Cochran's Q test overview

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Cochran's Q test	Paired sample $t$ test	Two way ANOVA
Independent/grouping variable	Independent variable	Independent/grouping variables
One within subject factor ($\geq 2$ related groups)	2 paired groups	Two categorical, the first with $I$ independent groups and the second with $J$ independent groups ($I \geqslant 2$, $J \geqslant 2$)
Dependent variable	Dependent variable	Dependent variable
One categorical with 2 independent groups	One quantitative of interval or ratio level	One quantitative of interval or ratio level
Null hypothesis	Null hypothesis	Null hypothesis
H₀: $\pi_1 = \pi_2 = \ldots = \pi_I$ 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 $I.$	H₀: $\mu = \mu_0$ Here $\mu$ is the population mean of the difference scores, and $\mu_0$ is the population mean of the difference scores according to the null hypothesis, which is usually 0. A difference score is the difference between the first score of a pair and the second score of a pair.	ANOVA $F$ tests: H₀ for main and interaction effects together (model): no main effects and interaction effect H₀ for independent variable A: no main effect for A H₀ for independent variable B: no main effect for B H₀ for the interaction term: no interaction effect between A and B Like in one way ANOVA, we can also perform $t$ tests for specific contrasts and multiple comparisons. This is more advanced stuff.
Alternative hypothesis	Alternative hypothesis	Alternative hypothesis
H₁: not all population proportions are equal	H₁ two sided: $\mu \neq \mu_0$ H₁ right sided: $\mu > \mu_0$ H₁ left sided: $\mu < \mu_0$	ANOVA $F$ tests: H₁ for main and interaction effects together (model): there is a main effect for A, and/or for B, and/or an interaction effect H₁ for independent variable A: there is a main effect for A H₁ for independent variable B: there is a main effect for B H₁ for the interaction term: there is an interaction effect between A and B
Assumptions	Assumptions	Assumptions
Sample of 'blocks' (usually the subjects) is a simple random sample from the population. That is, blocks are independent of one another	Difference scores are normally distributed in the population Sample of difference scores is a simple random sample from the population of difference scores. That is, difference scores are independent of one another	Within each of the $I \times J$ populations, the scores on the dependent variable are normally distributed The standard deviation of the scores on the dependent variable is the same in each of the $I \times J$ populations For each of the $I \times J$ groups, the sample is an independent and simple random sample from the population defined by that group. That is, within and between groups, observations are independent of one another Equal sample sizes for each group make the interpretation of the ANOVA output easier (unequal sample sizes result in overlap in the sum of squares; this is advanced stuff)
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.	$t = \dfrac{\bar{y} - \mu_0}{s / \sqrt{N}}$ Here $\bar{y}$ is the sample mean of the difference scores, $\mu_0$ is the population mean of the difference scores according to the null hypothesis, $s$ is the sample standard deviation of the difference scores, and $N$ is the sample size (number of difference scores). The denominator $s / \sqrt{N}$ is the standard error of the sampling distribution of $\bar{y}$. The $t$ value indicates how many standard errors $\bar{y}$ is removed from $\mu_0$.	For main and interaction effects together (model): $F = \dfrac{\mbox{mean square model}}{\mbox{mean square error}}$ For independent variable A: $F = \dfrac{\mbox{mean square A}}{\mbox{mean square error}}$ For independent variable B: $F = \dfrac{\mbox{mean square B}}{\mbox{mean square error}}$ For the interaction term: $F = \dfrac{\mbox{mean square interaction}}{\mbox{mean square error}}$ Note: mean square error is also known as mean square residual or mean square within.
n.a.	n.a.	Pooled standard deviation
-	-	$ \begin{aligned} s_p &= \sqrt{\dfrac{\sum\nolimits_{subjects} (\mbox{subject's score} - \mbox{its group mean})^2}{N - (I \times J)}}\\ &= \sqrt{\dfrac{\mbox{sum of squares error}}{\mbox{degrees of freedom error}}}\\ &= \sqrt{\mbox{mean square error}} \end{aligned} $
Sampling distribution of $Q$ if H₀ were true	Sampling distribution of $t$ if H₀ were true	Sampling distribution of $F$ if H₀ were true
If the number of blocks (usually the number of subjects) is large, approximately the chi-squared distribution with $k - 1$ degrees of freedom	$t$ distribution with $N - 1$ degrees of freedom	For main and interaction effects together (model): $F$ distribution with $(I - 1) + (J - 1) + (I - 1) \times (J - 1)$ (df model, numerator) and $N - (I \times J)$ (df error, denominator) degrees of freedom For independent variable A: $F$ distribution with $I - 1$ (df A, numerator) and $N - (I \times J)$ (df error, denominator) degrees of freedom For independent variable B: $F$ distribution with $J - 1$ (df B, numerator) and $N - (I \times J)$ (df error, denominator) degrees of freedom For the interaction term: $F$ distribution with $(I - 1) \times (J - 1)$ (df interaction, numerator) and $N - (I \times J)$ (df error, denominator) degrees of freedom Here $N$ is the total sample size.
Significant?	Significant?	Significant?
