# Wilcoxon signed-rank test - overview

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Wilcoxon signed-rank test | Logistic regression | Two sample $t$ test - equal variances assumed |
You cannot compare more than 3 methods |
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Independent variable | Independent variables | Independent/grouping variable | |

2 paired groups | One or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variables | One categorical with 2 independent groups | |

Dependent variable | Dependent variable | Dependent variable | |

One quantitative of interval or ratio level | One categorical with 2 independent groups | One quantitative of interval or ratio level | |

Null hypothesis | Null hypothesis | Null hypothesis | |

H_{0}: $m = 0$
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. | Model chi-squared test for the complete regression model:
- H
_{0}: $\beta_1 = \beta_2 = \ldots = \beta_K = 0$
- H
_{0}: $\beta_k = 0$ or in terms of odds ratio: - H
_{0}: $e^{\beta_k} = 1$
- H
_{0}: $\beta_k = 0$ or in terms of odds ratio: - H
_{0}: $e^{\beta_k} = 1$
| H_{0}: $\mu_1 = \mu_2$
Here $\mu_1$ is the population mean for group 1, and $\mu_2$ is the population mean for group 2. | |

Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |

H_{1} two sided: $m \neq 0$H _{1} right sided: $m > 0$H _{1} left sided: $m < 0$
| Model chi-squared test for the complete regression model:
- H
_{1}: not all population regression coefficients are 0
- H
_{1}: $\beta_k \neq 0$ or in terms of odds ratio: - H
_{1}: $e^{\beta_k} \neq 1$ If defined as Wald $ = \dfrac{b_k}{SE_{b_k}}$ (see 'Test statistic'), also one sided alternatives can be tested: - H
_{1}right sided: $\beta_k > 0$ - H
_{1}left sided: $\beta_k < 0$
- H
_{1}: $\beta_k \neq 0$ or in terms of odds ratio: - H
_{1}: $e^{\beta_k} \neq 1$
| H_{1} two sided: $\mu_1 \neq \mu_2$H _{1} right sided: $\mu_1 > \mu_2$H _{1} left sided: $\mu_1 < \mu_2$
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Assumptions | Assumptions | Assumptions | |

- The population distribution of the difference scores is symmetric
- Sample of difference scores is a simple random sample from the population of difference scores. That is, difference scores are independent of one another
ranked difference scores, we need to know whether a change in scores from, say, 6 to 7 is larger than/smaller than/equal to a change from 5 to 6. This is impossible to know for ordinal scales, since for these scales the size of the difference between values is meaningless.
| - In the population, the relationship between the independent variables and the log odds $\ln (\frac{\pi_{y=1}}{1 - \pi_{y=1}})$ is linear
- The residuals are independent of one another
- Variables are measured without error
- Multicollinearity
- Outliers
| - Within each population, the scores on the dependent variable are normally distributed
- The standard deviation of the scores on the dependent variable is the same in both populations: $\sigma_1 = \sigma_2$
- Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2. That is, within and between groups, observations are independent of one another
| |

