Marginal Homogeneity test / Stuart-Maxwell test - overview
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Marginal Homogeneity test / Stuart-Maxwell test | Cochran's Q 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 within subject factor ($\geq 2$ related 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 categorical with 2 independent groups | 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 $j.$ | H0: $\pi_1 = \pi_2 = \ldots = \pi_I$
Here $\pi_1$ is the population proportion of 'successes' for group 1, $\pi_2$ is the population proportion of 'successes' for group 2, and $\pi_I$ is the population proportion of 'successes' for group $I.$ | 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. | H1: not all population proportions are equal | 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. | If a failure is scored as 0 and a success is scored as 1:
$Q = k(k - 1) \dfrac{\sum_{groups} \Big (\mbox{group total} - \frac{\mbox{grand total}}{k} \Big)^2}{\sum_{blocks} \mbox{block total} \times (k - \mbox{block total})}$ Here $k$ is the number of related groups (usually the number of repeated measurements), a group total is the sum of the scores in a group, a block total is the sum of the scores in a block (usually a subject), and the grand total is the sum of all the scores. Before computing $Q$, first exclude blocks with equal scores in all $k$ groups. | $X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
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 $Q$ if H0 were true | Sampling distribution of $X^2$ if H0 were true | |
Approximately the chi-squared distribution with $J - 1$ degrees of freedom | If the number of blocks (usually the number of subjects) is large, approximately the chi-squared distribution with $k - 1$ degrees of freedom | Approximately the chi-squared distribution with $(I - 1) \times (J - 1)$ degrees of freedom | |
Significant? | Significant? | Significant? | |
If we denote the test statistic as $X^2$:
| If the number of blocks is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
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n.a. | Equivalent to | n.a. | |
- | Friedman test, with a categorical dependent variable consisting of two independent groups. | - | |
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 perform three different tasks, which they can either perform correctly or incorrectly. Is there a difference in task performance between the three different tasks? | 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 > K Related Samples...
| Analyze > Descriptive Statistics > Crosstabs...
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n.a. | Jamovi | Jamovi | |
- | Jamovi does not have a specific option for the Cochran's Q test. However, you can do the Friedman test instead. The $p$ value resulting from this Friedman test is equivalent to the $p$ value that would have resulted from the Cochran's Q test. Go to:
ANOVA > Repeated Measures ANOVA - Friedman
| Frequencies > Independent Samples - $\chi^2$ test of association
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Practice questions | Practice questions | Practice questions | |