02-eda-1.Rmd

# Exploratory Data Analysis 1

## Introduction {-}

This notebook covers the functionality of the [Exploratory Data Analysis 1](https://geodacenter.github.io/workbook/2a_eda/lab2a.html) section of the GeoDa workbook. We refer to that document for details on the methodology, references, etc. The goal of these notes is to approximate as closely as possible the operations carried out using GeoDa by means of a range of R packages.

The notes are written with R beginners in mind, more seasoned R users can probably skip most of the comments
on data structures and other R particulars. Also, as always in R, there are typically several ways to achieve a specific objective, so what is shown here is just one way that works, but there often are others (that may even be more elegant, work faster, or scale better).

For this notebook, we will use socioeconomic data for 55 New York City sub-boroughs from the GeoDa website. Our goal in this lab is show how to implement exploratory data analysis methods that
deal with one (univariate) and two (bivariate) variables. 

### Objectives {-}

After completing the notebook, you should know how to carry out the following tasks:

- Creating basic univariate plots, i.e., histogram and box plot

- Creating a scatter plot

- Implementing different smoothing methods in a scatter plot (linear, loess, and lowess)

- Showing linear fits for different subsets of the data (spatial heterogeneity)

- Testing the constancy of a regression slope (Chow test)


#### R Packages used {-}

- **tidyverse**: for general data wrangling (includes **readr** and **dplyr**)

- **ggplot2**: to make statistical plots; we use this rather than base R for increased functionality and more aesthetically pleasing plots (also part of **tidyverse**)

- **ggthemes**: additional themes for use with **ggplot**

- **Hmisc**: contains a LOWESS smoother for **ggplot**

- **gap**: to run the chow test 

#### R Commands used {-}

Below follows a list of the commands used in this notebook. For further details
and a comprehensive list of options, please consult the 
[R documentation](https://www.rdocumentation.org).

- **Base R**: `setwd`, `install.packages`, `library`, `head`, `names`, `summary`, `range`, `var`, `sd`,`pdf`,`dev.off`,`saveRDS`,`readRDS`, `function`, `lm`, `str`, `dim`

- **tidyverse**: `read_csv`, `rename`, `mutate`, `if_else`, `filter`

- **ggplot2**: `ggplot`, `geom_histogram`, `bins`, `theme_classic`, `theme_minimal`, `xlab`, `ylab`, `ggtitle`, `theme`, `layer_data`, `ggsave`, `geom_boxplot`, `stat_boxplot`, `geom_point`, `coord_fixed`, `geom_smooth`, `stat_smooth`, `labs`,
`scale_color_manual`

- **ggthemes**: `theme_tufte`

- **Hmisc**: `stat_plsmo`

- **gap**: `chow.test`

## Preliminaries {-}

Before starting, make sure to have the latest version of R and of packages that are compiled for the matching version of R (this document was created using R 3.5.1 of 2018-07-02). Also, make sure to set a working directory.^[Use `setwd(directorypath)` to specify the working directory.] We will use a relative path to the working directory to read the data set.

### Load packages {-}

First, we load all the required packages using the `library` command. If you don't have some of these in your system, make sure to install them first as well as
their dependencies.^[Use 
`install.packages(packagename)`.] You will get an error message if something is missing. If needed, just install the missing piece and everything will work after that. Note
that **ggplot2** does not need to be loaded separately since it is included
in the **tidyverse** package collection.


```{r}
library(tidyverse)
library(ggthemes)
library(Hmisc)
library(gap)
library(geodaData)
```

### Obtaining the data {-}

The data to implement the operations in this workbook are contained in [NYC Data](https://geodacenter.github.io/data-and-lab/nyc/) on the GeoDa support
web site. After the file is downloaded, it must be unzipped (e.g., double click 
on the file). The **nyc** folder should be moved to the current working directory
for the path names we use below to work correctly.

#### Creating an initial data frame {-}
We use the **tidyverse** function `read_csv` to read the data into a data frame
**nyc.data**. We could also have used the base R `read.csv`, but `read_csv` is a bit more robust and creates a **tibble**, a data frame with some additional information.
As usual, we check the contents of the data frame with a `head` command.

```{r}
nyc.data <- nyc
head(nyc.data)
```

#### Making the variable names compatible {-}

Note, that in contrast to GeoDa (where the dbf file is read), reading the csv file into a data frame results in almost all the variable names being in caps. We confirm this with a 
`names` command:

```{r}
names(nyc.data)
```

We now use the **tidyverse** `rename` function to turn the all-caps variables into
lower case for the examples we will use. As in the GeoDa workbook, we only use
three variables, **kids2009**, **kids2000**, and **pubast00**.

```{r}
nyc.data <- nyc.data %>% rename("kids2009" = "KIDS2009", "kids2000" = "KIDS2000",
                                "pubast00" = "PUBAST00")
names(nyc.data)
```

## Analyzing the Distribution of a Single Variable {-}
We follow the discussion in the GeoDa workbook and start with the common univariate
descriptive graphs, the histogram and box plot. Before covering the specifics, we
provide a brief overview of the principles behind the **ggplot** operations.