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:
- For each subject, compute the sign of the difference score $\mbox{sign}_d = \mbox{sgn}(\mbox{score}_2 - \mbox{score}_1)$. The sign is 1 if the difference is larger than zero, -1 if the diffence is smaller than zero, and 0 if the difference is equal to zero.
- For each subject, compute the absolute value of the difference score $|\mbox{score}_2 - \mbox{score}_1|$.
- Exclude subjects with a difference score of zero. This leaves us with a remaining number of difference scores equal to $N_r$.
- Assign ranks $R_d$ to the $N_r$ remaining
*absolute*difference scores. The smallest absolute difference score corresponds to a rank score of 1, and the largest absolute difference score corresponds to a rank score of $N_r$. If there are ties, assign them the average of the ranks they occupy.
- $W_1 = \sum\, R_d^{+}$
or $W_1 = \sum\, R_d^{-}$ That is, sum all ranks corresponding to a positive difference or sum all ranks corresponding to a negative difference. Theoratically, both definitions will result in the same test outcome. However:- tables with critical values for $W_1$ are usually based on the smaller of $\sum\, R_d^{+}$ and $\sum\, R_d^{-}$. So if you are using such a table, pick the smaller one.
- If you are using the normal approximation to find the $p$ value, it makes things most straightforward if you use $W_1 = \sum\, R_d^{+}$ (if you use $W_1 = \sum\, R_d^{-}$, the right and left sided alternative hypotheses 'flip').
- $W_2 = \sum\, \mbox{sign}_d \times R_d$
That is, for each remaining difference score, multiply the rank of the absolute difference score by the sign of the difference score, and then sum all of the products.
| Model chi-squared test for the complete regression model:
- $X^2 = D_{null} - D_K = \mbox{null deviance} - \mbox{model deviance} $
$D_{null}$, the null deviance, is conceptually similar to the total variance of the dependent variable in OLS regression analysis. $D_K$, the model deviance, is conceptually similar to the residual variance in OLS regression analysis.
The wald statistic can be defined in two ways: - Wald $ = \dfrac{b_k^2}{SE^2_{b_k}}$
- Wald $ = \dfrac{b_k}{SE_{b_k}}$
Likelihood ratio chi-squared test for individual $\beta_k$: - $X^2 = D_{K-1} - D_K$
$D_{K-1}$ is the model deviance, where independent variable $k$ is excluded from the model. $D_{K}$ is the model deviance, where independent variable $k$ is included in the model.
| $t = \dfrac{(\bar{y}_1 - \bar{y}_2) - 0}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}} = \dfrac{\bar{y}_1 - \bar{y}_2}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}}$
Here $\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $s_p$ is the pooled standard deviation, $n_1$ is the sample size of group 1, and $n_2$ is the sample size of group 2. The 0 represents the difference in population means according to the null hypothesis. The denominator $s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}$ is the standard error of the sampling distribution of $\bar{y}_1 - \bar{y}_2$. The $t$ value indicates how many standard errors $\bar{y}_1 - \bar{y}_2$ is removed from 0. Note: we could just as well compute $\bar{y}_2 - \bar{y}_1$ in the numerator, but then the left sided alternative becomes $\mu_2 < \mu_1$, and the right sided alternative becomes $\mu_2 > \mu_1$. | |

n.a. | n.a. | Pooled standard deviation | |

- | - | $s_p = \sqrt{\dfrac{(n_1 - 1) \times s^2_1 + (n_2 - 1) \times s^2_2}{n_1 + n_2 - 2}}$ | |

Sampling distribution of $W_1$ and of $W_2$ if H_{0} were true | Sampling distribution of $X^2$ and of the Wald statistic if H_{0} were true | Sampling distribution of $t$ if H_{0} 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. | Sampling distribution of $X^2$, as computed in the model chi-squared test for the complete model:
- chi-squared distribution with $K$ (number of independent variables) degrees of freedom
- If defined as Wald $ = \dfrac{b_k^2}{SE^2_{b_k}}$: approximately the chi-squared distribution with 1 degree of freedom
- If defined as Wald $ = \dfrac{b_k}{SE_{b_k}}$: approximately the standard normal distribution
- chi-squared distribution with 1 degree of freedom
| $t$ distribution with $n_1 + n_2 - 2$ degrees of freedom | |

Significant? | Significant? | Significant? | |

For large samples, the table for standard normal probabilities can be used: Two sided: - Check if $z$ observed in sample is at least as extreme as critical value $z^*$ or
- Find two sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
- Check if $z$ observed in sample is equal to or larger than critical value $z^*$ or
- Find right sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
- Check if $z$ observed in sample is equal to or smaller than critical value $z^*$ or
- Find left sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
| For the model chi-squared test for the complete regression model and likelihood ratio chi-squared test for individual $\beta_k$:
- Check if $X^2$ observed in sample is equal to or larger than critical value $X^{2*}$ or
- Find $p$ value corresponding to observed $X^2$ and check if it is equal to or smaller than $\alpha$
- If defined as Wald $ = \dfrac{b_k^2}{SE^2_{b_k}}$: same procedure as for the chi-squared tests. Wald can be interpret as $X^2$
- If defined as Wald $ = \dfrac{b_k}{SE_{b_k}}$: same procedure as for any $z$ test. Wald can be interpreted as $z$.
| Two sided:
- Check if $t$ observed in sample is at least as extreme as critical value $t^*$ or
- Find two sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$
- Check if $t$ observed in sample is equal to or larger than critical value $t^*$ or
- Find right sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$
- Check if $t$ observed in sample is equal to or smaller than critical value $t^*$ or
- Find left sided $p$ value corresponding to observed $t$ and check if it is equal to or smaller than $\alpha$
| |

n.a. | Wald-type approximate $C\%$ confidence interval for $\beta_k$ | $C\%$ confidence interval for $\mu_1 - \mu_2$ | |

