Cochran's Q test  overview
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Cochran's Q test  Friedman test  Cochran's Q test  Regression (OLS) 


Independent/grouping variable  Independent/grouping variable  Independent/grouping variable  Independent variables  
One within subject factor ($\geq 2$ related groups)  One within subject factor ($\geq 2$ related groups)  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  Dependent variable  Dependent variable  
One categorical with 2 independent groups  One of ordinal level  One categorical with 2 independent groups  One quantitative of interval or ratio level  
Null hypothesis  Null hypothesis  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 $I.$  H_{0}: the population scores in any of the related groups are not systematically higher or lower than the population scores in any of the other related groups
Usually the related groups are the different measurement points. Several different formulations of the null hypothesis can be found in the literature, and we do not agree with all of them. Make sure you (also) learn the one that is given in your text book or by your teacher.  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 $I.$  $F$ test for the complete regression model:
 
Alternative hypothesis  Alternative hypothesis  Alternative hypothesis  Alternative hypothesis  
H_{1}: not all population proportions are equal  H_{1}: the population scores in some of the related groups are systematically higher or lower than the population scores in other related groups  H_{1}: not all population proportions are equal  $F$ test for the complete regression model:
 
Assumptions  Assumptions  Assumptions  Assumptions  



 
Test statistic  Test statistic  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.  $Q = \dfrac{12}{N \times k(k + 1)} \sum R^2_i  3 \times N(k + 1)$
Here $N$ is the number of 'blocks' (usually the subjects  so if you have 4 repeated measurements for 60 subjects, $N$ equals 60), $k$ is the number of related groups (usually the number of repeated measurements), and $R_i$ is the sum of ranks in group $i$. Remember that multiplication precedes addition, so first compute $\frac{12}{N \times k(k + 1)} \times \sum R^2_i$ and then subtract $3 \times N(k + 1)$. Note: if ties are present in the data, the formula for $Q$ is more complicated.  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.  $F$ test for the complete regression model:
Note 2: if there is only one independent variable in the model ($K = 1$), the $F$ test for the complete regression model is equivalent to the two sided $t$ test for $\beta_1.$  
n.a.  n.a.  n.a.  Sample standard deviation of the residuals $s$  
      $\begin{aligned} s &= \sqrt{\dfrac{\sum (y_j  \hat{y}_j)^2}{N  K  1}}\\ &= \sqrt{\dfrac{\mbox{sum of squares error}}{\mbox{degrees of freedom error}}}\\ &= \sqrt{\mbox{mean square error}} \end{aligned} $  
Sampling distribution of $Q$ if H_{0} were true  Sampling distribution of $Q$ if H_{0} were true  Sampling distribution of $Q$ if H_{0} were true  Sampling distribution of $F$ and of $t$ 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  If the number of blocks $N$ is large, approximately the chisquared distribution with $k  1$ degrees of freedom.
For small samples, the exact distribution of $Q$ should be used.  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 $F$:
 
Significant?  Significant?  Significant?  Significant?  
If the number of blocks is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
 If the number of blocks $N$ is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
 If the number of blocks is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
 $F$ test:
 
n.a.  n.a.  n.a.  $C\%$ confidence interval for $\beta_k$ and for $\mu_y$, $C\%$ prediction interval for $y_{new}$  
      Confidence interval for $\beta_k$:
 
n.a.  n.a.  n.a.  Effect size  
      Complete model:
 
n.a.  n.a.  n.a.  Visual representation  
      Regression equations with:  
n.a.  n.a.  n.a.  ANOVA table  
      
Equivalent to  n.a.  Equivalent to  n.a.  
Friedman test, with a categorical dependent variable consisting of two independent groups.    Friedman test, with a categorical dependent variable consisting of two independent groups.    
Example context  Example context  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?  Is there a difference in depression level between measurement point 1 (preintervention), measurement point 2 (1 week postintervention), and measurement point 3 (6 weeks postintervention)?  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 mental health be predicted from fysical health, economic class, and gender?  
SPSS  SPSS  SPSS  SPSS  
Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
 Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
 Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
 Analyze > Regression > Linear...
 
Jamovi  Jamovi  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
 ANOVA > Repeated Measures ANOVA  Friedman
 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 > Linear Regression
 
Practice questions  Practice questions  Practice questions  Practice questions  