ANCOVA - overview

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ANCOVA
Ordinal logistic regression
Cochran's Q test
You cannot compare more than 3 methods
Independent variablesIndependent variablesIndependent/grouping 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 variablesOne within subject factor ($\geq 2$ related groups)
Dependent variableDependent variableDependent variable
One quantitative of interval or ratio levelOne of ordinal levelOne categorical with 2 independent groups
THIS TABLE IS YET TO BE COMPLETEDTHIS TABLE IS YET TO BE COMPLETEDNull hypothesis
--H0: $\pi_1 = \pi_2 = \ldots = \pi_I$

Here $\pi_1$ is the population proportion of 'successes' for group 1, $\pi_2$ is the population proportion of 'successes' for group 2, and $\pi_I$ is the population proportion of 'successes' for group $I.$
n.a.n.a.Alternative hypothesis
--H1: not all population proportions are equal
n.a.n.a.Assumptions
--
  • Sample of 'blocks' (usually the subjects) is a simple random sample from the population. That is, blocks are independent of one another
n.a.n.a.Test statistic
--If a failure is scored as 0 and a success is scored as 1:

$Q = k(k - 1) \dfrac{\sum_{groups} \Big (\mbox{group total} - \frac{\mbox{grand total}}{k} \Big)^2}{\sum_{blocks} \mbox{block total} \times (k - \mbox{block total})}$

Here $k$ is the number of related groups (usually the number of repeated measurements), a group total is the sum of the scores in a group, a block total is the sum of the scores in a block (usually a subject), and the grand total is the sum of all the scores.

Before computing $Q$, first exclude blocks with equal scores in all $k$ groups.
n.a.n.a.Sampling distribution of $Q$ if H0 were true
--If the number of blocks (usually the number of subjects) is large, approximately the chi-squared distribution with $k - 1$ degrees of freedom
n.a.n.a.Significant?
--If the number of blocks is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
  • 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.Equivalent to
--Friedman test, with a categorical dependent variable consisting of two independent groups.
n.a.n.a.Example context
--Subjects perform three different tasks, which they can either perform correctly or incorrectly. Is there a difference in task performance between the three different tasks?
n.a.n.a.SPSS
--Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
  • Put the $k$ variables containing the scores for the $k$ related groups in the white box below Test Variables
  • Under Test Type, select Cochran's Q test
n.a.n.a.Jamovi
--Jamovi does not have a specific option for the Cochran's Q test. However, you can do the Friedman test instead. The $p$ value resulting from this Friedman test is equivalent to the $p$ value that would have resulted from the Cochran's Q test. Go to:

ANOVA > Repeated Measures ANOVA - Friedman
  • Put the $k$ variables containing the scores for the $k$ related groups in the box below Measures
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