Marginal Homogeneity test / Stuart-Maxwell test overview

This page offers structured overviews of one or more selected methods. Add additional methods for comparisons by clicking on the dropdown button in the right-hand column. To practice with a specific method click the button at the bottom row of the table

Marginal Homogeneity test / Stuart-Maxwell test	Marginal Homogeneity test / Stuart-Maxwell test	Sign 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 of ordinal level
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 $j.$	H₀: P(first score of a pair exceeds second score of a pair) = P(second score of a pair exceeds first score of a pair) If the dependent variable is measured on a continuous scale, this can also be formulated as: H₀: the population median of the difference scores is equal to zero A difference score is the difference between the first score of a pair and the second score of a pair.
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.	H₁ two sided: P(first score of a pair exceeds second score of a pair) $\neq$ P(second score of a pair exceeds first score of a pair) H₁ right sided: P(first score of a pair exceeds second score of a pair) > P(second score of a pair exceeds first score of a pair) H₁ left sided: P(first score of a pair exceeds second score of a pair) < P(second score of a pair exceeds first score of a pair) If the dependent variable is measured on a continuous scale, this can also be formulated as: H₁ two sided: the population median of the difference scores is different from zero H₁ right sided: the population median of the difference scores is larger than zero H₁ left sided: the population median of the difference scores is smaller than zero
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.	$W = $ number of difference scores that is larger than 0
Sampling distribution of the test statistic if H₀ were true	Sampling distribution of the test statistic if H₀ were true	Sampling distribution of $W$ 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	The exact distribution of $W$ under the null hypothesis is the Binomial($n$, $P$) distribution, with $n =$ number of positive differences $+$ number of negative differences, and $P = 0.5$. If $n$ is large, $W$ is approximately normally distributed under the null hypothesis, with mean $nP = n \times 0.5$ and standard deviation $\sqrt{nP(1-P)} = \sqrt{n \times 0.5(1 - 0.5)}$. Hence, if $n$ is large, the standardized test statistic $$z = \frac{W - n \times 0.5}{\sqrt{n \times 0.5(1 - 0.5)}}$$ follows approximately the standard normal distribution if the null hypothesis were true.
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 $\alpha$	If $n$ is small, the table for the binomial distribution should be used: Two sided: Check if $W$ observed in sample is in the rejection region or Find two sided $p$ value corresponding to observed $W$ and check if it is equal to or smaller than $\alpha$ Right sided: Check if $W$ observed in sample is in the rejection region or Find right sided $p$ value corresponding to observed $W$ and check if it is equal to or smaller than $\alpha$ Left sided: Check if $W$ observed in sample is in the rejection region or Find left sided $p$ value corresponding to observed $W$ and check if it is equal to or smaller than $\alpha$ If $n$ is large, the table for standard normal probabilities can be used: Two sided: Check if $z$ observed in sample is at least as extreme as critical value $z^$ or Find two sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$ Right sided: Check if $z$ observed in sample is equal to or larger than critical value $z^$ or Find right sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$ Left sided: Check if $z$ observed in sample is equal to or smaller than critical value $z^*$ or Find left sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
n.a.	n.a.	Equivalent to
-	-	Two sided sign test is equivalent to McNemar's test: $W = b$, and $z^2 = X^2$ Friedman test with two related groups
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?	Do people tend to score higher on mental health after a mindfulness course?
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 Sign test
n.a.	n.a.	Jamovi
-	-	Jamovi does not have a specific option for the sign test. However, you can do the Friedman test instead. The $p$ value resulting from this Friedman test is equivalent to the two sided $p$ value that would have resulted from the sign test. Go to: ANOVA > Repeated Measures ANOVA - Friedman Put the two paired variables in the box below Measures
Practice questions	Practice questions	Practice questions

Marginal Homogeneity test / Stuart-Maxwell test - overview