of intact o-rings. We use the `cbind()` function to bind these two together into a matrix.\n\n:::\n\n## Model output\n\nThat's the easy part done! The trickier part is interpreting the output. First of all, we'll get some summary information.\n\n::: {.panel-tabset group=\"language\"}\n## R\n\nNext, we can have a closer look at the results:\n\n\n::: {.cell}\n\n```{.r .cell-code}\nsummary(glm_chl)\n```\n\n::: {.cell-output .cell-output-stdout}\n\n```\n\nCall:\nglm(formula = cbind(damage, intact) ~ temp, family = binomial, \n data = challenger)\n\nCoefficients:\n Estimate Std. Error z value Pr(>|z|) \n(Intercept) 11.66299 3.29626 3.538 0.000403 ***\ntemp -0.21623 0.05318 -4.066 4.78e-05 ***\n---\nSignif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n\n(Dispersion parameter for binomial family taken to be 1)\n\n Null deviance: 38.898 on 22 degrees of freedom\nResidual deviance: 16.912 on 21 degrees of freedom\nAIC: 33.675\n\nNumber of Fisher Scoring iterations: 6\n```\n\n\n:::\n:::\n\n\nWe can see that the p-values of the `intercept` and `temp` are significant. We can also use the intercept and `temp` coefficients to construct the logistic equation, which we can use to sketch the logistic curve.\n\n:::\n\n$$E(prop \\ failed\\ orings) = \\frac{\\exp{(11.66 - 0.22 \\times temp)}}{1 + \\exp{(11.66 - 0.22 \\times temp)}}$$\n\nLet's see how well our model would have performed if we would have fed it the data from the ill-fated Challenger launch.\n\n::: {.panel-tabset group=\"language\"}\n## R\n\n\n::: {.cell}\n\n```{.r .cell-code}\nggplot(challenger, aes(temp, prop_damaged)) +\n geom_point() +\n geom_smooth(method = \"glm\", se = FALSE, fullrange = TRUE, \n method.args = list(family = binomial)) +\n xlim(25,85)\n```\n\n::: {.cell-output .cell-output-stderr}\n\n```\nWarning in eval(family$initialize): non-integer #successes in a binomial glm!\n```\n\n\n:::\n\n::: {.cell-output-display}\n![](glm-practical-logistic-proportion_files/figure-html/unnamed-chunk-8-1.png){width=672}\n:::\n:::\n\n\n::: {.callout-note collapse=true}\n## Generating predicted values\n\n::: {.panel-tabset group=\"language\"}\n## R\n\nAnother way of doing this it to generate a table with data for a range of temperatures, from 25 to 85 degrees Fahrenheit, in steps of 1. We can then use these data to generate the logistic curve, based on the fitted model.\n\n\n::: {.cell}\n\n```{.r .cell-code}\n# create a table with sequential numbers ranging from 25 to 85\nmodel <- tibble(temp = seq(25, 85, by = 1)) %>% \n # add a new column containing the predicted values\n mutate(.pred = predict(glm_chl, newdata = ., type = \"response\"))\n\nggplot(model, aes(temp, .pred)) +\n geom_line()\n```\n\n::: {.cell-output-display}\n![](glm-practical-logistic-proportion_files/figure-html/unnamed-chunk-9-1.png){width=672}\n:::\n:::\n\n::: {.cell}\n\n```{.r .cell-code}\n# plot the curve and the original data\nggplot(model, aes(temp, .pred)) +\n geom_line(colour = \"blue\") +\n geom_point(data = challenger, aes(temp, prop_damaged)) +\n # add a vertical line at the disaster launch temperature\n geom_vline(xintercept = 31, linetype = \"dashed\")\n```\n\n::: {.cell-output-display}\n![](glm-practical-logistic-proportion_files/figure-html/unnamed-chunk-10-1.png){width=672}\n:::\n:::\n\n\nIt seems that there was a high probability of both o-rings failing at that launch temperature. One thing that the graph shows is that there is a lot of uncertainty involved in this model. We can tell, because the fit of the line is very poor at the lower temperature range. There is just very little data to work on, with the data point at 53 F having a large influence on the fit.\n:::\n:::\n:::\n\n## Exercises\n\n### Predicting failure {#sec-exr_failure}\n\n:::{.callout-exercise}\n\n\n{{< level 2 >}}\n\n\n\nThe data point at 53 degrees Fahrenheit is quite influential for the analysis. Remove this data point and repeat the analysis. Is there still a predicted link between launch temperature and o-ring failure?\n\n::: {.callout-answer collapse=\"true\"}\n\n::: {.panel-tabset group=\"language\"}\n## R\n\nFirst, we need to remove the influential data point:\n\n\n::: {.cell}\n\n```{.r .cell-code}\nchallenger_new <- challenger %>% filter(temp != 53)\n```\n:::\n\n\nWe can create a new generalised linear model, based on these data:\n\n\n::: {.cell}\n\n```{.r .cell-code}\nglm_chl_new <- glm(cbind(damage, intact) ~ temp,\n family = binomial,\n data = challenger_new)\n```\n:::\n\n\nWe can get the model parameters as follows:\n\n\n::: {.cell}\n\n```{.r .cell-code}\nsummary(glm_chl_new)\n```\n\n::: {.cell-output .cell-output-stdout}\n\n```\n\nCall:\nglm(formula = cbind(damage, intact) ~ temp, family = binomial, \n data = challenger_new)\n\nCoefficients:\n Estimate Std. Error z value Pr(>|z|) \n(Intercept) 5.68223 4.43138 1.282 0.1997 \ntemp -0.12817 0.06697 -1.914 0.0556 .\n---\nSignif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n\n(Dispersion parameter for binomial family taken to be 1)\n\n Null deviance: 16.375 on 21 degrees of freedom\nResidual deviance: 12.633 on 20 degrees of freedom\nAIC: 27.572\n\nNumber of Fisher Scoring iterations: 5\n```\n\n\n:::\n:::\n\n::: {.cell}\n\n```{.r .cell-code}\nggplot(challenger_new, aes(temp, prop_damaged)) +\n geom_point() +\n geom_smooth(method = \"glm\", se = FALSE, fullrange = TRUE, \n method.args = list(family = binomial)) +\n xlim(25,85) +\n # add a vertical line at 53 F temperature\n geom_vline(xintercept = 53, linetype = \"dashed\")\n```\n\n::: {.cell-output .cell-output-stderr}\n\n```\nWarning in eval(family$initialize): non-integer #successes in a binomial glm!\n```\n\n\n:::\n\n::: {.cell-output-display}\n![](glm-practical-logistic-proportion_files/figure-html/unnamed-chunk-14-1.png){width=672}\n:::\n:::\n\n\nThe prediction proportion of damaged o-rings is markedly less than what was observed.\n\nBefore we can make any firm conclusions, though, we need to check our model:\n\n\n::: {.cell}\n\n```{.r .cell-code}\n1- pchisq(12.633,20)\n```\n\n::: {.cell-output .cell-output-stdout}\n\n```\n[1] 0.8925695\n```\n\n\n:::\n:::\n\n\nWe get quite a high score (around 0.9) for this, which tells us that our goodness of fit is pretty rubbish – our points are not very close to our curve, overall.\n\nIs the model any better than the null though?\n\n\n::: {.cell}\n\n```{.r .cell-code}\n1 - pchisq(16.375 - 12.633, 1)\n```\n\n::: {.cell-output .cell-output-stdout}\n\n```\n[1] 0.0530609\n```\n\n\n:::\n\n```{.r .cell-code}\nanova(glm_chl_new, test = 'Chisq')\n```\n\n::: {.cell-output .cell-output-stdout}\n\n```\nAnalysis of Deviance Table\n\nModel: binomial, link: logit\n\nResponse: cbind(damage, intact)\n\nTerms added sequentially (first to last)\n\n Df Deviance Resid. Df Resid. Dev Pr(>Chi) \nNULL 21 16.375 \ntemp 1 3.7421 20 12.633 0.05306 .\n---\nSignif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n```\n\n\n:::\n:::\n\n\nHowever, the model is not significantly better than the null in this case, with a p-value here of just over 0.05 for both of these tests (they give a similar result since, yet again, we have just the one predictor variable).\n:::\n\nSo, could NASA have predicted what happened? This model is not significantly different from the null, i.e., temperature is not a significant predictor. Note that it’s only marginally non-significant, and we do have a high goodness-of-fit value.\n\nIt is possible that if more data points were available that followed a similar trend, the story might be different). Even if we did use our non-significant model to make a prediction, it doesn’t give us a value anywhere near 5 failures for a temperature of 53 degrees Fahrenheit. So overall, based on the model we’ve fitted with these data, there was no indication that a temperature just a few degrees cooler than previous missions could have been so disastrous for the Challenger.\n:::\n:::\n\n## Summary\n\n::: {.callout-tip}\n#### Key points\n\n- We can use a logistic model for proportion response variables\n\n:::\n", + "markdown": "---\ntitle: \"Proportional response\"\n---\n\n::: {.cell}\n\n:::\n\n::: {.cell}\n\n:::\n\n\n::: {.callout-tip}\n## Learning outcomes\n\n- How do I analyse proportion responses?\n- Be able to create a logistic model to test proportion response variables\n- Be able to plot the data and fitted curve\n- Assess the significance of the fit\n:::\n\n## Libraries and functions\n\n::: {.callout-note collapse=\"true\"}\n## Click to expand\n\n::: {.panel-tabset group=\"language\"}\n## R\n\n### Libraries\n### Functions\n\n## Python\n\n### Libraries\n\n\n::: {.cell}\n\n```{.python .cell-code}\n# A maths library\nimport math\n# A Python data analysis and manipulation tool\nimport pandas as pd\n\n# Python equivalent of `ggplot2`\nfrom plotnine import *\n\n# Statistical models, conducting tests and statistical data exploration\nimport statsmodels.api as sm\n\n# Convenience interface for specifying models using formula strings and DataFrames\nimport statsmodels.formula.api as smf\n\n# Needed for additional probability functionality\nfrom scipy.stats import *\n```\n:::\n\n\n### Functions\n:::\n:::\n\nThe example in this section uses the following data set:\n\n`data/challenger.csv`\n\nThese data, obtained from the [faraway package](https://www.rdocumentation.org/packages/faraway/versions/1.0.7) in R, contain information related to the explosion of the USA Space Shuttle Challenger on 28 January, 1986. An investigation after the disaster traced back to certain joints on one of the two solid booster rockets, each containing O-rings that ensured no exhaust gases could escape from the booster.\n\nThe night before the launch was unusually cold, with temperatures below freezing. The final report suggested that the cold snap during the night made the o-rings stiff, and unable to adjust to changes in pressure. As a result, exhaust gases leaked away from the solid booster rockets, causing one of them to break loose and rupture the main fuel tank, leading to the final explosion.\n\nThe question we're trying to answer in this session is: based on the data from the previous flights, would it have been possible to predict the failure of most o-rings on the Challenger flight?\n\n## Load and visualise the data\n\nFirst we load the data, then we visualise it.\n\n::: {.panel-tabset group=\"language\"}\n## R\n\n\n::: {.cell}\n\n```{.r .cell-code}\nchallenger <- read_csv(\"data/challenger.csv\")\n```\n\n::: {.cell-output .cell-output-stderr}\n\n```\nRows: 23 Columns: 2\n── Column specification ────────────────────────────────────────────────────────\nDelimiter: \",\"\ndbl (2): temp, damage\n\nℹ Use `spec()` to retrieve the full column specification for this data.\nℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.\n```\n\n\n:::\n:::\n\n\n## Python\n\n\n::: {.cell}\n\n```{.python .cell-code}\nchallenger_py = pd.read_csv(\"data/challenger.csv\")\n```\n:::\n\n\n:::\n\nThe data set contains several columns:\n\n1. `temp`, the launch temperature in degrees Fahrenheit\n2. `damage`, the number of o-rings that showed erosion\n\nBefore we have a further look at the data, let's calculate the proportion of damaged o-rings (`prop_damaged`) and the total number of o-rings (`total`) and update our data set.\n\n::: {.panel-tabset group=\"language\"}\n## R\n\n\n::: {.cell}\n\n```{.r .cell-code}\nchallenger <-\nchallenger %>%\n mutate(total = 6, # total number of o-rings\n intact = 6 - damage, # number of undamaged o-rings\n prop_damaged = damage / total) # proportion damaged o-rings\n\nchallenger\n```\n\n::: {.cell-output .cell-output-stdout}\n\n```\n# A tibble: 23 × 5\n temp damage total intact prop_damaged\n \n 1 53 5 6 1 0.833\n 2 57 1 6 5 0.167\n 3 58 1 6 5 0.167\n 4 63 1 6 5 0.167\n 5 66 0 6 6 0 \n 6 67 0 6 6 0 \n 7 67 0 6 6 0 \n 8 67 0 6 6 0 \n 9 68 0 6 6 0 \n10 69 0 6 6 0 \n# ℹ 13 more rows\n```\n\n\n:::\n:::\n\n\n## Python\n\n\n::: {.cell}\n\n```{.python .cell-code}\nchallenger_py['total'] = 6\nchallenger_py['intact'] = challenger_py['total'] - challenger_py['damage']\nchallenger_py['prop_damaged'] = challenger_py['damage'] / challenger_py['total']\n```\n:::\n\n\n:::\n\nPlotting the proportion of damaged o-rings against the launch temperature shows the following picture:\n\n::: {.panel-tabset group=\"language\"}\n## R\n\n\n::: {.cell}\n\n```{.r .cell-code}\nggplot(challenger, aes(x = temp, y = prop_damaged)) +\n geom_point()\n```\n\n::: {.cell-output-display}\n![](glm-practical-logistic-proportion_files/figure-html/unnamed-chunk-8-1.png){width=672}\n:::\n:::\n\n\n## Python\n\n\n::: {.cell}\n\n```{.python .cell-code}\n(ggplot(challenger_py,\n aes(x = \"temp\",\n y = \"prop_damaged\")) +\n geom_point())\n```\n\n::: {.cell-output-display}\n![](glm-practical-logistic-proportion_files/figure-html/unnamed-chunk-9-1.png){width=614}\n:::\n:::\n\n\n:::\n\nThe point on the left is the data point corresponding to the coldest flight experienced before the disaster, where five damaged o-rings were found. Fortunately, this did not result in a disaster.\n\nHere we'll explore if we could have reasonably predicted the failure of both o-rings on the Challenger flight, where the launch temperature was 31 degrees Fahrenheit.\n\n## Creating a suitable model\n\nWe only have 23 data points in total. So we're building a model on not that much data - we should keep this in mind when we draw our conclusions!\n\nWe are using a logistic regression for a proportion response in this case, since we're interested in the proportion of o-rings that are damaged.\n\nWe can define this as follows:\n\n::: {.panel-tabset group=\"language\"}\n## R\n\n\n::: {.cell}\n\n```{.r .cell-code}\nglm_chl <- glm(cbind(damage, intact) ~ temp,\n family = binomial,\n data = challenger)\n```\n:::\n\n\nDefining the relationship for proportion responses is a bit annoying, where you have to give the `glm` model a two-column matrix to specify the response variable.\n\nHere, the first column corresponds to the number of damaged o-rings, whereas the second column refers to the number of intact o-rings. We use the `cbind()` function to bind these two together into a matrix.\n\n## Python\n\n\n::: {.cell}\n\n```{.python .cell-code}\n# create a linear model\nmodel = smf.glm(formula = \"damage + intact ~ temp\",\n family = sm.families.Binomial(),\n data = challenger_py)\n# and get the fitted parameters of the model\nglm_chl_py = model.fit()\n```\n:::\n\n\n:::\n\n## Model output\n\nThat's the easy part done! The trickier part is interpreting the output. First of all, we'll get some summary information.\n\n::: {.panel-tabset group=\"language\"}\n## R\n\nNext, we can have a closer look at the results:\n\n\n::: {.cell}\n\n```{.r .cell-code}\nsummary(glm_chl)\n```\n\n::: {.cell-output .cell-output-stdout}\n\n```\n\nCall:\nglm(formula = cbind(damage, intact) ~ temp, family = binomial, \n data = challenger)\n\nCoefficients:\n Estimate Std. Error z value Pr(>|z|) \n(Intercept) 11.66299 3.29626 3.538 0.000403 ***\ntemp -0.21623 0.05318 -4.066 4.78e-05 ***\n---\nSignif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n\n(Dispersion parameter for binomial family taken to be 1)\n\n Null deviance: 38.898 on 22 degrees of freedom\nResidual deviance: 16.912 on 21 degrees of freedom\nAIC: 33.675\n\nNumber of Fisher Scoring iterations: 6\n```\n\n\n:::\n:::\n\n\nWe can see that the p-values of the `intercept` and `temp` are significant. We can also use the intercept and `temp` coefficients to construct the logistic equation, which we can use to sketch the logistic curve.\n\n## Python\n\n\n::: {.cell}\n\n```{.python .cell-code}\nprint(glm_chl_py.summary())\n```\n\n::: {.cell-output .cell-output-stdout}\n\n```\n Generalized Linear Model Regression Results \n================================================================================\nDep. Variable: ['damage', 'intact'] No. Observations: 23\nModel: GLM Df Residuals: 21\nModel Family: Binomial Df Model: 1\nLink Function: Logit Scale: 1.0000\nMethod: IRLS Log-Likelihood: -14.837\nDate: Thu, 01 Feb 2024 Deviance: 16.912\nTime: 09:00:44 Pearson chi2: 28.1\nNo. Iterations: 7 Pseudo R-squ. (CS): 0.6155\nCovariance Type: nonrobust \n==============================================================================\n coef std err z P>|z| [0.025 0.975]\n------------------------------------------------------------------------------\nIntercept 11.6630 3.296 3.538 0.000 5.202 18.124\ntemp -0.2162 0.053 -4.066 0.000 -0.320 -0.112\n==============================================================================\n```\n\n\n:::\n:::\n\n:::\n\n$$E(prop \\ failed\\ orings) = \\frac{\\exp{(11.66 - 0.22 \\times temp)}}{1 + \\exp{(11.66 - 0.22 \\times temp)}}$$\n\nLet's see how well our model would have performed if we would have fed it the data from the ill-fated Challenger launch.\n\n::: {.panel-tabset group=\"language\"}\n## R\n\n\n::: {.cell}\n\n```{.r .cell-code}\nggplot(challenger, aes(temp, prop_damaged)) +\n geom_point() +\n geom_smooth(method = \"glm\", se = FALSE, fullrange = TRUE, \n method.args = list(family = binomial)) +\n xlim(25,85)\n```\n\n::: {.cell-output .cell-output-stderr}\n\n```\nWarning in eval(family$initialize): non-integer #successes in a binomial glm!\n```\n\n\n:::\n\n::: {.cell-output-display}\n![](glm-practical-logistic-proportion_files/figure-html/unnamed-chunk-14-1.png){width=672}\n:::\n:::\n\n\n## Python\n\nWe can get the predicted values for the model as follows:\n\n\n::: {.cell}\n\n```{.python .cell-code}\nchallenger_py['predicted_values'] = glm_chl_py.predict()\n\nchallenger_py.head()\n```\n\n::: {.cell-output .cell-output-stdout}\n\n```\n temp damage total intact prop_damaged predicted_values\n0 53 5 6 1 0.833333 0.550479\n1 57 1 6 5 0.166667 0.340217\n2 58 1 6 5 0.166667 0.293476\n3 63 1 6 5 0.166667 0.123496\n4 66 0 6 6 0.000000 0.068598\n```\n\n\n:::\n:::\n\n\nThis would only give us the predicted values for the data we already have. Instead we want to extrapolate to what would have been predicted for a wider range of temperatures. Here, we use a range of $[25, 85]$ Fahrenheit.\n\n\n::: {.cell}\n\n```{.python .cell-code}\nmodel = pd.DataFrame({'temp': list(range(25, 86))})\n\nmodel[\"pred\"] = glm_chl_py.predict(model)\n\nmodel.head()\n```\n\n::: {.cell-output .cell-output-stdout}\n\n```\n temp pred\n0 25 0.998087\n1 26 0.997626\n2 27 0.997055\n3 28 0.996347\n4 29 0.995469\n```\n\n\n:::\n:::\n\n::: {.cell}\n\n```{.python .cell-code}\n(ggplot(challenger_py,\n aes(x = \"temp\",\n y = \"prop_damaged\")) +\n geom_point() +\n geom_line(model, aes(x = \"temp\", y = \"pred\"), colour = \"blue\"))\n```\n\n::: {.cell-output-display}\n![](glm-practical-logistic-proportion_files/figure-html/unnamed-chunk-17-1.png){width=614}\n:::\n:::\n\n\n\n::: {.callout-note collapse=true}\n## Generating predicted values\n\n::: {.panel-tabset group=\"language\"}\n## R\n\nAnother way of doing this it to generate a table with data for a range of temperatures, from 25 to 85 degrees Fahrenheit, in steps of 1. We can then use these data to generate the logistic curve, based on the fitted model.\n\n\n::: {.cell}\n\n```{.r .cell-code}\n# create a table with sequential numbers ranging from 25 to 85\nmodel <- tibble(temp = seq(25, 85, by = 1)) %>% \n # add a new column containing the predicted values\n mutate(.pred = predict(glm_chl, newdata = ., type = \"response\"))\n\nggplot(model, aes(temp, .pred)) +\n geom_line()\n```\n\n::: {.cell-output-display}\n![](glm-practical-logistic-proportion_files/figure-html/unnamed-chunk-18-3.png){width=672}\n:::\n:::\n\n::: {.cell}\n\n```{.r .cell-code}\n# plot the curve and the original data\nggplot(model, aes(temp, .pred)) +\n geom_line(colour = \"blue\") +\n geom_point(data = challenger, aes(temp, prop_damaged)) +\n # add a vertical line at the disaster launch temperature\n geom_vline(xintercept = 31, linetype = \"dashed\")\n```\n\n::: {.cell-output-display}\n![](glm-practical-logistic-proportion_files/figure-html/unnamed-chunk-19-1.png){width=672}\n:::\n:::\n\n\nIt seems that there was a high probability of both o-rings failing at that launch temperature. One thing that the graph shows is that there is a lot of uncertainty involved in this model. We can tell, because the fit of the line is very poor at the lower temperature range. There is just very little data to work on, with the data point at 53 F having a large influence on the fit.\n\n## Python\n\nWe already did this above, since this is the most straightforward way of plotting the model in Python.\n\n:::\n:::\n:::\n\n## Exercises\n\n### Predicting failure {#sec-exr_failure}\n\n:::{.callout-exercise}\n\n\n{{< level 2 >}}\n\n\n\nThe data point at 53 degrees Fahrenheit is quite influential for the analysis. Remove this data point and repeat the analysis. Is there still a predicted link between launch temperature and o-ring failure?\n\n::: {.callout-answer collapse=\"true\"}\n\n::: {.panel-tabset group=\"language\"}\n## R\n\nFirst, we need to remove the influential data point:\n\n\n::: {.cell}\n\n```{.r .cell-code}\nchallenger_new <- challenger %>% filter(temp != 53)\n```\n:::\n\n\nWe can create a new generalised linear model, based on these data:\n\n\n::: {.cell}\n\n```{.r .cell-code}\nglm_chl_new <- glm(cbind(damage, intact) ~ temp,\n family = binomial,\n data = challenger_new)\n```\n:::\n\n\nWe can get the model parameters as follows:\n\n\n::: {.cell}\n\n```{.r .cell-code}\nsummary(glm_chl_new)\n```\n\n::: {.cell-output .cell-output-stdout}\n\n```\n\nCall:\nglm(formula = cbind(damage, intact) ~ temp, family = binomial, \n data = challenger_new)\n\nCoefficients:\n Estimate Std. Error z value Pr(>|z|) \n(Intercept) 5.68223 4.43138 1.282 0.1997 \ntemp -0.12817 0.06697 -1.914 0.0556 .\n---\nSignif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n\n(Dispersion parameter for binomial family taken to be 1)\n\n Null deviance: 16.375 on 21 degrees of freedom\nResidual deviance: 12.633 on 20 degrees of freedom\nAIC: 27.572\n\nNumber of Fisher Scoring iterations: 5\n```\n\n\n:::\n:::\n\n::: {.cell}\n\n```{.r .cell-code}\nggplot(challenger_new, aes(temp, prop_damaged)) +\n geom_point() +\n geom_smooth(method = \"glm\", se = FALSE, fullrange = TRUE, \n method.args = list(family = binomial)) +\n xlim(25,85) +\n # add a vertical line at 53 F temperature\n geom_vline(xintercept = 53, linetype = \"dashed\")\n```\n\n::: {.cell-output .cell-output-stderr}\n\n```\nWarning in eval(family$initialize): non-integer #successes in a binomial glm!\n```\n\n\n:::\n\n::: {.cell-output-display}\n![](glm-practical-logistic-proportion_files/figure-html/unnamed-chunk-23-1.png){width=672}\n:::\n:::\n\n\nThe prediction proportion of damaged o-rings is markedly less than what was observed.\n\nBefore we can make any firm conclusions, though, we need to check our model:\n\n\n::: {.cell}\n\n```{.r .cell-code}\n1- pchisq(12.633,20)\n```\n\n::: {.cell-output .cell-output-stdout}\n\n```\n[1] 0.8925695\n```\n\n\n:::\n:::\n\n\nWe get quite a high score (around 0.9) for this, which tells us that our goodness of fit is pretty rubbish – our points are not very close to our curve, overall.