Kruskal-Wallis test - overview
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Kruskal-Wallis test | Spearman's rho | Sign test |
You cannot compare more than 3 methods |
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Independent/grouping variable | Variable 1 | Independent variable | |
One categorical with $I$ independent groups ($I \geqslant 2$) | One of ordinal level | 2 paired groups | |
Dependent variable | Variable 2 | Dependent variable | |
One of ordinal level | One of ordinal level | 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: $\rho_s = 0$
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. |
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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: $\rho_s \neq 0$ H1 right sided: $\rho_s > 0$ H1 left sided: $\rho_s < 0$ |
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Assumptions | Assumptions | Assumptions | |
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Test statistic | Test statistic | Test statistic | |
$H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i} - 3(N + 1)$ | $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. | $W = $ number of difference scores that is larger than 0 | |
Sampling distribution of $H$ if H0 were true | Sampling distribution of $t$ if H0 were true | Sampling distribution of $W$ 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. | Approximately the $t$ distribution with $N - 2$ degrees of freedom | The exact distribution of $W$ under the null hypothesis is the Binomial($n$, $P$) distribution, with $n =$ number of positive differences $+$ number of negative differences, and $P = 0.5$.
If $n$ is large, $W$ is approximately normally distributed under the null hypothesis, with mean $nP = n \times 0.5$ and standard deviation $\sqrt{nP(1-P)} = \sqrt{n \times 0.5(1 - 0.5)}$. Hence, if $n$ is large, the standardized test statistic $$z = \frac{W - n \times 0.5}{\sqrt{n \times 0.5(1 - 0.5)}}$$ follows approximately the standard normal distribution if the null hypothesis were true. | |
Significant? | Significant? | Significant? | |
For large samples, the table with critical $X^2$ values can be used. If we denote $X^2 = H$:
| Two sided:
| If $n$ is small, the table for the binomial distribution should be used: Two sided:
If $n$ is large, the table for standard normal probabilities can be used: Two sided:
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n.a. | n.a. | Equivalent to | |
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Two sided sign test is equivalent to
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Example context | Example context | Example context | |
Do people from different religions tend to score differently on social economic status? | Is there a monotonic relationship between physical health and mental health? | Do people tend to score higher on mental health after a mindfulness course? | |
SPSS | SPSS | SPSS | |
Analyze > Nonparametric Tests > Legacy Dialogs > K Independent Samples...
| Analyze > Correlate > Bivariate...
| Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
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Jamovi | Jamovi | Jamovi | |
ANOVA > One Way ANOVA - Kruskal-Wallis
| Regression > Correlation Matrix
| Jamovi does not have a specific option for the sign test. However, you can do the Friedman test instead. The $p$ value resulting from this Friedman test is equivalent to the two sided $p$ value that would have resulted from the sign test. Go to:
ANOVA > Repeated Measures ANOVA - Friedman
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Practice questions | Practice questions | Practice questions | |