Wilcoxon signed-rank test - overview
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Wilcoxon signed-rank test | Spearman's rho | McNemar's test |
You cannot compare more than 3 methods |
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Independent variable | Variable 1 | Independent variable | |
2 paired groups | One of ordinal level | 2 paired groups | |
Dependent variable | Variable 2 | Dependent variable | |
One quantitative of interval or ratio level | One of ordinal level | One categorical with 2 independent groups | |
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: $\rho_s = 0$
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. | Let's say that the scores on the dependent variable are scored 0 and 1. Then for each pair of scores, the data allow four options:
Other formulations of the null hypothesis are:
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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 two sided: $\rho_s \neq 0$ H1 right sided: $\rho_s > 0$ H1 left sided: $\rho_s < 0$ | The alternative hypothesis H1 is that for each pair of scores, P(first score of pair is 0 while second score of pair is 1) $\neq$ P(first score of pair is 1 while second score of pair is 0). That is, the probability that a pair of scores switches from 0 to 1 is not the same as the probability that a pair of scores switches from 1 to 0. Other formulations of the alternative hypothesis are:
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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:
| $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. | $X^2 = \dfrac{(b - c)^2}{b + c}$
Here $b$ is the number of pairs in the sample for which the first score is 0 while the second score is 1, and $c$ is the number of pairs in the sample for which the first score is 1 while the second score is 0. | |
Sampling distribution of $W_1$ and of $W_2$ if H0 were true | Sampling distribution of $t$ if H0 were true | Sampling distribution of $X^2$ 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 $t$ distribution with $N - 2$ degrees of freedom | If $b + c$ is large enough (say, > 20), approximately the chi-squared distribution with 1 degree of freedom. If $b + c$ is small, the Binomial($n$, $P$) distribution should be used, with $n = b + c$ and $P = 0.5$. In that case the test statistic becomes equal to $b$. | |
Significant? | Significant? | Significant? | |
For large samples, the table for standard normal probabilities can be used: Two sided:
| Two sided:
| For test statistic $X^2$:
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n.a. | n.a. | Equivalent to | |
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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 a monotonic relationship between physical health and mental health? | Does a tv documentary about spiders change whether people are afraid (yes/no) of spiders? | |
SPSS | SPSS | SPSS | |
Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
| Analyze > Correlate > Bivariate...
| Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
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Jamovi | Jamovi | Jamovi | |
T-Tests > Paired Samples T-Test
| Regression > Correlation Matrix
| Frequencies > Paired Samples - McNemar test
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Practice questions | Practice questions | Practice questions | |