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 | Chi-squared test for the relationship between two categorical variables |
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Independent/grouping variable | Independent variable | Independent variable | Independent variables | Independent /column variable | |
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 | One categorical with $I$ independent groups ($I \geqslant 2$) | |
Dependent variable | Dependent variable | Dependent variable | Dependent variable | Dependent /row 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 | One categorical with $J$ independent groups ($J \geqslant 2$) | |
Null hypothesis | 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: $\mu = \mu_0$
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 chi-squared test for the complete regression model:
| H0: there is no association between the row and column variable More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
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Alternative hypothesis | Alternative hypothesis | 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: $\mu < \mu_0$ | The alternative hypothesis H1 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 chi-squared test for the complete regression model:
| H1: there is an association between the row and column variable More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
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Assumptions | Assumptions | Assumptions | Assumptions | Assumptions | |
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Test statistic | 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 chi-squared test for the complete regression model:
The wald statistic can be defined in two ways:
Likelihood ratio chi-squared test for individual $\beta_k$:
| $X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
Here for each cell, the expected cell count = $\dfrac{\mbox{row total} \times \mbox{column total}}{\mbox{total sample size}}$, the observed cell count is the observed sample count in that same cell, and the sum is over all $I \times J$ cells. | |
Sampling distribution of $Q$ if H0 were true | Sampling distribution of $z$ if H0 were true | Sampling distribution of $X^2$ if H0 were true | Sampling distribution of $X^2$ and of the Wald statistic if H0 were true | Sampling distribution of $X^2$ 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 | Standard normal distribution | If $b + c$ is large enough (say, > 20), approximately the chi-squared 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 chi-squared test for the complete model:
| Approximately the chi-squared distribution with $(I - 1) \times (J - 1)$ degrees of freedom | |
Significant? | 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 chi-squared test for the complete regression model and likelihood ratio chi-squared test for individual $\beta_k$:
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n.a. | $C\%$ confidence interval for $\mu$ | n.a. | Wald-type approximate $C\%$ confidence interval for $\beta_k$ | 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. | - | $b_k \pm z^* \times SE_{b_k}$ 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 $R^2_L$ | 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 $\mu_0.$ | - | $R^2_L = \dfrac{D_{null} - D_K}{D_{null}}$ 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. | n.a. | |
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Equivalent to | n.a. | Equivalent to | n.a. | n.a. | |
Friedman test, with a categorical dependent variable consisting of two independent groups. | - |
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Example context | 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? | Is there an association between economic class and gender? Is the distribution of economic class different between men and women? | |
SPSS | n.a. | SPSS | SPSS | SPSS | |
Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
| - | Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
| Analyze > Regression > Binary Logistic...
| Analyze > Descriptive Statistics > Crosstabs...
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Jamovi | n.a. | 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
| - | Frequencies > Paired Samples - McNemar test
| Regression > 2 Outcomes - Binomial
| Frequencies > Independent Samples - $\chi^2$ test of association
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Practice questions | Practice questions | Practice questions | Practice questions | Practice questions | |