Marginal Homogeneity test / Stuart-Maxwell test - overview
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Marginal Homogeneity test / Stuart-Maxwell test | Mann-Whitney-Wilcoxon test | Chi-squared test for the relationship between two categorical variables |
You cannot compare more than 3 methods |
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Independent variable | Independent/grouping variable | Independent /column variable | |
2 paired groups | One categorical with 2 independent groups | One categorical with $I$ independent groups ($I \geqslant 2$) | |
Dependent variable | Dependent variable | Dependent /row variable | |
One categorical with $J$ independent groups ($J \geqslant 2$) | One of ordinal level | One categorical with $J$ independent groups ($J \geqslant 2$) | |
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 $j.$ | If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in both populations:
Formulation 1:
| H0: there is no association between the row and column variable More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
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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. | If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in both populations:
Formulation 1:
| H1: there is an association between the row and column variable More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
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Assumptions | Assumptions | Assumptions | |
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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. | Two different types of test statistics can be used; both will result in the same test outcome. The first is the Wilcoxon rank sum statistic $W$:
Note: we could just as well base W and U on group 2. This would only 'flip' the right and left sided alternative hypotheses. Also, tables with critical values for $U$ are often based on the smaller of $U$ for group 1 and for group 2. | $X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
Here for each cell, the expected cell count = $\dfrac{\mbox{row total} \times \mbox{column total}}{\mbox{total sample size}}$, the observed cell count is the observed sample count in that same cell, and the sum is over all $I \times J$ cells. | |
Sampling distribution of the test statistic if H0 were true | Sampling distribution of $W$ and of $U$ if H0 were true | Sampling distribution of $X^2$ if H0 were true | |
Approximately the chi-squared distribution with $J - 1$ degrees of freedom | Sampling distribution of $W$:
Sampling distribution of $U$: For small samples, the exact distribution of $W$ or $U$ should be used. Note: if ties are present in the data, the formula for the standard deviations $\sigma_W$ and $\sigma_U$ is more complicated. | Approximately the chi-squared distribution with $(I - 1) \times (J - 1)$ degrees of freedom | |
Significant? | Significant? | Significant? | |
If we denote the test statistic as $X^2$:
| For large samples, the table for standard normal probabilities can be used: Two sided:
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n.a. | Equivalent to | n.a. | |
- | If there are no ties in the data, the two sided Mann-Whitney-Wilcoxon test is equivalent to the Kruskal-Wallis test with an independent variable with 2 levels ($I$ = 2). | - | |
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? | Do men tend to score higher on social economic status than women? | Is there an association between economic class and gender? Is the distribution of economic class different between men and women? | |
SPSS | SPSS | SPSS | |
Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
| Analyze > Nonparametric Tests > Legacy Dialogs > 2 Independent Samples...
| Analyze > Descriptive Statistics > Crosstabs...
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n.a. | Jamovi | Jamovi | |
- | T-Tests > Independent Samples T-Test
| Frequencies > Independent Samples - $\chi^2$ test of association
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Practice questions | Practice questions | Practice questions | |