 Supporting Statistical Analysis for Research

## 3.2 Relationships between two continuous variables

### 3.2.1 Data concepts - Continuous variables

A continuous variable can take on infinitely many values. The "can take" part of the definition is important since our data sets have a finite number of observations.

Integer valued variables are considered continuous if they are measures on a continuous scale. For example a variable may have the height of subjects in inches. The measure of height is infinite on the inches scale so the variable would be treated as continuous.

### 3.2.2 Exploring - Scatter plots

One useful way to explore the relationship between two continuous variables is with a scatter plot. A scatter plot displays the observed values of a pair of variables as points on a coordinate grid. The values of one of the variables are aligned to the values of the horizontal axis and the other variable values to the vertical axis. A relationship between these two variables is seen as a pattern in the plotted points.

### 3.2.3 Programing - ggplot layers

This book uses ggplot to create graphs for both R and Python. While there are other plotting packages available in both languages, ggplot is particularly useful to quickly create graphs that explore relationships within data.

ggplot implements a layered grammar of graphics. This is an extension of
The Grammar of Graphics by Wilkinson, Anand, and Grossman (2005). Both of these approaches provide a structured method for specifying the transformation of data to graphical images (plots.)

Each of the layers in ggplot can be thought of as the contents of a single plot. The layers are stacked one on top of the another to create the completed graph. For example, one layer could be a scatter plot of data points and another could be a regression line. When stacked, these two layer display the points and the regression line through them. This layering allows for a nice stepwise approach to creating plots.

A layer is constructed from the following components

• data

This is either a data frame or an object that can be coerced to a data frame.

• geometric objects to be graphed

Examples are points, lines, bars, etc.

• aesthetics which map variable values to geometric characteristics

Variables can be mapped to an axis (to determine position on plot), color, shape for points, line type for lines, etc.

• statistical transformation

Examples are summary statistics generated for box plots and frequencies of occurrences for bar charts. When points or lines are drawn, there is no statistical transformation.

This allows the position of a geometric object to be adjusted. Examples include stacking the bars of a bar chart, or jittering the position of points in a scatter plot.

A layer is specified using a geometry function, such as geom_point(), geom_line(), geom_boxplox(), or geom_bar(). These geometry functions define the geometric object to be plotted. The other components of a layer (data, aesthetics mapping, statistical mapping, and position) are specified as parameters to the geometry. Each of the geometries has default settings for many of the these parameters. These defaults make it easy to quickly create plots.

The ggplot() function creates a new plot object. Layers are added to a plot using the + operator. For example, the following syntax template is used to create a scatter plot.

ggplot() + geom_points().

The parameters that identify the data frame to use and the aesthetics mapping would need to be included.

Data and aesthetics can be specified as parameters to either the ggplot() function or the geom_*() functions. When parameters are specified in the ggplot() function, the parameters are used for all geom_*() functions added to the plot. These parameters are said to have a global scope. It is good coding practice to specify data and most aesthetics globally in ggplot(). Any data or aesthetic that is unique to a layer is then specified as a parameter to the geom_*() function for the layer. This is called local scope. Locally scoped parameters override globally scoped parameters in ggplot. This allows global parameters to be replaced by local parameters when needed by a layer.

A ggplot object can be displayed upon its creation. It can also be saved (assigned a name) for later use. A saved plot can be displayed by printing the object (implicitly or explicitly). The saved ggplot object can also be modified. For example, a layer can be added by using the + operator.

### 3.2.4 Examples - R

These examples use the auto.csv data set.

1. We begin by using the same code as in the prior chapters to load the tidyverse and import the csv file.

library(tidyverse)
auto_path <- file.path("..", "datasets", "auto.csv")
auto <- read_csv(auto_path, col_types = cols())
Warning: Missing column names filled in: 'X1' 
glimpse(auto)
Observations: 392
Variables: 10
$X1 <dbl> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15...$ mpg          <dbl> 18, 15, 18, 16, 17, 15, 14, 14, 14, 15, 15, 14, 1...
$cylinders <dbl> 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 4, 6, 6...$ displacement <dbl> 307, 350, 318, 304, 302, 429, 454, 440, 455, 390,...
$horsepower <dbl> 130, 165, 150, 150, 140, 198, 220, 215, 225, 190,...$ weight       <dbl> 3504, 3693, 3436, 3433, 3449, 4341, 4354, 4312, 4...
$acceleration <dbl> 12.0, 11.5, 11.0, 12.0, 10.5, 10.0, 9.0, 8.5, 10....$ year         <dbl> 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 7...
$origin <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1...$ name         <chr> "chevrolet chevelle malibu", "buick skylark 320",...

