# Marginal Homogeneity test / Stuart-Maxwell test - overview

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Marginal Homogeneity test / Stuart-Maxwell test
Independent variable
2 paired groups
Dependent variable
One categorical with $J$ independent groups ($J \geqslant 2$)
Null hypothesis
For each category $j$ of the dependent variable:

$\pi_j$ in the first paired group = $\pi_j$ in the second paired group

Here $\pi_j$ is the population proportion for category $j$
Alternative hypothesis
For some categories of the dependent variable, $\pi_j$ in the first paired group $\neq$ $\pi_j$ in the second paired group
Assumptions
Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
Test statistic
Computing the test statistic is a bit complicated and involves matrix algebra. You probably won't need to calculate it by hand (unless you are following a technical course)
Sampling distribution of the test statistic if H0 were true
Approximately a chi-squared distribution with $J - 1$ degrees of freedom
Significant?
If we denote the test statistic as $X^2$:
• Check if $X^2$ observed in sample is equal to or larger than critical value $X^{2*}$ or
• Find $p$ value corresponding to observed $X^2$ and check if it is equal to or smaller than $\alpha$
Example context
Subjects are asked to taste three different types of mayonnaise, and to indicate which of the three types of mayonnaise they like best. They then have to drink a glass of beer, and taste and rate the three types of mayonnaise again. Does drinking a beer change which type of mayonnaise people like best?
SPSS
Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
• Put the two paired variables in the boxes below Variable 1 and Variable 2
• Under Test Type, select the Marginal Homogeneity test
Practice questions