Marginal Homogeneity test / StuartMaxwell test  overview
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Marginal Homogeneity test / StuartMaxwell test  Binomial test for a single proportion 


Independent variable  Independent variable  
2 paired groups  None  
Dependent variable  Dependent variable  
One categorical with $J$ independent groups ($J \geqslant 2$)  One categorical with 2 independent groups  
Null hypothesis  Null hypothesis  
H_{0}: 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 $j.$  H_{0}: $\pi = \pi_0$
Here $\pi$ is the population proportion of 'successes', and $\pi_0$ is the population proportion of successes according to the null hypothesis.  
Alternative hypothesis  Alternative hypothesis  
H_{1}: for some categories of the dependent variable, $\pi_j$ for the first paired group $\neq$ $\pi_j$ for the second paired group.  H_{1} two sided: $\pi \neq \pi_0$ H_{1} right sided: $\pi > \pi_0$ H_{1} left sided: $\pi < \pi_0$  
Assumptions  Assumptions  

 
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.  $X$ = number of successes in the sample  
Sampling distribution of the test statistic if H_{0} were true  Sampling distribution of $X$ if H0 were true  
Approximately the chisquared distribution with $J  1$ degrees of freedom  Binomial($n$, $P$) distribution.
Here $n = N$ (total sample size), and $P = \pi_0$ (population proportion according to the null hypothesis).  
Significant?  Significant?  
If we denote the test statistic as $X^2$:
 Two sided:
 
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?  Is the proportion of smokers amongst office workers different from $\pi_0 = 0.2$?  
SPSS  SPSS  
Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
 Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
 
n.a.  Jamovi  
  Frequencies > 2 Outcomes  Binomial test
 
Practice questions  Practice questions  