A Power Study of Goodness-of-Fit Tests for Categorical Data
School of Mathematical and Physical Sciences, James Cook University,
Australian School of Environmental Studies, Griffith University
School of Information Technology and Mathematical Sciences, University of Ballarat.
Goodness-of-fit (GOF) tests are used for the analysis of categorical data by applied
researchers from many disciplines however studies of their relative powers are limited. Although
the Chi-Square (2) test is a popular choice for many researchers, power studies show that this may
be at the expense of power in some instances. This paper compares the powers of two of the lesser
known GOF test statistics based on the empirical distribution function with the 2 test to determine
which is the more powerful for the investigated null and alternative distributions.
2. The test statistics used in the power study
The test statistics used are 2 (Pearson 1900), the discrete Kolmogorov-Smirnov KS (Pettitt
and Stephens 1977) and the discrete Cramér-von Mises W2 (Choulakian et al. 1994).
where k is the number of cells, N is the sample size, pi is the probability for cell i, Oi and
Ei and are the observed and expected frequencies for cell i, and Zi is the cummulative
sum of the differences between Oi and Ei up to and including cell i.
3. The power study
The power for each test statistic is approximated for a uniform null distribution over 10 cells
against the increasing trend and triangular ∨ or ’bath-tub’ type alternatives defined in Table 1. The
total sample sizes range from 10 to 200 which represents expected frquencies under the uniform
null distribution of 1 to 20 per cell. The power of each test statistic is estimated at the 5%
significance level from 10000 simulated ran