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I'm trying to use a statistical test to determine whether two different populations have the same distribution of a given categorical variable.

Note: In some cases, the categorical variable is ordered (e.g., short < medium < tall), like in the example below; but in other cases it may be unordered (e.g., apple, banana, or cherry).

To illustrate the problem that I am trying to solve, let's consider an example where we have Caucasian and Hispanic students, and we have the following counts of their heights, as belonging to one of 3 categories: short, medium, tall.

Short Medium Tall
Caucasian 200 400 600
Hispanic 300 200 100

Question: What statistical test should I use to test if Caucasians and Hispanics have the same distribution of (the categorical variable) height?

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  • $\begingroup$ This is the central use case of a Chi squared test $\endgroup$
    – rishi-k
    Commented Jul 14, 2021 at 19:23
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    $\begingroup$ One of your categories is ordered. This may be important if you're interested in alternatives like "tends to be taller". $\endgroup$
    – Glen_b
    Commented Jul 15, 2021 at 14:42

1 Answer 1

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A chi-squared test could assess whether proportions in the categories are homogeneous across the two populations. (This test treats categories as if nominal--without regard to order.)

For your (pretty obviously fictitious data) the test in R goes as shown below:

cau = c(200, 400, 600);  his = c(300, 200, 100)
TBL = rbind(cau, his);  TBL
    [,1] [,2] [,3]
cau  200  400  600
his  300  200  100

The null hypothesis of homogeneity is strongly rejected, with a P-value near $0.$

chisq.test(TBL)

        Pearson's Chi-squared test

data:  TBL
X-squared = 274.29, df = 2, p-value < 2.2e-16

Under the null hypothesis of homogeneity, the expected counts in the six cells of the table are:

chisq.test(TBL)$exp
        [,1] [,2]     [,3]
cau 333.3333  400 466.6667
his 166.6667  200 233.3333

The sum of the squared Pearson residuals is the chi-squared statistic $274.29.$ Looking at residuals with largest absolute values gives an idea which cells of the table contribute most to this significant result.

chisq.test(TBL)$res
         [,1] [,2]      [,3]
cau -7.302967    0  6.172134
his 10.327956    0 -8.728716

The largest discrepancies between observed and expected counts are in the short and tall categories.

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