Wilcoxon t-Test

The Wilcoxon signed-rank t-test is a non-parametric statistical hypothesis test used when comparing two related samples, matched samples, or repeated measurements on a single sample to assess whether their population mean ranks differ (i.e., it is a paired difference test). It can be used as an alternative to the paired Student's $t$-test, $t$-test for matched pairs, or the $t$-test for dependent samples when the population cannot be assumed to be normally distributed.

The test is named for Frank Wilcoxon who (in a single paper) proposed both the rank $t$-test and the rank-sum test for two independent samples. The test was popularized by Siegel in his influential text book on non-parametric statistics. Siegel used the symbol $T$ for the value defined below as $W$. In consequence, the test is sometimes referred to as the Wilcoxon $T$-test, and the test statistic is reported as a value of $T$. Other names may include the "$t$-test for matched pairs" or the "$t$-test for dependent samples."

Assumptions

1. Data are paired and come from the same population.
2. Each pair is chosen randomly and independent.
3. The data are measured at least on an ordinal scale, but need not be normal.

Test Procedure

Let $N$ be the sample size, the number of pairs. Thus, there are a total of $2N$ data points. For $i=1,\cdots,N$, let $x_{1,i}$ and $x_{2,i}$ denote the measurements.

$H_0$: The median difference between the pairs is zero.

$H_1$: The median difference is not zero.

1. For $i=1,\cdots,N$, calculate $\left| { x }_{ 2,i }-{ x }_{ 1,i } \right|$ and $\text{sgn}\left( { x }_{ 2,i }-{ x }_{ 1,i } \right)$, where $\text{sgn}$ is the sign function.

2. Exclude pairs with $\left|{ x }_{ 2,i }-{ x }_{ 1,i } \right|=0$. Let $N_r$ be the reduced sample size.

3. Order the remaining pairs from smallest absolute difference to largest absolute difference, $\left| { x }_{ 2,i }-{ x }_{ 1,i } \right|$.

4. Rank the pairs, starting with the smallest as 1. Ties receive a rank equal to the average of the ranks they span. Let $R_i$ denote the rank.

5. Calculate the test statistic $W$, the absolute value of the sum of the signed ranks:

$W= \left| \sum \left(\text{sgn}(x_{2,i}-x_{1,i}) \cdot R_i \right) \right|$

6. As $N_r$ increases, the sampling distribution of $W$ converges to a normal distribution. Thus, for $N_r \geq 10$, a $z$-score can be calculated as follows: 

$z=\dfrac{W-0.5}{\sigma_W}$

where

$\displaystyle{\sigma_W = \sqrt{\frac{N_r(N_r+1)(2N_r+1)}{6}}}$

If $z > z_{\text{critical}}$ then reject $H_0$.

For $N_r < 10$, $W$ is compared to a critical value from a reference table. If $W\ge { W }_{ \text{critical,}{ N }_{ r } }$ then reject $H_0$.

Alternatively, a $p$-value can be calculated from enumeration of all possible combinations of $W$ given $N_r$.