Cochran's Q test overview

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Cochran's Q test	Pearson correlation	One sample $t$ test for the mean	Binomial test for a single proportion
Independent/grouping variable	Variable 1	Independent variable	Independent variable
One within subject factor ($\geq 2$ related groups)	One quantitative of interval or ratio level	None	None
Dependent variable	Variable 2	Dependent variable	Dependent variable
One categorical with 2 independent groups	One quantitative of interval or ratio level	One quantitative of interval or ratio level	One categorical with 2 independent groups
Null hypothesis	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: $\rho = \rho_0$ Here $\rho$ is the Pearson correlation in the population, and $\rho_0$ is the Pearson correlation in the population according to the null hypothesis (usually 0). The Pearson correlation is a measure for the strength and direction of the linear relationship between two variables of at least interval measurement level.	H₀: $\mu = \mu_0$ Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis.	H₀: $\pi = \pi_0$ Here $\pi$ is the population proportion of 'successes', and $\pi_0$ is the population proportion of successes according to the null hypothesis.
Alternative hypothesis	Alternative hypothesis	Alternative hypothesis	Alternative hypothesis
H₁: not all population proportions are equal	H₁ two sided: $\rho \neq \rho_0$ H₁ right sided: $\rho > \rho_0$ H₁ left sided: $\rho < \rho_0$	H₁ two sided: $\mu \neq \mu_0$ H₁ right sided: $\mu > \mu_0$ H₁ left sided: $\mu < \mu_0$	H₁ two sided: $\pi \neq \pi_0$ H₁ right sided: $\pi > \pi_0$ H₁ left sided: $\pi < \pi_0$
Assumptions	Assumptions of test for correlation	Assumptions	Assumptions
Sample of 'blocks' (usually the subjects) is a simple random sample from the population. That is, blocks are independent of one another	In the population, the two variables are jointly normally distributed (this covers the normality, homoscedasticity, and linearity assumptions) Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another Note: these assumptions are only important for the significance test and confidence interval, not for the correlation coefficient itself. The correlation coefficient just measures the strength of the linear relationship between two variables.	Scores are normally distributed in the population Sample is a simple random sample from the population. That is, observations are independent of one another	Sample is a simple random sample from the population. That is, observations are independent of one another
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.	Test statistic for testing H0: $\rho = 0$: $t = \dfrac{r \times \sqrt{N - 2}}{\sqrt{1 - r^2}} $ where $r$ is the sample correlation $r = \frac{1}{N - 1} \sum_{j}\Big(\frac{x_{j} - \bar{x}}{s_x} \Big) \Big(\frac{y_{j} - \bar{y}}{s_y} \Big)$ and $N$ is the sample size Test statistic for testing values for $\rho$ other than $\rho = 0$: $z = \dfrac{r_{Fisher} - \rho_{0_{Fisher}}}{\sqrt{\dfrac{1}{N - 3}}}$ $r_{Fisher} = \dfrac{1}{2} \times \log\Bigg(\dfrac{1 + r}{1 - r} \Bigg )$, where $r$ is the sample correlation $\rho_{0_{Fisher}} = \dfrac{1}{2} \times \log\Bigg( \dfrac{1 + \rho_0}{1 - \rho_0} \Bigg )$, where $\rho_0$ is the population correlation according to H0	$t = \dfrac{\bar{y} - \mu_0}{s / \sqrt{N}}$ Here $\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to the null hypothesis, $s$ is the sample standard deviation, and $N$ is the sample size. 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 $\mu_0$.	$X$ = number of successes in the sample
Sampling distribution of $Q$ if H₀ were true	Sampling distribution of $t$ and of $z$ if H₀ were true	Sampling distribution of $t$ if H₀ were true	Sampling distribution of $X$ 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	Sampling distribution of $t$: $t$ distribution with $N - 2$ degrees of freedom Sampling distribution of $z$: Approximately the standard normal distribution	$t$ distribution with $N - 1$ degrees of freedom	Binomial($n$, $P$) distribution. Here $n = N$ (total sample size), and $P = \pi_0$ (population proportion according to the null hypothesis).
Significant?	Significant?	Significant?	Significant?