Note that linking and brushing between a plot and a map is not (yet) readily
implemented in R, so that our discussion will focus primarily on static graphs.

### A quick introduction to **ggplot** {-}
We will be using the commands in the **ggplot2** package for the descriptive statistics plots. There are many options to create nice looking graphs in R, including the functionality in base R, but we chose **ggplot2** for its clean logic and its 
similarity to the **tmap** package that we already encountered (in fact, **tmap** uses
the same layered logic as **ggplot**).^[Note that, strictly speaking, the package is **ggplot2**, i.e., the second iteration of the **ggplot** package, but the commands use **ggplot**. From now on, we will use **ggplot** to refer to both.]

An in-depth introduction to **ggplot** is beyond our scope, but a quick overview
can be found in the [Data Visualization](http://r4ds.had.co.nz/data-visualisation.html) chapter of Wickham and Grolemund's *R for Data Science* book, and full details
are covered in Wickham's *ggplot2: elegant graphics for data analysis (2nd Edition)* (Springer Verlag, 2016).

The logic behind **ggplot** is an implementation of Wilkinson's *grammar for graphics*,
using the concept of *layers*. These are the components that make up a plot, such as a *data set*, *aesthetic mappings* (variables for different aspects of the graph, such
as the x and y-axes, colors, shapes, etc.), *statistical transformations*, a
*geometric object* and position adjustments. Several layers can be drawn on top of each other, providing the ability to create incredibly complex graphs.

For now, the main parts to concentrate on are the data set and the aesthetics,
or `aes`. The latter are typically (at least) the variables to be plotted. These
are usually declared in the main **ggplot** command, e.g., `ggplot(dataset,aes(x=var1,y=var2))` and apply to all the following layers. However,
they can also be specified for each layer individually.

Next follow one or more geometric objects, `geom_*` and various adjustments, 
added to the first command by means of a plus sign, just as we saw how a **tmap**
choropleth map was constructed.

The terminology may seem a little unfamiliar at first, but as long as you remember that
`aes` are the variables and the `geom_*` are the plot types, you will be on your way.

### Histogram {-}
We start with the simple histogram command. As in the GeoDa workbook, we
will use the **kids2009** variable.

The `geom` for a histogram is `geom_histogram`. In contrast to most plots in
**ggplot**, only one variable needs to be passed. The general setup for **ggplot** is to think of the graph as a two-dimensional representation, with the x variable for the x axis and the y variable for the y-axis. In a histogram, the vertical axis is by default taken to be the **count** of the observations in each bin.^[In order to obtain
the frequency on the vertical axis, the y variable needs to be set to `..density..`,
as in `aes(y = ..density..)`.]

The three pieces we need to create the plot are the data set (`data`), **nyc.data**, the
aesthetic (`aes`), **kids2009**, and the geom, `geom_histogram`. The command is as follows,
with all the other settings left to their default:

```{r}
ggplot(data=nyc.data,aes(kids2009)) +
  geom_histogram()
```

The resulting histogram is not very informative, and the first thing we will do
is heed the warning to pick a better bin width.

#### Selecting the number of histogram bins {-}
The standard way in **ggplot** is to adjust the number of bins indirectly, by
means of the `binwidth` option, i.e., the range of values that make up a bin,
in the units of the variable under consideration. To keep the parallel with the
GeoDa workbook, we instead use the option `bins`, which sets the number of
bins directly. The resulting histogram now matches the one in GeoDa (except for the
lack of color, which is immaterial).


```{r}
ggplot(data=nyc.data,aes(kids2009)) +
  geom_histogram(bins=7)
```

As in the GeoDa workbook, we can now change the number of bins to 5, which yields
the following histogram.

```{r}
ggplot(data=nyc.data,aes(kids2009)) +
  geom_histogram(bins=5)
```

#### Spiffing up the graph {-}
The graph as shown is just rudimentary. There are many options in **ggplot** to
change the appearance of the graph, too many to cover here. But to illustrate
some basic features, below, we add a label for the x and y axes using `xlab`
and `ylab`, and a title for the graph with `ggtitle`. An unfortunate aspect
of the latter is that it left aligns the text, whereas we would typically want it to be centered over the graph. 

