Weighted Mann-Whitney U test, Part 1

Previously, I have discussed how to build weighted versions of various statistical methods. I have already covered weighted versions of various quantile estimators and the Hodges-Lehmann location estimator. Such methods can be useful in various tasks like the support of weighted mixture distributions or exponential smoothing. In this post, I suggest a way to build a weighted version of the Mann-Whitney U test.

We consider the one-sided Mann-Whitney U test that compares two samples \(\mathbf{x}\) and \(\mathbf{y}\):

\[\mathbf{x} = ( x_1, x_2, \ldots, x_n ), \quad \mathbf{y} = ( y_1, y_2, \ldots, y_m ). \]

The U statistic for this test is defined as follows:

\[U(x, y) = \sum_{i=1}^n \sum_{j=1}^m S(x_i, y_j),\quad S(a,b) = \begin{cases} 1, & \text{if } a > b, \\ 0.5, & \text{if } a = b, \\ 0, & \text{if } a < b. \end{cases} \]

It is easy to see that \(U(x, y) \in [0; n \cdot m]\). For further discussion, it is convenient to also consider a “normalized” version of \(U\) that we denote by \(U_\circ\):

\[U_\circ(x, y) = \frac{U(x, y)}{n \cdot m} \in [0; 1]. \]

For the weighted version of the test, we assign vectors of weights \(\mathbf{w}\) and \(\mathbf{v}\) for \(\mathbf{x}\) and \(\mathbf{y}\) respectively.

\[\mathbf{w} = ( w_1, w_2, \ldots, w_n ), \quad \mathbf{v} = ( v_1, v_2, \ldots, v_m ), \]

where \(w_i \geq 0\), \(v_j \geq 0\), \(\sum w_i > 0\), \(\sum v_j > 0\).

Let us consider the normalized versions of the weight vectors:

\[\overline{\mathbf{w}} = ( \overline{w}_1, \overline{w}_2, \ldots, \overline{w}_n ), \quad \overline{\mathbf{v}} = ( \overline{v}_1, \overline{v}_2, \ldots, \overline{v}_m ), \]

\[\overline{w}_i = \frac{w_i}{\sum_{k=1}^n w_k},\quad \overline{v}_j = \frac{v_j}{\sum_{k=1}^m v_k}. \]

Let us introduce the weighted version of the \(U\) statistic. Similarly to the weighted Hodges-Lehmann location estimator, I suggest using the following approach for the normalized weighted version:

\[U_\circ^\star(\mathbf{x}, \mathbf{y}, \mathbf{w}, \mathbf{v}) = \sum_{i=1}^n \sum_{j=1}^m S(x_i, y_j) \cdot \overline{w}_i \cdot \overline{v}_j. \]

It seems that denormalizing \(U_\circ^\star\) does not make a lot of sense. Firstly, the sample size in the weighted case is not unambiguously defined.1 Secondly, the denormalized version is not much more useful than the normalized one since we cannot reuse the distribution of the classic non-weighted \(U\) statistic for the weighted case. Therefore, we can continue with the normalized statistic \(U_\circ^\star\).

To convert the statistic value to the p-value, we can approximate the distribution of \(U_\circ^\star\) via bootstrap.

  1. For example, we can use the Huggins-Roy family of effective sample sizes that provides a class of equations to define the weighted versions of \(n\) and \(m\). This family includes Kish’s effective sample size↩︎