If the number of blocks is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$: 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 $\alpha$	$t$ Test 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 $\alpha$ $t$ Test 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 $\alpha$ $t$ Test 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 $\alpha$ $z$ Test two sided: Check if $z$ observed in sample is at least as extreme as critical value $z^$ or Find two sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$ $z$ Test right sided: Check if $z$ observed in sample is equal to or larger than critical value $z^$ or Find right sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$ $z$ Test left sided: Check if $z$ observed in sample is equal to or smaller than critical value $z^$ or Find left sided $p$ value corresponding to observed $z$ 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 $\alpha$	Two sided: Check if $X$ observed in sample is in the rejection region or Find two sided $p$ value corresponding to observed $X$ and check if it is equal to or smaller than $\alpha$ Right sided: Check if $X$ observed in sample is in the rejection region or Find right sided $p$ value corresponding to observed $X$ and check if it is equal to or smaller than $\alpha$ Left sided: Check if $X$ observed in sample is in the rejection region or Find left sided $p$ value corresponding to observed $X$ and check if it is equal to or smaller than $\alpha$
n.a.	Approximate $C$% confidence interval for $\rho$	$C\%$ confidence interval for $\mu$	n.a.
-	First compute the approximate $C$% confidence interval for $\rho_{Fisher}$: $lower_{Fisher} = r_{Fisher} - z^* \times \sqrt{\dfrac{1}{N - 3}}$ $upper_{Fisher} = r_{Fisher} + z^* \times \sqrt{\dfrac{1}{N - 3}}$ where $r_{Fisher} = \frac{1}{2} \times \log\Bigg(\dfrac{1 + r}{1 - r} \Bigg )$ and 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). Then transform back to get the approximate $C$% confidence interval for $\rho$: lower bound = $\dfrac{e^{2 \times lower_{Fisher}} - 1}{e^{2 \times lower_{Fisher}} + 1}$ upper bound = $\dfrac{e^{2 \times upper_{Fisher}} - 1}{e^{2 \times upper_{Fisher}} + 1}$	$\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$ 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.	Properties of the Pearson correlation coefficient	Effect size	n.a.
-	The Pearson correlation coefficient is a measure for the linear relationship between two quantitative variables. The Pearson correlation coefficient squared reflects the proportion of variance explained in one variable by the other variable. The Pearson correlation coefficient can take on values between -1 (perfect negative relationship) and 1 (perfect positive relationship). A value of 0 means no linear relationship. The absolute size of the Pearson correlation coefficient is not affected by any linear transformation of the variables. However, the sign of the Pearson correlation will flip when the scores on one of the two variables are multiplied by a negative number (reversing the direction of measurement of that variable). For example: the correlation between $x$ and $y$ is equivalent to the correlation between $3x + 5$ and $2y - 6$. the absolute value of the correlation between $x$ and $y$ is equivalent to the absolute value of the correlation between $-3x + 5$ and $2y - 6$. However, the signs of the two correlation coefficients will be in opposite directions, due to the multiplication of $x$ by $-3$. The Pearson correlation coefficient does not say anything about causality. The Pearson correlation coefficient is sensitive to outliers.	Cohen's $d$: Standardized difference between the sample mean and $\mu_0$: $$d = \frac{\bar{y} - \mu_0}{s}$$ Cohen's $d$ indicates how many standard deviations $s$ the sample mean $\bar{y}$ is removed from $\mu_0.$	-
n.a.	n.a.	Visual representation	n.a.
-	-		-
Equivalent to	Equivalent to	n.a.	n.a.
Friedman test, with a categorical dependent variable consisting of two independent groups.	OLS regression with one independent variable: $b_1 = r \times \frac{s_y}{s_x}$ Results significance test ($t$ and $p$ value) testing $H_0$: $\beta_1 = 0$ are equivalent to results significance test testing $H_0$: $\rho = 0$	-	-
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 linear relationship between physical health and mental health?	Is the average mental health score of office workers different from $\mu_0 = 50$?	Is the proportion of smokers amongst office workers different from $\pi_0 = 0.2$?
SPSS	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 > Correlate > Bivariate... Put your two variables in the box below Variables	Analyze > Compare Means > One-Sample T Test... Put your variable in the box below Test Variable(s) Fill in the value for $\mu_0$ in the box next to Test Value	Analyze > Nonparametric Tests > Legacy Dialogs > Binomial... Put your dichotomous variable in the box below Test Variable List Fill in the value for $\pi_0$ in the box next to Test Proportion
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 Put the $k$ variables containing the scores for the $k$ related groups in the box below Measures	Regression > Correlation Matrix Put your two variables in the white box at the right Under Correlation Coefficients, select Pearson (selected by default) Under Hypothesis, select your alternative hypothesis	T-Tests > One Sample T-Test Put your variable in the box below Dependent Variables Under Hypothesis, fill in the value for $\mu_0$ in the box next to Test Value, and select your alternative hypothesis	Frequencies > 2 Outcomes - Binomial test Put your dichotomous variable in the white box at the right Fill in the value for $\pi_0$ in the box next to Test value Under Hypothesis, select your alternative hypothesis
Practice questions	Practice questions	Practice questions	Practice questions

Cochran's Q test - overview