We can adjust this using the very powerful `theme`
option. But first the basics. Every graph has a theme, which sets the main parameters for its appearance.
The default theme with the grey grids, separated by white lines is
`theme_grey( )`. If we want to change this, we can specify one of the other
themes. For example, a classic graph a la base R plot, without background
shading or grid lines is `theme_classic( )`. In order to obtain this specialized *look*, we set the associated `theme` command.
Our histogram in this theme
looks as follows, with a label on the x and y axis, and a title (and back to 7 bins). 

```{r}
ggplot(data=nyc.data,aes(kids2009)) +
  geom_histogram(bins=7) +
  xlab("Percent kids in 2009") +
  ylab("Frequency") +
  ggtitle("Example Histogram") +
  theme_classic()
```

There are seven built-in themes as well as several contributed
ones. Another built-in example is `theme_minimal( )`, shown next.


```{r}
ggplot(data=nyc.data,aes(kids2009)) +
  geom_histogram(bins=7) +
  xlab("Percent kids in 2009") +
  ylab("Frequency") +
  ggtitle("Example Histogram") +
  theme_minimal()
```





In addition, the package **ggthemes** contains several additional themes that look
extremely professional. For example, `theme_tufte( )`.

```{r}
ggplot(data=nyc.data,aes(kids2009)) +
  geom_histogram(bins=7) +
  xlab("Percent kids in 2009") +
  ylab("Frequency") +
  ggtitle("Example Histogram") +
  theme_tufte()
```



Besides selecting a different default `theme`, we can also
override the basic settings associated with the current
theme. For example, we adjust the `plot.title` (of course, you need to know
what everything is called). Specifically, we set the `element_text` property's
horizontal justification (`hjust`) to 0.5. This centers the title. The number
of other refinements is near infinite. Again, using the default `theme_grey( )`:


```{r}
ggplot(data=nyc.data,aes(kids2009)) +
  geom_histogram(bins=7) +
  xlab("Percent kids in 2009") +
  ylab("Frequency") +
  ggtitle("Example Histogram") +
  theme(plot.title = element_text(hjust = 0.5))
```

#### Histogram summary statistics {-}
The histogram in GeoDa has an option to display the contents of
each bin as well as some descriptive statistics.

As anything in R, the plot created by **ggplot** is nothing but an object. When we enter the commands as above, starting with **ggplot**, the result is drawn directly to the screen. But we can also assign the plot object to a variable. This variable will then contain all the information needed to draw the graph, which includes the `count` of observations in each bin, the min and max values for each bin, etc. For example, we can assign our histogram plot to the **plot.data** object, and then extract the
information using the `layer_data` function. The result is a data frame.

```{r}
plot.data <- ggplot(data=nyc.data,aes(kids2009)) +
  geom_histogram(bins=7) +
  xlab("Percent kids in 2009") +
  ylab("Frequency") +
  ggtitle("Example Histogram") +
  theme(plot.title = element_text(hjust = 0.5))
layer_data(plot.data)
```

The convention used to create the histogram in **ggplot** is slightly different from that in GeoDa, hence small differences in the bounds of the bins. The summary statistics give the number of observations in each bin (**count**), the mid-point of the bin (**x**), and the lower and upper bound for the bin (**xmin** and **xmax**). Unlike GeoDa, where the histogram starts at the minimum value and ends at the maximum value,
the histogram in **ggplot** starts with the minimum value at the *mid-point* of the lowest bin, and the maximum value at the *mid-point* of the upper bin.

#### Assigning (part of) a graph to an object {-}

Any subset of **ggplot** commands can be assigned to an object, which
can save on some typing if the same data set and variables are used for 
several plots. For example, we assign the main **ggplot** command with the
`geom_histogram` to the
object **baseplt**. As such, this does not draw anything. Next, we add the different
options to the **baseplt** object, and the graph appears.

```{r}
baseplt <- ggplot(data=nyc.data,aes(kids2009)) +
  geom_histogram(bins=7) 
baseplt +
  xlab("Percent kids in 2009") +
  ylab("Frequency") +
  ggtitle("Example Histogram") +
  theme(plot.title = element_text(hjust = 0.5))
```

#### Other descriptive statistics {-}
The usual descriptive statistics can be displayed by means of the base R
`summary` command. In principle, we could assign these to an object and then
add them to the plot using the `geom_text` geom, but that is beyond the
current scope. We can easily obtain the descriptive statistics provided by 
GeoDa that are not contained in the R `summary` command, by means
of `range`, `var`, and `sd`, for the range, variance and standard
deviation.

```{r}
summary(nyc.data$kids2009)
range(nyc.data$kids2009)
var(nyc.data$kids2009)
sd(nyc.data$kids2009)
```


#### Writing the graph to a file {-}
In our discussion so far, the graphs are drawn to the screen and then disappear.
To save a **ggplot** graph to a file for publication, there are two ways to proceed.
One is the classic R approach, in which first a **device** is opened, e.g., by means
of a `pdf` command, then the plot commands are entered, and finally the device
is turned off by means of `dev.off()`.

