Cochran's Q test - overview
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Cochran's Q test | Mann-Whitney-Wilcoxon test | $z$ test for the difference between two proportions | Chi-squared test for the relationship between two categorical variables |
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Independent/grouping variable | Independent/grouping variable | Independent/grouping variable | Independent /column variable | |
One within subject factor ($\geq 2$ related groups) | One categorical with 2 independent groups | One categorical with 2 independent groups | One categorical with $I$ independent groups ($I \geqslant 2$) | |
Dependent variable | Dependent variable | Dependent variable | Dependent /row variable | |
One categorical with 2 independent groups | One of ordinal level | One categorical with 2 independent groups | One categorical with $J$ independent groups ($J \geqslant 2$) | |
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 $I.$ | If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in both populations:
Formulation 1:
| H0: $\pi_1 = \pi_2$
Here $\pi_1$ is the population proportion of 'successes' for group 1, and $\pi_2$ is the population proportion of 'successes' for group 2. | 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 | |
H1: not all population proportions are equal | If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in both populations:
Formulation 1:
| H1 two sided: $\pi_1 \neq \pi_2$ H1 right sided: $\pi_1 > \pi_2$ H1 left sided: $\pi_1 < \pi_2$ | 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 | |
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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. | Two different types of test statistics can be used; both will result in the same test outcome. The first is the Wilcoxon rank sum statistic $W$:
Note: we could just as well base W and U on group 2. This would only 'flip' the right and left sided alternative hypotheses. Also, tables with critical values for $U$ are often based on the smaller of $U$ for group 1 and for group 2. | $z = \dfrac{p_1 - p_2}{\sqrt{p(1 - p)\Bigg(\dfrac{1}{n_1} + \dfrac{1}{n_2}\Bigg)}}$
Here $p_1$ is the sample proportion of successes in group 1: $\dfrac{X_1}{n_1}$, $p_2$ is the sample proportion of successes in group 2: $\dfrac{X_2}{n_2}$, $p$ is the total proportion of successes in the sample: $\dfrac{X_1 + X_2}{n_1 + n_2}$, $n_1$ is the sample size of group 1, and $n_2$ is the sample size of group 2. Note: we could just as well compute $p_2 - p_1$ in the numerator, but then the left sided alternative becomes $\pi_2 < \pi_1$, and the right sided alternative becomes $\pi_2 > \pi_1.$ | $X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
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 $W$ and of $U$ if H0 were true | Sampling distribution of $z$ 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 | Sampling distribution of $W$:
Sampling distribution of $U$: For small samples, the exact distribution of $W$ or $U$ should be used. Note: if ties are present in the data, the formula for the standard deviations $\sigma_W$ and $\sigma_U$ is more complicated. | Approximately the standard normal distribution | Approximately the chi-squared distribution with $(I - 1) \times (J - 1)$ degrees of freedom | |
Significant? | Significant? | Significant? | Significant? | |
If the number of blocks is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
| For large samples, the table for standard normal probabilities can be used: Two sided:
| Two sided:
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n.a. | n.a. | Approximate $C\%$ confidence interval for $\pi_1 - \pi_2$ | n.a. | |
- | - | Regular (large sample):
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Equivalent to | Equivalent to | Equivalent to | n.a. | |
Friedman test, with a categorical dependent variable consisting of two independent groups. | If there are no ties in the data, the two sided Mann-Whitney-Wilcoxon test is equivalent to the Kruskal-Wallis test with an independent variable with 2 levels ($I$ = 2). | When testing two sided: chi-squared test for the relationship between two categorical variables, where both categorical variables have 2 levels. | - | |
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? | Do men tend to score higher on social economic status than women? | Is the proportion of smokers different between men and women? Use the normal approximation for the sampling distribution of the test statistic. | Is there an association between economic class and gender? Is the distribution of economic class different between men and women? | |
SPSS | SPSS | SPSS | SPSS | |
Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
| Analyze > Nonparametric Tests > Legacy Dialogs > 2 Independent Samples...
| SPSS does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chi-squared test instead. The $p$ value resulting from this chi-squared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to:
Analyze > Descriptive Statistics > Crosstabs...
| Analyze > Descriptive Statistics > Crosstabs...
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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
| T-Tests > Independent Samples T-Test
| Jamovi does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chi-squared test instead. The $p$ value resulting from this chi-squared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to:
Frequencies > Independent Samples - $\chi^2$ test of association
| Frequencies > Independent Samples - $\chi^2$ test of association
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Practice questions | Practice questions | Practice questions | Practice questions | |