ANCOVA - overview

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ANCOVA
Multilevel ordinal logistic regression
Marginal Homogeneity test / Stuart-Maxwell test
You cannot compare more than 3 methods
Independent variablesIndependent variablesIndependent variable
One or more categorical with independent groups, and one or more quantitative control variables of interval or ratio level (covariates)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 variableDependent variable
One quantitative of interval or ratio levelOne of ordinal levelOne categorical with $J$ independent groups ($J \geqslant 2$)
THIS TABLE IS YET TO BE COMPLETEDTHIS 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.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.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.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.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.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.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.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
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