\n\nIs the model any better than the null though?\n\n\n::: {.cell}\n\n```{.r .cell-code}\n1 - pchisq(16.375 - 12.633, 1)\n```\n\n::: {.cell-output .cell-output-stdout}\n\n```\n[1] 0.0530609\n```\n\n\n:::\n\n```{.r .cell-code}\nanova(glm_chl_new, test = 'Chisq')\n```\n\n::: {.cell-output .cell-output-stdout}\n\n```\nAnalysis of Deviance Table\n\nModel: binomial, link: logit\n\nResponse: cbind(damage, intact)\n\nTerms added sequentially (first to last)\n\n Df Deviance Resid. Df Resid. Dev Pr(>Chi) \nNULL 21 16.375 \ntemp 1 3.7421 20 12.633 0.05306 .\n---\nSignif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n```\n\n\n:::\n:::\n\n\nHowever, the model is not significantly better than the null in this case, with a p-value here of just over 0.05 for both of these tests (they give a similar result since, yet again, we have just the one predictor variable).\n:::\n\nSo, could NASA have predicted what happened? This model is not significantly different from the null, i.e., temperature is not a significant predictor. Note that it’s only marginally non-significant, and we do have a high goodness-of-fit value.\n\nIt is possible that if more data points were available that followed a similar trend, the story might be different). Even if we did use our non-significant model to make a prediction, it doesn’t give us a value anywhere near 5 failures for a temperature of 53 degrees Fahrenheit. So overall, based on the model we’ve fitted with these data, there was no indication that a temperature just a few degrees cooler than previous missions could have been so disastrous for the Challenger.\n:::\n:::\n\n## Summary\n\n::: {.callout-tip}\n#### Key points\n\n- We can use a logistic model for proportion response variables\n\n:::\n", "supporting": [ "glm-practical-logistic-proportion_files" ], diff --git a/_freeze/materials/glm-practical-logistic-proportion/figure-html/unnamed-chunk-14-1.png b/_freeze/materials/glm-practical-logistic-proportion/figure-html/unnamed-chunk-14-1.png index 5ea53a3..7c097f1 100644 Binary files a/_freeze/materials/glm-practical-logistic-proportion/figure-html/unnamed-chunk-14-1.png and b/_freeze/materials/glm-practical-logistic-proportion/figure-html/unnamed-chunk-14-1.png differ diff --git a/_freeze/materials/glm-practical-logistic-proportion/figure-html/unnamed-chunk-17-1.png b/_freeze/materials/glm-practical-logistic-proportion/figure-html/unnamed-chunk-17-1.png new file mode 100644 index 0000000..284065c Binary files /dev/null and b/_freeze/materials/glm-practical-logistic-proportion/figure-html/unnamed-chunk-17-1.png differ diff --git a/_freeze/materials/glm-practical-logistic-proportion/figure-html/unnamed-chunk-18-3.png b/_freeze/materials/glm-practical-logistic-proportion/figure-html/unnamed-chunk-18-3.png new file mode 100644 index 0000000..101c3bb Binary files /dev/null and b/_freeze/materials/glm-practical-logistic-proportion/figure-html/unnamed-chunk-18-3.png differ diff --git a/_freeze/materials/glm-practical-logistic-proportion/figure-html/unnamed-chunk-19-1.png b/_freeze/materials/glm-practical-logistic-proportion/figure-html/unnamed-chunk-19-1.png new file mode 100644 index 0000000..9b3e566 Binary files /dev/null and b/_freeze/materials/glm-practical-logistic-proportion/figure-html/unnamed-chunk-19-1.png differ diff --git a/_freeze/materials/glm-practical-logistic-proportion/figure-html/unnamed-chunk-23-1.png b/_freeze/materials/glm-practical-logistic-proportion/figure-html/unnamed-chunk-23-1.png new file mode 100644 index 0000000..5ea53a3 Binary files /dev/null and b/_freeze/materials/glm-practical-logistic-proportion/figure-html/unnamed-chunk-23-1.png differ diff --git a/_freeze/materials/glm-practical-logistic-proportion/figure-html/unnamed-chunk-8-1.png b/_freeze/materials/glm-practical-logistic-proportion/figure-html/unnamed-chunk-8-1.png index 7c097f1..cd17d72 100644 Binary files a/_freeze/materials/glm-practical-logistic-proportion/figure-html/unnamed-chunk-8-1.png and b/_freeze/materials/glm-practical-logistic-proportion/figure-html/unnamed-chunk-8-1.png differ diff --git a/_freeze/materials/glm-practical-logistic-proportion/figure-html/unnamed-chunk-9-1.png b/_freeze/materials/glm-practical-logistic-proportion/figure-html/unnamed-chunk-9-1.png index 101c3bb..e2b5db3 100644 Binary files a/_freeze/materials/glm-practical-logistic-proportion/figure-html/unnamed-chunk-9-1.png and b/_freeze/materials/glm-practical-logistic-proportion/figure-html/unnamed-chunk-9-1.png differ diff --git a/_site/search.json b/_site/search.json index d38010e..699b35f 100644 --- a/_site/search.json +++ b/_site/search.json @@ -279,7 +279,7 @@ "href": "materials/glm-practical-logistic-proportion.html", "title": "\n8  Proportional response\n", "section": "", - "text": "8.1 Libraries and functions\nThe example in this section uses the following data set:\ndata/challenger.csv\nThese data, obtained from the faraway package, contain information related to the explosion of the USA Space Shuttle Challenger on 28 January, 1986. An investigation after the disaster traced back to certain joints on one of the two solid booster rockets, each containing O-rings that ensured no exhaust gases could escape from the booster.\nThe night before the launch was unusually cold, with temperatures below freezing. The final report suggested that the cold snap during the night made the o-rings stiff, and unable to adjust to changes in pressure. As a result, exhaust gases leaked away from the solid booster rockets, causing one of them to break loose and rupture the main fuel tank, leading to the final explosion.\nThe question we’re trying to answer in this session is: based on the data from the previous flights, would it have been possible to predict the failure of most o-rings on the Challenger flight?", + "text": "8.1 Libraries and functions\nThe example in this section uses the following data set:\ndata/challenger.csv\nThese data, obtained from the faraway package in R, contain information related to the explosion of the USA Space Shuttle Challenger on 28 January, 1986. An investigation after the disaster traced back to certain joints on one of the two solid booster rockets, each containing O-rings that ensured no exhaust gases could escape from the booster.\nThe night before the launch was unusually cold, with temperatures below freezing. The final report suggested that the cold snap during the night made the o-rings stiff, and unable to adjust to changes in pressure. As a result, exhaust gases leaked away from the solid booster rockets, causing one of them to break loose and rupture the main fuel tank, leading to the final explosion.\nThe question we’re trying to answer in this session is: based on the data from the previous flights, would it have been possible to predict the failure of most o-rings on the Challenger flight?", "crumbs": [ "Binary and proportional data", "8  Proportional response" @@ -290,7 +290,7 @@ "href": "materials/glm-practical-logistic-proportion.html#libraries-and-functions", "title": "\n8  Proportional response\n", "section": "", - "text": "Click to expand\n\n\n\n\n\n\n\nR\nPython\n\n\n\n\n8.1.1 Libraries\n\n8.1.2 Functions\n\n\n\n\n8.1.3 Libraries\n\n# A maths library\nimport math\n# A Python data analysis and manipulation tool\nimport pandas as pd\n\n# Python equivalent of `ggplot2`\nfrom plotnine import *\n\n# Statistical models, conducting tests and statistical data exploration\nimport statsmodels.api as sm\n\n# Convenience interface for specifying models using formula strings and DataFrames\nimport statsmodels.formula.api as smf\n\n\n8.1.4 Functions", + "text": "Click to expand\n\n\n\n\n\n\n\nR\nPython\n\n\n\n\n8.1.1 Libraries\n\n8.1.2 Functions\n\n\n\n\n8.1.3 Libraries\n\n# A maths library\nimport math\n# A Python data analysis and manipulation tool\nimport pandas as pd\n\n# Python equivalent of `ggplot2`\nfrom plotnine import *\n\n# Statistical models, conducting tests and statistical data exploration\nimport statsmodels.api as sm\n\n# Convenience interface for specifying models using formula strings and DataFrames\nimport statsmodels.formula.api as smf\n\n# Needed for additional probability functionality\nfrom scipy.stats import *\n\n\n8.1.4 Functions", "crumbs": [ "Binary and proportional data", "8  Proportional response" @@ -301,7 +301,7 @@ "href": "materials/glm-practical-logistic-proportion.html#load-and-visualise-the-data", "title": "\n8  Proportional response\n", "section": "\n8.2 Load and visualise the data", - "text": "8.2 Load and visualise the data\nFirst we load the data, then we visualise it.\n\nR\n\n\n\nchallenger <- read_csv(\"data/challenger.csv\")\n\nRows: 23 Columns: 2\n── Column specification ────────────────────────────────────────────────────────\nDelimiter: \",\"\ndbl (2): temp, damage\n\nℹ Use `spec()` to retrieve the full column specification for this data.\nℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.\n\n\n\n\n\nThe data set contains several columns:\n\n\ntemp, the launch temperature in degrees Fahrenheit\n\ndamage, the number of o-rings that showed erosion\n\nBefore we have a further look at the data, let’s calculate the proportion of damaged o-rings (prop_damaged) and the total number of o-rings (total) and update our data set.\n\nR\n\n\n\nchallenger <-\nchallenger %>%\n mutate(total = 6, # total number of o-rings\n intact = 6 - damage, # number of undamaged o-rings\n prop_damaged = damage / total) # proportion damaged o-rings\n\nchallenger\n\n# A tibble: 23 × 5\n temp damage total intact prop_damaged\n <dbl> <dbl> <dbl> <dbl> <dbl>\n 1 53 5 6 1 0.833\n 2 57 1 6 5 0.167\n 3 58 1 6 5 0.167\n 4 63 1 6 5 0.167\n 5 66 0 6 6 0 \n 6 67 0 6 6 0 \n 7 67 0 6 6 0 \n 8 67 0 6 6 0 \n 9 68 0 6 6 0 \n10 69 0 6 6 0 \n# ℹ 13 more rows\n\n\n\n\n\nPlotting the proportion of damaged o-rings against the launch temperature shows the following picture:\n\nR\n\n\n\nggplot(challenger, aes(x = temp, y = prop_damaged)) +\n geom_point()\n\n\n\n\n\n\n\n\n\n\nThe point on the left is the data point corresponding to the coldest flight experienced before the disaster, where five damaged o-rings were found. Fortunately, this did not result in a disaster.\nHere we’ll explore if we could have predicted the failure of both o-rings on the Challenger flight, where the launch temperature was 31 degrees Fahrenheit.", + "text": "8.2 Load and visualise the data\nFirst we load the data, then we visualise it.\n\n\nR\nPython\n\n\n\n\nchallenger <- read_csv(\"data/challenger.csv\")\n\nRows: 23 Columns: 2\n── Column specification ────────────────────────────────────────────────────────\nDelimiter: \",\"\ndbl (2): temp, damage\n\nℹ Use `spec()` to retrieve the full column specification for this data.\nℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.\n\n\n\n\n\nchallenger_py = pd.read_csv(\"data/challenger.csv\")\n\n\n\n\nThe data set contains several columns:\n\n\ntemp, the launch temperature in degrees Fahrenheit\n\ndamage, the number of o-rings that showed erosion\n\nBefore we have a further look at the data, let’s calculate the proportion of damaged o-rings (prop_damaged) and the total number of o-rings (total) and update our data set.\n\n\nR\nPython\n\n\n\n\nchallenger <-\nchallenger %>%\n mutate(total = 6, # total number of o-rings\n intact = 6 - damage, # number of undamaged o-rings\n prop_damaged = damage / total) # proportion damaged o-rings\n\nchallenger\n\n# A tibble: 23 × 5\n temp damage total intact prop_damaged\n <dbl> <dbl> <dbl> <dbl> <dbl>\n 1 53 5 6 1 0.833\n 2 57 1 6 5 0.167\n 3 58 1 6 5 0.167\n 4 63 1 6 5 0.167\n 5 66 0 6 6 0 \n 6 67 0 6 6 0 \n 7 67 0 6 6 0 \n 8 67 0 6 6 0 \n 9 68 0 6 6 0 \n10 69 0 6 6 0 \n# ℹ 13 more rows\n\n\n\n\n\nchallenger_py['total'] = 6\nchallenger_py['intact'] = challenger_py['total'] - challenger_py['damage']\nchallenger_py['prop_damaged'] = challenger_py['damage'] / challenger_py['total']\n\n\n\n\nPlotting the proportion of damaged o-rings against the launch temperature shows the following picture:\n\n\nR\nPython\n\n\n\n\nggplot(challenger, aes(x = temp, y = prop_damaged)) +\n geom_point()\n\n\n\n\n\n\n\n\n\n\n(ggplot(challenger_py,\n aes(x = \"temp\",\n y = \"prop_damaged\")) +\n geom_point())\n\n\n\n\n\n\n\n\n\n\nThe point on the left is the data point corresponding to the coldest flight experienced before the disaster, where five damaged o-rings were found. Fortunately, this did not result in a disaster.\nHere we’ll explore if we could have reasonably predicted the failure of both o-rings on the Challenger flight, where the launch temperature was 31 degrees Fahrenheit.", "crumbs": [ "Binary and proportional data", "8  Proportional response" @@ -312,7 +312,7 @@ "href": "materials/glm-practical-logistic-proportion.html#creating-a-suitable-model", "title": "\n8  Proportional response\n", "section": "\n8.3 Creating a suitable model", - "text": "8.3 Creating a suitable model\nWe only have 23 data points in total. So we’re building a model on not that much data - we should keep this in mind when we draw our conclusions!\n\nR\n\n\nWe are using a logistic regression for a proportion response in this case, since we’re interested in the proportion of o-rings that are damaged.\nWe can define this as follows:\n\nglm_chl <- glm(cbind(damage, intact) ~ temp,\n family = binomial,\n data = challenger)\n\nDefining the relationship for proportion responses is a bit annoying, where you have to give the glm model a two-column matrix to specify the response variable.\nHere, the first column corresponds to the number of damaged o-rings, whereas the second column refers to the number of intact o-rings. We use the cbind() function to bind these two together into a matrix.", + "text": "8.3 Creating a suitable model\nWe only have 23 data points in total. So we’re building a model on not that much data - we should keep this in mind when we draw our conclusions!\nWe are using a logistic regression for a proportion response in this case, since we’re interested in the proportion of o-rings that are damaged.\nWe can define this as follows:\n\n\nR\nPython\n\n\n\n\nglm_chl <- glm(cbind(damage, intact) ~ temp,\n family = binomial,\n data = challenger)\n\nDefining the relationship for proportion responses is a bit annoying, where you have to give the glm model a two-column matrix to specify the response variable.\nHere, the first column corresponds to the number of damaged o-rings, whereas the second column refers to the number of intact o-rings. We use the cbind() function to bind these two together into a matrix.\n\n\n\n# create a linear model\nmodel = smf.glm(formula = \"damage + intact ~ temp\",\n family = sm.families.Binomial(),\n data = challenger_py)\n# and get the fitted parameters of the model\nglm_chl_py = model.fit()", "crumbs": [ "Binary and proportional data", "8  Proportional response" @@ -323,7 +323,7 @@ "href": "materials/glm-practical-logistic-proportion.html#model-output", "title": "\n8  Proportional response\n", "section": "\n8.4 Model output", - "text": "8.4 Model output\nThat’s the easy part done! The trickier part is interpreting the output. First of all, we’ll get some summary information.\n\nR\n\n\nNext, we can have a closer look at the results:\n\nsummary(glm_chl)\n\n\nCall:\nglm(formula = cbind(damage, intact) ~ temp, family = binomial, \n data = challenger)\n\nCoefficients:\n Estimate Std. Error z value Pr(>|z|) \n(Intercept) 11.66299 3.29626 3.538 0.000403 ***\ntemp -0.21623 0.05318 -4.066 4.78e-05 ***\n---\nSignif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n\n(Dispersion parameter for binomial family taken to be 1)\n\n Null deviance: 38.898 on 22 degrees of freedom\nResidual deviance: 16.912 on 21 degrees of freedom\nAIC: 33.675\n\nNumber of Fisher Scoring iterations: 6\n\n\nWe can see that the p-values of the intercept and temp are significant. We can also use the intercept and temp coefficients to construct the logistic equation, which we can use to sketch the logistic curve.\n\n\n\n\\[E(prop \\ failed\\ orings) = \\frac{\\exp{(11.66 - 0.22 \\times temp)}}{1 + \\exp{(11.66 - 0.22 \\times temp)}}\\]\nLet’s see how well our model would have performed if we would have fed it the data from the ill-fated Challenger launch.\n\nR\n\n\n\nggplot(challenger, aes(temp, prop_damaged)) +\n geom_point() +\n geom_smooth(method = \"glm\", se = FALSE, fullrange = TRUE, \n method.args = list(family = binomial)) +\n xlim(25,85)\n\nWarning in eval(family$initialize): non-integer #successes in a binomial glm!\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nGenerating predicted values\n\n\n\n\n\n\nR\n\n\nAnother way of doing this it to generate a table with data for a range of temperatures, from 25 to 85 degrees Fahrenheit, in steps of 1. We can then use these data to generate the logistic curve, based on the fitted model.\n\n# create a table with sequential numbers ranging from 25 to 85\nmodel <- tibble(temp = seq(25, 85, by = 1)) %>% \n # add a new column containing the predicted values\n mutate(.pred = predict(glm_chl, newdata = ., type = \"response\"))\n\nggplot(model, aes(temp, .pred)) +\n geom_line()\n\n\n\n\n\n\n\n\n# plot the curve and the original data\nggplot(model, aes(temp, .pred)) +\n geom_line(colour = \"blue\") +\n geom_point(data = challenger, aes(temp, prop_damaged)) +\n # add a vertical line at the disaster launch temperature\n geom_vline(xintercept = 31, linetype = \"dashed\")\n\n\n\n\n\n\n\nIt seems that there was a high probability of both o-rings failing at that launch temperature. One thing that the graph shows is that there is a lot of uncertainty involved in this model. We can tell, because the fit of the line is very poor at the lower temperature range. There is just very little data to work on, with the data point at 53 F having a large influence on the fit.", + "text": "8.4 Model output\nThat’s the easy part done! The trickier part is interpreting the output. First of all, we’ll get some summary information.\n\n\nR\nPython\n\n\n\nNext, we can have a closer look at the results:\n\nsummary(glm_chl)\n\n\nCall:\nglm(formula = cbind(damage, intact) ~ temp, family = binomial, \n data = challenger)\n\nCoefficients:\n Estimate Std. Error z value Pr(>|z|) \n(Intercept) 11.66299 3.29626 3.538 0.000403 ***\ntemp -0.21623 0.05318 -4.066 4.78e-05 ***\n---\nSignif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n\n(Dispersion parameter for binomial family taken to be 1)\n\n Null deviance: 38.898 on 22 degrees of freedom\nResidual deviance: 16.912 on 21 degrees of freedom\nAIC: 33.675\n\nNumber of Fisher Scoring iterations: 6\n\n\nWe can see that the p-values of the intercept and temp are significant. We can also use the intercept and temp coefficients to construct the logistic equation, which we can use to sketch the logistic curve.\n\n\n\nprint(glm_chl_py.summary())\n\n Generalized Linear Model Regression Results \n================================================================================\nDep. Variable: ['damage', 'intact'] No. Observations: 23\nModel: GLM Df Residuals: 21\nModel Family: Binomial Df Model: 1\nLink Function: Logit Scale: 1.0000\nMethod: IRLS Log-Likelihood: -14.837\nDate: Thu, 01 Feb 2024 Deviance: 16.912\nTime: 09:00:44 Pearson chi2: 28.1\nNo. Iterations: 7 Pseudo R-squ. (CS): 0.6155\nCovariance Type: nonrobust \n==============================================================================\n coef std err z P>|z| [0.025 0.975]\n------------------------------------------------------------------------------\nIntercept 11.6630 3.296 3.538 0.000 5.202 18.124\ntemp -0.2162 0.053 -4.066 0.000 -0.320 -0.112\n==============================================================================\n\n\n\n\n\n\\[E(prop \\ failed\\ orings) = \\frac{\\exp{(11.66 - 0.22 \\times temp)}}{1 + \\exp{(11.66 - 0.22 \\times temp)}}\\]\nLet’s see how well our model would have performed if we would have fed it the data from the ill-fated Challenger launch.\n\n\nR\nPython\n\n\n\n\nggplot(challenger, aes(temp, prop_damaged)) +\n geom_point() +\n geom_smooth(method = \"glm\", se = FALSE, fullrange = TRUE, \n method.args = list(family = binomial)) +\n xlim(25,85)\n\nWarning in eval(family$initialize): non-integer #successes in a binomial glm!\n\n\n\n\n\n\n\n\n\n\nWe can get the predicted values for the model as follows:\n\nchallenger_py['predicted_values'] = glm_chl_py.predict()\n\nchallenger_py.head()\n\n temp damage total intact prop_damaged predicted_values\n0 53 5 6 1 0.833333 0.550479\n1 57 1 6 5 0.166667 0.340217\n2 58 1 6 5 0.166667 0.293476\n3 63 1 6 5 0.166667 0.123496\n4 66 0 6 6 0.000000 0.068598\n\n\nThis would only give us the predicted values for the data we already have. Instead we want to extrapolate to what would have been predicted for a wider range of temperatures. Here, we use a range of \\([25, 85]\\) Fahrenheit.\n\nmodel = pd.DataFrame({'temp': list(range(25, 86))})\n\nmodel[\"pred\"] = glm_chl_py.predict(model)\n\nmodel.head()\n\n temp pred\n0 25 0.998087\n1 26 0.997626\n2 27 0.997055\n3 28 0.996347\n4 29 0.995469\n\n\n\n(ggplot(challenger_py,\n aes(x = \"temp\",\n y = \"prop_damaged\")) +\n geom_point() +\n geom_line(model, aes(x = \"temp\", y = \"pred\"), colour = \"blue\"))\n\n\n\n\n\n\n\n\n\n\n\n\n\nGenerating predicted values\n\n\n\n\n\n\n\nR\nPython\n\n\n\nAnother way of doing this it to generate a table with data for a range of temperatures, from 25 to 85 degrees Fahrenheit, in steps of 1. We can then use these data to generate the logistic curve, based on the fitted model.\n\n# create a table with sequential numbers ranging from 25 to 85\nmodel <- tibble(temp = seq(25, 85, by = 1)) %>% \n # add a new column containing the predicted values\n mutate(.pred = predict(glm_chl, newdata = ., type = \"response\"))\n\nggplot(model, aes(temp, .pred)) +\n geom_line()\n\n\n\n\n\n\n\n\n# plot the curve and the original data\nggplot(model, aes(temp, .pred)) +\n geom_line(colour = \"blue\") +\n geom_point(data = challenger, aes(temp, prop_damaged)) +\n # add a vertical line at the disaster launch temperature\n geom_vline(xintercept = 31, linetype = \"dashed\")\n\n\n\n\n\n\n\nIt seems that there was a high probability of both o-rings failing at that launch temperature. One thing that the graph shows is that there is a lot of uncertainty involved in this model. We can tell, because the fit of the line is very poor at the lower temperature range. There is just very little data to work on, with the data point at 53 F having a large influence on the fit.\n\n\nWe already did this above, since this is the most straightforward way of plotting the model in Python.", "crumbs": [ "Binary and proportional data", "8  Proportional response" diff --git a/materials/glm-practical-logistic-binary-original.html b/materials/glm-practical-logistic-binary-original.html deleted file mode 100644 index f63913e..0000000 --- a/materials/glm-practical-logistic-binary-original.html +++ /dev/null @@ -1,820 +0,0 @@ - - - - - - - - - -Binary response - - - - - - - - - - - - - - - - - - - - -
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Binary response

