One sample Wilcoxon signed-rank test - overview
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One sample Wilcoxon signed-rank test | One sample $z$ test for the mean | Pearson correlation |
You cannot compare more than 3 methods |
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Independent variable | Independent variable | Variable 1 | |
None | None | One quantitative of interval or ratio level | |
Dependent variable | Dependent variable | Variable 2 | |
One of ordinal level | One quantitative of interval or ratio level | One quantitative of interval or ratio level | |
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: $\mu = \mu_0$
Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis. | H0: $\rho = \rho_0$
Here $\rho$ is the Pearson correlation in the population, and $\rho_0$ is the Pearson correlation in the population according to the null hypothesis (usually 0). The Pearson correlation is a measure for the strength and direction of the linear relationship between two variables of at least interval measurement level. | |
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: $\mu \neq \mu_0$ H1 right sided: $\mu > \mu_0$ H1 left sided: $\mu < \mu_0$ | H1 two sided: $\rho \neq \rho_0$ H1 right sided: $\rho > \rho_0$ H1 left sided: $\rho < \rho_0$ | |
Assumptions | Assumptions | Assumptions of test for correlation | |
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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:
| $z = \dfrac{\bar{y} - \mu_0}{\sigma / \sqrt{N}}$
Here $\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to the null hypothesis, $\sigma$ is the population standard deviation, and $N$ is the sample size. The denominator $\sigma / \sqrt{N}$ is the standard deviation of the sampling distribution of $\bar{y}$. The $z$ value indicates how many of these standard deviations $\bar{y}$ is removed from $\mu_0$. | Test statistic for testing H0: $\rho = 0$:
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Sampling distribution of $W_1$ and of $W_2$ if H0 were true | Sampling distribution of $z$ if H0 were true | Sampling distribution of $t$ and of $z$ 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. | Standard normal distribution | Sampling distribution of $t$:
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Significant? | Significant? | Significant? | |
For large samples, the table for standard normal probabilities can be used: Two sided:
| Two sided:
| $t$ Test two sided:
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n.a. | $C\%$ confidence interval for $\mu$ | Approximate $C$% confidence interval for $\rho$ | |
- | $\bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}}$
where the critical value $z^*$ is the value under the normal curve with the area $C / 100$ between $-z^*$ and $z^*$ (e.g. $z^*$ = 1.96 for a 95% confidence interval). The confidence interval for $\mu$ can also be used as significance test. | First compute the approximate $C$% confidence interval for $\rho_{Fisher}$:
Then transform back to get the approximate $C$% confidence interval for $\rho$:
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n.a. | Effect size | Properties of the Pearson correlation coefficient | |
- | Cohen's $d$: Standardized difference between the sample mean and $\mu_0$: $$d = \frac{\bar{y} - \mu_0}{\sigma}$$ Cohen's $d$ indicates how many standard deviations $\sigma$ the sample mean $\bar{y}$ is removed from $\mu_0.$ |
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n.a. | Visual representation | n.a. | |
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n.a. | n.a. | Equivalent to | |
- | - | OLS regression with one independent variable:
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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 the average mental health score of office workers different from $\mu_0 = 50$? Assume that the standard deviation of the mental health scores in the population is $\sigma = 3.$ | Is there a linear relationship between physical health and mental health? | |
SPSS | n.a. | 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...
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Jamovi | n.a. | Jamovi | |
T-Tests > One Sample T-Test
| - | Regression > Correlation Matrix
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Practice questions | Practice questions | Practice questions | |