Marginal Homogeneity test / StuartMaxwell test  overview
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Marginal Homogeneity test / StuartMaxwell test  Marginal Homogeneity test / StuartMaxwell test  Sign test 
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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  
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}: 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 $j.$ 
 
Alternative 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}: for some categories of the dependent variable, $\pi_j$ for the first paired group $\neq$ $\pi_j$ for the second paired group. 
 
Assumptions  Assumptions  Assumptions  


 
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 H_{0} were true  Sampling distribution of the test statistic if H_{0} were true  Sampling distribution of $W$ if H_{0} were true  
Approximately the chisquared distribution with $J  1$ degrees of freedom  Approximately the chisquared 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 $\sqrt{nP(1P)} = \sqrt{n \times 0.5(1  0.5)}$. 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 $X^2$:
 If we denote the test statistic as $X^2$:
 If $n$ is small, the table for the binomial distribution should be used: Two sided:
If $n$ is large, the table for standard normal probabilities can be used: Two sided:
 
n.a.  n.a.  Equivalent to  
   
Two sided sign test is equivalent to
 
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...
 Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
 Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
 
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
 
Practice questions  Practice questions  Practice questions  