Note that it is always a good idea to specify the dimension of the graph (in inches).
If not, the results can be unexpected.

```{r eval=FALSE}
pdf("hist.pdf",height=3,width=3)
ggplot(data=nyc.data,aes(kids2009)) +
  geom_histogram(bins=7) +
  xlab("Percent kids in 2009") +
  ylab("Frequency") +
  ggtitle("Example Histogram") +
  theme(plot.title = element_text(hjust = 0.5))
dev.off()
```

```
## quartz_off_screen 
##                 2
```

In addiiton to the standard R approach, **ggplot** also has the `ggsave` command,
which does the same thing. It requires the name for the output file, but derives
the proper format from the file extension. For example, an output file with a **png** file extension will create a png file, and similarly for pdf, etc.

The second argument specifies the **plot**. It is optional, and when not specified, the last plot is saved. Again, it is a good idea to specify the `width` and `height` (in inches). In addition, for raster files, the dots per inch (`dpi`) can be set as well. The default is 300, which is fine for most use cases, but for high resolution graphs, one can set the dpi to 600, as in the example below.


```{r eval=FALSE}
hist.plot <- ggplot(data=nyc.data,aes(kids2009)) +
  geom_histogram(bins=7) +
  xlab("Percent kids in 2009") +
  ylab("Frequency") +
  ggtitle("Example Histogram") +
  theme(plot.title = element_text(hjust = 0.5))
ggsave("hist.png",plot=hist.plot,width=3,height=3,dpi=600)
```

#### Saving the graph object {-}
An alternative approach to keep a plot object is to assign the plot commands
to a variable and then save this to disk, using the standard R command `saveRDS`. This
can later be brought back into an R session using `readRDS`. To save the plot,
we need to specify a file name with an **.rds** file extension.

```{r eval=FALSE}
saveRDS(hist.plot,"hist.rds")
```

At some later point (or in a different R session), we can then read the object and plot it. Note that we do not need to assign it to the same variable name as before.
For example, here we call the graph object **newplot**.

```{r eval=FALSE}
newplot <- readRDS("hist.rds")
newplot
```

```{r echo=FALSE}
hist.plot <- ggplot(data=nyc.data,aes(kids2009)) +
  geom_histogram(bins=7) +
  xlab("Percent kids in 2009") +
  ylab("Frequency") +
  ggtitle("Example Histogram") +
  theme(plot.title = element_text(hjust = 0.5))
hist.plot
```

### Box plot {-}
The box plot, also referred to as Tukey's box and whisker plot, is an alternative way to visualize the distribution of a single variable, with a focus on descriptive statistics such as quartiles and the median.^[For a fuller technical description, see the GeoDa workbook.] We continue our example using the **kids2009** variable. We first consider the default option, then move on to various optional settings.

#### Default settings {-}
The minimal arguments to create a boxplot are the `data` set and the x and y
variables passed to `aes`. As mentioned above, the logic behind the graphs in **ggplot** is two-dimensional, so both x and y need to be specified. The x variable is used to create separate box plots for different subsets of the data. In our simple example, we don't need this feature, so we set the x variable to empty, i.e., " ".
The y variable is the actual variable of interest, **kids2009**. The resulting graph is shown below.


```{r}
ggplot(data=nyc.data,aes(x="",y=kids2009)) +
  geom_boxplot()
```

The box encloses the first and third quartile and shows the median as a thick horizontal bar. The vertical lines show the range of data, not the extent of the fences, as is the case in GeoDa. The single dot at value 0 is the lower outlier.

#### Box plot statistics {-}
Unlike what is the case in GeoDa, there is no straightforward way to show the descriptive statistics on the graph. However, we can access the statistics, extract them, and then use text labeling techniques to add them to the plot. We will not pursue that here, but we will illustrate the type of statistics associated with the box plot.

As we did for the histogram, we will assign the **ggplot** object to a variable and then use the `layer_data` function to extract the information.

```{r}
box.plt <- ggplot(data=nyc.data,aes(x="",y=kids2009)) +
  geom_boxplot()
box.dta <- layer_data(box.plt)
box.dta
```

The result is a data frame that contains all the information needed to create the graph.

The descriptive statistics require a little clarification. The values for **lower** and **upper** are, respectively, the values for the first and third quartile, and
**middle** is the median. **ymin** (8.6623) and **ymax** (48.1308) are *not* the smallest and largest values overall, but the smallest and largest values that fall *inside* the fences.^[This can be checked in GeoDa by selecting the corresponding points and checking their value in the Table.] They are the begin and end points of the vertical lines in the plot. **outliers** contains a list with the outlier values. In our example, there is just one, the value 0 (compare to Figure 18 in the GeoDa workbook). The fences, also sometimes called whiskers, are not contained among the statistics, but they can be easily computed.

