z test for a single proportion - overview
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$z$ test for a single proportion | Kruskal-Wallis test | Cochran's Q test |
You cannot compare more than 3 methods |
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Independent variable | Independent/grouping variable | Independent/grouping variable | |
None | One categorical with $I$ independent groups ($I \geqslant 2$) | One within subject factor ($\geq 2$ related groups) | |
Dependent variable | Dependent variable | Dependent variable | |
One categorical with 2 independent groups | One of ordinal level | One categorical with 2 independent groups | |
Null hypothesis | Null hypothesis | 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. | If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in all $I$ populations:
Formulation 1:
| 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 $I.$ | |
Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |
H1 two sided: $\pi \neq \pi_0$ H1 right sided: $\pi > \pi_0$ H1 left sided: $\pi < \pi_0$ | If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in all $I$ populations:
Formulation 1:
| H1: not all population proportions are equal | |
Assumptions | Assumptions | Assumptions | |
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Test statistic | Test statistic | Test statistic | |
$z = \dfrac{p - \pi_0}{\sqrt{\dfrac{\pi_0(1 - \pi_0)}{N}}}$
Here $p$ is the sample proportion of successes: $\dfrac{X}{N}$, $N$ is the sample size, and $\pi_0$ is the population proportion of successes according to the null hypothesis. | $H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i} - 3(N + 1)$ | 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. | |
Sampling distribution of $z$ if H0 were true | Sampling distribution of $H$ if H0 were true | Sampling distribution of $Q$ if H0 were true | |
Approximately the standard normal distribution | For large samples, approximately the chi-squared distribution with $I - 1$ degrees of freedom. For small samples, the exact distribution of $H$ should be used. | If the number of blocks (usually the number of subjects) is large, approximately the chi-squared distribution with $k - 1$ degrees of freedom | |
Significant? | Significant? | Significant? | |
Two sided:
| For large samples, the table with critical $X^2$ values can be used. If we denote $X^2 = H$:
| If the number of blocks is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
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Approximate $C\%$ confidence interval for $\pi$ | n.a. | n.a. | |
Regular (large sample):
| - | - | |
Equivalent to | n.a. | Equivalent to | |
| - | Friedman test, with a categorical dependent variable consisting of two independent groups. | |
Example context | Example context | Example context | |
Is the proportion of smokers amongst office workers different from $\pi_0 = 0.2$? Use the normal approximation for the sampling distribution of the test statistic. | Do people from different religions tend to score differently on social economic status? | 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? | |
SPSS | SPSS | SPSS | |
Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
| Analyze > Nonparametric Tests > Legacy Dialogs > K Independent Samples...
| Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
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Jamovi | Jamovi | Jamovi | |
Frequencies > 2 Outcomes - Binomial test
| ANOVA > One Way ANOVA - Kruskal-Wallis
| 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
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Practice questions | Practice questions | Practice questions | |