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Questions

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  • How do we analyse data with a binary outcome?
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  • Can we test if our model is any good?
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  • Be able to perform a logistic regression with a binary outcome
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  • Predict outcomes of new data, based on a defined model
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Objectives

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  • Be able to analyse binary outcome data
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  • Understand different methods of testing model fit
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Libraries and functions

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Libraries

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Functions

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Libraries

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# A Python data analysis and manipulation tool
-import pandas as pd
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-# Python equivalent of `ggplot2`
-from plotnine import *
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-# Statistical models, conducting tests and statistical data exploration
-import statsmodels.api as sm
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-# Convenience interface for specifying models using formula strings and DataFrames
-import statsmodels.formula.api as smf
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The example in this section uses the following data set:

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data/diabetes.csv

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This is a data set comprising 768 observations of three variables (one dependent and two predictor variables). This records the results of a diabetes test result as a binary variable (1 is a positive result, 0 is a negative result), along with the result of a glucose test and the diastolic blood pressure for each of 767 women. The variables are called test_result, glucose and diastolic.

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Load and visualise the data

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First we load the data, then we visualise it.

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diabetes <- read_csv("data/diabetes.csv")
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diabetes_py = pd.read_csv("data/diabetes.csv")
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Looking at the data, we can see that the test_result column contains zeros and ones. These are yes/no test result outcomes and not actually numeric representations.

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We’ll have to deal with this soon. For now, we can plot the data, by outcome:

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diabetes %>% 
-  ggplot(aes(x = factor(test_result),
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We could just give Python the test_result data directly, but then it would view the values as numeric. Which doesn’t really work, because we have two groups as such: those with a negative diabetes test result, and those with a positive one.

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We can force Python to temporarily covert the data to a factor, by making the test_result column an object type. We can do this directly inside the ggplot() function.

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(ggplot(diabetes_py,
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It looks as though the patients with a positive diabetes test have slightly higher glucose levels than those with a negative diabetes test.

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We can visualise that differently by plotting all the data points as a classic binary response plot:

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diabetes %>% 
-  ggplot(aes(x = glucose, y = test_result)) +
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(ggplot(diabetes_py,
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This presents us with a bit of an issue. We could fit a linear regression model to these data, although we already know that this is a bad idea…

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diabetes %>% 
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(ggplot(diabetes_py,
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Of course this is rubbish - we can’t have test results outside the range of [0, 1].

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But for the sake of exploration, let’s look at the assumptions:

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lm_dia <- lm(test_result ~ glucose,
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-lm_dia %>% 
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`geom_smooth()` using formula = 'y ~ x'
-`geom_smooth()` using formula = 'y ~ x'
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First, we create a linear model:

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-model = smf.ols(formula= "test_result ~ glucose",
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-lm_dia_py = model.fit()
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Next, we can create the diagnostic plots:

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dgplots(lm_dia_py)
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They’re pretty extremely bad.

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  • The response is not linear (Residual Plot, binary response plot, common sense).
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Another way of viewing the residuals (apart from the Q-Q plot) is as a dot-plot. In R, the ggdist and distributional packages are extremely useful for this kind of stuff.

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What I’m doing here is:

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diabetes %>% 
-  lm(test_result ~ glucose, data = .) %>%
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-                    orientation = "horizontal") +
-  stat_dotsinterval(aes(x = value),
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-  labs(title = "Linear model (diabetes)", y = "probability", x = NULL)
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This again shows us that the residuals are really not normally distributed. If they were, then they would overlap much more closely with the distribution (in grey).

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Creating a suitable model

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So far we’ve established that using a simple linear model to describe a potential relationship between glucose levels and the probability of getting a positive test result is not a good idea. So, what can we do?

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One of the ways we can deal with binary outcome data is by performing a logistic regression. Instead of fitting a straight line to our data, and performing a regression on that, we fit a line that has an S shape. This avoids the model making predictions outside the \([0, 1]\) range.

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There are many mathematical functions that produce S-shaped graphs. The logistic function is one of them and well-suited to these kind of data.

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In the most simple form a logistic function is written like this:

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\(Y = \frac{\exp(X)}{1 + \exp(X)}\)

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In mathematics, \(\rm e\) represents a constant of around 2.718. Another notation is \(\exp\), which is often used when notations become a bit cumbersome. Here, I exclusively use the \(\exp\) notation for consistency.

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We can generalise this, by writing it as follows:

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\(Y = \frac{\exp(X)}{1 + \exp(\beta_0 + \beta_1X)}\)

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Note that the \(\beta_0 + \beta_1X\) part is identical to the formula of a straight line. We’ve come across this before when we were doing simple linear regression!

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The rest of the function is what makes the straight line curve into its characteristic S shape. The middle of the S (where \(Y = 0.5\)) occurs when \(X = \frac{-a}{b}\).

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The shape of the logistic function is hugely influenced by the different parameters, in particular \(\beta_1\). The plots below show different situations, where \(\beta_0 = 0\) in all cases, but \(\beta_1\) varies.

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The first plot shows the logistic function in its simplest form, with the others showing the effect of varying \(\beta_1\).

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We can fit such an S-shaped curve to our diabetes data set, by creating a generalised linear model.

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In R we have a few options to do this, and by far the most familiar function would be glm(). Here we save the model in an object called glm_dia:

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glm_dia <- glm(test_result ~ glucose,
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The format of this function is similar to that used by the lm() function for linear models. The important difference is that we must specify the family of error distribution to use. For logistic regression we must set the family to binomial.

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If you forget to set the family argument, then the glm() function will perform a standard linear model fit, identical to what the lm() function would do.

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Model output

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That’s the easy part done! The trickier part is interpreting the output. First of all, we’ll get some summary information.

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summary(glm_dia)
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-Coefficients:
-             Estimate Std. Error z value Pr(>|z|)    
-(Intercept) -5.611732   0.442289  -12.69   <2e-16 ***
-glucose      0.039510   0.003398   11.63   <2e-16 ***
----
-Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
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-(Dispersion parameter for binomial family taken to be 1)
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-    Null deviance: 936.6  on 727  degrees of freedom
-Residual deviance: 752.2  on 726  degrees of freedom
-AIC: 756.2
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                 Generalized Linear Model Regression Results                  
-==============================================================================
-Dep. Variable:            test_result   No. Observations:                  728
-Model:                            GLM   Df Residuals:                      726
-Model Family:                Binomial   Df Model:                            1
-Link Function:                  Logit   Scale:                          1.0000
-Method:                          IRLS   Log-Likelihood:                -376.10
-Date:                Mon, 07 Aug 2023   Deviance:                       752.20
-Time:                        14:28:30   Pearson chi2:                     713.
-No. Iterations:                     4   Pseudo R-squ. (CS):             0.2238
-Covariance Type:            nonrobust                                         
-==============================================================================
-                 coef    std err          z      P>|z|      [0.025      0.975]
-------------------------------------------------------------------------------
-Intercept     -5.6117      0.442    -12.688      0.000      -6.479      -4.745
-glucose        0.0395      0.003     11.628      0.000       0.033       0.046
-==============================================================================
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There’s a lot to unpack here so take a deep breath (or make sure you have a coffee) before continuing…

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(Intercept)  -5.611732
-glucose      0.039510
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The top part of table contains a lot of information about the model we’ve just created at a high level. The left-hand column just tells us a lot of stuff that we really don’t care about like:

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The right-hand column starts to give us some more useful summary statistics that we’ll end up using. In particular we care about the Deviance value. For linear models the residuals were calculated for each data point and then squared and added up to get the SS (sum of squares), which is used to fit the model. For generalised linear models we don’t use SS to fit the model and instead we use an entirely different method called maximum likelihood. This fitting procedure generates a different quantity, called Deviance, which is the analogue of SS. A deviance of zero indicates the best model we could hope for (basically that the model completely matches the data) and bigger values indicate a model that doesn’t fit quite as well. We’ll also need the Df residuals value here (degrees of freedom) for some later calculations.

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Right at the bottom is a table showing the model coefficients. The main numbers to extract from the output are the two numbers in the coef column:

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-Intercept     -5.6117
-glucose        0.0395
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These are the coefficients of the logistic model equation and need to be placed in the correct equation if we want to be able to actually calculate the probability of having a positive diabetes test for a given glucose level.

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The p values (Pr(>|z|) at the end of each coefficient row merely show whether that particular coefficient is significantly different from zero. This is similar to the p-values obtained in the summary output of a linear model, and as before, for continuous predictors in simple models these p-values can be used to decide whether that predictor is important (so in this case glucose appears to be significant). However, these p-values aren’t great when we have multiple predictor variables, or when we have categorical predictors with multiple levels (since the output will give us a p-value for each level rather than for the predictor as a whole).

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We can use the coefficients to actually calculate the probability of having a positive diabetes test for a given glucose level:

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\[ P(positive \ test\ result) = \frac{\exp(-5.61 + 0.04 \times glucose)}{1 + \exp(-5.61 + 0.04 \times glucose)} \]

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\(\beta_0\)-5.6117317
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Which means that we can write the linear predictor equation as follows:

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\[logit(p) = -5.61 + 0.04 \times glucose\] We still have to take into account our link function. Combining the two equations gives us:

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\[log(\frac{p}{1 - p}) = -5.61 + 0.04 \times glucose\]

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To get our \(p\) (the probability of a person having a positive diabetes test result), we need to exponentiate our equation:

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\[\frac{p}{1 - p} = \exp{(-5.61 + 0.04 \times glucose)}\]

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leading to…

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\[p = \frac{\exp{(-5.61 + 0.04 \times glucose)}}{1 + \exp{(-5.61 + 0.04 \times glucose)}}\]

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The last three lines relate to quantities called deviance and AIC (Akaike Information Criterion).

