Marginal Homogeneity test / Stuart-Maxwell test - overview
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Marginal Homogeneity test / Stuart-Maxwell test | Wilcoxon signed-rank test | Mann-Whitney-Wilcoxon test |
You cannot compare more than 3 methods |
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Independent variable | Independent variable | Independent/grouping variable | |
2 paired groups | 2 paired groups | One categorical with 2 independent groups | |
Dependent variable | Dependent variable | Dependent variable | |
One categorical with $J$ independent groups ($J \geqslant 2$) | One quantitative of interval or ratio level | One of ordinal level | |
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: $m = 0$
Here $m$ is the population median of the difference scores. A difference score is the difference between the first score of a pair and the second score of a pair. Several different formulations of the null hypothesis can be found in the literature, and we do not agree with all of them. Make sure you (also) learn the one that is given in your text book or by your teacher. | 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:
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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 two sided: $m \neq 0$ H1 right sided: $m > 0$ H1 left sided: $m < 0$ | 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:
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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, but both will result in the same test outcome. We will denote the first option the $W_1$ statistic (also known as the $T$ statistic), and the second option the $W_2$ statistic.
In order to compute each of the test statistics, follow the steps below:
| 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. | |
Sampling distribution of the test statistic if H0 were true | Sampling distribution of $W_1$ and of $W_2$ if H0 were true | Sampling distribution of $W$ and of $U$ if H0 were true | |
Approximately the chi-squared distribution with $J - 1$ degrees of freedom | Sampling distribution of $W_1$:
If $N_r$ is large, $W_1$ is approximately normally distributed with mean $\mu_{W_1}$ and standard deviation $\sigma_{W_1}$ if the null hypothesis were true. Here $$\mu_{W_1} = \frac{N_r(N_r + 1)}{4}$$ $$\sigma_{W_1} = \sqrt{\frac{N_r(N_r + 1)(2N_r + 1)}{24}}$$ Hence, if $N_r$ is large, the standardized test statistic $$z = \frac{W_1 - \mu_{W_1}}{\sigma_{W_1}}$$ follows approximately the standard normal distribution if the null hypothesis were true. Sampling distribution of $W_2$: If $N_r$ is large, $W_2$ is approximately normally distributed with mean $0$ and standard deviation $\sigma_{W_2}$ if the null hypothesis were true. Here $$\sigma_{W_2} = \sqrt{\frac{N_r(N_r + 1)(2N_r + 1)}{6}}$$ Hence, if $N_r$ is large, the standardized test statistic $$z = \frac{W_2}{\sigma_{W_2}}$$ follows approximately the standard normal distribution if the null hypothesis were true. If $N_r$ is small, the exact distribution of $W_1$ or $W_2$ should be used. Note: if ties are present in the data, the formula for the standard deviations $\sigma_{W_1}$ and $\sigma_{W_2}$ is more complicated. | 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. | |
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:
| For large samples, the table for standard normal probabilities can be used: Two sided:
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n.a. | n.a. | Equivalent to | |
- | - | 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? | Is the median of the differences between the mental health scores before and after an intervention different from 0? | Do men tend to score higher on social economic status than women? | |
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 Independent Samples...
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n.a. | Jamovi | Jamovi | |
- | T-Tests > Paired Samples T-Test
| T-Tests > Independent Samples T-Test
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Practice questions | Practice questions | Practice questions | |