One sample Wilcoxon signed-rank test - overview
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One sample Wilcoxon signed-rank test | Spearman's rho | Binomial test for a single proportion |
You cannot compare more than 3 methods |
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Independent variable | Variable 1 | Independent variable | |
None | One of ordinal level | None | |
Dependent variable | Variable 2 | Dependent variable | |
One of ordinal level | One of ordinal level | One categorical with 2 independent groups | |
Null hypothesis | Null hypothesis | Null hypothesis | |
H0: $m = m_0$
Here $m$ is the population median, and $m_0$ is the population median 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. | 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. | |
Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |
H1 two sided: $m \neq m_0$ H1 right sided: $m > m_0$ H1 left sided: $m < m_0$ | H1 two sided: $\rho_s \neq 0$ H1 right sided: $\rho_s > 0$ H1 left sided: $\rho_s < 0$ | H1 two sided: $\pi \neq \pi_0$ H1 right sided: $\pi > \pi_0$ H1 left sided: $\pi < \pi_0$ | |
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. | $X$ = number of successes in the sample | |
Sampling distribution of $W_1$ and of $W_2$ if H0 were true | Sampling distribution of $t$ if H0 were true | Sampling distribution of $X$ 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 | Binomial($n$, $P$) distribution.
Here $n = N$ (total sample size), and $P = \pi_0$ (population proportion according to the null hypothesis). | |
Significant? | Significant? | Significant? | |
For large samples, the table for standard normal probabilities can be used: Two sided:
| Two sided:
| Two sided:
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Example context | Example context | Example context | |
Is the median mental health score of office workers different from $m_0 = 50$? | Is there a monotonic relationship between physical health and mental health? | Is the proportion of smokers amongst office workers different from $\pi_0 = 0.2$? | |
SPSS | SPSS | SPSS | |
Specify the measurement level of your variable on the Variable View tab, in the column named Measure. Then go to:
Analyze > Nonparametric Tests > One Sample...
| Analyze > Correlate > Bivariate...
| Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
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Jamovi | Jamovi | Jamovi | |
T-Tests > One Sample T-Test
| Regression > Correlation Matrix
| Frequencies > 2 Outcomes - Binomial test
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Practice questions | Practice questions | Practice questions | |