Cochran's Q test  overview
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Cochran's Q test  One sample $z$ test for the mean  McNemar's test  Logistic regression 


Independent/grouping variable  Independent variable  Independent variable  Independent variables  
One within subject factor ($\geq 2$ related groups)  None  2 paired 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 quantitative of interval or ratio level  One categorical with 2 independent groups  One categorical with 2 independent groups  
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}: $\mu = \mu_0$
Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis.  Let's say that the scores on the dependent variable are scored 0 and 1. Then for each pair of scores, the data allow four options:
Other formulations of the null hypothesis are:
 Model chisquared test for the complete regression model:
 
Alternative hypothesis  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: $\mu < \mu_0$  The alternative hypothesis H_{1} is that for each pair of scores, P(first score of pair is 0 while second score of pair is 1) $\neq$ P(first score of pair is 1 while second score of pair is 0). That is, the probability that a pair of scores switches from 0 to 1 is not the same as the probability that a pair of scores switches from 1 to 0. Other formulations of the alternative hypothesis are:
 Model chisquared 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.  $z = \dfrac{\bar{y}  \mu_0}{\sigma / \sqrt{N}}$
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 $\mu_0$.  $X^2 = \dfrac{(b  c)^2}{b + c}$
Here $b$ is the number of pairs in the sample for which the first score is 0 while the second score is 1, and $c$ is the number of pairs in the sample for which the first score is 1 while the second score is 0.  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 $z$ if H_{0} were true  Sampling distribution of $X^2$ 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  Standard normal distribution  If $b + c$ is large enough (say, > 20), approximately the chisquared distribution with 1 degree of freedom. If $b + c$ is small, the Binomial($n$, $P$) distribution should be used, with $n = b + c$ and $P = 0.5$. In that case the test statistic becomes equal to $b$.  Sampling distribution of $X^2$, as computed in the model chisquared test for the complete model:
 
Significant?  Significant?  Significant?  Significant?  
If the number of blocks is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
 Two sided:
 For test statistic $X^2$:
 For the model chisquared test for the complete regression model and likelihood ratio chisquared test for individual $\beta_k$:
 
n.a.  $C\%$ confidence interval for $\mu$  n.a.  Waldtype approximate $C\%$ confidence interval for $\beta_k$  
  $\bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}}$
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.    $b_k \pm z^* \times SE_{b_k}$ 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.  Effect size  n.a.  Goodness of fit measure $R^2_L$  
  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 $\mu_0.$    $R^2_L = \dfrac{D_{null}  D_K}{D_{null}}$ 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.  
n.a.  Visual representation  n.a.  n.a.  
      
Equivalent to  n.a.  Equivalent to  n.a.  
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 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 $\sigma = 3.$  Does a tv documentary about spiders change whether people are afraid (yes/no) of spiders?  Can body mass index, stress level, and gender predict whether people get diagnosed with diabetes?  
SPSS  n.a.  SPSS  SPSS  
Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
   Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
 Analyze > Regression > Binary Logistic...
 
Jamovi  n.a.  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
   Frequencies > Paired Samples  McNemar test
 Regression > 2 Outcomes  Binomial
 
Practice questions  Practice questions  Practice questions  Practice questions  