Course Outline

segmentGetting Started (Don't Skip This Part)

segmentStatistics and Data Science: A Modeling Approach

segmentPART I: EXPLORING VARIATION

segmentChapter 1  Welcome to Statistics: A Modeling Approach

segmentChapter 2  Understanding Data

segmentChapter 3  Examining Distributions

segmentChapter 4  Explaining Variation

segmentPART II: MODELING VARIATION

segmentChapter 5  A Simple Model

segmentChapter 6  Quantifying Error

segmentChapter 7  Adding an Explanatory Variable to the Model

segmentChapter 8  Models with a Quantitative Explanatory Variable

8.8 Correlation

segmentPART III: EVALUATING MODELS

segmentChapter 9  The Logic of Inference

segmentChapter 10  Model Comparison with F

segmentChapter 11  Parameter Estimation and Confidence Intervals

segmentPART IV: MULTIVARIATE MODELS

segmentChapter 12  Introduction to Multivariate Models

segmentChapter 13  Multivariate Model Comparisons

segmentFinishing Up (Don't Skip This Part!)

segmentResources
list College / Advanced Statistics and Data Science (ABCD)
8.8 Correlation
You might have heard of Pearson’s r, often referred to as a “correlation coefficient.” Correlation is just a special case of regression in which both the outcome and explanatory variables are transformed into z scores prior to analysis.
Let’s see what happens when we transform the two variables we have been working with: Thumb length and Height. Because both variables are transformed into z scores, the mean of each distribution will be 0, and the standard deviation will be 1. The function zscore()
will convert all the values in a variable to z scores.
require(coursekata)
Fingers < filter(Fingers, Thumb >= 33 & Thumb <= 100)
Height.model < lm(Thumb ~ Height, data = Fingers)
Fingers$Height.resid < resid(Height.model)
# this transforms all Thumb lengths into zscores
Fingers$zThumb < zscore(Fingers$Thumb)
# modify this to do the same for Height
Fingers$zHeight <
# this transforms all Thumb lengths into zscores
Fingers$zThumb < zscore(Fingers$Thumb)
# modify this to do the same for Height
Fingers$zHeight < zscore(Fingers$Height)
ex() %>% check_object("Fingers") %>% {
check_column(., "zThumb") %>% check_equal()
check_column(., "zHeight") %>% check_equal()
}
Let’s make a scatterplot of zThumb and zHeight and look at the distribution. Then also make (again) a scatterplot of Thumb and Height, and compare the two scatterplots.
Make two scatterplots by modifying the code below.
require(coursekata)
Fingers < Fingers %>%
filter(Thumb >= 33 & Thumb <= 100) %>%
mutate(
zThumb = zscore(Thumb),
zHeight = zscore(Height)
)
Height.model < lm(Thumb ~ Height, data = Fingers)
# this makes a scatterplot of the raw scores
# size makes the points bigger or smaller
gf_point(Thumb ~ Height, data = Fingers, size = 4, color = "black")
# modify this to make a scatterplot of the zscores
# feel free to change the colors
gf_point( , data = Fingers, size = 4, color = "firebrick")
# this makes a scatterplot of the raw scores
# size makes the points bigger or smaller
gf_point(Thumb ~ Height, data = Fingers, size = 4, color = "black")
# modify this to make a scatterplot of the zscores
# feel free to change the colors
gf_point(zThumb ~ zHeight, data = Fingers, size = 4, color = "firebrick")
ex() %>% {
check_function(., "gf_point", index = 1) %>% {
check_arg(., "object") %>% check_equal()
check_arg(., "data") %>% check_equal()
}
check_function(., "gf_point", index = 2) %>% {
check_arg(., "object") %>% check_equal()
check_arg(., "data") %>% check_equal()
}
}
Fitting the Regression Model to the Two Distributions
In the code window below we’ve provided the code to fit a regression line for Thumb
based on Height
. Below that code, fit a regression model to the two transformed variables, using zThumb
as the outcome variable (instead of Thumb
) and zHeight
as the explanatory variable (instead of Height
). Save the model in an R object called zHeight_model
.
Then, print the model estimates for both the zHeight_model
and the Height_model
.
require(coursekata)
Fingers < Fingers %>%
filter(Thumb >= 33 & Thumb <= 100) %>%
mutate(
zThumb = zscore(Thumb),
zHeight = zscore(Height)