If the number of blocks is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$: Check if $X^2$ observed in sample is equal to or larger than critical value $X^{2*}$ or Find $p$ value corresponding to observed $X^2$ and check if it is equal to or smaller than $\alpha$	Two sided: Check if $t$ observed in sample is at least as extreme as critical value $t^$ or Find two sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$ Right sided: Check if $t$ observed in sample is equal to or larger than critical value $t^$ or Find right sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$ Left sided: Check if $t$ observed in sample is equal to or smaller than critical value $t^*$ or Find left sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$	Check if $F$ observed in sample is equal to or larger than critical value $F^*$ or Find $p$ value corresponding to observed $F$ and check if it is equal to or smaller than $\alpha$
n.a.	$C\%$ confidence interval for $\mu$	n.a.
-	$\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$ where the critical value $t^$ is the value under the $t_{N-1}$ distribution with the area $C / 100$ between $-t^$ and $t^$ (e.g. $t^$ = 2.086 for a 95% confidence interval when df = 20). The confidence interval for $\mu$ can also be used as significance test.	-
n.a.	Effect size	Effect size
-	Cohen's $d$: Standardized difference between the sample mean of the difference scores and $\mu_0$: $$d = \frac{\bar{y} - \mu_0}{s}$$ Cohen's $d$ indicates how many standard deviations $s$ the sample mean of the difference scores $\bar{y}$ is removed from $\mu_0.$	Proportion variance explained $R^2$: Proportion variance of the dependent variable $y$ explained by the independent variables and the interaction effect together: $$ \begin{align} R^2 &= \dfrac{\mbox{sum of squares model}}{\mbox{sum of squares total}} \end{align} $$ $R^2$ is the proportion variance explained in the sample. It is a positively biased estimate of the proportion variance explained in the population. Proportion variance explained $\eta^2$: Proportion variance of the dependent variable $y$ explained by an independent variable or interaction effect: $$ \begin{align} \eta^2_A &= \dfrac{\mbox{sum of squares A}}{\mbox{sum of squares total}}\\ \\ \eta^2_B &= \dfrac{\mbox{sum of squares B}}{\mbox{sum of squares total}}\\ \\ \eta^2_{int} &= \dfrac{\mbox{sum of squares int}}{\mbox{sum of squares total}} \end{align} $$ $\eta^2$ is the proportion variance explained in the sample. It is a positively biased estimate of the proportion variance explained in the population. Proportion variance explained $\omega^2$: Corrects for the positive bias in $\eta^2$ and is equal to: $$ \begin{align} \omega^2_A &= \dfrac{\mbox{sum of squares A} - \mbox{degrees of freedom A} \times \mbox{mean square error}}{\mbox{sum of squares total} + \mbox{mean square error}}\\ \\ \omega^2_B &= \dfrac{\mbox{sum of squares B} - \mbox{degrees of freedom B} \times \mbox{mean square error}}{\mbox{sum of squares total} + \mbox{mean square error}}\\ \\ \omega^2_{int} &= \dfrac{\mbox{sum of squares int} - \mbox{degrees of freedom int} \times \mbox{mean square error}}{\mbox{sum of squares total} + \mbox{mean square error}}\\ \end{align} $$ $\omega^2$ is a better estimate of the explained variance in the population than $\eta^2$. Only for balanced designs (equal sample sizes). Proportion variance explained $\eta^2_{partial}$: $$ \begin{align} \eta^2_{partial\,A} &= \frac{\mbox{sum of squares A}}{\mbox{sum of squares A} + \mbox{sum of squares error}}\\ \\ \eta^2_{partial\,B} &= \frac{\mbox{sum of squares B}}{\mbox{sum of squares B} + \mbox{sum of squares error}}\\ \\ \eta^2_{partial\,int} &= \frac{\mbox{sum of squares int}}{\mbox{sum of squares int} + \mbox{sum of squares error}} \end{align} $$
n.a.	Visual representation	n.a.
-		-
n.a.	n.a.	ANOVA table
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Equivalent to	Equivalent to	Equivalent to
Friedman test, with a categorical dependent variable consisting of two independent groups.	One sample $t$ test on the difference scores. Repeated measures ANOVA with one dichotomous within subjects factor.	OLS regression with two categorical independent variables and the interaction term, transformed into $(I - 1)$ + $(J - 1)$ + $(I - 1) \times (J - 1)$ code variables.
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 difference between the mental health scores before and after an intervention different from $\mu_0 = 0$?	Is the average mental health score different between people from a low, moderate, and high economic class? And is the average mental health score different between men and women? And is there an interaction effect between economic class and gender?
SPSS	SPSS	SPSS
Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples... Put the $k$ variables containing the scores for the $k$ related groups in the white box below Test Variables Under Test Type, select Cochran's Q test	Analyze > Compare Means > Paired-Samples T Test... Put the two paired variables in the boxes below Variable 1 and Variable 2	Analyze > General Linear Model > Univariate... Put your dependent (quantitative) variable in the box below Dependent Variable and your two independent (grouping) variables in the box below Fixed Factor(s)
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 Put the $k$ variables containing the scores for the $k$ related groups in the box below Measures	T-Tests > Paired Samples T-Test Put the two paired variables in the box below Paired Variables, one on the left side of the vertical line and one on the right side of the vertical line Under Hypothesis, select your alternative hypothesis	ANOVA > ANOVA Put your dependent (quantitative) variable in the box below Dependent Variable and your two independent (grouping) variables in the box below Fixed Factors
Practice questions	Practice questions	Practice questions

Cochran's Q test - overview