We illustrate a simple function to accomplish this (note that this function does not implement any error checking and is purely illustrative of the concepts involved). As anything else in R, a function is an object that is assigned to a name. It takes arguments and returns the result. For example, the function `box.desc` given below takes the layer_data object as the argument **box.lyr**, extracts the quartiles to compute the interquartile range and to calculate the fences. We also pass the multiplier as **mult**, with
a default value of 1.5 (the default value is set by the equal sign).

```{r}
box.desc <- function(box.lyr,mult=1.5) {
  # function to computer lower and upper fence in a box plot
  # box.lyr: a box plot layer_data object
  # mult: the multiplier for the fence calculation, default = 1.5
  iqr <- box.lyr$upper - box.lyr$lower # inter-quartile range
  upfence <- box.lyr$upper + mult * iqr  # upper fence
  lofence <- box.lyr$lower - mult * iqr  # lower fence
  return(c(lofence,upfence))
}
```

We can now pass the **box.dta** results to this function to obtain the lower and upper
fences.

```{r}
box.desc(box.dta)
```

These results match the values in the GeoDa workbook.

#### Changing the fence {-}
As we did in the GeoDa workbook, we can change the multiplier value to compute the fences. The default is 1.5, but we can set this to 3.0 by means of the `coef` option.
We again assign the plot to an object to both illustrate the graph and the associated statistics.

```{r}
box.plt3 <- ggplot(data=nyc.data,aes(x="",y=kids2009)) +
  geom_boxplot(coef=3)
box.plt3
```

As in the GeoDa example, there are no longer any outliers. In the graph, the lowest point on the vertical line now corresponds with the value of 0.

We extract the statistics as before.

```{r}
box.dta3 <- layer_data(box.plt3)
box.dta3
```

The statistics are the same, except that the value for **ymin** is now 0. We double check the results for the fences using our function **box.desc**, but now pass
**box.dta3** and set **mult=3.0**.

```{r}
box.desc(box.dta3,mult=3.0)
```

Since the lower fence is negative, the value of 0 is no longer an outlier.

#### Fancier options {-}
As is, the default box plot is pretty rudimentary. We will illustrate the power of **ggplot** by adding a number of features to the plot in order to mimic the visual representation given in GeoDa. First, we will remove the label for the x-axis by setting it to "", and add a title using `ggtitle`. As we did for the histogram, we will center the title over the graph. To save on some typing, we will assign the **ggplot** command with its arguments to the variable **base.plt** and then build the graph by adding layers. First, just the labels.

```{r}
base.plt <- ggplot(data=nyc.data,aes(x="",y=kids2009))
base.plt + geom_boxplot() +
  xlab("") +
  ggtitle("Example Box Plot") +
  theme(plot.title = element_text(hjust=0.5))
```

Next, we want to give the box plot a color, as in GeoDa. This is accomplished with the 
`color` (for the outlines) and `fill` (for the inside of the box) options to `geom_boxplot`. In addition, we can give the outlier point a different color by means of `outlier.color`. For example, setting the `color` to **black**, with the `fill` to **purple** (if we set both to the same color, we can no longer distinguish the median), and the `outlier.color` to **red**, we obtain:

```{r}
base.plt + 
  geom_boxplot(color="black",fill="purple",outlier.color="red") +
  xlab("") +
  ggtitle("Example Box Plot") +
  theme(plot.title = element_text(hjust=0.5))
```

So far, the box plots do not show the fences, the way they do in GeoDa. This can
be remedied, but not quite in the same way as in GeoDa. In **ggplot**, the fences are
drawn at the location of the extreme values, the **ymin** and **ymax** we saw before,
and not at the location of the fence cut-off values, as in GeoDa. The fences are
obtained from the `stat_boxplot` function, by passing the `geom` as `errorbar`.
The result is as shown below.

```{r}
base.plt + 
  geom_boxplot(color="black",fill="purple",outlier.color="red") +
  stat_boxplot(geom="errorbar") +
  xlab("") +
  ggtitle("Example Box Plot") +
  theme(plot.title = element_text(hjust=0.5))
```


One final refinement. In GeoDa, the box plot also shows the locations of the actual observations as points on the central axis. We obtain the same effect by adding
`geom_point` with `color` **blue**. We draw the points first, and the box plot on
top of it, using the layers logic. However, we want to make sure that the central box
doesn't mask the points, which it does when the *transparency* is kept as the default.
To accomplish this, we set the `alpha` level for both points and box plot at **0.5**.

The result comes as close to the GeoDa visualization as we can get without going
overboard.

```{r}
base.plt + 
  geom_point(color="blue",alpha=0.5) +
  geom_boxplot(color="black",fill="purple",outlier.color="red",alpha=0.5) +
  stat_boxplot(geom="errorbar") +
  xlab("") +
  ggtitle("Example Box Plot") +
  theme(plot.title = element_text(hjust=0.5))
```


As before, we can write this plot to a file using `ggsave`, or save the object for future use with `saveRDS`.

