How to get the correlation between two categorical variables and a categorical variable and a continuous variable?

Correlation between Two Categorical Variables

Checking if two categorical variables are independent can be done with the Chi-Squared test of independence. This is a typical Chi-Square test: if we assume that two variables are independent, then the values of the contingency table for these variables should be distributed uniformly. And then we check how far away from uniform the actual values are. There also exists a Crammer's V that is a measure of correlation that follows from this test

Example

Suppose we have two variables

gender: male and female

city: Blois and Tours

We observed the following data:

Are gender and city independent? Let's perform a Chi-Square test. Null hypothesis: they are independent, Alternative hypothesis is that they are correlated in some way.

Under the Null hypothesis, we assume uniform distribution. So our expected values are the following

So we run the chi-squared test and the resulting p-value here can be seen as a measure of correlation between these two variables.

To compute Crammer's V we first find the normalizing factor chi-squared-max which is typically the size of the sample, divide the chi-square by it and take a square root

R

tbl = matrix(data=c(55, 45, 20, 30), nrow=2, ncol=2, byrow=T)

dimnames(tbl) = list(City=c('B', 'T'), Gender=c('M', 'F'))

chi2 = chisq.test(tbl, correct=F)

c(chi2$statistic, chi2$p.value)

Here the p value is 0.08 - quite small, but still not enough to reject the hypothesis of independence. So we can say that the "correlation" here is 0.08

We also compute V:

sqrt(chi2$statistic / sum(tbl))

And get 0.14 (the smaller v, the lower the correlation)

Consider another dataset

    Gender

City  M  F

   B 51 49

   T 24 26

For this, it would give the following

tbl = matrix(data=c(51, 49, 24, 26), nrow=2, ncol=2, byrow=T)

dimnames(tbl) = list(City=c('B', 'T'), Gender=c('M', 'F'))

chi2 = chisq.test(tbl, correct=F)

c(chi2$statistic, chi2$p.value)

sqrt(chi2$statistic / sum(tbl))

The p-value is 0.72 which is far closer to 1, and v is 0.03 - very close to 0

Categorical vs Numerical Variables

For this type we typically perform a One-way ANOVA test: we calculate in-group variance and intra-group variance and then compare them.

Example

We want to study the relationship between absorbed fat from donuts vs the type of fat used to produce donuts 

Is there any dependence between the variables? For that we conduct ANOVA test and see that the p-value is just 0.007 - there's no correlation between these variables.

R

t1 = c(164, 172, 168, 177, 156, 195)

t2 = c(178, 191, 197, 182, 185, 177)

t3 = c(175, 193, 178, 171, 163, 176)

t4 = c(155, 166, 149, 164, 170, 168)

val = c(t1, t2, t3, t4)

fac = gl(n=4, k=6, labels=c('type1', 'type2', 'type3', 'type4'))

aov1 = aov(val ~ fac)

summary(aov1)

Output is

            Df Sum Sq Mean Sq F value  Pr(>F)   

fac          3   1636   545.5   5.406 0.00688 **

Residuals   20   2018   100.9                   

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Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

So we can take the p-value as the measure of correlation here as well.