Cochran's Q test  overview
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Cochran's Q test  Logistic regression 


Independent/grouping variable  Independent variables  
One within subject factor ($\geq 2$ related groups)  One or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variables  
Dependent variable  Dependent variable  
One categorical with 2 independent groups  One categorical with 2 independent groups  
Null hypothesis  Null hypothesis  
H_{0}: $\pi_1 = \pi_2 = \ldots = \pi_I$
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 $I.$  Model chisquared test for the complete regression model:
 
Alternative hypothesis  Alternative hypothesis  
H_{1}: not all population proportions are equal  Model chisquared test for the complete regression model:
 
Assumptions  Assumptions  

 
Test statistic  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.  Model chisquared test for the complete regression model:
The wald statistic can be defined in two ways:
Likelihood ratio chisquared test for individual $\beta_k$:
 
Sampling distribution of $Q$ if H_{0} were true  Sampling distribution of $X^2$ and of the Wald statistic if H_{0} were true  
If the number of blocks (usually the number of subjects) is large, approximately the chisquared distribution with $k  1$ degrees of freedom  Sampling distribution of $X^2$, as computed in the model chisquared test for the complete model:
 
Significant?  Significant?  
If the number of blocks is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
 For the model chisquared test for the complete regression model and likelihood ratio chisquared test for individual $\beta_k$:
 
n.a.  Waldtype approximate $C\%$ confidence interval for $\beta_k$  
  $b_k \pm z^* \times SE_{b_k}$ where the critical value $z^*$ is the value under the normal curve with the area $C / 100$ between $z^*$ and $z^*$ (e.g. $z^*$ = 1.96 for a 95% confidence interval).  
n.a.  Goodness of fit measure $R^2_L$  
  $R^2_L = \dfrac{D_{null}  D_K}{D_{null}}$ There are several other goodness of fit measures in logistic regression. In logistic regression, there is no single agreed upon measure of goodness of fit.  
Equivalent to  n.a.  
Friedman test, with a categorical dependent variable consisting of two independent groups.    
Example context  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?  Can body mass index, stress level, and gender predict whether people get diagnosed with diabetes?  
SPSS  SPSS  
Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
 Analyze > Regression > Binary Logistic...
 
Jamovi  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
 Regression > 2 Outcomes  Binomial
 
Practice questions  Practice questions  