- | $b_k \pm z^* \times SE_{b_k}$ 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). | $(\bar{y}_1 - \bar{y}_2) \pm t^* \times s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}$
where the critical value $t^*$ is the value under the $t_{n_1 + n_2 - 2}$ distribution with the area $C / 100$ between $-t^*$ and $t^*$ (e.g. $t^*$ = 2.086 for a 95% confidence interval when df = 20). The confidence interval for $\mu_1 - \mu_2$ can also be used as significance test. | |

n.a. | Goodness of fit measure $R^2_L$ | Effect size | |

- | $R^2_L = \dfrac{D_{null} - D_K}{D_{null}}$ There are several other goodness of fit measures in logistic regression. In logistic regression, there is no single agreed upon measure of goodness of fit. | Cohen's $d$:Standardized difference between the mean in group $1$ and in group $2$: $$d = \frac{\bar{y}_1 - \bar{y}_2}{s_p}$$ Cohen's $d$ indicates how many standard deviations $s_p$ the two sample means are removed from each other. | |

n.a. | n.a. | Visual representation | |

- | - | ||

n.a. | n.a. | Equivalent to | |

- | - | One way ANOVA with an independent variable with 2 levels ($I$ = 2):
- two sided two sample $t$ test is equivalent to ANOVA $F$ test when $I$ = 2
- two sample $t$ test is equivalent to $t$ test for contrast when $I$ = 2
- two sample $t$ test is equivalent to $t$ test multiple comparisons when $I$ = 2
- two sided two sample $t$ test is equivalent to $F$ test regression model
- two sample $t$ test is equivalent to $t$ test for regression coefficient $\beta_1$
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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? | Can body mass index, stress level, and gender predict whether people get diagnosed with diabetes? | Is the average mental health score different between men and women? Assume that in the population, the standard deviation of mental health scores is equal amongst men and women. | |

SPSS | SPSS | SPSS | |

Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
- Put the two paired variables in the boxes below Variable 1 and Variable 2
- Under Test Type, select the Wilcoxon test
| Analyze > Regression > Binary Logistic...
- Put your dependent variable in the box below Dependent and your independent (predictor) variables in the box below Covariate(s)
| Analyze > Compare Means > Independent-Samples T Test...
- Put your dependent (quantitative) variable in the box below Test Variable(s) and your independent (grouping) variable in the box below Grouping Variable
- Click on the Define Groups... button. If you can't click on it, first click on the grouping variable so its background turns yellow
- Fill in the value you have used to indicate your first group in the box next to Group 1, and the value you have used to indicate your second group in the box next to Group 2
- Continue and click OK
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Jamovi | Jamovi | Jamovi | |

T-Tests > Paired Samples T-Test
- Put the two paired variables in the box below Paired Variables, one on the left side of the vertical line and one on the right side of the vertical line
- Under Tests, select Wilcoxon rank
- Under Hypothesis, select your alternative hypothesis
| Regression > 2 Outcomes - Binomial
- Put your dependent variable in the box below Dependent Variable and your independent variables of interval/ratio level in the box below Covariates
- If you also have code (dummy) variables as independent variables, you can put these in the box below Covariates as well
- Instead of transforming your categorical independent variable(s) into code variables, you can also put the untransformed categorical independent variables in the box below Factors. Jamovi will then make the code variables for you 'behind the scenes'
| T-Tests > Independent Samples T-Test
- Put your dependent (quantitative) variable in the box below Dependent Variables and your independent (grouping) variable in the box below Grouping Variable
- Under Tests, select Student's (selected by default)
- Under Hypothesis, select your alternative hypothesis
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Practice questions | Practice questions | Practice questions | |