Marginal Homogeneity test / Stuart-Maxwell test overview

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Marginal Homogeneity test / Stuart-Maxwell test	Marginal Homogeneity test / Stuart-Maxwell test	McNemar's test
Independent variable	Independent variable	Independent variable
2 paired groups	2 paired groups	2 paired groups
Dependent variable	Dependent variable	Dependent variable
One categorical with $J$ independent groups ($J \geqslant 2$)	One categorical with $J$ independent groups ($J \geqslant 2$)	One categorical with 2 independent groups
Null hypothesis	Null hypothesis	Null hypothesis
H₀: for each category $j$ of the dependent variable, $\pi_j$ for the first paired group = $\pi_j$ for the second paired group. Here $\pi_j$ is the population proportion in category $j.$	H₀: for each category $j$ of the dependent variable, $\pi_j$ for the first paired group = $\pi_j$ for the second paired group. Here $\pi_j$ is the population proportion in category $j.$	Let's say that the scores on the dependent variable are scored 0 and 1. Then for each pair of scores, the data allow four options: First score of pair is 0, second score of pair is 0 First score of pair is 0, second score of pair is 1 (switched) First score of pair is 1, second score of pair is 0 (switched) First score of pair is 1, second score of pair is 1 The null hypothesis H₀ is that for each pair of scores, P(first score of pair is 0 while second score of pair is 1) = P(first score of pair is 1 while second score of pair is 0). That is, the probability that a pair of scores switches from 0 to 1 is the same as the probability that a pair of scores switches from 1 to 0. Other formulations of the null hypothesis are: H₀: $\pi_1 = \pi_2$, where $\pi_1$ is the population proportion of ones for the first paired group and $\pi_2$ is the population proportion of ones for the second paired group H₀: for each pair of scores, P(first score of pair is 1) = P(second score of pair is 1)
Alternative hypothesis	Alternative hypothesis	Alternative hypothesis
H₁: for some categories of the dependent variable, $\pi_j$ for the first paired group $\neq$ $\pi_j$ for the second paired group.	H₁: for some categories of the dependent variable, $\pi_j$ for the first paired group $\neq$ $\pi_j$ for the second paired group.	The alternative hypothesis H₁ is that for each pair of scores, P(first score of pair is 0 while second score of pair is 1) $\neq$ P(first score of pair is 1 while second score of pair is 0). That is, the probability that a pair of scores switches from 0 to 1 is not the same as the probability that a pair of scores switches from 1 to 0. Other formulations of the alternative hypothesis are: H₁: $\pi_1 \neq \pi_2$ H₁: for each pair of scores, P(first score of pair is 1) $\neq$ P(second score of pair is 1)
Assumptions	Assumptions	Assumptions
Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another	Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another	Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
Test statistic	Test statistic	Test statistic
Computing the test statistic is a bit complicated and involves matrix algebra. Unless you are following a technical course, you probably won't need to calculate it by hand.	Computing the test statistic is a bit complicated and involves matrix algebra. Unless you are following a technical course, you probably won't need to calculate it by hand.	$X^2 = \dfrac{(b - c)^2}{b + c}$ Here $b$ is the number of pairs in the sample for which the first score is 0 while the second score is 1, and $c$ is the number of pairs in the sample for which the first score is 1 while the second score is 0.
Sampling distribution of the test statistic if H₀ were true	Sampling distribution of the test statistic if H₀ were true	Sampling distribution of $X^2$ if H₀ were true
Approximately the chi-squared distribution with $J - 1$ degrees of freedom	Approximately the chi-squared distribution with $J - 1$ degrees of freedom	If $b + c$ is large enough (say, > 20), approximately the chi-squared distribution with 1 degree of freedom. If $b + c$ is small, the Binomial($n$, $P$) distribution should be used, with $n = b + c$ and $P = 0.5$. In that case the test statistic becomes equal to $b$.
Significant?	Significant?	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 $\alpha$	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 $\alpha$	For test statistic $X^2$: 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 $\alpha$ If $b + c$ is small, the table for the binomial distribution should be used, with as test statistic $b$: Check if $b$ observed in sample is in the rejection region or Find two sided $p$ value corresponding to observed $b$ and check if it is equal to or smaller than $\alpha$
n.a.	n.a.	Equivalent to
-	-	Stuart-Maxwell test, with a categorical dependent variable consisting of two independent groups Cochran's Q test, with two related groups Two sided sign test: $b = W$, and $X^2 = z^2$
Example context	Example context	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?	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?	Does a tv documentary about spiders change whether people are afraid (yes/no) of spiders?
SPSS	SPSS	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	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	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 McNemar test
n.a.	n.a.	Jamovi
-	-	Frequencies > Paired Samples - McNemar test Put one of the two paired variables in the box below Rows and the other paired variable in the box below Columns
Practice questions	Practice questions	Practice questions

Marginal Homogeneity test / Stuart-Maxwell test - overview