ANCOVA - overview

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ANCOVA
Cochran's Q test
Repeated measures ANCOVA
You cannot compare more than 3 methods
Independent variablesIndependent/grouping variableIndependent 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)One or more within subjects factors (related groups), one or more quantitative control variables of interval or ratio level (covariates), and possibly one or more between subjects factors (independent groups)
Dependent variableDependent variableDependent variables
One quantitative of interval or ratio levelOne categorical with 2 independent groupsOne quantitative of interval or ratio level
THIS TABLE IS YET TO BE COMPLETEDNull hypothesisTHIS TABLE IS YET TO BE COMPLETED
-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 $I.$
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n.a.Alternative hypothesisn.a.
-H1: not all population proportions are equal-
n.a.Assumptionsn.a.
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  • Sample of 'blocks' (usually the subjects) is a simple random sample from the population. That is, blocks are independent of one another
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n.a.Test statisticn.a.
-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.
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n.a.Sampling distribution of $Q$ if H0 were truen.a.
-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?n.a.
-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 $\alpha$
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n.a.Equivalent ton.a.
-Friedman test, with a categorical dependent variable consisting of two independent groups.-
n.a.Example contextn.a.
-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.SPSSn.a.
-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
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n.a.Jamovin.a.
-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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Practice questionsPractice questionsPractice questions