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As we said just above, the deviance values are the equivalent of Sums of Squares values in linear models (and are a product of the technique used to fit the curve to the data). They can be used as a metric of goodness of fit for the model, with a deviance of 0 indicating a perfect fitting model. The deviance for the null model (i.e. the model without any predictors, basically saying that the probability of getting a positive diabetes score is constant and doesn’t depend on glucose level) is given by the first line and the deviance for the actual model is given by the residual deviance line. We will see how we can use the deviance to do two things.

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It’s important to realise that these two things can be independent of each other; we can have a model that is significantly better than a null model whilst still being rubbish overall (the null model will have been even more rubbish in comparison), and we can have a model that is brilliant yet still not be better than the null model (in this case the null model was already brilliant).

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As we found in the model comparisons practical, the AIC value is meaningless by itself, but it will allow us to compare this model to another model with different terms (with the model with the smaller AIC value being the better fitting model).

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There are some additional parameters in the right-hand column, but we’ll first focus on how we can assess significance.

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- - - - \ No newline at end of file diff --git a/materials/glm-practical-logistic-binary-original.qmd b/materials/glm-practical-logistic-binary-original.qmd deleted file mode 100644 index f374cf5..0000000 --- a/materials/glm-practical-logistic-binary-original.qmd +++ /dev/null @@ -1,508 +0,0 @@ ---- -title: "Binary response" ---- - -```{r} -#| echo: false -#| message: false -#| results: hide -source(file = "setup_files/setup.R") -``` - -```{python} -#| echo: false -#| message: false -import shutup;shutup.please() -exec(open('setup_files/setup.py').read()) -``` - -::: {.callout-tip} -## Learning outcomes - -**Questions** - -- How do we analyse data with a binary outcome? -- Can we test if our model is any good? -- Be able to perform a logistic regression with a binary outcome -- Predict outcomes of new data, based on a defined model - -**Objectives** - -- Be able to analyse binary outcome data -- Understand different methods of testing model fit -- Be able to make model predictions -::: - -## Libraries and functions - -::: {.callout-note collapse="true"} -## Click to expand - -::: {.panel-tabset group="language"} -## R - -### Libraries -### Functions - -## Python - -### Libraries - -```{python} -#| eval: false -# A Python data analysis and manipulation tool -import pandas as pd - -# Python equivalent of `ggplot2` -from plotnine import * - -# Statistical models, conducting tests and statistical data exploration -import statsmodels.api as sm - -# Convenience interface for specifying models using formula strings and DataFrames -import statsmodels.formula.api as smf -``` - -### Functions -::: -::: - -The example in this section uses the following data set: - -`data/diabetes.csv` - -This is a data set comprising 768 observations of three variables (one dependent and two predictor variables). This records the results of a diabetes test result as a binary variable (1 is a positive result, 0 is a negative result), along with the result of a glucose test and the diastolic blood pressure for each of 767 women. The variables are called `test_result`, `glucose` and `diastolic`. - -## Load and visualise the data - -First we load the data, then we visualise it. - -::: {.panel-tabset group="language"} -## R - -```{r} -#| message: false -#| warning: false -diabetes <- read_csv("data/diabetes.csv") -``` - -## Python - -```{python} -diabetes_py = pd.read_csv("data/diabetes.csv") -``` - -::: - -Looking at the data, we can see that the `test_result` column contains zeros and ones. These are yes/no test result outcomes and not actually numeric representations. - -We'll have to deal with this soon. For now, we can plot the data, by outcome: - -::: {.panel-tabset group="language"} -## R - -```{r} -diabetes %>% - ggplot(aes(x = factor(test_result), - y = glucose)) + - geom_boxplot() -``` - -## Python - -We could just give Python the `test_result` data directly, but then it would view the values as numeric. Which doesn't really work, because we have two groups as such: those with a negative diabetes test result, and those with a positive one. - -We can force Python to temporarily covert the data to a factor, by making the `test_result` column an `object` type. We can do this directly inside the `ggplot()` function. - -```{python} -#| results: hide -(ggplot(diabetes_py, - aes(x = diabetes_py.test_result.astype(object), - y = "glucose")) + - geom_boxplot()) -``` -::: - -It looks as though the patients with a positive diabetes test have slightly higher glucose levels than those with a negative diabetes test. - -We can visualise that differently by plotting all the data points as a classic binary response plot: - -::: {.panel-tabset group="language"} -## R - -```{r} -diabetes %>% - ggplot(aes(x = glucose, y = test_result)) + - geom_point() -``` - -## Python - -```{python} -#| results: hide -(ggplot(diabetes_py, - aes(x = "glucose", - y = "test_result")) + - geom_point()) -``` - -::: - -This presents us with a bit of an issue. We could fit a linear regression model to these data, although we already know that this is a bad idea... - -::: {.panel-tabset group="language"} -## R - -```{r} -#| message: false -diabetes %>% - ggplot(aes(x = glucose, y = test_result)) + - geom_point() + - geom_smooth(method = "lm") -``` - -## Python - -```{python} -#| results: hide -(ggplot(diabetes_py, - aes(x = "glucose", - y = "test_result")) + - geom_point() + - geom_smooth(method = "lm", - colour = "blue")) -``` - -::: - -Of course this is rubbish - we can't have test results outside the range of \[0, 1\]. - -But for the sake of exploration, let's look at the assumptions: - -::: {.panel-tabset group="language"} -## R - -```{r} -lm_dia <- lm(test_result ~ glucose, - data = diabetes) - -lm_dia %>% - resid_panel(plots = c("resid", "qq", "ls", "cookd"), - smoother = TRUE) -``` - -## Python - -First, we create a linear model: - -```{python} -# create a linear model -model = smf.ols(formula= "test_result ~ glucose", - data = diabetes_py) -# and get the fitted parameters of the model -lm_dia_py = model.fit() -``` - -Next, we can create the diagnostic plots: -```{python} -#| eval: false -dgplots(lm_dia_py) -``` - -```{python} -#| echo: false -# load dgplots function for knitr -exec(open('setup_files/dgplots_knitr.py').read()) -# create rendered diagnostic plots image -# and save the file link -dgplot = dgplotsknitr(lm_dia_py) -``` - -```{r} -#| echo: false -library(reticulate) -# display image -knitr::include_graphics(py$dgplot) -``` - -::: - -They're ~~pretty~~ extremely bad. - -- The response is not linear (Residual Plot, binary response plot, common sense). -- The residuals are not distributed normally (Q-Q Plot) -- The variance is not homogeneous across the predicted values (Location-Scale Plot) -- But - there is always a silver lining - we don't have influential data points. - -::: {.callout-note} -## Viewing residuals - -Another way of viewing the residuals (apart from the Q-Q plot) is as a dot-plot. In R, the `ggdist` and `distributional` packages are extremely useful for this kind of stuff. - -What I'm doing here is: - -- define the model -- create a normal distribution with $\mu = 0$ and $\sigma = 0.415$ (I've extracted the $\sigma$ value from the residuals with `rstatix::get_summary_stats()`) -- plot the residuals - -```{r} -diabetes %>% - lm(test_result ~ glucose, data = .) %>% - resid() %>% - as_tibble() %>% - # rstatix::get_summary_stats() - ggplot(aes(x = value)) + - stat_dist_halfeye(aes(dist = dist_normal(0, 0.415)), - orientation = "horizontal") + - stat_dotsinterval(aes(x = value), - orientation = "horizontal", - fill = "firebrick", scale = 1) + - labs(title = "Linear model (diabetes)", y = "probability", x = NULL) -``` - -This again shows us that the residuals are really not normally distributed. If they were, then they would overlap much more closely with the distribution (in grey). -::: - -## Creating a suitable model - -So far we've established that using a simple linear model to describe a potential relationship between `glucose` levels and the probability of getting a positive test result is not a good idea. So, what _can_ we do? - -One of the ways we can deal with binary outcome data is by performing a logistic regression. Instead of fitting a straight line to our data, and performing a regression on that, we fit a line that has an S shape. This avoids the model making predictions outside the $[0, 1]$ range. - -There are many mathematical functions that produce S-shaped graphs. The **logistic function** is one of them and well-suited to these kind of data. - -In the most simple form a logistic function is written like this: - -$Y = \frac{\exp(X)}{1 + \exp(X)}$ - -:::{.callout-important} -## Euler's number - -In mathematics, $\rm e$ represents a constant of around 2.718. Another notation is $\exp$, which is often used when notations become a bit cumbersome. Here, I exclusively use the $\exp$ notation for consistency. -::: - -We can _generalise_ this, by writing it as follows: - -$Y = \frac{\exp(X)}{1 + \exp(\beta_0 + \beta_1X)}$ - -Note that the $\beta_0 + \beta_1X$ part is identical to the formula of a straight line. We've come across this before when we were doing simple linear regression! - -The rest of the function is what makes the straight line curve into its characteristic S shape. The middle of the S (where $Y = 0.5$) occurs when $X = \frac{-a}{b}$. - -::: {.callout-important} -## The logistic function - -The shape of the logistic function is hugely influenced by the different parameters, in particular $\beta_1$. The plots below show different situations, where $\beta_0 = 0$ in all cases, but $\beta_1$ varies. - -The first plot shows the logistic function in its simplest form, with the others showing the effect of varying $\beta_1$. - -```{r} -#| echo: false -#| warning: false -X <- seq(-10, 10 ,0.1) - -b0 <- c(0, 0, 0, 0) -b1 <- c(1, 0, -1.5, 10) - -cdf = function(b0, b1){ - return(tibble(X = X, - Y = exp(b0 + b1 * X)/(1 + exp(b0 + b1 * X)))) - } - -df <- map2(b0, b1, cdf) - -df %>% - bind_rows(.id = "keys") %>% - mutate(keys = str_replace(keys, "1", "logistic function")) %>% - mutate(keys = str_replace(keys, "2", "when \u03b21 = 0")) %>% - mutate(keys = str_replace(keys, "3", "when \u03b21 is negative")) %>% - mutate(keys = str_replace(keys, "4", "when \u03b21 is large")) %>% - ggplot(aes(x = X, y = Y)) + - geom_line() + - facet_wrap(vars(keys)) -``` - -* when $\beta_1 = 1$, this gives the simplest logistic function -* when $\beta_1 = 0$ gives a horizontal line, with $Y = \frac{\exp(\beta_0)}{1+\exp(\beta_0)}$ -* when $\beta_1$ is negative flips the curve around, so it slopes down -* when $\beta_1$ is very large then the curve becomes extremely steep -::: - -We can fit such an S-shaped curve to our `diabetes` data set, by creating a generalised linear model. - -::: {.panel-tabset group="language"} -## R - -In R we have a few options to do this, and by far the most familiar function would be `glm()`. Here we save the model in an object called `glm_dia`: - -```{r} -glm_dia <- glm(test_result ~ glucose, - family = binomial, - data = diabetes) -``` - -The format of this function is similar to that used by the `lm()` function for linear models. The important difference is that we must specify the _family_ of error distribution to use. For logistic regression we must set the family to **binomial**. - -If you forget to set the `family` argument, then the `glm()` function will perform a standard linear model fit, identical to what the `lm()` function would do. - -## Python - -```{python} -# create a linear model -model = smf.glm(formula= "test_result ~ glucose", - family = sm.families.Binomial(), - data = diabetes_py) -# and get the fitted parameters of the model -glm_dia_py = model.fit() -``` - -::: - -## Model output - -That's the easy part done! The trickier part is interpreting the output. First of all, we'll get some summary information. - -::: {.panel-tabset group="language"} -## R - -```{r} -summary(glm_dia) -``` - - -## Python - -```{python} -print(glm_dia_py.summary()) -``` - -::: - -There’s a lot to unpack here so take a deep breath (or make sure you have a coffee) before continuing... - -::: {.panel-tabset group="language"} -## R - -* The first lines just confirm which model we’ve been fitting (trust me, this can be useful when you’re in the middle of a load of analysis and you’ve lost track of what the hell is going on!) -* The next block is called Deviance Residuals. This isn’t particularly useful, but just so you know: for linear models the residuals were calculated for each data point and then squared and added up to get the `SS` (sum of squares), which is used to fit the model. For generalised linear models we don’t use SS to fit the model and instead we use an entirely different method called maximum likelihood. This fitting procedure generates a different quantity, called Deviance, which is the analogue of SS. A deviance of zero indicates the best model we could hope for and bigger values indicate a model that doesn’t fit quite as well. The deviance residuals are then values associated with each data point, that when squared and summed give the deviance for the model (an exact analogy to normal residuals). You’re unlikely to ever need to know this, but I had some time on my hands and decided to share this little nugget with you :-) -* The `Coefficients` block is next. The main numbers to extract from the output are the two numbers underneath `Estimate.Std`: - -``` -(Intercept) -5.611732 -glucose 0.039510 -``` - -## Python - -The top part of table contains a lot of information about the model we’ve just created at a high level. The left-hand column just tells us a lot of stuff that we really don’t care about like: - -* the name of the dependent variable (which we already know because we entered it ourselves!); -* the type of model (a binomial glm which we again already knew). -* the type of link function (in this case a logit link – we don’t really care -about this either). -* Generalized Linear Models don’t have exact analytical methods for -fitting to the data and they all use some form of iterative algorithm. In this case we used the standard method called Iteratively Reweighted Least Squares (IRLS), -* and the model took 4 iterations to converge. -* This left-hand column also helpfully tells us the time and date when the -analysis was carried out – how helpful! - -The right-hand column starts to give us some more useful summary statistics -that we’ll end up using. In particular we care about the Deviance value. For linear models the residuals were calculated for each data point and then squared and added up to get the SS (sum of squares), which is used to fit the model. For generalised linear models we don’t use SS to fit the model and instead we use an entirely different method called maximum likelihood. This fitting procedure generates a different quantity, called Deviance, which is the analogue of SS. A deviance of zero indicates the best model we could hope for (basically that the model completely matches the data) and bigger values indicate a model that doesn’t fit quite as well. We’ll also need the Df residuals value here (degrees of freedom) for some later calculations. - -Right at the bottom is a table showing the model coefficients. The main numbers to extract from the output are the two numbers in the `coef` column: - -``` -====================== - coef ----------------------- -Intercept -5.6117 -glucose 0.0395 -====================== -``` - -::: - -These are the coefficients of the logistic model equation and need to be placed in the correct equation if we want to be able to actually calculate the probability of having a positive diabetes test for a given glucose level. - -The p values (`Pr(>|z|`) at the end of each coefficient row merely show whether that particular coefficient is significantly different from zero. This is similar to the p-values obtained in the summary output of a linear model, and as before, for continuous predictors in simple models these p-values can be used to decide whether that predictor is important (so in this case glucose appears to be significant). However, these p-values aren’t great when we have multiple predictor variables, or when we have categorical predictors with multiple levels (since the output will give us a p-value for each level rather than for the predictor as a whole). - -We can use the coefficients to actually calculate the probability of having a positive diabetes test for a given glucose level: - -$$ P(positive \ test\ result) = \frac{\exp(-5.61 + 0.04 \times glucose)}{1 + \exp(-5.61 + 0.04 \times glucose)} $$ - -::: {.callout-note collapse="true"} -## Expanded explanation - -We started with: - -- Our linear predictor equation $$logit(p) = \beta_0 + \beta_1 x_1 + ... + \beta_p x_p$$ - -- and link function $$logit(p) = log(\frac{p}{1 - p})$$ - -Our coefficients are as follows: - -| coefficient | value | -|:-----------------------------------|:-----------------------------------| -| $\beta_0$ | `r glm_dia %>% tidy() %>% filter(term == "(Intercept)") %>% pull(estimate)` | -| $\beta_1$ | `r glm_dia %>% tidy() %>% filter(term == "glucose") %>% pull(estimate)` | - -Which means that we can write the linear predictor equation as follows: - -$$logit(p) = -5.61 + 0.04 \times glucose$$ We still have to take into account our link function. Combining the two equations gives us: - -$$log(\frac{p}{1 - p}) = -5.61 + 0.04 \times glucose$$ - -To get our $p$ (the probability of a person having a positive diabetes test result), we need to exponentiate our equation: - -$$\frac{p}{1 - p} = \exp{(-5.61 + 0.04 \times glucose)}$$ - -leading to... - -$$p = \frac{\exp{(-5.61 + 0.04 \times glucose)}}{1 + \exp{(-5.61 + 0.04 \times glucose)}}$$ -::: - -::: {.panel-tabset group="language"} -## R - -* The next line tells us that the dispersion parameter was assumed to be 1 for this binomial model. Dispersion is a property that says whether the data were more or less spread out around the logistic curve than we would expect. A dispersion parameter of 1 says that the data are spread out exactly as we would expect. Greater than 1 is called over-dispersion and less than 1 is called under-dispersion. Here this line is saying that when we fitted this model, we were assuming that the dispersion