ANCOVA: overview
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ANCOVA  Cochran's Q test 


Independent variables  Independent 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 variable  Dependent variable  
One quantitative of interval or ratio level  One categorical with 2 independent groups  
THIS TABLE IS YET TO BE COMPLETED  Null 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 chisquared 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 $X^2 = Q$:
 
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...
 
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
 
Practice questions  Practice questions  