# How to check if a correlation exists between a continuous independent and a binary dependent variable

So this has been a headache for me. None of the resources I found could explain it in a manner I understand.

Having two sets of data:

x <- c(2, 4, 5, 6, 7, 8, 9, 9, 4, 5)
y <- c(0, 1, 0, 0, 0, 1, 1, 0, 1, 0)
df <- data.frame(x, y)


The y variable is binary and the x variable is continuous as you can see.

What I understand so far is that some machine learning models, like logistic regression, expect the log odds of the y variable to have a linear relationship with the y variable.

How would I make a plot that shows if they abide to this assumption or not and how would I read that plot? I prefer examples in R but any explanation you feel might contribute to my understanding is more than welcome.

• You need a two-sample procedure with groups 0 and 1 determined by y and continuous variable in x. – BruceET May 25 '20 at 22:56

One way to look at the data:

x <- c(2, 4, 5, 6, 7, 8, 9, 9, 4, 5)
y <- c(0, 1, 0, 0, 0, 1, 1, 0, 1, 0)

stripchart(x ~ y, meth="stack", ylim=c(.5,2.5), pch=19)
abline(h=1.5, col="green2")


From the stripcharts, we can see that the data values for the two groups are not much different. If integer values given in x are rounded continuous variables, then a 2-sample t-test would be appropriate, but with so much overlap between values for the two groups, I would not expect significance. Output from running this test in R shows a P-value greater than 5%, so the two sample means, 5.67 and 6.25, are not significantly different from each other.

t.test(x ~ y)

Welch Two Sample t-test

data:  x by y
t = -0.359, df = 5.9956, p-value = 0.7319
alternative hypothesis:
true difference in means is not equal to 0
95 percent confidence interval:
-4.560011  3.393344
sample estimates:
mean in group 0 mean in group 1
5.666667        6.250000


Here are somewhat similar data for which there is a highly significant difference between groups.

stripchart(x1 ~ y, meth="stack", ylim=c(.5,2.5), pch=19)
abline(h=1.5, col="green2")


t.test(x1 ~ y)

Welch Two Sample t-test

data:  x1 by y
t = -5.2824, df = 5.9956, p-value = 0.001865
alternative hypothesis:
true difference in means is not equal to 0
95 percent confidence interval:
-12.560011  -4.606656
sample estimates:
mean in group 0 mean in group 1
5.666667       14.250000

• You my friend are a hero. Could you by accident know anything about Wald's Test as well? I just learned this is a test that could be done to figure out if a independent variable can be used in a machine learning model to predict a dependent variable. – Ruud Verhoef May 26 '20 at 6:02
• Sorry not sure about that reference. Wald was very prolific, so I'm not sure how to narrow it down by searching. Maybe include search word asymptotic. – BruceET May 26 '20 at 6:20
• No problem. Just out of curiousity. Does using jitter/points plots difference from using boxplots to see if they split up nicely? As that is my simplistic interpretation of what you are doing. – Ruud Verhoef May 26 '20 at 8:03