-
Notifications
You must be signed in to change notification settings - Fork 0
/
logic_hyopthesis_testing.qmd
226 lines (174 loc) · 11.6 KB
/
logic_hyopthesis_testing.qmd
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
# The logic of hyothesis testing
```{r}
#| results: "asis"
#| echo: false
source("_common.R")
status("complete")
```
## What is Hypothesis testing?
Hypothesis testing is a statistical technique which allows us to make inferences about the characteristics of a populations based on a sample. We almost always have to use samples because we are very rarely able to measure every observation in a population. For example, if we are interested in knowing if maternal poverty influences birth weight we would not be able to measure the birth weight of every baby in our population. Instead we would take a sample of babies born to mothers in poverty and determine if their weight, on average, was different to the national average. A statistical procedure is needed because even if maternal poverty has no effect, our sample average will differ from the national average just by chance.
## Samples and populations
## Logic of hypothesis testing
The logic behind hypothesis testing follows these general steps:
1. Formulating a "Null Hypothesis" denoted $H_0$. The null hypothesis is what we expect to happen if nothing interesting is happening. It states that there is no difference between groups or no relationship between variables. In contrast, the "Alternative Hypothesis" ($H_1$) states that there is a significant difference between groups or a relationship between variables.
2. Designing an experiment that generates data to test the null hypothesis.
3. Finding the probability (the *p*-value) of getting our experimental data, or data more extreme, if $H_0$ is true.
4. Deciding whether to reject or not reject the $H_0$ based on that probability:
- If p ≤ 0.05 we reject $H_0$
- If p \> 0.05 do not reject $H_0$
If the null hypothesis is rejected it means we have evidence that $H_0$ is untrue and support for $H_1$. If the null hypothesis is not rejected, it means there is insufficient evidence to support the alternative hypothesis. Note that it does not mean that $H_0$ is true, just that it cannot be discounted. Since there is a real state to $H_0$, that is, $H_0$ is either true or not true, and we make a decision to reject or not reject it, we can be wrong.
### Type I and type II errors
Type I and type II errors describe the cases when we make the wrong decision about the null hypothesis. These errors are inherent in the approach rather than mistakes you can prevent.
- A type I error occurs when we reject a null hypothesis that is true. This can be thought of as a false positive. It is a real error in that we have a real difference or effect. Since we use a probability of 0.05 to reject the null hypothesis, we will make a type I error 5% of the time.
- A type II error occurs when we do not reject a null hypothesis that is false. This is a false negative. It is not a real error in the sense that we only conclude we do not have enough evidence to reject the null hypothesis.
- If we reject a null hypothesis that is false we have not made an error.
- If we do not reject a null hypothesis that is true we have not made an error.
{fig-alt="tabular representation of type I and II error. If we reject a true H0 we have made a type I error; if we do not reject a true H0 we have not made an error. If we do not reject a false H0 we have not made an error; if we do reject a false H0 we have made a type II error"}
We can decrease our chance of making a type I error by reducing the the *p*-value required to reject the null hypothesis. However, this will increase our chance of making a type II error. We can decrease our chance of making a type II error by collecting enough data. The amount of data needed will depend on the the size of the effect relative to the random variation in the data.
## Sampling distribution of the mean
The sampling distribution of the mean is a fundamental concept in hypothesis testing and constructing confidence intervals. Parametric tests such as regression, two-sample tests and ANOVA (all applied with `lm()`) are based on the sampling distribution of the mean. It is a theoretical distribution that describes the distribution of the sample means if an infinite number of samples were taken.
The key characteristics of the sampling distribution of the mean are:
- The mean of the sampling distribution of the mean is equal to the population mean
- The standard deviation of the sampling distribution of the mean is known the standard error of the mean and is always smaller than the standard deviation of the values. There is a fixed relationship between the standard deviation of a sample or population and the standard error of the mean: $s.e. = \frac{s.d.}{\sqrt{n}}$
When we are determining the probability of getting our experimental data, or data more extreme, if $H_0$ is true, it is the sampling distribution of the mean that matters. We ask what is the probability of getting a sample mean like this.
```{r}
#| echo: false
#| label: fig-sample-dist-means
#| fig-cap: "The sampling distribution of the means is a theoretical distribution that describes the distribution of the sample means if an infinite number of samples of size n were taken. The sampling distribution of the mean has a standard devation which is smaller than the population and is called the standard error."
mu <- 3300
sd <- 900
n <- 12
se <- sd/sqrt(n)
mean <- 3000
ggplot(data = data.frame(mass = c(mu - 3 * sd, mu + 3 * sd)),
aes(mass)) +
stat_function(fun = dnorm, n = 101,
args = list(mean = mu, sd = sd)) +
stat_function(fun = dnorm, n = 101,
args = list(mean = mu, sd = se)) +
geom_segment(x = mu, xend = mu, y = 0, yend = dnorm(mu, mu, se)) +
scale_y_continuous(breaks = NULL, name = "",
expand = c(0, 0),
limits = c(0, 0.0016)) +
scale_x_continuous(breaks = NULL, name = "",
expand = c(0, 0),
limits = c(1000, 6000)) +
annotate("text", size = 5,
x = 1800, y = 0.00055,
label = "Distribution\nof the values") +
annotate("segment",
x = 1800, y = 0.0005,
xend = 2000, yend = dnorm(2000, mu, sd),
arrow = arrow(type = "closed",
length = unit(0.03, "npc"))) +
annotate("text", size = 5,
x = 4600, y = 0.0011,
label = "Distribution of the\nsample means") +
annotate("segment",
x = 4600, y = 0.001,
xend = 3600, yend = dnorm(3600, mu, se),
arrow = arrow(type = "closed",
length = unit(0.03, "npc"))) +
theme_classic() +
theme(axis.line.y = element_blank())
```
### Example
Let's work through this logic using an example.