of the data was exactly 1. For binary data, like we have here, the data cannot be over or under-dispersed but this is something that we’ll need to check for other sorts of glms. - -The last three lines relate to quantities called deviance and AIC (Akaike Information Criterion). - -As we said just above, the deviance values are the equivalent of Sums of Squares values in linear models (and are a product of the technique used to fit the curve to the data). They can be used as a metric of goodness of fit for the model, with a deviance of 0 indicating a perfect fitting model. The deviance for the null model (i.e. the model without any predictors, basically saying that the probability of getting a positive diabetes score is constant and doesn’t depend on glucose level) is given by the first line and the deviance for the actual model is given by the residual deviance line. We will see how we can use the deviance to do two things. - -1. to check of whether the model is actually any good (i.e. does it in any way look like it’s close to the data). This is akin to what we were doing with $R^2$ values in linear models. -2. to check if the model we’ve specified is better than the null model. - -It’s important to realise that these two things can be independent of each other; we can have a model that is significantly better than a null model whilst still being rubbish overall (the null model will have been even more rubbish in comparison), and we can have a model that is brilliant yet still not be better than the null model (in this case the null model was already brilliant). - -As we found in the [model comparisons practical](https://cambiotraining.github.io/corestats/materials/cs5_practical_model-comparisons.html), the AIC value is meaningless by itself, but it will allow us to compare this model to another model with different terms (with the model with the smaller AIC value being the better fitting model). - -## Python - -There are some additional parameters in the right-hand column, but we'll first focus on how we can assess 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} -.bi-grid::before { content: "\f3fc"; } -.bi-grip-horizontal::before { content: "\f3fd"; } -.bi-grip-vertical::before { content: "\f3fe"; } -.bi-hammer::before { content: "\f3ff"; } -.bi-hand-index-fill::before { content: "\f400"; } -.bi-hand-index-thumb-fill::before { content: "\f401"; } -.bi-hand-index-thumb::before { content: "\f402"; } -.bi-hand-index::before { content: "\f403"; } -.bi-hand-thumbs-down-fill::before { content: "\f404"; } -.bi-hand-thumbs-down::before { content: "\f405"; } -.bi-hand-thumbs-up-fill::before { content: "\f406"; } -.bi-hand-thumbs-up::before { content: "\f407"; } -.bi-handbag-fill::before { content: "\f408"; } -.bi-handbag::before { content: "\f409"; } -.bi-hash::before { content: "\f40a"; } -.bi-hdd-fill::before { content: "\f40b"; } -.bi-hdd-network-fill::before { content: "\f40c"; } -.bi-hdd-network::before { content: "\f40d"; } -.bi-hdd-rack-fill::before { content: "\f40e"; } -.bi-hdd-rack::before { content: "\f40f"; } -.bi-hdd-stack-fill::before { content: "\f410"; } -.bi-hdd-stack::before { content: "\f411"; } -.bi-hdd::before { content: "\f412"; } -.bi-headphones::before { content: "\f413"; } -.bi-headset::before { content: "\f414"; } -.bi-heart-fill::before { content: "\f415"; } -.bi-heart-half::before { content: "\f416"; } -.bi-heart::before { content: "\f417"; } -.bi-heptagon-fill::before { content: "\f418"; } -.bi-heptagon-half::before { content: "\f419"; } -.bi-heptagon::before { content: "\f41a"; } -.bi-hexagon-fill::before { content: "\f41b"; } -.bi-hexagon-half::before { content: "\f41c"; } -.bi-hexagon::before { content: "\f41d"; } -.bi-hourglass-bottom::before { content: "\f41e"; } -.bi-hourglass-split::before { content: "\f41f"; } -.bi-hourglass-top::before { content: "\f420"; } -.bi-hourglass::before { content: "\f421"; } -.bi-house-door-fill::before { content: "\f422"; } -.bi-house-door::before { content: "\f423"; } -.bi-house-fill::before { content: "\f424"; } -.bi-house::before { content: "\f425"; } -.bi-hr::before { content: "\f426"; } -.bi-hurricane::before { content: "\f427"; } -.bi-image-alt::before { content: "\f428"; } -.bi-image-fill::before { content: "\f429"; } -.bi-image::before { content: "\f42a"; } -.bi-images::before { content: "\f42b"; } -.bi-inbox-fill::before { content: "\f42c"; } -.bi-inbox::before { content: "\f42d"; } -.bi-inboxes-fill::before { content: "\f42e"; } -.bi-inboxes::before { content: "\f42f"; } -.bi-info-circle-fill::before { content: "\f430"; } -.bi-info-circle::before { content: "\f431"; } -.bi-info-square-fill::before { content: "\f432"; } -.bi-info-square::before { content: "\f433"; } -.bi-info::before { content: "\f434"; } -.bi-input-cursor-text::before { content: "\f435"; } -.bi-input-cursor::before { content: "\f436"; } -.bi-instagram::before { content: "\f437"; } -.bi-intersect::before { content: "\f438"; } -.bi-journal-album::before { content: "\f439"; } -.bi-journal-arrow-down::before { content: "\f43a"; } -.bi-journal-arrow-up::before { content: "\f43b"; } -.bi-journal-bookmark-fill::before { content: "\f43c"; } -.bi-journal-bookmark::before { content: "\f43d"; } -.bi-journal-check::before { content: "\f43e"; } -.bi-journal-code::before { content: "\f43f"; } -.bi-journal-medical::before { content: "\f440"; } -.bi-journal-minus::before { content: "\f441"; } -.bi-journal-plus::before { content: "\f442"; } -.bi-journal-richtext::before { content: "\f443"; } -.bi-journal-text::before { content: "\f444"; } -.bi-journal-x::before { content: "\f445"; } -.bi-journal::before { content: "\f446"; } -.bi-journals::before { content: "\f447"; } -.bi-joystick::before { content: "\f448"; } -.bi-justify-left::before { content: "\f449"; } -.bi-justify-right::before { content: "\f44a"; } -.bi-justify::before { content: "\f44b"; } -.bi-kanban-fill::before { content: "\f44c"; } -.bi-kanban::before { content: "\f44d"; } -.bi-key-fill::before { content: "\f44e"; } -.bi-key::before { content: "\f44f"; } -.bi-keyboard-fill::before { content: "\f450"; } -.bi-keyboard::before { content: "\f451"; } -.bi-ladder::before { content: "\f452"; } -.bi-lamp-fill::before { content: "\f453"; } -.bi-lamp::before { content: "\f454"; } -.bi-laptop-fill::before { content: "\f455"; } -.bi-laptop::before { content: "\f456"; } -.bi-layer-backward::before { content: "\f457"; } -.bi-layer-forward::before { content: "\f458"; } -.bi-layers-fill::before { content: "\f459"; } -.bi-layers-half::before { content: "\f45a"; } -.bi-layers::before { content: "\f45b"; } -.bi-layout-sidebar-inset-reverse::before { content: "\f45c"; } -.bi-layout-sidebar-inset::before { content: "\f45d"; } -.bi-layout-sidebar-reverse::before { content: "\f45e"; } -.bi-layout-sidebar::before { content: "\f45f"; } -.bi-layout-split::before { content: "\f460"; } -.bi-layout-text-sidebar-reverse::before { content: "\f461"; } -.bi-layout-text-sidebar::before { content: "\f462"; } -.bi-layout-text-window-reverse::before { content: "\f463"; } -.bi-layout-text-window::before { content: "\f464"; } -.bi-layout-three-columns::before { content: "\f465"; } -.bi-layout-wtf::before { content: "\f466"; } -.bi-life-preserver::before { content: "\f467"; } -.bi-lightbulb-fill::before { content: "\f468"; } -.bi-lightbulb-off-fill::before { content: "\f469"; } -.bi-lightbulb-off::before { content: "\f46a"; } -.bi-lightbulb::before { content: "\f46b"; } -.bi-lightning-charge-fill::before { content: "\f46c"; } -.bi-lightning-charge::before { content: "\f46d"; } -.bi-lightning-fill::before { content: "\f46e"; } -.bi-lightning::before { content: "\f46f"; } -.bi-link-45deg::before { content: "\f470"; } -.bi-link::before { content: "\f471"; } -.bi-linkedin::before { content: "\f472"; } -.bi-list-check::before { content: "\f473"; } -.bi-list-nested::before { content: "\f474"; } -.bi-list-ol::before { content: "\f475"; } -.bi-list-stars::before { content: "\f476"; } -.bi-list-task::before { content: "\f477"; } -.bi-list-ul::before { content: "\f478"; } -.bi-list::before { content: "\f479"; } -.bi-lock-fill::before { content: "\f47a"; } -.bi-lock::before { content: "\f47b"; } -.bi-mailbox::before { content: "\f47c"; } -.bi-mailbox2::before { content: "\f47d"; } -.bi-map-fill::before { content: "\f47e"; } -.bi-map::before { content: "\f47f"; } -.bi-markdown-fill::before { content: "\f480"; } -.bi-markdown::before { content: "\f481"; } -.bi-mask::before { content: "\f482"; } -.bi-megaphone-fill::before { content: "\f483"; } -.bi-megaphone::before { content: "\f484"; } -.bi-menu-app-fill::before { content: "\f485"; } -.bi-menu-app::before { content: "\f486"; } -.bi-menu-button-fill::before { content: "\f487"; } -.bi-menu-button-wide-fill::before { content: "\f488"; } -.bi-menu-button-wide::before { content: "\f489"; } -.bi-menu-button::before { content: "\f48a"; } -.bi-menu-down::before { content: "\f48b"; } -.bi-menu-up::before { content: "\f48c"; } -.bi-mic-fill::before { content: "\f48d"; } -.bi-mic-mute-fill::before { content: "\f48e"; } -.bi-mic-mute::before { content: "\f48f"; } -.bi-mic::before { content: "\f490"; } -.bi-minecart-loaded::before { content: "\f491"; } -.bi-minecart::before { content: "\f492"; } -.bi-moisture::before { content: "\f493"; } -.bi-moon-fill::before { content: "\f494"; } -.bi-moon-stars-fill::before { content: "\f495"; } -.bi-moon-stars::before { content: "\f496"; } -.bi-moon::before { content: "\f497"; } -.bi-mouse-fill::before { content: "\f498"; } -.bi-mouse::before { content: "\f499"; } -.bi-mouse2-fill::before { content: "\f49a"; } -.bi-mouse2::before { content: "\f49b"; } -.bi-mouse3-fill::before { content: "\f49c"; } -.bi-mouse3::before { content: "\f49d"; } -.bi-music-note-beamed::before { content: "\f49e"; } -.bi-music-note-list::before { content: "\f49f"; } -.bi-music-note::before { content: "\f4a0"; } -.bi-music-player-fill::before { content: "\f4a1"; } -.bi-music-player::before { content: "\f4a2"; } -.bi-newspaper::before { content: "\f4a3"; } -.bi-node-minus-fill::before { content: "\f4a4"; } -.bi-node-minus::before { content: "\f4a5"; } -.bi-node-plus-fill::before { content: "\f4a6"; } -.bi-node-plus::before { content: "\f4a7"; } -.bi-nut-fill::before { content: "\f4a8"; } -.bi-nut::before { content: "\f4a9"; } -.bi-octagon-fill::before { content: "\f4aa"; } -.bi-octagon-half::before { content: "\f4ab"; } -.bi-octagon::before { content: "\f4ac"; } -.bi-option::before { content: "\f4ad"; } -.bi-outlet::before { content: "\f4ae"; } -.bi-paint-bucket::before { content: "\f4af"; } -.bi-palette-fill::before { content: "\f4b0"; } -.bi-palette::before { content: "\f4b1"; } -.bi-palette2::before { content: "\f4b2"; } -.bi-paperclip::before { content: "\f4b3"; } -.bi-paragraph::before { content: "\f4b4"; } -.bi-patch-check-fill::before { content: "\f4b5"; } -.bi-patch-check::before { content: "\f4b6"; } -.bi-patch-exclamation-fill::before { content: "\f4b7"; } -.bi-patch-exclamation::before { content: "\f4b8"; } -.bi-patch-minus-fill::before { content: "\f4b9"; } -.bi-patch-minus::before { content: "\f4ba"; } -.bi-patch-plus-fill::before { content: "\f4bb"; } -.bi-patch-plus::before { content: "\f4bc"; } -.bi-patch-question-fill::before { content: "\f4bd"; } -.bi-patch-question::before { content: "\f4be"; } -.bi-pause-btn-fill::before { content: "\f4bf"; } -.bi-pause-btn::before { content: "\f4c0"; } -.bi-pause-circle-fill::before { content: "\f4c1"; } -.bi-pause-circle::before { content: "\f4c2"; } -.bi-pause-fill::before { content: "\f4c3"; } -.bi-pause::before { content: "\f4c4"; } -.bi-peace-fill::before { content: "\f4c5"; } -.bi-peace::before { content: "\f4c6"; } -.bi-pen-fill::before { content: "\f4c7"; } -.bi-pen::before { content: "\f4c8"; } -.bi-pencil-fill::before { content: "\f4c9"; } -.bi-pencil-square::before { content: "\f4ca"; } -.bi-pencil::before { content: "\f4cb"; } -.bi-pentagon-fill::before { content: "\f4cc"; } -.bi-pentagon-half::before { content: "\f4cd"; } -.bi-pentagon::before { content: "\f4ce"; } -.bi-people-fill::before { content: "\f4cf"; } -.bi-people::before { content: "\f4d0"; } -.bi-percent::before { content: "\f4d1"; } -.bi-person-badge-fill::before { content: "\f4d2"; } -.bi-person-badge::before { content: "\f4d3"; } -.bi-person-bounding-box::before { content: "\f4d4"; } -.bi-person-check-fill::before { content: "\f4d5"; } -.bi-person-check::before { content: "\f4d6"; } -.bi-person-circle::before { content: "\f4d7"; } -.bi-person-dash-fill::before { content: "\f4d8"; } -.bi-person-dash::before { content: "\f4d9"; } -.bi-person-fill::before { content: "\f4da"; } -.bi-person-lines-fill::before { content: "\f4db"; } -.bi-person-plus-fill::before { content: "\f4dc"; } -.bi-person-plus::before { content: "\f4dd"; } -.bi-person-square::before { content: "\f4de"; } -.bi-person-x-fill::before { content: "\f4df"; } -.bi-person-x::before { content: "\f4e0"; } -.bi-person::before { content: "\f4e1"; } -.bi-phone-fill::before { content: "\f4e2"; } -.bi-phone-landscape-fill::before { content: "\f4e3"; } -.bi-phone-landscape::before { content: "\f4e4"; } -.bi-phone-vibrate-fill::before { content: "\f4e5"; } -.bi-phone-vibrate::before { content: "\f4e6"; } -.bi-phone::before { content: "\f4e7"; } -.bi-pie-chart-fill::before { content: "\f4e8"; } -.bi-pie-chart::before { content: "\f4e9"; } -.bi-pin-angle-fill::before { content: "\f4ea"; } -.bi-pin-angle::before { content: "\f4eb"; } -.bi-pin-fill::before { content: "\f4ec"; } -.bi-pin::before { content: "\f4ed"; } -.bi-pip-fill::before { content: "\f4ee"; } -.bi-pip::before { content: "\f4ef"; } -.bi-play-btn-fill::before { content: "\f4f0"; } -.bi-play-btn::before { content: "\f4f1"; } -.bi-play-circle-fill::before { content: "\f4f2"; } -.bi-play-circle::before { content: "\f4f3"; } -.bi-play-fill::before { content: "\f4f4"; } -.bi-play::before { content: "\f4f5"; } -.bi-plug-fill::before { content: "\f4f6"; } -.bi-plug::before { content: "\f4f7"; } -.bi-plus-circle-dotted::before { content: "\f4f8"; } -.bi-plus-circle-fill::before { content: "\f4f9"; } -.bi-plus-circle::before { content: "\f4fa"; } -.bi-plus-square-dotted::before { content: "\f4fb"; } -.bi-plus-square-fill::before { content: "\f4fc"; } -.bi-plus-square::before { content: "\f4fd"; } -.bi-plus::before { content: "\f4fe"; } -.bi-power::before { content: "\f4ff"; } -.bi-printer-fill::before { content: "\f500"; } -.bi-printer::before { content: "\f501"; } -.bi-puzzle-fill::before { content: "\f502"; } -.bi-puzzle::before { content: "\f503"; } -.bi-question-circle-fill::before { content: "\f504"; } -.bi-question-circle::before { content: "\f505"; } -.bi-question-diamond-fill::before { content: "\f506"; } -.bi-question-diamond::before { content: "\f507"; } -.bi-question-octagon-fill::before { content: "\f508"; } -.bi-question-octagon::before { content: "\f509"; } -.bi-question-square-fill::before { content: "\f50a"; } -.bi-question-square::before { content: "\f50b"; } -.bi-question::before { content: "\f50c"; } -.bi-rainbow::before { content: "\f50d"; } -.bi-receipt-cutoff::before { content: "\f50e"; } -.bi-receipt::before { content: "\f50f"; } -.bi-reception-0::before { content: "\f510"; } -.bi-reception-1::before { content: "\f511"; } -.bi-reception-2::before { content: "\f512"; } -.bi-reception-3::before { content: "\f513"; } -.bi-reception-4::before { content: "\f514"; } -.bi-record-btn-fill::before { content: "\f515"; } -.bi-record-btn::before { content: "\f516"; } -.bi-record-circle-fill::before { content: "\f517"; } -.bi-record-circle::before { content: "\f518"; } -.bi-record-fill::before { content: "\f519"; } -.bi-record::before { content: "\f51a"; } -.bi-record2-fill::before { content: "\f51b"; } -.bi-record2::before { content: "\f51c"; } -.bi-reply-all-fill::before { content: "\f51d"; } -.bi-reply-all::before { content: "\f51e"; } -.bi-reply-fill::before { content: "\f51f"; } -.bi-reply::before { content: "\f520"; } -.bi-rss-fill::before { content: "\f521"; } -.bi-rss::before { content: "\f522"; } -.bi-rulers::before { content: "\f523"; } -.bi-save-fill::before { content: "\f524"; } -.bi-save::before { content: "\f525"; } -.bi-save2-fill::before { content: "\f526"; } -.bi-save2::before { content: "\f527"; } -.bi-scissors::before { content: "\f528"; } -.bi-screwdriver::before { content: "\f529"; } -.bi-search::before { content: "\f52a"; } -.bi-segmented-nav::before { content: "\f52b"; } -.bi-server::before { content: "\f52c"; } -.bi-share-fill::before { content: "\f52d"; } -.bi-share::before { content: "\f52e"; } -.bi-shield-check::before { content: "\f52f"; } -.bi-shield-exclamation::before { content: "\f530"; } -.bi-shield-fill-check::before { content: "\f531"; } -.bi-shield-fill-exclamation::before { content: "\f532"; } -.bi-shield-fill-minus::before { content: "\f533"; } -.bi-shield-fill-plus::before { content: "\f534"; } -.bi-shield-fill-x::before { content: "\f535"; } -.bi-shield-fill::before { content: "\f536"; } -.bi-shield-lock-fill::before { content: "\f537"; } -.bi-shield-lock::before { content: "\f538"; } -.bi-shield-minus::before { content: "\f539"; } -.bi-shield-plus::before { content: "\f53a"; } -.bi-shield-shaded::before { content: "\f53b"; } -.bi-shield-slash-fill::before { content: "\f53c"; } -.bi-shield-slash::before { content: "\f53d"; } -.bi-shield-x::before { content: "\f53e"; } -.bi-shield::before { content: "\f53f"; } -.bi-shift-fill::before { content: "\f540"; } -.bi-shift::before { content: "\f541"; } -.bi-shop-window::before { content: "\f542"; } -.bi-shop::before { content: "\f543"; } -.bi-shuffle::before { content: "\f544"; } -.bi-signpost-2-fill::before { content: "\f545"; } -.bi-signpost-2::before { content: "\f546"; } -.bi-signpost-fill::before { content: "\f547"; } -.bi-signpost-split-fill::before { content: "\f548"; } -.bi-signpost-split::before { content: "\f549"; } -.bi-signpost::before { content: "\f54a"; } -.bi-sim-fill::before { content: "\f54b"; } -.bi-sim::before { content: "\f54c"; } 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"\f55f"; } -.bi-skip-start-btn::before { content: "\f560"; } -.bi-skip-start-circle-fill::before { content: "\f561"; } -.bi-skip-start-circle::before { content: "\f562"; } -.bi-skip-start-fill::before { content: "\f563"; } -.bi-skip-start::before { content: "\f564"; } -.bi-slack::before { content: "\f565"; } -.bi-slash-circle-fill::before { content: "\f566"; } -.bi-slash-circle::before { content: "\f567"; } -.bi-slash-square-fill::before { content: "\f568"; } -.bi-slash-square::before { content: "\f569"; } -.bi-slash::before { content: "\f56a"; } -.bi-sliders::before { content: "\f56b"; } -.bi-smartwatch::before { content: "\f56c"; } -.bi-snow::before { content: "\f56d"; } -.bi-snow2::before { content: "\f56e"; } -.bi-snow3::before { content: "\f56f"; } -.bi-sort-alpha-down-alt::before { content: "\f570"; } -.bi-sort-alpha-down::before { content: "\f571"; } -.bi-sort-alpha-up-alt::before { content: "\f572"; } -.bi-sort-alpha-up::before { content: "\f573"; } -.bi-sort-down-alt::before { content: "\f574"; } -.bi-sort-down::before { content: "\f575"; } -.bi-sort-numeric-down-alt::before { content: "\f576"; } -.bi-sort-numeric-down::before { content: "\f577"; } -.bi-sort-numeric-up-alt::before { content: "\f578"; } -.bi-sort-numeric-up::before { content: "\f579"; } -.bi-sort-up-alt::before { content: "\f57a"; } -.bi-sort-up::before { content: "\f57b"; } -.bi-soundwave::before { content: "\f57c"; } -.bi-speaker-fill::before { content: "\f57d"; } -.bi-speaker::before { content: "\f57e"; } -.bi-speedometer::before { content: "\f57f"; } -.bi-speedometer2::before { content: "\f580"; } -.bi-spellcheck::before { content: "\f581"; } -.bi-square-fill::before { content: "\f582"; } -.bi-square-half::before { content: "\f583"; } -.bi-square::before { content: "\f584"; } -.bi-stack::before { content: "\f585"; } -.bi-star-fill::before { content: "\f586"; } -.bi-star-half::before { content: "\f587"; } -.bi-star::before { content: "\f588"; } -.bi-stars::before { content: "\f589"; } -.bi-stickies-fill::before { content: "\f58a"; } -.bi-stickies::before { content: "\f58b"; } -.bi-sticky-fill::before { content: "\f58c"; } -.bi-sticky::before { content: "\f58d"; } -.bi-stop-btn-fill::before { content: "\f58e"; } -.bi-stop-btn::before { content: "\f58f"; } -.bi-stop-circle-fill::before { content: "\f590"; } -.bi-stop-circle::before { content: "\f591"; } -.bi-stop-fill::before { content: "\f592"; } -.bi-stop::before { content: "\f593"; } -.bi-stoplights-fill::before { content: "\f594"; } -.bi-stoplights::before { content: "\f595"; } -.bi-stopwatch-fill::before { content: "\f596"; } -.bi-stopwatch::before { content: "\f597"; } -.bi-subtract::before { content: "\f598"; } -.bi-suit-club-fill::before { content: "\f599"; } -.bi-suit-club::before { content: "\f59a"; } -.bi-suit-diamond-fill::before { content: "\f59b"; } -.bi-suit-diamond::before { content: "\f59c"; } -.bi-suit-heart-fill::before { content: "\f59d"; } -.bi-suit-heart::before { content: "\f59e"; } 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1.5em [body-end body-end-outset page-end-inset page-end] 5fr [screen-end-inset] 1.5em [screen-end]}body.floating .page-columns{display:grid;gap:0;grid-template-columns:[screen-start] 1.5em [screen-start-inset] 5fr [page-start page-start-inset body-start-outset body-start] 1em [body-content-start] minmax(500px, calc(750px - 3em)) [body-content-end] 1.5em [body-end] 50px [body-end-outset] minmax(75px, 150px) [page-end-inset] 25px [page-end] 5fr [screen-end-inset] 1.5em [screen-end]}body.docked .page-columns{display:grid;gap:0;grid-template-columns:[screen-start] 1.5em [screen-start-inset page-start page-start-inset body-start-outset body-start body-content-start] minmax(500px, calc(750px - 3em)) [body-content-end] 1.5em [body-end] 50px [body-end-outset] minmax(25px, 50px) [page-end-inset] 50px [page-end] 5fr [screen-end-inset] 1.5em [screen-end]}body.docked.fullcontent .page-columns{display:grid;gap:0;grid-template-columns:[screen-start] 1.5em [screen-start-inset page-start page-start-inset body-start-outset body-start body-content-start] minmax(500px, calc( 1000px - 3em )) [body-content-end] 1.5em [body-end body-end-outset page-end-inset page-end] 5fr [screen-end-inset] 1.5em [screen-end]}body.floating.fullcontent .page-columns{display:grid;gap:0;grid-template-columns:[screen-start] 1.5em [screen-start-inset] 5fr [page-start page-start-inset body-start-outset body-start] 1em [body-content-start] minmax(500px, calc(800px - 3em)) [body-content-end] 1.5em [body-end body-end-outset page-end-inset page-end] 5fr [screen-end-inset] 1.5em [screen-end]}body.docked.slimcontent .page-columns{display:grid;gap:0;grid-template-columns:[screen-start] 1.5em [screen-start-inset page-start page-start-inset body-start-outset body-start body-content-start] minmax(500px, calc(750px - 3em)) [body-content-end] 1.5em [body-end] 35px [body-end-outset] minmax(75px, 145px) [page-end-inset] 35px [page-end] 5fr [screen-end-inset] 1.5em [screen-end]}body.docked.listing .page-columns{display:grid;gap:0;grid-template-columns:[screen-start] 1.5em [screen-start-inset page-start page-start-inset body-start-outset body-start body-content-start] minmax(500px, calc(750px - 3em)) [body-content-end] 1.5em [body-end] 50px [body-end-outset] minmax(25px, 50px) [page-end-inset] 50px [page-end] 5fr [screen-end-inset] 1.5em [screen-end]}body.floating.slimcontent .page-columns{display:grid;gap:0;grid-template-columns:[screen-start] 1.5em [screen-start-inset] 5fr [page-start page-start-inset body-start-outset body-start] 1em [body-content-start] minmax(500px, calc(750px - 3em)) [body-content-end] 1.5em [body-end] 35px [body-end-outset] minmax(75px, 145px) [page-end-inset] 35px [page-end] 4fr [screen-end-inset] 1.5em [screen-end]}body.floating.listing .page-columns{display:grid;gap:0;grid-template-columns:[screen-start] 1.5em [screen-start-inset] 5fr [page-start page-start-inset body-start-outset body-start] 1em [body-content-start] minmax(500px, calc(750px - 3em)) [body-content-end] 1.5em [body-end] 50px [body-end-outset] minmax(75px, 150px) [page-end-inset] 25px [page-end] 5fr [screen-end-inset] 1.5em [screen-end]}}@media(max-width: 767.98px){body .page-columns,body.fullcontent:not(.floating):not(.docked) .page-columns,body.slimcontent:not(.floating):not(.docked) .page-columns,body.docked .page-columns,body.docked.slimcontent .page-columns,body.docked.fullcontent .page-columns,body.floating .page-columns,body.floating.slimcontent .page-columns,body.floating.fullcontent .page-columns{display:grid;gap:0;grid-template-columns:[screen-start] 1.5em [screen-start-inset page-start page-start-inset body-start-outset body-start body-content-start] minmax(0px, 1fr) [body-content-end body-end body-end-outset page-end-inset page-end screen-end-inset] 1.5em [screen-end]}body:not(.floating):not(.docked) .page-columns.toc-left{display:grid;gap:0;grid-template-columns:[screen-start] 1.5em [screen-start-inset page-start page-start-inset body-start-outset body-start body-content-start] minmax(0px, 1fr) [body-content-end body-end body-end-outset page-end-inset page-end screen-end-inset] 1.5em [screen-end]}body:not(.floating):not(.docked) .page-columns.toc-left .page-columns{display:grid;gap:0;grid-template-columns:[screen-start] 1.5em [screen-start-inset page-start page-start-inset body-start-outset body-start body-content-start] minmax(0px, 1fr) [body-content-end body-end body-end-outset page-end-inset page-end screen-end-inset] 1.5em [screen-end]}nav[role=doc-toc]{display:none}}body,.page-row-navigation{grid-template-rows:[page-top] max-content [contents-top] max-content [contents-bottom] max-content [page-bottom]}.page-rows-contents{grid-template-rows:[content-top] minmax(max-content, 1fr) [content-bottom] minmax(60px, max-content) [page-bottom]}.page-full{grid-column:screen-start/screen-end !important}.page-columns>*{grid-column:body-content-start/body-content-end}.page-columns.column-page>*{grid-column:page-start/page-end}.page-columns.column-page-left>*{grid-column:page-start/body-content-end}.page-columns.column-page-right>*{grid-column:body-content-start/page-end}.page-rows{grid-auto-rows:auto}.header{grid-column:screen-start/screen-end;grid-row:page-top/contents-top}#quarto-content{padding:0;grid-column:screen-start/screen-end;grid-row:contents-top/contents-bottom}body.floating .sidebar.sidebar-navigation{grid-column:page-start/body-start;grid-row:content-top/page-bottom}body.docked .sidebar.sidebar-navigation{grid-column:screen-start/body-start;grid-row:content-top/page-bottom}.sidebar.toc-left{grid-column:page-start/body-start;grid-row:content-top/page-bottom}.sidebar.margin-sidebar{grid-column:body-end/page-end;grid-row:content-top/page-bottom}.page-columns .content{grid-column:body-content-start/body-content-end;grid-row:content-top/content-bottom;align-content:flex-start}.page-columns .page-navigation{grid-column:body-content-start/body-content-end;grid-row:content-bottom/page-bottom}.page-columns .footer{grid-column:screen-start/screen-end;grid-row:contents-bottom/page-bottom}.page-columns .column-body{grid-column:body-content-start/body-content-end}.page-columns .column-body-fullbleed{grid-column:body-start/body-end}.page-columns .column-body-outset{grid-column:body-start-outset/body-end-outset;z-index:998;transform:translate3d(0, 0, 0)}.page-columns .column-body-outset table{background:#fff}.page-columns .column-body-outset-left{grid-column:body-start-outset/body-content-end;z-index:998;transform:translate3d(0, 0, 0)}.page-columns .column-body-outset-left table{background:#fff}.page-columns .column-body-outset-right{grid-column:body-content-start/body-end-outset;z-index:998;transform:translate3d(0, 0, 0)}.page-columns .column-body-outset-right table{background:#fff}.page-columns .column-page{grid-column:page-start/page-end;z-index:998;transform:translate3d(0, 0, 0)}.page-columns .column-page table{background:#fff}.page-columns .column-page-inset{grid-column:page-start-inset/page-end-inset;z-index:998;transform:translate3d(0, 0, 0)}.page-columns .column-page-inset table{background:#fff}.page-columns .column-page-inset-left{grid-column:page-start-inset/body-content-end;z-index:998;transform:translate3d(0, 0, 0)}.page-columns .column-page-inset-left table{background:#fff}.page-columns .column-page-inset-right{grid-column:body-content-start/page-end-inset;z-index:998;transform:translate3d(0, 0, 0)}.page-columns .column-page-inset-right figcaption table{background:#fff}.page-columns .column-page-left{grid-column:page-start/body-content-end;z-index:998;transform:translate3d(0, 0, 0)}.page-columns .column-page-left table{background:#fff}.page-columns 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Be={placement:"bottom",modifiers:[],strategy:"absolute"};function Re(){for(var t=arguments.length,e=new Array(t),i=0;ij.on(t,"mouseover",d))),this._element.focus(),this._element.setAttribute("aria-expanded",!0),this._menu.classList.add(Je),this._element.classList.add(Je),j.trigger(this._element,"shown.bs.dropdown",t)}hide(){if(c(this._element)||!this._isShown(this._menu))return;const t={relatedTarget:this._element};this._completeHide(t)}dispose(){this._popper&&this._popper.destroy(),super.dispose()}update(){this._inNavbar=this._detectNavbar(),this._popper&&this._popper.update()}_completeHide(t){j.trigger(this._element,"hide.bs.dropdown",t).defaultPrevented||("ontouchstart"in document.documentElement&&[].concat(...document.body.children).forEach((t=>j.off(t,"mouseover",d))),this._popper&&this._popper.destroy(),this._menu.classList.remove(Je),this._element.classList.remove(Je),this._element.setAttribute("aria-expanded","false"),U.removeDataAttribute(this._menu,"popper"),j.trigger(this._element,"hidden.bs.dropdown",t))}_getConfig(t){if(t={...this.constructor.Default,...U.getDataAttributes(this._element),...t},a(Ue,t,this.constructor.DefaultType),"object"==typeof t.reference&&!o(t.reference)&&"function"!=typeof t.reference.getBoundingClientRect)throw new TypeError(`${Ue.toUpperCase()}: Option "reference" provided type "object" without a required "getBoundingClientRect" method.`);return t}_createPopper(t){if(void 0===Fe)throw new TypeError("Bootstrap's dropdowns require Popper (https://popper.js.org)");let e=this._element;"parent"===this._config.reference?e=t:o(this._config.reference)?e=r(this._config.reference):"object"==typeof this._config.reference&&(e=this._config.reference);const i=this._getPopperConfig(),n=i.modifiers.find((t=>"applyStyles"===t.name&&!1===t.enabled));this._popper=qe(e,this._menu,i),n&&U.setDataAttribute(this._menu,"popper","static")}_isShown(t=this._element){return t.classList.contains(Je)}_getMenuElement(){return V.next(this._element,ei)[0]}_getPlacement(){const t=this._element.parentNode;if(t.classList.contains("dropend"))return ri;if(t.classList.contains("dropstart"))return ai;const e="end"===getComputedStyle(this._menu).getPropertyValue("--bs-position").trim();return t.classList.contains("dropup")?e?ni:ii:e?oi:si}_detectNavbar(){return null!==this._element.closest(".navbar")}_getOffset(){const{offset:t}=this._config;return"string"==typeof t?t.split(",").map((t=>Number.parseInt(t,10))):"function"==typeof t?e=>t(e,this._element):t}_getPopperConfig(){const t={placement:this._getPlacement(),modifiers:[{name:"preventOverflow",options:{boundary:this._config.boundary}},{name:"offset",options:{offset:this._getOffset()}}]};return"static"===this._config.display&&(t.modifiers=[{name:"applyStyles",enabled:!1}]),{...t,..."function"==typeof this._config.popperConfig?this._config.popperConfig(t):this._config.popperConfig}}_selectMenuItem({key:t,target:e}){const i=V.find(".dropdown-menu .dropdown-item:not(.disabled):not(:disabled)",this._menu).filter(l);i.length&&v(i,e,t===Ye,!i.includes(e)).focus()}static jQueryInterface(t){return this.each((function(){const e=hi.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t])throw new TypeError(`No method named "${t}"`);e[t]()}}))}static clearMenus(t){if(t&&(2===t.button||"keyup"===t.type&&"Tab"!==t.key))return;const e=V.find(ti);for(let i=0,n=e.length;ie+t)),this._setElementAttributes(di,"paddingRight",(e=>e+t)),this._setElementAttributes(ui,"marginRight",(e=>e-t))}_disableOverFlow(){this._saveInitialAttribute(this._element,"overflow"),this._element.style.overflow="hidden"}_setElementAttributes(t,e,i){const n=this.getWidth();this._applyManipulationCallback(t,(t=>{if(t!==this._element&&window.innerWidth>t.clientWidth+n)return;this._saveInitialAttribute(t,e);const s=window.getComputedStyle(t)[e];t.style[e]=`${i(Number.parseFloat(s))}px`}))}reset(){this._resetElementAttributes(this._element,"overflow"),this._resetElementAttributes(this._element,"paddingRight"),this._resetElementAttributes(di,"paddingRight"),this._resetElementAttributes(ui,"marginRight")}_saveInitialAttribute(t,e){const i=t.style[e];i&&U.setDataAttribute(t,e,i)}_resetElementAttributes(t,e){this._applyManipulationCallback(t,(t=>{const i=U.getDataAttribute(t,e);void 0===i?t.style.removeProperty(e):(U.removeDataAttribute(t,e),t.style[e]=i)}))}_applyManipulationCallback(t,e){o(t)?e(t):V.find(t,this._element).forEach(e)}isOverflowing(){return this.getWidth()>0}}const pi={className:"modal-backdrop",isVisible:!0,isAnimated:!1,rootElement:"body",clickCallback:null},mi={className:"string",isVisible:"boolean",isAnimated:"boolean",rootElement:"(element|string)",clickCallback:"(function|null)"},gi="show",_i="mousedown.bs.backdrop";class bi{constructor(t){this._config=this._getConfig(t),this._isAppended=!1,this._element=null}show(t){this._config.isVisible?(this._append(),this._config.isAnimated&&u(this._getElement()),this._getElement().classList.add(gi),this._emulateAnimation((()=>{_(t)}))):_(t)}hide(t){this._config.isVisible?(this._getElement().classList.remove(gi),this._emulateAnimation((()=>{this.dispose(),_(t)}))):_(t)}_getElement(){if(!this._element){const t=document.createElement("div");t.className=this._config.className,this._config.isAnimated&&t.classList.add("fade"),this._element=t}return this._element}_getConfig(t){return(t={...pi,..."object"==typeof t?t:{}}).rootElement=r(t.rootElement),a("backdrop",t,mi),t}_append(){this._isAppended||(this._config.rootElement.append(this._getElement()),j.on(this._getElement(),_i,(()=>{_(this._config.clickCallback)})),this._isAppended=!0)}dispose(){this._isAppended&&(j.off(this._element,_i),this._element.remove(),this._isAppended=!1)}_emulateAnimation(t){b(t,this._getElement(),this._config.isAnimated)}}const vi={trapElement:null,autofocus:!0},yi={trapElement:"element",autofocus:"boolean"},wi=".bs.focustrap",Ei="backward";class Ai{constructor(t){this._config=this._getConfig(t),this._isActive=!1,this._lastTabNavDirection=null}activate(){const{trapElement:t,autofocus:e}=this._config;this._isActive||(e&&t.focus(),j.off(document,wi),j.on(document,"focusin.bs.focustrap",(t=>this._handleFocusin(t))),j.on(document,"keydown.tab.bs.focustrap",(t=>this._handleKeydown(t))),this._isActive=!0)}deactivate(){this._isActive&&(this._isActive=!1,j.off(document,wi))}_handleFocusin(t){const{target:e}=t,{trapElement:i}=this._config;if(e===document||e===i||i.contains(e))return;const n=V.focusableChildren(i);0===n.length?i.focus():this._lastTabNavDirection===Ei?n[n.length-1].focus():n[0].focus()}_handleKeydown(t){"Tab"===t.key&&(this._lastTabNavDirection=t.shiftKey?Ei:"forward")}_getConfig(t){return t={...vi,..."object"==typeof t?t:{}},a("focustrap",t,yi),t}}const Ti="modal",Oi="Escape",Ci={backdrop:!0,keyboard:!0,focus:!0},ki={backdrop:"(boolean|string)",keyboard:"boolean",focus:"boolean"},Li="hidden.bs.modal",xi="show.bs.modal",Di="resize.bs.modal",Si="click.dismiss.bs.modal",Ni="keydown.dismiss.bs.modal",Ii="mousedown.dismiss.bs.modal",Pi="modal-open",ji="show",Mi="modal-static";class Hi extends B{constructor(t,e){super(t),this._config=this._getConfig(e),this._dialog=V.findOne(".modal-dialog",this._element),this._backdrop=this._initializeBackDrop(),this._focustrap=this._initializeFocusTrap(),this._isShown=!1,this._ignoreBackdropClick=!1,this._isTransitioning=!1,this._scrollBar=new fi}static get Default(){return Ci}static get NAME(){return Ti}toggle(t){return this._isShown?this.hide():this.show(t)}show(t){this._isShown||this._isTransitioning||j.trigger(this._element,xi,{relatedTarget:t}).defaultPrevented||(this._isShown=!0,this._isAnimated()&&(this._isTransitioning=!0),this._scrollBar.hide(),document.body.classList.add(Pi),this._adjustDialog(),this._setEscapeEvent(),this._setResizeEvent(),j.on(this._dialog,Ii,(()=>{j.one(this._element,"mouseup.dismiss.bs.modal",(t=>{t.target===this._element&&(this._ignoreBackdropClick=!0)}))})),this._showBackdrop((()=>this._showElement(t))))}hide(){if(!this._isShown||this._isTransitioning)return;if(j.trigger(this._element,"hide.bs.modal").defaultPrevented)return;this._isShown=!1;const t=this._isAnimated();t&&(this._isTransitioning=!0),this._setEscapeEvent(),this._setResizeEvent(),this._focustrap.deactivate(),this._element.classList.remove(ji),j.off(this._element,Si),j.off(this._dialog,Ii),this._queueCallback((()=>this._hideModal()),this._element,t)}dispose(){[window,this._dialog].forEach((t=>j.off(t,".bs.modal"))),this._backdrop.dispose(),this._focustrap.deactivate(),super.dispose()}handleUpdate(){this._adjustDialog()}_initializeBackDrop(){return new bi({isVisible:Boolean(this._config.backdrop),isAnimated:this._isAnimated()})}_initializeFocusTrap(){return new Ai({trapElement:this._element})}_getConfig(t){return t={...Ci,...U.getDataAttributes(this._element),..."object"==typeof t?t:{}},a(Ti,t,ki),t}_showElement(t){const e=this._isAnimated(),i=V.findOne(".modal-body",this._dialog);this._element.parentNode&&this._element.parentNode.nodeType===Node.ELEMENT_NODE||document.body.append(this._element),this._element.style.display="block",this._element.removeAttribute("aria-hidden"),this._element.setAttribute("aria-modal",!0),this._element.setAttribute("role","dialog"),this._element.scrollTop=0,i&&(i.scrollTop=0),e&&u(this._element),this._element.classList.add(ji),this._queueCallback((()=>{this._config.focus&&this._focustrap.activate(),this._isTransitioning=!1,j.trigger(this._element,"shown.bs.modal",{relatedTarget:t})}),this._dialog,e)}_setEscapeEvent(){this._isShown?j.on(this._element,Ni,(t=>{this._config.keyboard&&t.key===Oi?