Multilevel ordinal logistic regression - overview

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Multilevel ordinal logistic regression
Marginal Homogeneity test / Stuart-Maxwell test
Independent variablesIndependent variable
One or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variables, plus at least one random factor2 paired groups
Dependent variableDependent variable
One of ordinal levelOne categorical with $J$ independent groups ($J \geqslant 2$)
THIS TABLE IS YET TO BE COMPLETEDNull 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 $j.$
n.a.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.
n.a.Assumptions
-
  • Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
n.a.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.
n.a.Sampling distribution of the test statistic if H0 were true
-Approximately the chi-squared distribution with $J - 1$ degrees of freedom
n.a.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 $\alpha$
n.a.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?
n.a.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
Practice questionsPractice questions