Exploring the power curve of the Ansari-Bradley test


The Ansari-Bradley test is a popular rank-based nonparametric test for a difference in scale/dispersion parameters. In this post, we explore its power curve in a numerical simulation.

The Ansari-Bradley test

Let \(\mathbf{x} = (x_1, x_2, \ldots, x_n)\) and \(\mathbf{y} = (y_1, y_2, \ldots, y_m)\) be random samples. Let \(N=n+m\). Let \(\mathbf{V} = (V_1, V_2, \ldots, V_N)\) be a boolean vector of zeros and ones, where \(V_i = 1\) if the \(i^\textrm{th}\) order statistic of the pooled sample \((\mathbf{x}, \mathbf{y})\) is come from \(\mathbf{x}\).

With this notation, we can define the Ansari-Bradley statistic \(\operatorname{AB}\) as follows:

\[\operatorname{AB} = \frac{1}{2} n (N + 1) - \sum_{i=1}^N \left| i - \frac{1}{2}(N+1) \right| \cdot V_i. \]

The asymptotic approximation is defined by the following normal distribution:

\[\mu = \begin{cases} \frac{1}{4} n(N + 2), & \;\textrm{if}\;N\;\textrm{is even,}\\ \frac{1}{4} n(N + 1)^2/N, & \;\textrm{if}\;N\;\textrm{is odd,} \end{cases} \]

\[\sigma^2 = \begin{cases} nm(N^2-4) / (48(N-1)) & \;\textrm{if}\;N\;\textrm{is even,}\\ nm(N+1)(N^2+3) / (48N^2) & \;\textrm{if}\;N\;\textrm{is odd.} \end{cases} \]

Power curve

For exploring the power curve, we compare \(\mathcal{N}(0, 1)\) and \(\mathcal{N}(0, \sigma^2)\). We enumerate \(\sigma\) values (aka “ratio”) in \([1; 10]\). The sample size are in \(n \in \{5, 10, 20 \}\), the statistical level \(\alpha = 0.05\), all the statistical tests are two-sided. Just for fun, in addition to the Ansari-Bradley test, we also consider Welch’s t-test and Mann-Whitney U test. Here are the power curves:

We can make the following observation:

  • Welch’s t-test: while this modification of the Student’s t-test is specifically designed for normal distributions with unequal variances, it is still a location test. Since the distribution locations in this simulation are the same, the test maintains its statistical power at \(\alpha = 0.05\) regardless of the values of \(n\) and \(\sigma\).
  • Mann-Whitney U test: while this is one of the most popular nonparametric tests, it is not always suitable for all kinds of distribution changes. This test is often mistakenly declared as a test for medians or a test for stochastic greatness. Note that while in the above simulation, both distributions are stochastic equal, the statistical power of the Mann-Whitney U test is higher than \(\alpha = 0.05\) for large \(n\), \(\sigma\). We will discuss this phenomenon in one of the future posts.
  • Ansari-Bradley test: this is the only actually suitable test for our problem. The statistical power starts at \(\alpha = 0.05\) for \(\sigma = 1\) (as it should) and increases while \(\sigma\) increases. Expectedly, the higher sample size \(n\) gives a better power increase rate and, therefore, higher statistical power for the same value of \(\sigma\).