Lecture 16: Comparing two continuous variables

BIOS 600 - Spring 2026

Announcements

  • HW 6 Due Thursday at 11:59pm. Will be graded by the end of the weekend.

  • No new lab this week, the session at 3:30pm is just TA Office Hours.

  • Exam 02 is in-class on Tuesday, March 31.

  • Thursday’s class will be review for exam.

  • Next new lab will be after the exam.

Exam 02 Info

  • Practice problems will be posted today

  • Formula sheet, topic list on course website

Nonparametric methods vs parametric methods

Parametric test Nonparametric test
Paired/one-sample t-test Sign test
Paired/one-sample t-test Signed rank test
Two-sample t-test Rank sum test (aka Mann Whitney U test)
ANOVA Kruskal-Wallis

Reading

  • P&G Chapter 17

  • OI: Section 8.1

Motivation: Comparisons of Interest

Predictor Type Outcome Type Common Tests / Topics
Categorical Categorical Fisher’s exact test, \(\chi^2\) test
Categorical Continuous t-tests, ANOVA, nonparametric alternatives
Continuous Continuous Correlation*, regression **
Continuous Categorical Logistic regression, classification **
Other / Complex Various (e.g. survival, counts) Advanced or “exotic” methods **

* = covering today

** = covering in upcoming lectures

Remember this plot?

Some questions of interest may include:

  • Direction of relationship: are variables positively or negatively related?

  • Form: is any relationship linear or more complex?

  • Strength of relationship: how accurately can one variable predict the other?

  • Influential points: are one or a few points driving the relationship we see?

Correlation

  • The correlation coefficient \(\rho\) quantifies the linear relationship between two random variables.

  • In statistics, a correlation coefficient implies a very specific type of association.

  • A correlation coefficient of zero does NOT imply no relationship between two variables, as we shall see in some further examples.

Correlation

\(\rho\) ranges from -1 to 1

  • \(\rho>0\) implies positive correlation

  • \(\rho < 0\) implies negative correlation

  • \(\rho = 0\) is consistent with no linear relationship between variables (again, this does not imply that no relationship exists!)

What does it mean to have a correlation of -1 or 1?

Visualizing \(\rho\)

Visualizing \(\rho\)

Visualizing \(\rho\)

Pearson’s correlation coefficient

Pearson’s correlation \(r\) gives and estimate of \(\rho\) as follows. Assuming our observed data are the pairs \((x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)\), we can calculate \(r\) as

\[r = \frac{1}{n} \sum_{i=1}^n \left(\frac{x_i - \bar{X}}{S_x}\right)\left(\frac{y_i - \bar{Y}}{S_y}\right)\]

\[= \frac{\sum_{i=1}^n (x_i - \bar{X})(y_i - \bar{Y})}{\sqrt{\sum_{i=1}^n (x_i - \bar{X})^2\sum_{i=1}^n(y_i - \bar{Y})^2}}\]

No need to memorize this!…we’ll just use the cor() function to calculate it in R.

Anscombe’s quartet

Anscombe’s quartet

In each of the datasets the following statistical summaries hold:

  • mean of x: 9

  • variance of x: 11

  • mean of y: 7.5

  • variance of y: 4.125

  • correlation between x and y: 0.816

Takeaway

Takeaway: Visualizing your data is important! Summary statistics alone cannot capture the full relationship between x and y.

Also, Datasaurus Dozen!

Correlation does not imply causation

Source: Tyler Vigen, Spurious Correlations

Confounding

  • Many of these spurious correlations are due to confounding - when a third lurking variable is responsible for the observed relationship.

  • Example: A near perfect negative correlation (r = -0.99) was seen between cholera mortality and elevation above sea level during a 19th century epidemic.

  • The observed relationship between cholera and elevation was confounded by a lurking variable, proximity to polluted water.

ggcorrplot

  • ggcorrplot is a fantastic function for making correlation plots in R.

  • This function is in the ggcorrplot package.

library(ggcorrplot)

Example

  • mtcars is a built-in R dataset, taken from the 1974 Motor Trend US magazine. It has fuel consumption and 10 aspects of automobile design/performance for 32 automobiles.
# Compute a correlation matrix
data(mtcars)
corr <- round(cor(mtcars), 1) # round to 1 decimal
head(corr[, 1:6])
      mpg  cyl disp   hp drat   wt
mpg   1.0 -0.9 -0.8 -0.8  0.7 -0.9
cyl  -0.9  1.0  0.9  0.8 -0.7  0.8
disp -0.8  0.9  1.0  0.8 -0.7  0.9
hp   -0.8  0.8  0.8  1.0 -0.4  0.7
drat  0.7 -0.7 -0.7 -0.4  1.0 -0.7
wt   -0.9  0.8  0.9  0.7 -0.7  1.0

Plotting the correlation: code

# Visualize the correlation matrix
# --------------------------------
# method = "square" (default)
ggcorrplot(corr)

Plotting the correlation: output

Testing the correlation

  • We can also test whether or not each correlation is statistically equal to zero.

\[H_0: \rho = 0 \quad \text{vs.} \quad H_A: \rho \neq 0\]

# Compute a matrix of correlation p-values
p.mat <- cor_pmat(mtcars)
head(p.mat[, 1:4])
              mpg          cyl         disp           hp
mpg  0.000000e+00 6.112687e-10 9.380327e-10 1.787835e-07
cyl  6.112687e-10 0.000000e+00 1.802838e-12 3.477861e-09
disp 9.380327e-10 1.802838e-12 0.000000e+00 7.142679e-08
hp   1.787835e-07 3.477861e-09 7.142679e-08 0.000000e+00
drat 1.776240e-05 8.244636e-06 5.282022e-06 9.988772e-03
wt   1.293959e-10 1.217567e-07 1.222320e-11 4.145827e-05

Example: code

ggcorrplot(p.mat)

Application Exercise: taking a closer look at mtcars

On course website.

Next class

  • Review for Exam 02