## Bivariate Analysis: The Scatter Plot {-}
The scatter plot shows the relationship between two variables as points with cartesian 
(x, y) coordinates matching the value for each variable, one on the x-axis, the
other on the y-axis. In **ggplot** the scatter plot is constructed by means
of a `geom_point`. The aesthetics are mapped to the variables for the x and y axis.

We mimic the example in the GeoDa workbook and use **kids2000** for `x`, and 
**pubast00** for `y`. The bare bones scatter plot is obtained as follows:

```{r}
ggplot(data=nyc.data,aes(x=kids2000,y=pubast00)) +
   geom_point()
```

The graph looks slightly different from the one in GeoDa because of a different
aspect ratio (the ratio between the scale used for the y-axis to that for the
x-axis). In **ggplot**, the aspect ratio is set as an argument to the
`coord_fixed` command. We can obtain a scatter plot that more closely mimics
the shape of the one in GeoDa by setting the aspect `ratio` to 55/25, i.e., the
ratio of the range on the x-axis over the range of the y-axis in the example
in the GeoDa workbook (Figure 24).

```{r}
ggplot(data=nyc.data,aes(x=kids2000,y=pubast00)) +
   geom_point() +
   coord_fixed(ratio=55.0/25.0)
```


As we saw before, we can customize the graph further with a title and labels,
but we will not pursue that here. The main interest is in bringing out the overall
pattern of bivariate association by means of a scatter plot *smoother*, to which we turn next.

### Smoothing the scatter plot {-}
Scatter plot smoothers are implemented through the `geom_smooth` command in 
**ggplot**. Options include a linear smoother, as `method = lm`, and a 
nonlinear *loess* smoother, as `method = loess`. Note that the loess smoother is not quite the same as the LOWESS smoother implemented in GeoDa. The latter is not
included in **ggplot**, but can be implemented by means of the `stat_plsmo` function
from the **Hmisc** package. We consider each smoother in turn.

#### Linear smoother {-}
The linear smoother is added to the plot by including the `geom_smooth` command
after the `geom_point` call. The `method` is set as **lm**. In order to better
distinguish the fitted line from the points, we set its `color` to **blue**,
and add a centered title to specify the smoothing algorithm using `ggtitle`
(also, in what follows, we ignore the aspect ratio issue).


```{r}
ggplot(data=nyc.data,aes(x=kids2000,y=pubast00)) +
   geom_point() +
   geom_smooth(method=lm, color="blue") +
   ggtitle("Linear Smoother") +
   theme(plot.title = element_text(hjust=0.5))
```

By default, the linear smoother includes a 95% confidence interval band (the grey band
around the blue line). To turn this off, we need to set the option `se` to
**FALSE**, as below.

```{r}
ggplot(data=nyc.data,aes(x=kids2000,y=pubast00)) +
   geom_point() +
   geom_smooth(method=lm, color="blue",se=FALSE) +
   ggtitle("Linear Smoother") +
   theme(plot.title = element_text(hjust=0.5))
```

#### Extracting the linear regression results {-}
In GeoDa, a linear fit to the scatter plot also yields the results of the 
regression, such as the coefficients, their standard errors, p-values and
an R^2^ measure of fit. This is not the case in **ggplot** (by design).

However, we can readily obtain these results from the ``lm`` function in base R.
In its bare minimum, this function takes as arguments a `formula` and a 
data set (actually, the latter is not absolutely necessary, depending on how
the variables are referred to). In our example, we specify the regression
formula as **pubast00 ~ kids2000** and the data set as **nyc.data**.^[A **formula** in R is the generic way to specify a functional relationship. The dependent variable is on the left hand side of the **~** sign, with an expression for the explanatory variables on the right hand side. The latter are typically separated by **+**, but in our example, we only have a bivariate relationship. Also, a constant term is included by default.]

Rather than just printing the results of the regression, we assign it to an object,
**reg1** in our example below. We then apply the `summary` command to this object,
which we assign to yet another object (**reg1.sum**). When we list the latter, we
see a summary of the regression results.


```{r}
reg1 <- lm(pubast00 ~ kids2000, data=nyc.data)
reg1.sum <- summary(reg1)
reg1.sum
```

The reason for taking what may seem like a circuitous route is that the
summary object is simply a list of elements that correspond to aspects of the
regression result. Of course, one needs to know what these are called, but, in
doubt, a `str` command will reveal the full *structure*.

```{r}
str(reg1.sum)
```

For example, we see that the coefficients are in the list element `coefficients`.
They can be extracted by means of the standard `$` notation.

```{r}
reg1.sum$coefficients
```

Similarly, we can extract the R^2^ and adjusted R^2^.

```{r}
c(reg1.sum$r.squared,reg1.sum$adj.r.squared)
```

We can now use the text and labeling functionality of **ggplot** to place these
results on the graph. We don't pursue this any further.

