Kruskal-Wallis test - overview
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Kruskal-Wallis test | Binomial test for a single proportion | Spearman's rho |
You cannot compare more than 3 methods |
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Independent/grouping variable | Independent variable | Variable 1 | |
One categorical with $I$ independent groups ($I \geqslant 2$) | None | One of ordinal level | |
Dependent variable | Dependent variable | Variable 2 | |
One of ordinal level | One categorical with 2 independent groups | One of ordinal level | |
Null hypothesis | Null hypothesis | 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: $\pi = \pi_0$
Here $\pi$ is the population proportion of 'successes', and $\pi_0$ is the population proportion of successes according to the null hypothesis. | H0: $\rho_s = 0$
Here $\rho_s$ is the Spearman correlation in the population. The Spearman correlation is a measure for the strength and direction of the monotonic relationship between two variables of at least ordinal measurement level. In words, the null hypothesis would be: H0: there is no monotonic relationship between the two variables in the population. | |
Alternative hypothesis | Alternative hypothesis | Alternative 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:
| H1 two sided: $\pi \neq \pi_0$ H1 right sided: $\pi > \pi_0$ H1 left sided: $\pi < \pi_0$ | H1 two sided: $\rho_s \neq 0$ H1 right sided: $\rho_s > 0$ H1 left sided: $\rho_s < 0$ | |
Assumptions | Assumptions | Assumptions | |
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Test statistic | Test statistic | Test statistic | |
$H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i} - 3(N + 1)$ | $X$ = number of successes in the sample | $t = \dfrac{r_s \times \sqrt{N - 2}}{\sqrt{1 - r_s^2}} $ Here $r_s$ is the sample Spearman correlation and $N$ is the sample size. The sample Spearman correlation $r_s$ is equal to the Pearson correlation applied to the rank scores. | |
Sampling distribution of $H$ if H0 were true | Sampling distribution of $X$ if H0 were true | Sampling distribution of $t$ if H0 were true | |
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. | Binomial($n$, $P$) distribution.
Here $n = N$ (total sample size), and $P = \pi_0$ (population proportion according to the null hypothesis). | Approximately the $t$ distribution with $N - 2$ degrees of freedom | |
Significant? | Significant? | Significant? | |
For large samples, the table with critical $X^2$ values can be used. If we denote $X^2 = H$:
| Two sided:
| Two sided:
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Example context | Example context | Example context | |
Do people from different religions tend to score differently on social economic status? | Is the proportion of smokers amongst office workers different from $\pi_0 = 0.2$? | Is there a monotonic relationship between physical health and mental health? | |
SPSS | SPSS | SPSS | |
Analyze > Nonparametric Tests > Legacy Dialogs > K Independent Samples...
| Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
| Analyze > Correlate > Bivariate...
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Jamovi | Jamovi | Jamovi | |
ANOVA > One Way ANOVA - Kruskal-Wallis
| Frequencies > 2 Outcomes - Binomial test
| Regression > Correlation Matrix
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Practice questions | Practice questions | Practice questions | |