Marascuilo procedure
Proportions (counts) · Multiple comparisons · reference distribution: χ²(k−1)
When to use it
After a significant chi-square test of homogeneity across three or more groups, find out which pairs of proportions differ.
Null hypothesis
For each pair: pᵢ = pⱼ.
Assumptions
- The same as the chi-square test of homogeneity
- Binary response in each group
Test statistic
|\hat{p}_i - \hat{p}_j| > \sqrt{\chi^2_{\alpha;\,k-1}} \, \sqrt{\dfrac{\hat{p}_i(1 - \hat{p}_i)}{n_i} + \dfrac{\hat{p}_j(1 - \hat{p}_j)}{n_j}}How to report it
A − C: |0.42 − 0.25| = 0.17, above the critical value of 0.15: they differ at 5%
In R and Python
# no standard function: compare each |p[i] - p[j]| with
sqrt(qchisq(0.95, k - 1)) * sqrt(p[i]*(1-p[i])/n[i] + p[j]*(1-p[j])/n[j])# no standard function: compare each |p[i] - p[j]| with
np.sqrt(stats.chi2.ppf(0.95, k - 1)) * np.sqrt(p[i]*(1-p[i])/n[i] + p[j]*(1-p[j])/n[j])In Python, stats is scipy.stats and np is numpy.
Variants and alternatives
- With repeated measures, pairwise McNemar with adjusted p-values: McNemar
Where it sits in the catalog
Proportions (counts). For categorical answers, such as yes or no, hit or miss: the data are how often each category appears, and the test compares proportions.
Multiple comparisons. Post hoc tests: after rejecting that all groups are equal, they point out which pairs differ, controlling the error of the whole set of comparisons.
In the decision tree
It has no path of its own in the tree, but appears alongside other tests:
- As a post hoc test, if H₀ is rejected, after Chi-square of homogeneity.