#### Loess smoother {-}
The default nonlinear smoother in **ggplot** uses the **loess** algorithm as a locally
weighted regression model. This is similar in spirit to the **LOWESS** method used in GeoDa, but not the same.^[See the GeoDa workbook for further discussion] The implementation is along the same lines as the linear smoother, using
`geom_smooth`, with the only difference that the `method` is now `loess`, as shown below.

```{r}
ggplot(data=nyc.data,aes(x=kids2000,y=pubast00)) +
   geom_point() +
   geom_smooth(method=loess, color="blue") +
   ggtitle("Loess Smoother") +
   theme(plot.title = element_text(hjust=0.5))
```

As in any local regression method, an important parameter is how much of the data is used in the local fit, the so-called `span`. This is typically set to 2/3 of the data by default. A narrower span will yield a smoother that emphasizes local changes. 
In order to set the `span` parameter, we use the `stat_smooth` command. It is 
in all respects equivalent to `geom_smooth`, but allows for somewhat more
flexibility. 

For example, we see the difference between the default and a smoother with a `span = 0.4`. We also turn off the confidence
interval by setting `se = FALSE`.


```{r}
ggplot(data=nyc.data,aes(x=kids2000,y=pubast00)) +
   stat_smooth(method="loess",span=0.4,color="blue",se=FALSE) +
   geom_point() +
   ggtitle("Loess Smoother - Span=0.4") +
   theme(plot.title = element_text(hjust=0.5))
```

And even more with a `span = 0.2`.


```{r}
ggplot(data=nyc.data,aes(x=kids2000,y=pubast00)) +
   stat_smooth(method="loess",span=0.2,color="blue",se=FALSE) +
   geom_point() +
   ggtitle("Loess Smoother - Span=0.2") +
   theme(plot.title = element_text(hjust=0.5))
```


#### LOWESS smoother {-}
The LOWESS smoother is not implemented in **ggplot**, but can be found
in the **Hmisc** package. The approach taken illustrates an alternative way to
compute a smoother, similar to the `stat_smooth` function in **ggplot**. The 
latter is equivalent to `geom_smooth`, but allows for non-standard geoms to visualize the results. We won't pursue that here, but want to point out that this is the 
spirit in which the function `stat_plsmo` is implemented. The default is to
use a span of 2/3 of the observations (the `stat_plsmo` command does not have a 
confidence interval option, so that we do not need to set `se`).


```{r}
ggplot(data=nyc.data,aes(x=kids2000,y=pubast00)) +
   stat_plsmo(color="blue") +
   geom_point() +
   ggtitle("LOWESS Smoother") +
   theme(plot.title = element_text(hjust=0.5))
```

When compared to the `loess` results, we can distinguish minor differences. These 
become more pronounced when setting the `span=0.4`.


```{r}
ggplot(data=nyc.data,aes(x=kids2000,y=pubast00)) +
   stat_plsmo(span=0.4,color="blue") +
   geom_point() +
   ggtitle("LOWESS Smoother - Span=0.4") +
   theme(plot.title = element_text(hjust=0.5))
```

And even more when setting the `span=0.2`.


```{r}
ggplot(data=nyc.data,aes(x=kids2000,y=pubast00)) +
   stat_plsmo(span=0.2,color="blue") +
   geom_point() +
   ggtitle("LOWESS Smoother - Span=0.2") +
   theme(plot.title = element_text(hjust=0.5))
```


#### Putting it all together {-}
In GeoDa, it is easy to show both the linear and LOWESS smoother on the
same plot. In order to accomplish the same in **ggplot**, we 
will use some of the really powerful customization options to 
create a graph that contains all three smoothing methods. We distinguish
between them by setting the `color` argument in `aes` to the name of the method, in essence a constant (not a variable).

In other words, the color argument to `aes` is not set to a variable, where it would take a different color depending on the value of that variable, but to a constant, where the color is fixed.
This will yield a different color for each of the methods. In order to 
highlight the curves themselves, we turn off `se` for `lm` and `loess`.
The plot will include a legend by default, showing the color with the name
of the method. The default title of the legend will be `color`, i.e., the
argument used to reflect the categories. We override this by means of the `labs` command (for legend labeling), and set `color="Method"`. The result is as shown below.

```{r}
ggplot(data=nyc.data,aes(x=kids2000,y=pubast00)) +
   stat_plsmo(aes(color="lowess")) +
   geom_point() +
   geom_smooth(aes(color="lm"),method=lm,se=FALSE) +
   geom_smooth(aes(color="loess"),method=loess,se=FALSE) +
   ggtitle("Comparison of Smoothing Methods") +
   theme(plot.title = element_text(hjust=0.5)) +
   labs(color="Method")
```



## Spatial heterogeneity {-}
In GeoDa, it is relatively straightforward to assess the structural stability of the regression coefficients across selected and unselected observations. The selection can be carried out interactively, from the scatter plot or from any other open view, through linking and brushing. The corresponding regression lines and coefficient
information are updated dynamically. In R, this is not (yet) quite possible.

