Marginal Homogeneity test / Stuart-Maxwell test overview

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Marginal Homogeneity test / Stuart-Maxwell test	Marginal Homogeneity test / Stuart-Maxwell test	Logistic regression
Independent variable	Independent variable	Independent variables
2 paired groups	2 paired groups	One or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variables
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 categorical with 2 independent groups
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: 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 $j.$	Model chi-squared test for the complete regression model: H₀: $\beta_1 = \beta_2 = \ldots = \beta_K = 0$ Wald test for individual regression coefficient $\beta_k$: H₀: $\beta_k = 0$ or in terms of odds ratio: H₀: $e^{\beta_k} = 1$ Likelihood ratio chi-squared test for individual regression coefficient $\beta_k$: H₀: $\beta_k = 0$ or in terms of odds ratio: H₀: $e^{\beta_k} = 1$ in the regression equation $ \ln \big(\frac{\pi_{y = 1}}{1 - \pi_{y = 1}} \big) = \beta_0 + \beta_1 \times x_1 + \beta_2 \times x_2 + \ldots + \beta_K \times x_K $. Here $ x_i$ represents independent variable $ i$, $\beta_i$ is the regression weight for independent variable $ x_i$, and $\pi_{y = 1}$ represents the true probability that the dependent variable $ y = 1$ (or equivalently, the proportion of $ y = 1$ in the population) given the scores on the independent variables.
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: for some categories of the dependent variable, $\pi_j$ for the first paired group $\neq$ $\pi_j$ for the second paired group.	Model chi-squared test for the complete regression model: H₁: not all population regression coefficients are 0 Wald test for individual regression coefficient $\beta_k$: H₁: $\beta_k \neq 0$ or in terms of odds ratio: H₁: $e^{\beta_k} \neq 1$ If defined as Wald $ = \dfrac{b_k}{SE_{b_k}}$ (see 'Test statistic'), also one sided alternatives can be tested: H₁ right sided: $\beta_k > 0$ H₁ left sided: $\beta_k < 0$ Likelihood ratio chi-squared test for individual regression coefficient $\beta_k$: H₁: $\beta_k \neq 0$ or in terms of odds ratio: H₁: $e^{\beta_k} \neq 1$
Assumptions	Assumptions	Assumptions
Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another	Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another	In the population, the relationship between the independent variables and the log odds $\ln (\frac{\pi_{y=1}}{1 - \pi_{y=1}})$ is linear The residuals are independent of one another Often ignored additional assumption: Variables are measured without error Also pay attention to: Multicollinearity Outliers
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.	Model chi-squared test for the complete regression model: $X^2 = D_{null} - D_K = \mbox{null deviance} - \mbox{model deviance} $ $D_{null}$, the null deviance, is conceptually similar to the total variance of the dependent variable in OLS regression analysis. $D_K$, the model deviance, is conceptually similar to the residual variance in OLS regression analysis. Wald test for individual $\beta_k$: The wald statistic can be defined in two ways: Wald $ = \dfrac{b_k^2}{SE^2_{b_k}}$ Wald $ = \dfrac{b_k}{SE_{b_k}}$ SPSS uses the first definition. Likelihood ratio chi-squared test for individual $\beta_k$: $X^2 = D_{K-1} - D_K$ $D_{K-1}$ is the model deviance, where independent variable $k$ is excluded from the model. $D_{K}$ is the model deviance, where independent variable $k$ is included in the model.
Sampling distribution of the test statistic if H₀ were true	Sampling distribution of the test statistic if H₀ were true	Sampling distribution of $X^2$ and of the Wald statistic if H₀ were true
Approximately the chi-squared distribution with $J - 1$ degrees of freedom	Approximately the chi-squared distribution with $J - 1$ degrees of freedom	Sampling distribution of $X^2$, as computed in the model chi-squared test for the complete model: chi-squared distribution with $K$ (number of independent variables) degrees of freedom Sampling distribution of the Wald statistic: If defined as Wald $ = \dfrac{b_k^2}{SE^2_{b_k}}$: approximately the chi-squared distribution with 1 degree of freedom If defined as Wald $ = \dfrac{b_k}{SE_{b_k}}$: approximately the standard normal distribution Sampling distribution of $X^2$, as computed in the likelihood ratio chi-squared test for individual $\beta_k$: chi-squared distribution with 1 degree of freedom
Significant?	Significant?	Significant?
If we denote the test statistic as $X^2$: Check if $X^2$ observed in sample is equal to or larger than critical value $X^{2*}$ or Find $p$ value corresponding to observed $X^2$ and check if it is equal to or smaller than $\alpha$	If we denote the test statistic as $X^2$: Check if $X^2$ observed in sample is equal to or larger than critical value $X^{2*}$ or Find $p$ value corresponding to observed $X^2$ and check if it is equal to or smaller than $\alpha$	For the model chi-squared test for the complete regression model and likelihood ratio chi-squared test for individual $\beta_k$: Check if $X^2$ observed in sample is equal to or larger than critical value $X^{2*}$ or Find $p$ value corresponding to observed $X^2$ and check if it is equal to or smaller than $\alpha$ For the Wald test: If defined as Wald $ = \dfrac{b_k^2}{SE^2_{b_k}}$: same procedure as for the chi-squared tests. Wald can be interpret as $X^2$ If defined as Wald $ = \dfrac{b_k}{SE_{b_k}}$: same procedure as for any $z$ test. Wald can be interpreted as $z$.
n.a.	n.a.	Wald-type approximate $C\%$ confidence interval for $\beta_k$
-	-	$b_k \pm z^* \times SE_{b_k}$ where the critical value $z^$ is the value under the normal curve with the area $C / 100$ between $-z^$ and $z^$ (e.g. $z^$ = 1.96 for a 95% confidence interval).
n.a.	n.a.	Goodness of fit measure $R^2_L$
-	-	$R^2_L = \dfrac{D_{null} - D_K}{D_{null}}$ There are several other goodness of fit measures in logistic regression. In logistic regression, there is no single agreed upon measure of goodness of fit.
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?	Can body mass index, stress level, and gender predict whether people get diagnosed with diabetes?
SPSS	SPSS	SPSS
Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples... Put the two paired variables in the boxes below Variable 1 and Variable 2 Under Test Type, select the Marginal Homogeneity test	Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples... Put the two paired variables in the boxes below Variable 1 and Variable 2 Under Test Type, select the Marginal Homogeneity test	Analyze > Regression > Binary Logistic... Put your dependent variable in the box below Dependent and your independent (predictor) variables in the box below Covariate(s)
n.a.	n.a.	Jamovi
-	-	Regression > 2 Outcomes - Binomial Put your dependent variable in the box below Dependent Variable and your independent variables of interval/ratio level in the box below Covariates If you also have code (dummy) variables as independent variables, you can put these in the box below Covariates as well Instead of transforming your categorical independent variable(s) into code variables, you can also put the untransformed categorical independent variables in the box below Factors. Jamovi will then make the code variables for you 'behind the scenes'
Practice questions	Practice questions	Practice questions

Marginal Homogeneity test / Stuart-Maxwell test - overview