(t.preventDefault(),this.hide()):this._config.keyboard||t.key!==Oi||this._triggerBackdropTransition()})):j.off(this._element,Ni)}_setResizeEvent(){this._isShown?j.on(window,Di,(()=>this._adjustDialog())):j.off(window,Di)}_hideModal(){this._element.style.display="none",this._element.setAttribute("aria-hidden",!0),this._element.removeAttribute("aria-modal"),this._element.removeAttribute("role"),this._isTransitioning=!1,this._backdrop.hide((()=>{document.body.classList.remove(Pi),this._resetAdjustments(),this._scrollBar.reset(),j.trigger(this._element,Li)}))}_showBackdrop(t){j.on(this._element,Si,(t=>{this._ignoreBackdropClick?this._ignoreBackdropClick=!1:t.target===t.currentTarget&&(!0===this._config.backdrop?this.hide():"static"===this._config.backdrop&&this._triggerBackdropTransition())})),this._backdrop.show(t)}_isAnimated(){return this._element.classList.contains("fade")}_triggerBackdropTransition(){if(j.trigger(this._element,"hidePrevented.bs.modal").defaultPrevented)return;const{classList:t,scrollHeight:e,style:i}=this._element,n=e>document.documentElement.clientHeight;!n&&"hidden"===i.overflowY||t.contains(Mi)||(n||(i.overflowY="hidden"),t.add(Mi),this._queueCallback((()=>{t.remove(Mi),n||this._queueCallback((()=>{i.overflowY=""}),this._dialog)}),this._dialog),this._element.focus())}_adjustDialog(){const t=this._element.scrollHeight>document.documentElement.clientHeight,e=this._scrollBar.getWidth(),i=e>0;(!i&&t&&!m()||i&&!t&&m())&&(this._element.style.paddingLeft=`${e}px`),(i&&!t&&!m()||!i&&t&&m())&&(this._element.style.paddingRight=`${e}px`)}_resetAdjustments(){this._element.style.paddingLeft="",this._element.style.paddingRight=""}static jQueryInterface(t,e){return this.each((function(){const i=Hi.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===i[t])throw new TypeError(`No method named "${t}"`);i[t](e)}}))}}j.on(document,"click.bs.modal.data-api",'[data-bs-toggle="modal"]',(function(t){const e=n(this);["A","AREA"].includes(this.tagName)&&t.preventDefault(),j.one(e,xi,(t=>{t.defaultPrevented||j.one(e,Li,(()=>{l(this)&&this.focus()}))}));const i=V.findOne(".modal.show");i&&Hi.getInstance(i).hide(),Hi.getOrCreateInstance(e).toggle(this)})),R(Hi),g(Hi);const Bi="offcanvas",Ri={backdrop:!0,keyboard:!0,scroll:!1},Wi={backdrop:"boolean",keyboard:"boolean",scroll:"boolean"},$i="show",zi=".offcanvas.show",qi="hidden.bs.offcanvas";class Fi extends B{constructor(t,e){super(t),this._config=this._getConfig(e),this._isShown=!1,this._backdrop=this._initializeBackDrop(),this._focustrap=this._initializeFocusTrap(),this._addEventListeners()}static get NAME(){return Bi}static get Default(){return Ri}toggle(t){return this._isShown?this.hide():this.show(t)}show(t){this._isShown||j.trigger(this._element,"show.bs.offcanvas",{relatedTarget:t}).defaultPrevented||(this._isShown=!0,this._element.style.visibility="visible",this._backdrop.show(),this._config.scroll||(new fi).hide(),this._element.removeAttribute("aria-hidden"),this._element.setAttribute("aria-modal",!0),this._element.setAttribute("role","dialog"),this._element.classList.add($i),this._queueCallback((()=>{this._config.scroll||this._focustrap.activate(),j.trigger(this._element,"shown.bs.offcanvas",{relatedTarget:t})}),this._element,!0))}hide(){this._isShown&&(j.trigger(this._element,"hide.bs.offcanvas").defaultPrevented||(this._focustrap.deactivate(),this._element.blur(),this._isShown=!1,this._element.classList.remove($i),this._backdrop.hide(),this._queueCallback((()=>{this._element.setAttribute("aria-hidden",!0),this._element.removeAttribute("aria-modal"),this._element.removeAttribute("role"),this._element.style.visibility="hidden",this._config.scroll||(new fi).reset(),j.trigger(this._element,qi)}),this._element,!0)))}dispose(){this._backdrop.dispose(),this._focustrap.deactivate(),super.dispose()}_getConfig(t){return t={...Ri,...U.getDataAttributes(this._element),..."object"==typeof t?t:{}},a(Bi,t,Wi),t}_initializeBackDrop(){return new bi({className:"offcanvas-backdrop",isVisible:this._config.backdrop,isAnimated:!0,rootElement:this._element.parentNode,clickCallback:()=>this.hide()})}_initializeFocusTrap(){return new Ai({trapElement:this._element})}_addEventListeners(){j.on(this._element,"keydown.dismiss.bs.offcanvas",(t=>{this._config.keyboard&&"Escape"===t.key&&this.hide()}))}static jQueryInterface(t){return this.each((function(){const e=Fi.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t]||t.startsWith("_")||"constructor"===t)throw new TypeError(`No method named "${t}"`);e[t](this)}}))}}j.on(document,"click.bs.offcanvas.data-api",'[data-bs-toggle="offcanvas"]',(function(t){const e=n(this);if(["A","AREA"].includes(this.tagName)&&t.preventDefault(),c(this))return;j.one(e,qi,(()=>{l(this)&&this.focus()}));const i=V.findOne(zi);i&&i!==e&&Fi.getInstance(i).hide(),Fi.getOrCreateInstance(e).toggle(this)})),j.on(window,"load.bs.offcanvas.data-api",(()=>V.find(zi).forEach((t=>Fi.getOrCreateInstance(t).show())))),R(Fi),g(Fi);const Ui=new Set(["background","cite","href","itemtype","longdesc","poster","src","xlink:href"]),Vi=/^(?:(?:https?|mailto|ftp|tel|file|sms):|[^#&/:?]*(?:[#/?]|$))/i,Ki=/^data:(?:image\/(?:bmp|gif|jpeg|jpg|png|tiff|webp)|video\/(?:mpeg|mp4|ogg|webm)|audio\/(?:mp3|oga|ogg|opus));base64,[\d+/a-z]+=*$/i,Xi=(t,e)=>{const i=t.nodeName.toLowerCase();if(e.includes(i))return!Ui.has(i)||Boolean(Vi.test(t.nodeValue)||Ki.test(t.nodeValue));const n=e.filter((t=>t instanceof RegExp));for(let t=0,e=n.length;t{Xi(t,r)||i.removeAttribute(t.nodeName)}))}return n.body.innerHTML}const Qi="tooltip",Gi=new Set(["sanitize","allowList","sanitizeFn"]),Zi={animation:"boolean",template:"string",title:"(string|element|function)",trigger:"string",delay:"(number|object)",html:"boolean",selector:"(string|boolean)",placement:"(string|function)",offset:"(array|string|function)",container:"(string|element|boolean)",fallbackPlacements:"array",boundary:"(string|element)",customClass:"(string|function)",sanitize:"boolean",sanitizeFn:"(null|function)",allowList:"object",popperConfig:"(null|object|function)"},Ji={AUTO:"auto",TOP:"top",RIGHT:m()?"left":"right",BOTTOM:"bottom",LEFT:m()?"right":"left"},tn={animation:!0,template:'',trigger:"hover focus",title:"",delay:0,html:!1,selector:!1,placement:"top",offset:[0,0],container:!1,fallbackPlacements:["top","right","bottom","left"],boundary:"clippingParents",customClass:"",sanitize:!0,sanitizeFn:null,allowList:{"*":["class","dir","id","lang","role",/^aria-[\w-]*$/i],a:["target","href","title","rel"],area:[],b:[],br:[],col:[],code:[],div:[],em:[],hr:[],h1:[],h2:[],h3:[],h4:[],h5:[],h6:[],i:[],img:["src","srcset","alt","title","width","height"],li:[],ol:[],p:[],pre:[],s:[],small:[],span:[],sub:[],sup:[],strong:[],u:[],ul:[]},popperConfig:null},en={HIDE:"hide.bs.tooltip",HIDDEN:"hidden.bs.tooltip",SHOW:"show.bs.tooltip",SHOWN:"shown.bs.tooltip",INSERTED:"inserted.bs.tooltip",CLICK:"click.bs.tooltip",FOCUSIN:"focusin.bs.tooltip",FOCUSOUT:"focusout.bs.tooltip",MOUSEENTER:"mouseenter.bs.tooltip",MOUSELEAVE:"mouseleave.bs.tooltip"},nn="fade",sn="show",on="show",rn="out",an=".tooltip-inner",ln=".modal",cn="hide.bs.modal",hn="hover",dn="focus";class un extends B{constructor(t,e){if(void 0===Fe)throw new TypeError("Bootstrap's tooltips require Popper (https://popper.js.org)");super(t),this._isEnabled=!0,this._timeout=0,this._hoverState="",this._activeTrigger={},this._popper=null,this._config=this._getConfig(e),this.tip=null,this._setListeners()}static get Default(){return tn}static get NAME(){return Qi}static get Event(){return en}static get DefaultType(){return Zi}enable(){this._isEnabled=!0}disable(){this._isEnabled=!1}toggleEnabled(){this._isEnabled=!this._isEnabled}toggle(t){if(this._isEnabled)if(t){const e=this._initializeOnDelegatedTarget(t);e._activeTrigger.click=!e._activeTrigger.click,e._isWithActiveTrigger()?e._enter(null,e):e._leave(null,e)}else{if(this.getTipElement().classList.contains(sn))return void this._leave(null,this);this._enter(null,this)}}dispose(){clearTimeout(this._timeout),j.off(this._element.closest(ln),cn,this._hideModalHandler),this.tip&&this.tip.remove(),this._disposePopper(),super.dispose()}show(){if("none"===this._element.style.display)throw new Error("Please use show on visible elements");if(!this.isWithContent()||!this._isEnabled)return;const t=j.trigger(this._element,this.constructor.Event.SHOW),e=h(this._element),i=null===e?this._element.ownerDocument.documentElement.contains(this._element):e.contains(this._element);if(t.defaultPrevented||!i)return;"tooltip"===this.constructor.NAME&&this.tip&&this.getTitle()!==this.tip.querySelector(an).innerHTML&&(this._disposePopper(),this.tip.remove(),this.tip=null);const n=this.getTipElement(),s=(t=>{do{t+=Math.floor(1e6*Math.random())}while(document.getElementById(t));return t})(this.constructor.NAME);n.setAttribute("id",s),this._element.setAttribute("aria-describedby",s),this._config.animation&&n.classList.add(nn);const o="function"==typeof this._config.placement?this._config.placement.call(this,n,this._element):this._config.placement,r=this._getAttachment(o);this._addAttachmentClass(r);const{container:a}=this._config;H.set(n,this.constructor.DATA_KEY,this),this._element.ownerDocument.documentElement.contains(this.tip)||(a.append(n),j.trigger(this._element,this.constructor.Event.INSERTED)),this._popper?this._popper.update():this._popper=qe(this._element,n,this._getPopperConfig(r)),n.classList.add(sn);const l=this._resolvePossibleFunction(this._config.customClass);l&&n.classList.add(...l.split(" ")),"ontouchstart"in document.documentElement&&[].concat(...document.body.children).forEach((t=>{j.on(t,"mouseover",d)}));const c=this.tip.classList.contains(nn);this._queueCallback((()=>{const t=this._hoverState;this._hoverState=null,j.trigger(this._element,this.constructor.Event.SHOWN),t===rn&&this._leave(null,this)}),this.tip,c)}hide(){if(!this._popper)return;const t=this.getTipElement();if(j.trigger(this._element,this.constructor.Event.HIDE).defaultPrevented)return;t.classList.remove(sn),"ontouchstart"in document.documentElement&&[].concat(...document.body.children).forEach((t=>j.off(t,"mouseover",d))),this._activeTrigger.click=!1,this._activeTrigger.focus=!1,this._activeTrigger.hover=!1;const e=this.tip.classList.contains(nn);this._queueCallback((()=>{this._isWithActiveTrigger()||(this._hoverState!==on&&t.remove(),this._cleanTipClass(),this._element.removeAttribute("aria-describedby"),j.trigger(this._element,this.constructor.Event.HIDDEN),this._disposePopper())}),this.tip,e),this._hoverState=""}update(){null!==this._popper&&this._popper.update()}isWithContent(){return Boolean(this.getTitle())}getTipElement(){if(this.tip)return this.tip;const t=document.createElement("div");t.innerHTML=this._config.template;const e=t.children[0];return 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fn={...un.Default,placement:"right",offset:[0,8],trigger:"click",content:"",template:''},pn={...un.DefaultType,content:"(string|element|function)"},mn={HIDE:"hide.bs.popover",HIDDEN:"hidden.bs.popover",SHOW:"show.bs.popover",SHOWN:"shown.bs.popover",INSERTED:"inserted.bs.popover",CLICK:"click.bs.popover",FOCUSIN:"focusin.bs.popover",FOCUSOUT:"focusout.bs.popover",MOUSEENTER:"mouseenter.bs.popover",MOUSELEAVE:"mouseleave.bs.popover"};class gn extends un{static get Default(){return fn}static get NAME(){return"popover"}static get Event(){return mn}static get DefaultType(){return pn}isWithContent(){return this.getTitle()||this._getContent()}setContent(t){this._sanitizeAndSetContent(t,this.getTitle(),".popover-header"),this._sanitizeAndSetContent(t,this._getContent(),".popover-body")}_getContent(){return this._resolvePossibleFunction(this._config.content)}_getBasicClassPrefix(){return"bs-popover"}static jQueryInterface(t){return this.each((function(){const 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Dn="toast",Sn="hide",Nn="show",In="showing",Pn={animation:"boolean",autohide:"boolean",delay:"number"},jn={animation:!0,autohide:!0,delay:5e3};class Mn extends B{constructor(t,e){super(t),this._config=this._getConfig(e),this._timeout=null,this._hasMouseInteraction=!1,this._hasKeyboardInteraction=!1,this._setListeners()}static get DefaultType(){return Pn}static get Default(){return jn}static get NAME(){return Dn}show(){j.trigger(this._element,"show.bs.toast").defaultPrevented||(this._clearTimeout(),this._config.animation&&this._element.classList.add("fade"),this._element.classList.remove(Sn),u(this._element),this._element.classList.add(Nn),this._element.classList.add(In),this._queueCallback((()=>{this._element.classList.remove(In),j.trigger(this._element,"shown.bs.toast"),this._maybeScheduleHide()}),this._element,this._config.animation))}hide(){this._element.classList.contains(Nn)&&(j.trigger(this._element,"hide.bs.toast").defaultPrevented||(this._element.classList.add(In),this._queueCallback((()=>{this._element.classList.add(Sn),this._element.classList.remove(In),this._element.classList.remove(Nn),j.trigger(this._element,"hidden.bs.toast")}),this._element,this._config.animation)))}dispose(){this._clearTimeout(),this._element.classList.contains(Nn)&&this._element.classList.remove(Nn),super.dispose()}_getConfig(t){return t={...jn,...U.getDataAttributes(this._element),..."object"==typeof 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b/materials/glm-practical-logistic-binary-original_files/libs/quarto-html/quarto-syntax-highlighting.css deleted file mode 100644 index 36cb328..0000000 --- a/materials/glm-practical-logistic-binary-original_files/libs/quarto-html/quarto-syntax-highlighting.css +++ /dev/null @@ -1,171 +0,0 @@ -/* quarto syntax highlight colors */ -:root { - --quarto-hl-ot-color: #003B4F; - --quarto-hl-at-color: #657422; - --quarto-hl-ss-color: #20794D; - --quarto-hl-an-color: #5E5E5E; - --quarto-hl-fu-color: #4758AB; - --quarto-hl-st-color: #20794D; - --quarto-hl-cf-color: #003B4F; - --quarto-hl-op-color: #5E5E5E; - --quarto-hl-er-color: #AD0000; - --quarto-hl-bn-color: #AD0000; - --quarto-hl-al-color: #AD0000; - --quarto-hl-va-color: #111111; - --quarto-hl-bu-color: inherit; - --quarto-hl-ex-color: inherit; - --quarto-hl-pp-color: #AD0000; - --quarto-hl-in-color: #5E5E5E; - --quarto-hl-vs-color: #20794D; - --quarto-hl-wa-color: #5E5E5E; - --quarto-hl-do-color: #5E5E5E; - --quarto-hl-im-color: #00769E; 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heading.addEventListener("mouseout", handleMouseout); - } - } - }); - }); - - const sections = tocLinks.map((link) => { - const target = link.getAttribute("data-scroll-target"); - if (target.startsWith("#")) { - return window.document.getElementById(decodeURI(`${target.slice(1)}`)); - } else { - return window.document.querySelector(decodeURI(`${target}`)); - } - }); - - const sectionMargin = 200; - let currentActive = 0; - // track whether we've initialized state the first time - let init = false; - - const updateActiveLink = () => { - // The index from bottom to top (e.g. reversed list) - let sectionIndex = -1; - if ( - window.innerHeight + window.pageYOffset >= - window.document.body.offsetHeight - ) { - sectionIndex = 0; - } else { - sectionIndex = [...sections].reverse().findIndex((section) => { - if (section) { - return window.pageYOffset >= section.offsetTop - sectionMargin; - } else { - return false; - } - }); - } - if (sectionIndex > -1) { - const current = sections.length - sectionIndex - 1; - if (current !== currentActive) { - removeAllActive(); - currentActive = current; - makeActive(current); - if (init) { - window.dispatchEvent(sectionChanged); - } - init = true; - } - } - }; - - const inHiddenRegion = (top, bottom, hiddenRegions) => { - for (const region of hiddenRegions) { - if (top <= region.bottom && bottom >= region.top) { - return true; - } - } - return false; - }; - - const categorySelector = "header.quarto-title-block .quarto-category"; - const activateCategories = (href) => { - // Find any categories - // Surround them with a link pointing back to: - // #category=Authoring - try { - const categoryEls = window.document.querySelectorAll(categorySelector); - for (const categoryEl of categoryEls) { - const categoryText = categoryEl.textContent; - if (categoryText) { - const link = `${href}#category=${encodeURIComponent(categoryText)}`; - const linkEl = window.document.createElement("a"); - linkEl.setAttribute("href", link); - for (const child of categoryEl.childNodes) { - linkEl.append(child); - } - categoryEl.appendChild(linkEl); - } - } - } catch { - // Ignore errors - } - }; - function hasTitleCategories() { - return window.document.querySelector(categorySelector) !