To illustrate the visualization and assessment of spatial heterogeneity, we will assume we have a way to express the selection as a logical statement. As it turns out, the sub-boroughs in Manhattan have a **CODE** from 301 to 310, and the sub-boroughs
in the Bronx have a **CODE** from 101 to 110. While this is not exactly the example given in the GeoDa workbook, it is easy enough to replicate.

We will proceed by using the `mutate` command to create a new variable **manbronx** that matches this selection. For all practical purposes, this gives the same result as if we had selected these observations in a map. Then we will use this classification to create a scatterplot with a separate regression line for the selected and unselected observations, as well as for all the observations, mimicking the behavior of GeoDa (but in a static fashion).


### Structural breaks in the scatter plot {-}
The first step in our process is to create the new variable **manbronx** using `mutate`. We also use the `if_else` command from **dplyr** to create values for the new variable of **Select** when the condition is true, and **Rest** when the condition is false. The condition checks whether the values for **CODE** are between 300 and 311 (using the symbol `&` for the logical **and**), **or** (using the symbol `|`) between
100 and 111, the codes for Manhattan and the Bronx. As a check, we list the values
for our new variable (since there are only 55 observations, this is not too onerous).

```{r}
nyc.data <- nyc.data %>% mutate(manbronx = if_else((CODE > 300 & CODE < 311) | (CODE > 100 & CODE < 111),"Select","Rest"))
nyc.data$manbronx
```

Next, we create the plot. There are several new elements that we introduce, highlighting the flexibility of **ggplot**. First, we
set the color of the points to correspond with the two categories in the **manbronx**
variable by specifying `aes(color=manbronx)` for `geom_point` (i.e., the aesthetic **color** is mapped to the values of the variable **manbronx**). Then, we create
two separate linear regression lines, one for each category, again by setting
`aes(color=manbronx)` in the `geom_smooth` command. In order not to overwhelm the graph, we turn the confidence band off (`se=FALSE`). To construct a regression line for the full sample, we again use `geom_smooth`, but now we set the color explicitly to **black**. Since this is outside the `aes` setting, the regression line is for the full sample.
We next set the colors for selected and unselected observations to **red** and **blue**, to match the color code in GeoDa (the default setting will have **Rest** colored red,
and **Select** blue, which is the opposite of the behavior in GeoDa). To accomplish
this, we use `scale_color_manual` to set the **values** to **blue** and **red**, in this order (unselected comes first, since it matches FALSE in the logical statement). 
Finally, we use `labs` as before to specify the title for the legend to **Selection**,
and add a centered title.

Except for the aspect ratio, the result looks like what one would obtain in GeoDa.


```{r}
ggplot(nyc.data,aes(x=kids2000,y=pubast00)) +
  geom_point(aes(color=manbronx)) +
  geom_smooth(aes(color=manbronx),method=lm,se=FALSE) +
  geom_smooth(method=lm,se=FALSE,color="black") +
  scale_color_manual(values=c("blue","red")) +
  ggtitle("Spatial Heterogeneity") +
  theme(plot.title = element_text(hjust=0.5)) +
  labs(color="Selection")
```


### Chow test {-}
In GeoDa, a Chow test on the equality of the regression coefficients between the selected and unselected observations is calculated on the fly and shown at the
bottom of the scatter plot. This is not supported by **ggplot**, but we can
run separate regressions for each subset using `lm`. We can also run the Chow test itself, using the `chow.test` command from the **gap** package.

First, we create two subsets from the **nyc.data** by means of `filter`, based
on the value of the **manbronx** variable. We call the two resulting subsets
**nyc.select** and **nyc.rest**. We double check their size (there should be
20 selected observations and 35 unselected ones) using the `dim` command.


```{r}
nyc.select <- nyc.data %>% filter(manbronx == "Select")
nyc.rest <- nyc.data %>% filter(manbronx == "Rest")
dim(nyc.select)
dim(nyc.rest)
```

Next, we carry out two separate regressions, one for each subset, and
list the results.


```{r}
reg.select <- lm(pubast00 ~ kids2000,data=nyc.select)
reg.rest <- lm(pubast00 ~ kids2000,data=nyc.rest)
summary(reg.select)
summary(reg.rest)
```

These values match what we would have obtained in GeoDa with the same observations selected.

Finally, we implement the Chow test as `chow.test`. We pass the y and x variables for each subset, in turn. Again, the results are the same as what one would have obtained in GeoDa and suggest a significant difference between the slopes of the two regression lines.

```{r}
chow <- chow.test(nyc.select$pubast00,nyc.select$kids2000,
                  nyc.rest$pubast00,nyc.rest$kids2000)
chow
```