#### 3.2.4.1 Exploring - Scatter plots

1. We will explore the relationship between the weight and mpg variables. (There are a number of other relationships we could explore as well.) They are both continuous variables. This example creates a scatter plot with the weight on the horizontal axis and mpg on the vertical axis.

The data frame and aesthetics are specified globally in the ggplot() function. Note that the variable names do not need to be quoted here. R knows to interpret these as names of variables in the data frame.

ggplot(data=auto, mapping = aes(x = weight, y = mpg)) +
geom_point() +
theme_bw() In the above graph, one can see observations that are aligned horizontally. This occurs most noticeably in the graph where weight is between 3500 pounds and 5000 pounds. This is a result of the mpg variable being recorded as an integer. The true gas mileage for these automobiles was likely rounded to these values.

The above example used named parameters to the ggplot() function. The data and mapping are well understood using their position and the code is more readable without these parameter names. These parameter names will be dropped in future examples.

2. This example adds a loess (locally estimated scatterplot smoothing) line to the same scatter plot as was created the prior example. A loess line can be an aid in determining the pattern in a graph.

ggplot(auto, aes(x = weight, y = mpg)) +
geom_point() +
geom_smooth(color = "blue") +
theme_bw()
geom_smooth() using method = 'loess' and formula 'y ~ x' The loess line in the above graph suggests there may be a slight non-linear relationship between the weight and mpg variables.

### 3.2.5 Examples - Python

ggplot() was written in R and is part of the ggplot2 R package. There are several ports of the ggplot2 package to Python. We will be using the version in the plotnine package. This is a nice implementation and works very similarly to the R ggplot2.

These examples use the auto.csv data set. It is a record of air accidents.

1. We begin by loading the pandas and os package and then importing the csv file.

from pathlib import Path
import pandas as pd
import plotnine as p9

The above code imports the plotnine package. This include the ggplot functions and methods. Python name space management requires the use of the p9 name when using the functions and methods of ggplot from the plotnine package. This is the same as with functions from pandas.

auto_path = Path('..') / 'datasets' / 'Auto.csv'

print(auto.dtypes)
Unnamed: 0        int64
mpg             float64
cylinders         int64
displacement    float64
horsepower        int64
weight            int64
acceleration    float64
year              int64
origin            int64
name             object
dtype: object

#### 3.2.5.1 Exploring - Scatter plots

1. We will explore the relationship between the weight and mpg variables. (There are a number of other relationships we could explore as well.) They are both continuous variables. This example creates a scatter plot with the weight on the horizontal axis and mpg on the vertical axis.

The data frame and aesthetics are specified globally in the ggplot() function. Note that the variable names need to be quoted when used as parameters in the ggplot functions.

print(
p9.ggplot(data=auto, mapping=p9.aes(x='weight', y='mpg')) +
p9.geom_point() +
p9.theme_bw())
<ggplot: (-9223371893263534350)> In the above graph, one can see observations that are aligned horizontally. This occurs most noticeably in the graph where weight is between 3500 pounds and 5000 pounds. This is a result of the mpg variable being recorded as an integer. The true gas mileage for these automobiles was likely rounded to these values.

The above example used named parameters to the ggplot() function. The data and mapping are well understood using their position and the code is more readable without these parameter names. These parameter names will be dropped in future examples.

2. This example adds a loess (locally estimated scatterplot smoothing) line to the same scatter plot as was created the prior example. A loess line can be an aid in determining the pattern in a graph.

print(
p9.ggplot(auto, p9.aes(x='weight', y='mpg')) +
p9.geom_point() +
p9.geom_smooth(color = "blue") +
p9.theme_bw())
<ggplot: (143602231030)>

C:\PROGRA~3\ANACON~1\lib\site-packages\plotnine\stats\smoothers.py:146: UserWarning: Confidence intervals are not yet implementedfor lowess smoothings.
warnings.warn("Confidence intervals are not yet implemented" The loess line in the above graph suggests there may be a slight non-linear relationship between the weight and mpg variables.

### 3.2.6 Exercises

1. Import the Mroz.csv data set.

2. Plot inc against lwg.

3. Plot age against lwg. Add a loess line to the plot.