dixonstat

dixonstat implements numerical evaluation of critical values for Dixon’s \(r_{j,i}\) statistics [Dix50] following the methods introduced by McBane [McB06] and Verma and Suarez [VS14].

Given a set of \(n\) observations \(x_i\) ordered such that \(x_1\leq x_2\leq \dotsb \leq x_n\), the statistics are defined by

\[r_{j,i-1} = \frac{x_n-x_{n-j}}{x_n-x_i}\]

Dixon’s \(r_{1,0}\) statistic (i.e., \(i=j=1\)) is often called \(Q\) and the corresponding outlier rejection test which uses this ratio is called the \(Q\) test.

The ratio \(r_{1,0}\), for instance, simply compares the difference between a single suspected outlier (\(x_1\) or \(x_n\)) and its nearest-neighboring value to the overall range of values in the sample. In other words, the ratio determines the fraction of the total range that is attributable to one suspected outlier.

Different numerical approaches exist for generating the critical values of Dixon’s \(r\) statistics. A straightforward method is to interpolate new confidence levels using cubic regression from previously tabulated data as suggested by Rorabacher [Ror91]. Given critical values were originally tabulated for relatively small sample sizes (i.e., \(3\leq n\leq 30\)), interpolation might not always be feasible.

Without interpolation, determining critical values boils down to integrating the probability density function to obtain the cumulative distribution function. The integration can be performed either using a stochastic approach [Efs92] (e.g., by means of Monte Carlo simulation) or using Gaussian quadrature. The latter numerical evaluation of the corresponding integral is employed by dixonstat.