Wilcoxon signed-rank test - overview
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Wilcoxon signed-rank test | Chi-squared test for the relationship between two categorical variables | Kruskal-Wallis test |
You cannot compare more than 3 methods |
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Independent variable | Independent /column variable | Independent/grouping variable | |
2 paired groups | One categorical with $I$ independent groups ($I \geqslant 2$) | One categorical with $I$ independent groups ($I \geqslant 2$) | |
Dependent variable | Dependent /row variable | Dependent variable | |
One quantitative of interval or ratio level | One categorical with $J$ independent groups ($J \geqslant 2$) | One of ordinal level | |
Null hypothesis | Null hypothesis | Null hypothesis | |
H0: $m = 0$
Here $m$ is the population median of the difference scores. A difference score is the difference between the first score of a pair and the second score of a pair. Several different formulations of the null hypothesis can be found in the literature, and we do not agree with all of them. Make sure you (also) learn the one that is given in your text book or by your teacher. | 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:
| 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:
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Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |
H1 two sided: $m \neq 0$ H1 right sided: $m > 0$ H1 left sided: $m < 0$ | 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:
| 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:
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Assumptions | Assumptions | Assumptions | |
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Test statistic | Test statistic | Test statistic | |
Two different types of test statistics can be used, but both will result in the same test outcome. We will denote the first option the $W_1$ statistic (also known as the $T$ statistic), and the second option the $W_2$ statistic.
In order to compute each of the test statistics, follow the steps below:
| $X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
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. | $H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i} - 3(N + 1)$ | |
Sampling distribution of $W_1$ and of $W_2$ if H0 were true | Sampling distribution of $X^2$ if H0 were true | Sampling distribution of $H$ if H0 were true | |
Sampling distribution of $W_1$:
If $N_r$ is large, $W_1$ is approximately normally distributed with mean $\mu_{W_1}$ and standard deviation $\sigma_{W_1}$ if the null hypothesis were true. Here $$\mu_{W_1} = \frac{N_r(N_r + 1)}{4}$$ $$\sigma_{W_1} = \sqrt{\frac{N_r(N_r + 1)(2N_r + 1)}{24}}$$ Hence, if $N_r$ is large, the standardized test statistic $$z = \frac{W_1 - \mu_{W_1}}{\sigma_{W_1}}$$ follows approximately the standard normal distribution if the null hypothesis were true. Sampling distribution of $W_2$: If $N_r$ is large, $W_2$ is approximately normally distributed with mean $0$ and standard deviation $\sigma_{W_2}$ if the null hypothesis were true. Here $$\sigma_{W_2} = \sqrt{\frac{N_r(N_r + 1)(2N_r + 1)}{6}}$$ Hence, if $N_r$ is large, the standardized test statistic $$z = \frac{W_2}{\sigma_{W_2}}$$ follows approximately the standard normal distribution if the null hypothesis were true. If $N_r$ is small, the exact distribution of $W_1$ or $W_2$ should be used. Note: if ties are present in the data, the formula for the standard deviations $\sigma_{W_1}$ and $\sigma_{W_2}$ is more complicated. | Approximately the chi-squared distribution with $(I - 1) \times (J - 1)$ degrees of freedom | 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. | |
Significant? | Significant? | Significant? | |
For large samples, the table for standard normal probabilities can be used: Two sided:
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| For large samples, the table with critical $X^2$ values can be used. If we denote $X^2 = H$:
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Example context | Example context | Example context | |
Is the median of the differences between the mental health scores before and after an intervention different from 0? | Is there an association between economic class and gender? Is the distribution of economic class different between men and women? | Do people from different religions tend to score differently on social economic status? | |
SPSS | SPSS | SPSS | |
Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
| Analyze > Descriptive Statistics > Crosstabs...
| Analyze > Nonparametric Tests > Legacy Dialogs > K Independent Samples...
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Jamovi | Jamovi | Jamovi | |
T-Tests > Paired Samples T-Test
| Frequencies > Independent Samples - $\chi^2$ test of association
| ANOVA > One Way ANOVA - Kruskal-Wallis
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Practice questions | Practice questions | Practice questions | |