Question: National average birth weight is 3300 grams with an s.d. = 900 grams. Does maternal poverty influence birth weight?
1. Set up the null hypothesis There is no effect of maternal poverty on birth weight so the mean of a sample of babies born into poverty is equal to the national average (@fig-null-example). This is written as $H_0: \bar{x} = 3300$. The alternative hypothesis is that the sample mean is not equal to the national average. This is written as $H_1: \bar{x} \neq 3300$.
```{r}
#| echo: false
#| label: fig-null-example
#| fig-cap: "Distribution of the population of birth weights has a mean of 3300 g and a standard deviation of 900. The null hypothesis is that the mean birth weight of babies born to women in poverty is the same as the national average of 3300 g."
mu <- 3300
sd <- 900
ggplot(data = data.frame(mass = c(mu - 3 * sd, mu + 3 * sd)),
aes(mass)) +
stat_function(fun = dnorm, n = 101,
args = list(mean = mu, sd = sd)) +
geom_segment(x = mu, xend = mu, y = 0, yend = dnorm(mu, mu, sd)) +
scale_y_continuous(breaks = NULL, name = "",
expand = c(0, 0),
limits = c(0, 0.0016)) +
scale_x_continuous(breaks = c(1000, 2000, 3000, 4000, 5000, 6000),
expand = c(0, 0),
limits = c(1000, 6000)) +
annotate("text", x = 3300, y = 0.0006, label = glue::glue("mean = {mu}")) +
theme_classic() +
theme(axis.line.y = element_blank())
```
2. We take a sample of $n = 12$ women who live in poverty and determine the mean birth weight of their babies. We calculate $\bar{x} = 3000 g$. This is lower than the national average but might we get a sample like that even if the null hypothesis is true?
3. Determine the probability (the *p*-value) of getting our experimental data, or more extreme data, if $H_0$ is true. (@fig-sample-dist-means-example)
```{r}
#| echo: false
#| label: fig-sample-dist-means-example
#| fig-cap: "The probability of getting a sample mean of 3000 or less is the area under the sampling distribution of the mean to the left of 3000."
n <- 12
se <- sd/sqrt(n)
mean <- 3000
# function for shading under a normal distribution curve
dnorm_limit <- function(x) {
y <- dnorm(x, mu, se)
y[x < 2000 | x > mean] <- NA
return(y)
}
ggplot(data = data.frame(mass = c(mu - 3 * sd, mu + 3 * sd)),
aes(mass)) +
stat_function(fun = dnorm, n = 101,
args = list(mean = mu, sd = sd)) +
stat_function(fun = dnorm_limit,
geom = "area",
fill = pal2[1]) +
stat_function(fun = dnorm, n = 101,
args = list(mean = mu, sd = se)) +
geom_segment(x = mu, xend = mu, y = 0, yend = dnorm(mu, mu, se)) +
annotate("text", size = 5,
x = 1800, y = 0.00055,
label = glue::glue("Probability of {mean}\nor less"),
colour = pal4[1]) +
scale_y_continuous(breaks = NULL, name = "",
expand = c(0, 0),
limits = c(0, 0.0016)) +
scale_x_continuous(breaks = c(1000, 2000, 3000, 4000, 5000, 6000),
expand = c(0, 0),
limits = c(1000, 6000)) +
theme_classic() +
theme(axis.line.y = element_blank())
```
```{r}
#| echo: false
#| label: fig-sample-dist-means-example2
#| fig-cap: "The probability of getting a sample mean of 3000 or more extreme is the area under the sampling distribution of the mean to the left of 3000 plus that to the right of 3600. This is because 3600 is as extreme (as far away from the mean) as 3000."
# function for shading the other side
dnorm_limit2 <- function(x) {
y <- dnorm(x, mu, se)
y[x < 3600 | x > 4500] <- NA
return(y)
}
ggplot(data = data.frame(mass = c(mu - 3 * sd, mu + 3 * sd)),
aes(mass)) +
stat_function(fun = dnorm, n = 101,
args = list(mean = mu, sd = sd)) +
stat_function(fun = dnorm_limit,
geom = "area",
fill = pal2[1]) +
stat_function(fun = dnorm_limit2,
geom = "area",
fill = pal2[1]) +
annotate("text", size = 5,
x = 1800, y = 0.00055,
label = glue::glue("Probability of {mean}\nor more extreme"),
colour = pal4[1]) +
stat_function(fun = dnorm, n = 101,
args = list(mean = mu, sd = se)) +
geom_segment(x = mu, xend = mu, y = 0, yend = dnorm(mu, mu, se)) +
scale_y_continuous(breaks = NULL, name = "",
expand = c(0, 0),
limits = c(0, 0.0016)) +
scale_x_continuous(breaks = c(1000, 2000, 3000, 4000, 5000, 6000),
expand = c(0, 0),
limits = c(1000, 6000)) +
theme_classic() +
theme(axis.line.y = element_blank())
```
4. Deciding whether to reject or not reject the $H_0$ based on that probability. If the shaded area is less than 0.05 we reject the null hypothesis and conclude maternal poverty does influence birth weight. If the shaded area is more than 0.05 we do not reject the null hypothesis.