# Marginal Homogeneity test / Stuart-Maxwell test - overview

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Marginal Homogeneity test / Stuart-Maxwell test
McNemar's test
Independent variableIndependent variable
2 paired groups2 paired groups
Dependent variableDependent variable
One categorical with $J$ independent groups ($J \geqslant 2$)One categorical with 2 independent groups
Null hypothesisNull 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 $j$

For each pair of scores, the data allow four options:

1. First score of pair is 0, second score of pair is 0
2. First score of pair is 0, second score of pair is 1 (switched)
3. First score of pair is 1, second score of pair is 0 (switched)
4. First score of pair is 1, second score of pair is 1
Null hypothesis 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 :

• $\pi_1 = \pi_2$, where $\pi_1$ is the population proportion of ones in the first paired group and $\pi_2$ is the population proportion of ones in the second paired group
• For each pair of scores, P(first score of pair is 1) = P(second score of pair is 1)

Alternative hypothesisAlternative hypothesis
For some categories of the dependent variable, $\pi_j$ in the first paired group $\neq$ $\pi_j$ in the second paired group

Alternative hypothesis 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 that, for each pair of scores:

• $\pi_1 \neq \pi_2$
• For each pair of scores, P(first score of pair is 1) $\neq$ P(second score of pair is 1)

AssumptionsAssumptions
Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one anotherSample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
Test statisticTest 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)$X^2 = \dfrac{(b - c)^2}{b + c}$
$b$ 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 trueSampling distribution of $X^2$ if H0 were true
Approximately a chi-squared distribution with $J - 1$ degrees of freedom

If $b + c$ is large enough (say, > 20), approximately a 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?
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.Equivalent to
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Example contextExample 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?Does a tv documentary about spiders change whether people are afraid (yes/no) of spiders?
SPSSSPSS
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.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 questionsPractice questions