Marginal Homogeneity test / StuartMaxwell test  overview
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Marginal Homogeneity test / StuartMaxwell test  Goodness of fit test 


Independent variable  Independent variable  
2 paired groups  None  
Dependent variable  Dependent variable  
One categorical with $J$ independent groups ($J \geqslant 2$)  One categorical with $J$ independent groups ($J \geqslant 2$)  
Null hypothesis  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  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  Assumptions  
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  
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)  $X^2 = \sum{\frac{(\mbox{observed cell count}  \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
where the expected cell count for one cell = $N \times \pi_j$, the observed cell count is the observed sample count in that same cell, and the sum is over all $J$ cells  
Sampling distribution of the test statistic if H0 were true  Sampling distribution of $X^2$ if H0 were true  
Approximately a chisquared distribution with $J  1$ degrees of freedom  Approximately a chisquared distribution with $J  1$ degrees of freedom  
Significant?  Significant?  
If we denote the test statistic as $X^2$:

 
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 people with a low, moderate, and high social economic status in the population different from $\pi_{low}$ = .2, $\pi_{moderate}$ = .6, and $\pi_{high}$ = .2?  
SPSS  SPSS  
Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
 Analyze > Nonparametric Tests > Legacy Dialogs > Chisquare...
 
n.a.  Jamovi  
  Frequencies > N Outcomes  $\chi^2$ Goodness of fit
 
Practice questions  Practice questions  