)
Height_model < lm(Thumb ~ Height, data = Fingers)
# this fits a regression model of Thumb by Height
Height_model < lm(Thumb ~ Height, data = Fingers)
# modify this to fit a regression model predicting zThumb with zHeight
zHeight_model < lm()
# this prints the estimates
Height_model
zHeight_model
zHeight_model < lm(zThumb ~ zHeight, data = Fingers)
ex() %>% check_object("zHeight_model") %>% check_equal()
Next, redo the two scatterplots, this time overlaying the bestfitting regression line for each one.
require(coursekata)
Fingers < Fingers %>%
filter(Thumb >= 33 & Thumb <= 100) %>%
mutate(
zThumb = zscore(Thumb),
zHeight = zscore(Height)
)
Height.model < lm(Thumb ~ Height, data = Fingers)
# this overlays the bestfitting regression model on this scatterplot
gf_point(Thumb ~ Height, data = Fingers, size = 4, color = "black") %>%
gf_lm()
# modify this to overlay the bestfitting regression model on this scatterplot
gf_point(zThumb ~ zHeight, data = Fingers, size = 4, color = "firebrick")
# this overlays the bestfitting regression model on this scatterplot
gf_point(Thumb ~ Height, data = Fingers, size = 4, color = "black") %>%
gf_lm()
# modify this to overlay the bestfitting regression model on this scatterplot
gf_point(zThumb ~ zHeight, data = Fingers, size = 4, color = "firebrick") %>%
gf_lm()
ex() %>% check_function("gf_lm", index = 2)
Below we’ve organized the results of all this in a table: the two scatterplots, bestfitting regression lines, and estimated model parameters.
Note that R will sometimes express parameter estimates in scientific notation. Thus, 1.801e16 means that the decimal point is shifted 16 digits to the left. So, the actual yintercept of the bestfitting regression line is .00000000000000018. Which is, for all practical purposes, 0.
We know from earlier that the bestfitting regression line passes through the mean of both the outcome and explanatory variables. Note that in the case of zThumb
and zHeight
, the middle of the scatterplot is at 0 on both the x and yaxes. The yintercept is 0 in this model because when x is 0, y is also 0.
Comparing the Fit of the Two Models
Let’s now run supernova()
on the two models, and compare their fit to the data.
require(coursekata)
Fingers < Fingers %>%
filter(Thumb >= 33 & Thumb <= 100) %>%
mutate(
zThumb = zscore(Thumb),
zHeight = zscore(Height)
)
Height_model < lm(Thumb ~ Height, data = Fingers)
zHeight_model < lm(zThumb ~ zHeight, data = Fingers)
# this quantifies error from Height_model
supernova(Height_model)
# modify this to quantify error from zHeight_model
supernova()
# this quantifies error from Height_model
supernova(Height_model)
# modify this to quantify error from zHeight_model
supernova(zHeight_model)
ex() %>% check_output_expr("supernova(zHeight_model)")
Analysis of Variance Table (Type III SS)
Model: Thumb ~ Height
SS df MS F PRE p
        
Model (error reduced)  1816.862 1 1816.862 27.984 0.1529 .0000
Error (from model)  10063.349 155 64.925
        
Total (empty model)  11880.211 156 76.155
Analysis of Variance Table (Type III SS)
Model: zThumb ~ zHeight
SS df MS F PRE p
        
Model (error reduced)  23.857 1 23.857 27.984 0.1529 .0000
Error (from model)  132.143 155 0.853
        
Total (empty model)  156.000 156 1.000
The fit of the models is identical because all we have changed is the unit in which we measure the outcome and explanatory variables. We saw when we first introduced z scores that transforming an entire distribution into z scores did not change the shape of the distribution, but only the mean and standard deviation (to 0 and 1).
The same thing is true when we transform both the outcome and explanatory variables. The z transformation does not change the shape of the bivariate distribution, as represented in the scatterplot, at all. It simply changes the scale on both axes to standard deviations instead of inches and millimeters.
Unlike PRE, which is a proportion of the total, SS are expressed in the units of the measurement. So if we converted the mm (for Thumb
length) and inches (for Height
) into cm, feet, etc, the SS would change to reflect those new units.