== null; - } - - function offsetRelativeUrl(url) { - const offset = getMeta("quarto:offset"); - return offset ? offset + url : url; - } - - function offsetAbsoluteUrl(url) { - const offset = getMeta("quarto:offset"); - const baseUrl = new URL(offset, window.location); - - const projRelativeUrl = url.replace(baseUrl, ""); - if (projRelativeUrl.startsWith("/")) { - return projRelativeUrl; - } else { - return "/" + projRelativeUrl; - } - } - - // read a meta tag value - function getMeta(metaName) { - const metas = window.document.getElementsByTagName("meta"); - for (let i = 0; i < metas.length; i++) { - if (metas[i].getAttribute("name") === metaName) { - return metas[i].getAttribute("content"); - } - } - return ""; - } - - async function findAndActivateCategories() { - const currentPagePath = offsetAbsoluteUrl(window.location.href); - const response = await fetch(offsetRelativeUrl("listings.json")); - if (response.status == 200) { - return response.json().then(function (listingPaths) { - const listingHrefs = []; - for (const listingPath of listingPaths) { - const pathWithoutLeadingSlash = listingPath.listing.substring(1); - for (const item of listingPath.items) { - if ( - item === currentPagePath || - item === currentPagePath + "index.html" - ) { - // Resolve this path against the offset to be sure - // we already are using the correct path to the listing - // (this adjusts the listing urls to be rooted against - // whatever root the page is actually running against) - const relative = offsetRelativeUrl(pathWithoutLeadingSlash); - const baseUrl = window.location; - const resolvedPath = new URL(relative, baseUrl); - listingHrefs.push(resolvedPath.pathname); - break; - } - } - } - - // Look up the tree for a nearby linting and use that if we find one - const nearestListing = findNearestParentListing( - offsetAbsoluteUrl(window.location.pathname), - listingHrefs - ); - if (nearestListing) { - activateCategories(nearestListing); - } else { - // See if the referrer is a listing page for this item - const referredRelativePath = offsetAbsoluteUrl(document.referrer); - const referrerListing = listingHrefs.find((listingHref) => { - const isListingReferrer = - listingHref === referredRelativePath || - listingHref === referredRelativePath + "index.html"; - return isListingReferrer; - }); - - if (referrerListing) { - // Try to use the referrer if possible - activateCategories(referrerListing); - } else if (listingHrefs.length > 0) { - // Otherwise, just fall back to the first listing - activateCategories(listingHrefs[0]); - } - } - }); - } - } - if (hasTitleCategories()) { - findAndActivateCategories(); - } - - const findNearestParentListing = (href, listingHrefs) => { - if (!href || !listingHrefs) { - return undefined; - } - // Look up the tree for a nearby linting and use that if we find one - const relativeParts = href.substring(1).split("/"); - while (relativeParts.length > 0) { - const path = relativeParts.join("/"); - for (const listingHref of listingHrefs) { - if (listingHref.startsWith(path)) { - return listingHref; - } - } - relativeParts.pop(); - } - - return undefined; - }; - - const manageSidebarVisiblity = (el, placeholderDescriptor) => { - let isVisible = true; - - return (hiddenRegions) => { - if (el === null) { - return; - } - - // Find the last element of the TOC - const lastChildEl = el.lastElementChild; - - if (lastChildEl) { - // Find the top and bottom o the element that is being managed - const elTop = el.offsetTop; - const elBottom = - elTop + lastChildEl.offsetTop + lastChildEl.offsetHeight; - - // Converts the sidebar to a menu - const convertToMenu = () => { - for (const child of el.children) { - child.style.opacity = 0; - child.style.overflow = "hidden"; - } - - const toggleContainer = window.document.createElement("div"); - toggleContainer.style.width = "100%"; - toggleContainer.classList.add("zindex-over-content"); - toggleContainer.classList.add("quarto-sidebar-toggle"); - toggleContainer.classList.add("headroom-target"); // Marks this to be managed by headeroom - toggleContainer.id = placeholderDescriptor.id; - toggleContainer.style.position = "fixed"; - - const toggleIcon = window.document.createElement("i"); - toggleIcon.classList.add("quarto-sidebar-toggle-icon"); - toggleIcon.classList.add("bi"); - toggleIcon.classList.add("bi-caret-down-fill"); - - const toggleTitle = window.document.createElement("div"); - const titleEl = window.document.body.querySelector( - placeholderDescriptor.titleSelector - ); - if (titleEl) { - toggleTitle.append(titleEl.innerText, toggleIcon); - } - toggleTitle.classList.add("zindex-over-content"); - toggleTitle.classList.add("quarto-sidebar-toggle-title"); - toggleContainer.append(toggleTitle); - - const toggleContents = window.document.createElement("div"); - toggleContents.classList = el.classList; - toggleContents.classList.add("zindex-over-content"); - toggleContents.classList.add("quarto-sidebar-toggle-contents"); - for (const child of el.children) { - if (child.id === "toc-title") { - continue; - } - - const clone = child.cloneNode(true); - clone.style.opacity = 1; - clone.style.display = null; - toggleContents.append(clone); - } - toggleContents.style.height = "0px"; - toggleContainer.append(toggleContents); - el.parentElement.prepend(toggleContainer); - - // Process clicks - let tocShowing = false; - // Allow the caller to control whether this is dismissed - // when it is clicked (e.g. sidebar navigation supports - // opening and closing the nav tree, so don't dismiss on click) - const clickEl = placeholderDescriptor.dismissOnClick - ? toggleContainer - : toggleTitle; - - const closeToggle = () => { - if (tocShowing) { - toggleContainer.classList.remove("expanded"); - toggleContents.style.height = "0px"; - tocShowing = false; - } - }; - - const positionToggle = () => { - // position the element (top left of parent, same width as parent) - const elRect = el.getBoundingClientRect(); - toggleContainer.style.left = `${elRect.left}px`; - toggleContainer.style.top = `${elRect.top}px`; - toggleContainer.style.width = `${elRect.width}px`; - }; - - // Get rid of any expanded toggle if the user scrolls - window.document.addEventListener( - "scroll", - throttle(() => { - closeToggle(); - }, 50) - ); - - // Handle positioning of the toggle - window.addEventListener( - "resize", - throttle(() => { - positionToggle(); - }, 50) - ); - positionToggle(); - - // Process the click - clickEl.onclick = () => { - if (!tocShowing) { - toggleContainer.classList.add("expanded"); - toggleContents.style.height = null; - tocShowing = true; - } else { - closeToggle(); - } - }; - }; - - // Converts a sidebar from a menu back to a sidebar - const convertToSidebar = () => { - for (const child of el.children) { - child.style.opacity = 1; - child.style.overflow = null; - } - - const placeholderEl = window.document.getElementById( - placeholderDescriptor.id - ); - if (placeholderEl) { - placeholderEl.remove(); - } - - el.classList.remove("rollup"); - }; - - if (isReaderMode()) { - convertToMenu(); - isVisible = false; - } else { - if (!isVisible) { - // If the element is current not visible reveal if there are - // no conflicts with overlay regions - if (!inHiddenRegion(elTop, elBottom, hiddenRegions)) { - convertToSidebar(); - isVisible = true; - } - } else { - // If the element is visible, hide it if it conflicts with overlay regions - // and insert a placeholder toggle (or if we're in reader mode) - if (inHiddenRegion(elTop, elBottom, hiddenRegions)) { - convertToMenu(); - isVisible = false; - } - } - } - } - }; - }; - - // Find any conflicting margin elements and add margins to the - // top to prevent overlap - const marginChildren = window.document.querySelectorAll( - ".column-margin.column-container > * " - ); - - nexttick(() => { - let lastBottom = 0; - for (const marginChild of marginChildren) { - const top = marginChild.getBoundingClientRect().top; - if (top < lastBottom) { - const margin = lastBottom - top; - marginChild.style.marginTop = `${margin}px`; - } - const styles = window.getComputedStyle(marginChild); - const marginTop = parseFloat(styles["marginTop"]); - - lastBottom = top + marginChild.getBoundingClientRect().height + marginTop; - } - }); - - // Manage the visibility of the toc and the sidebar - const marginScrollVisibility = manageSidebarVisiblity(marginSidebarEl, { - id: "quarto-toc-toggle", - titleSelector: "#toc-title", - dismissOnClick: true, - }); - const sidebarScrollVisiblity = manageSidebarVisiblity(sidebarEl, { - id: "quarto-sidebarnav-toggle", - titleSelector: ".title", - dismissOnClick: false, - }); - let tocLeftScrollVisibility; - if (leftTocEl) { - tocLeftScrollVisibility = manageSidebarVisiblity(leftTocEl, { - id: "quarto-lefttoc-toggle", - titleSelector: "#toc-title", - dismissOnClick: true, - }); - } - - // Find the first element that uses formatting in special columns - const conflictingEls = window.document.body.querySelectorAll( - '[class^="column-"], [class*=" column-"], aside, [class*="margin-caption"], [class*=" margin-caption"], [class*="margin-ref"], [class*=" margin-ref"]' - ); - - // Filter all the possibly conflicting elements into ones - // the do conflict on the left or ride side - const arrConflictingEls = Array.from(conflictingEls); - const leftSideConflictEls = arrConflictingEls.filter((el) => { - if (el.tagName === "ASIDE") { - return false; - } - return Array.from(el.classList).find((className) => { - return ( - className !== "column-body" && - className.startsWith("column-") && - !className.endsWith("right") && - !className.endsWith("container") && - className !== "column-margin" - ); - }); - }); - const rightSideConflictEls = arrConflictingEls.filter((el) => { - if (el.tagName === "ASIDE") { - return true; - } - - const hasMarginCaption = Array.from(el.classList).find((className) => { - return className == "margin-caption"; - }); - if (hasMarginCaption) { - return true; - } - - return Array.from(el.classList).find((className) => { - return ( - className !== "column-body" && - !className.endsWith("container") && - className.startsWith("column-") && - !className.endsWith("left") - ); - }); - }); - - const kOverlapPaddingSize = 10; - function toRegions(els) { - return els.map((el) => { - const top = - el.getBoundingClientRect().top + - document.documentElement.scrollTop - - kOverlapPaddingSize; - return { - top, - bottom: top + el.scrollHeight + 2 * kOverlapPaddingSize, - }; - }); - } - - const hideOverlappedSidebars = () => { - marginScrollVisibility(toRegions(rightSideConflictEls)); - sidebarScrollVisiblity(toRegions(leftSideConflictEls)); - if (tocLeftScrollVisibility) { - tocLeftScrollVisibility(toRegions(leftSideConflictEls)); - } - }; - - window.quartoToggleReader = () => { - // Applies a slow class (or removes it) - // to update the transition speed - const slowTransition = (slow) => { - const manageTransition = (id, slow) => { - const el = document.getElementById(id); - if (el) { - if (slow) { - el.classList.add("slow"); - } else { - el.classList.remove("slow"); - } - } - }; - - manageTransition("TOC", slow); - manageTransition("quarto-sidebar", slow); - }; - - const readerMode = !isReaderMode(); - setReaderModeValue(readerMode); - - // If we're entering reader mode, slow the transition - if (readerMode) { - slowTransition(readerMode); - } - highlightReaderToggle(readerMode); - hideOverlappedSidebars(); - - // If we're exiting reader mode, restore the non-slow transition - if (!readerMode) { - slowTransition(!readerMode); - } - }; - - const highlightReaderToggle = (readerMode) => { - const els = document.querySelectorAll(".quarto-reader-toggle"); - if (els) { - els.forEach((el) => { - if (readerMode) { - el.classList.add("reader"); - } else { - el.classList.remove("reader"); - } - }); - } - }; - - const setReaderModeValue = (val) => { - if (window.location.protocol !== "file:") { - window.localStorage.setItem("quarto-reader-mode", val); - } else { - localReaderMode = val; - } - }; - - const isReaderMode = () => { - if (window.location.protocol !== "file:") { - return window.localStorage.getItem("quarto-reader-mode") === "true"; - } else { - return localReaderMode; - } - }; - let localReaderMode = null; - - // Walk the TOC and collapse/expand nodes - // Nodes are expanded if: - // - they are top level - // - they have children that are 'active' links - // - they are directly below an link that is 'active' - const walk = (el, depth) => { - // Tick depth when we enter a UL - if (el.tagName === "UL") { - depth = depth + 1; - } - - // It this is active link - let isActiveNode = false; - if (el.tagName === "A" && el.classList.contains("active")) { - isActiveNode = true; - } - - // See if there is an active child to this element - let hasActiveChild = false; - for (child of el.children) { - hasActiveChild = walk(child, depth) || hasActiveChild; - } - - // Process the collapse state if this is an UL - if (el.tagName === "UL") { - if (depth === 1 || hasActiveChild || prevSiblingIsActiveLink(el)) { - el.classList.remove("collapse"); - } else { - el.classList.add("collapse"); - } - - // untick depth when we leave a UL - depth = depth - 1; - } - return hasActiveChild || isActiveNode; - }; - - // walk the TOC and expand / collapse any items that should be shown - - if (tocEl) { - walk(tocEl, 0); - updateActiveLink(); - } - - // Throttle the scroll event and walk peridiocally - window.document.addEventListener( - "scroll", - throttle(() => { - if (tocEl) { - updateActiveLink(); - walk(tocEl, 0); - } - if (!isReaderMode()) { - hideOverlappedSidebars(); - } - }, 5) - ); - window.addEventListener( - "resize", - throttle(() => { - if (!isReaderMode()) { - hideOverlappedSidebars(); - } - }, 10) - ); - hideOverlappedSidebars(); - highlightReaderToggle(isReaderMode()); -}); - -// grouped tabsets -window.addEventListener("pageshow", (_event) => { - function getTabSettings() { - const data = localStorage.getItem("quarto-persistent-tabsets-data"); - if (!data) { - localStorage.setItem("quarto-persistent-tabsets-data", "{}"); - return {}; - } - if (data) { - return JSON.parse(data); - } - } - - function setTabSettings(data) { - localStorage.setItem( - "quarto-persistent-tabsets-data", - JSON.stringify(data) - ); - } - - function setTabState(groupName, groupValue) { - const data = getTabSettings(); - data[groupName] = groupValue; - setTabSettings(data); - } - - function toggleTab(tab, active) { - const tabPanelId = tab.getAttribute("aria-controls"); - const tabPanel = document.getElementById(tabPanelId); - if (active) { - tab.classList.add("active"); - tabPanel.classList.add("active"); - } 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b/materials/glm-practical-logistic-proportion.qmd index dd2faaa..9e3820b 100644 --- a/materials/glm-practical-logistic-proportion.qmd +++ b/materials/glm-practical-logistic-proportion.qmd @@ -9,6 +9,13 @@ title: "Proportional response" source(file = "setup_files/setup.R") ``` +```{python} +#| echo: false +#| message: false +import shutup;shutup.please() +exec(open('setup_files/setup.py').read()) +``` + ::: {.callout-tip} ## Learning outcomes @@ -48,6 +55,9 @@ import statsmodels.api as sm # Convenience interface for specifying models using formula strings and DataFrames import statsmodels.formula.api as smf + +# Needed for additional probability functionality +from scipy.stats import * ``` ### Functions @@ -58,7 +68,7 @@ The example in this section uses the following data set: `data/challenger.csv` -These data, obtained from the [faraway package](https://www.rdocumentation.org/packages/faraway/versions/1.0.7), contain information related to the explosion of the USA Space Shuttle Challenger on 28 January, 1986. An investigation after the disaster traced back to certain joints on one of the two solid booster rockets, each containing O-rings that ensured no exhaust gases could escape from the booster. +These data, obtained from the [faraway package](https://www.rdocumentation.org/packages/faraway/versions/1.0.7) in R, contain information related to the explosion of the USA Space Shuttle Challenger on 28 January, 1986. An investigation after the disaster traced back to certain joints on one of the two solid booster rockets, each containing O-rings that ensured no exhaust gases could escape from the booster. The night before the launch was unusually cold, with temperatures below freezing. The final report suggested that the cold snap during the night made the o-rings stiff, and unable to adjust to changes in pressure. As a result, exhaust gases leaked away from the solid booster rockets, causing one of them to break loose and rupture the main fuel tank, leading to the final explosion. @@ -74,6 +84,13 @@ First we load the data, then we visualise it. ```{r} challenger <- read_csv("data/challenger.csv") ``` + +## Python + +```{python} +challenger_py = pd.read_csv("data/challenger.csv") +``` + ::: The data set contains several columns: @@ -95,6 +112,15 @@ challenger %>% challenger ``` + +## Python + +```{python} +challenger_py['total'] = 6 +challenger_py['intact'] = challenger_py['total'] - challenger_py['damage'] +challenger_py['prop_damaged'] = challenger_py['damage'] / challenger_py['total'] +``` + ::: Plotting the proportion of damaged o-rings against the launch temperature shows the following picture: @@ -106,23 +132,34 @@ Plotting the proportion of damaged o-rings against the launch temperature shows ggplot(challenger, aes(x = temp, y = prop_damaged)) + geom_point() ``` + +## Python + +```{python} +#| results: hide +(ggplot(challenger_py, + aes(x = "temp", + y = "prop_damaged")) + + geom_point()) +``` + ::: The point on the left is the data point corresponding to the coldest flight experienced before the disaster, where five damaged o-rings were found. Fortunately, this did not result in a disaster. -Here we'll explore if we could have predicted the failure of both o-rings on the Challenger flight, where the launch temperature was 31 degrees Fahrenheit. +Here we'll explore if we could have reasonably predicted the failure of both o-rings on the Challenger flight, where the launch temperature was 31 degrees Fahrenheit. ## Creating a suitable model We only have 23 data points in total. So we're building a model on not that much data - we should keep this in mind when we draw our conclusions! -::: {.panel-tabset group="language"} -## R - We are using a logistic regression for a proportion response in this case, since we're interested in the proportion of o-rings that are damaged. We can define this as follows: +::: {.panel-tabset group="language"} +## R + ```{r} glm_chl <- glm(cbind(damage, intact) ~ temp, family = binomial, @@ -133,6 +170,17 @@ Defining the relationship for proportion responses is a bit annoying, where you Here, the first column corresponds to the number of damaged o-rings, whereas the second column refers to the number of intact o-rings. We use the `cbind()` function to bind these two together into a matrix. +## Python + +```{python} +# create a linear model +model = smf.glm(formula = "damage + intact ~ temp", + family = sm.families.Binomial(), + data = challenger_py) +# and get the fitted parameters of the model +glm_chl_py = model.fit() +``` + ::: ## Model output @@ -150,6 +198,11 @@ summary(glm_chl) We can see that the p-values of the `intercept` and `temp` are significant. We can also use the intercept and `temp` coefficients to construct the logistic equation, which we can use to sketch the logistic curve. +## Python + +```{python} +print(glm_chl_py.summary()) +``` ::: $$E(prop \ failed\ orings) = \frac{\exp{(11.66 - 0.22 \times temp)}}{1 + \exp{(11.66 - 0.22 \times temp)}}$$ @@ -168,6 +221,36 @@ ggplot(challenger, aes(temp, prop_damaged)) + xlim(25,85) ``` +## Python + +We can get the predicted values for the model as follows: + +```{python} +challenger_py['predicted_values'] = glm_chl_py.predict() + +challenger_py.head() +``` + +This would only give us the predicted values for the data we already have. Instead we want to extrapolate to what would have been predicted for a wider range of temperatures. Here, we use a range of $[25, 85]$ Fahrenheit. + +```{python} +model = pd.DataFrame({'temp': list(range(25, 86))}) + +model["pred"] = glm_chl_py.predict(model) + +model.head() +``` + +```{python} +#| results: hide +(ggplot(challenger_py, + aes(x = "temp", + y = "prop_damaged")) + + geom_point() + + geom_line(model, aes(x = "temp", y = "pred"), colour = "blue")) +``` + + ::: {.callout-note collapse=true} ## Generating predicted values @@ -196,6 +279,11 @@ ggplot(model, aes(temp, .pred)) + ``` It seems that there was a high probability of both o-rings failing at that launch temperature. One thing that the graph shows is that there is a lot of uncertainty involved in this model. We can tell, because the fit of the line is very poor at the lower temperature range. There is just very little data to work on, with the data point at 53 F having a large influence on the fit. + +## Python + +We already did this above, since this is the most straightforward way of plotting the model in Python. + ::: ::: :::