# ANCOVA - overview

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ANCOVA
Cochran's Q test
Independent variablesIndependent variable
One or more categorical with independent groups, and one or more quantitative control variables of interval or ratio level (covariates)One within subject factor ($\geq 2$ related groups)
Dependent variableDependent variable
One quantitative of interval or ratio levelOne categorical with 2 independent groups
THIS TABLE IS YET TO BE COMPLETEDNull hypothesis
-$\pi_1 = \pi_2 = \ldots = \pi_I$
$\pi_1$ is the population proportion of 'successes' in group 1; $\pi_2$ is the population proportion of 'successes' in group 2; $\pi_I$ is the population proportion of 'successes' in group $I$
n.a.Alternative hypothesis
-Not all population proportions are equal
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.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.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.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.Equivalent to
-Friedman test, with a categorical dependent variable consisting of two independent groups
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.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.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
Practice questionsPractice questions