Marginal Homogeneity test / Stuart-Maxwell test - overview
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Marginal Homogeneity test / Stuart-Maxwell test
Binomial test for a single proportion
|Independent variable||Independent variable|
|2 paired groups||None|
|Dependent variable||Dependent variable|
|One categorical with $J$ independent groups ($J \geqslant 2$)||One categorical with 2 independent groups|
|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: $\pi = \pi_0$
Here $\pi$ is the population proportion of 'successes', and $\pi_0$ is the population proportion of successes according to the null 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: $\pi \neq \pi_0$|
H1 right sided: $\pi > \pi_0$
H1 left sided: $\pi < \pi_0$
|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.||$X$ = number of successes in the sample|
|Sampling distribution of the test statistic if H0 were true||Sampling distribution of $X$ if H0 were true|
|Approximately the chi-squared distribution with $J - 1$ degrees of freedom||Binomial($n$, $P$) distribution.
Here $n = N$ (total sample size), and $P = \pi_0$ (population proportion according to the null hypothesis).
|If we denote the test statistic as $X^2$:||Two sided:
|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 smokers amongst office workers different from $\pi_0 = 0.2$?|
|Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
||Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
|-||Frequencies > 2 Outcomes - Binomial test
|Practice questions||Practice questions|