Cochran's Q test  overview
This page offers structured overviews of one or more selected methods. Add additional methods for comparisons by clicking on the dropdown button in the righthand column. To practice with a specific method click the button at the bottom row of the table
Cochran's Q test  One sample $z$ test for the mean  Spearman's rho 


Independent/grouping variable  Independent variable  Variable 1  
One within subject factor ($\geq 2$ related groups)  None  One of ordinal level  
Dependent variable  Dependent variable  Variable 2  
One categorical with 2 independent groups  One quantitative of interval or ratio level  One of ordinal level  
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}: $\mu = \mu_0$
Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis.  H_{0}: $\rho_s = 0$
Here $\rho_s$ is the Spearman correlation in the population. The Spearman correlation is a measure for the strength and direction of the monotonic relationship between two variables of at least ordinal measurement level. In words, the null hypothesis would be: H_{0}: there is no monotonic relationship between the two variables in the population.  
Alternative hypothesis  Alternative hypothesis  Alternative hypothesis  
H_{1}: not all population proportions are equal  H_{1} two sided: $\mu \neq \mu_0$ H_{1} right sided: $\mu > \mu_0$ H_{1} left sided: $\mu < \mu_0$  H_{1} two sided: $\rho_s \neq 0$ H_{1} right sided: $\rho_s > 0$ H_{1} left sided: $\rho_s < 0$  
Assumptions  Assumptions  Assumptions  


 
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.  $z = \dfrac{\bar{y}  \mu_0}{\sigma / \sqrt{N}}$
Here $\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to the null hypothesis, $\sigma$ is the population standard deviation, and $N$ is the sample size. The denominator $\sigma / \sqrt{N}$ is the standard deviation of the sampling distribution of $\bar{y}$. The $z$ value indicates how many of these standard deviations $\bar{y}$ is removed from $\mu_0$.  $t = \dfrac{r_s \times \sqrt{N  2}}{\sqrt{1  r_s^2}} $ Here $r_s$ is the sample Spearman correlation and $N$ is the sample size. The sample Spearman correlation $r_s$ is equal to the Pearson correlation applied to the rank scores.  
Sampling distribution of $Q$ if H_{0} were true  Sampling distribution of $z$ if H_{0} were true  Sampling distribution 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  Standard normal distribution  Approximately the $t$ distribution with $N  2$ degrees of freedom  
Significant?  Significant?  Significant?  
If the number of blocks is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
 Two sided:
 Two sided:
 
n.a.  $C\%$ confidence interval for $\mu$  n.a.  
  $\bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}}$
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). The confidence interval for $\mu$ can also be used as significance test.    
n.a.  Effect size  n.a.  
  Cohen's $d$: Standardized difference between the sample mean and $\mu_0$: $$d = \frac{\bar{y}  \mu_0}{\sigma}$$ Cohen's $d$ indicates how many standard deviations $\sigma$ the sample mean $\bar{y}$ is removed from $\mu_0.$    
n.a.  Visual representation  n.a.  
    
Equivalent to  n.a.  n.a.  
Friedman test, with a categorical dependent variable consisting of two independent groups.      
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 the average mental health score of office workers different from $\mu_0 = 50$? Assume that the standard deviation of the mental health scores in the population is $\sigma = 3.$  Is there a monotonic relationship between physical health and mental health?  
SPSS  n.a.  SPSS  
Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
   Analyze > Correlate > Bivariate...
 
Jamovi  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
   Regression > Correlation Matrix
 
Practice questions  Practice questions  Practice questions  