A better alternative to Wald's test-statistic for simple goodness-of-fit tests under one-stage cluster sampling

Significance levels of the popular Wald's Chi-squared statistic for simple goodness-of-fit (GOF) tests under one-stage cluster sampling are often unreliable. A large number of alternatives to Wald's GOF test with Type-I error rates more closely matching the nominal level of significance have been proposed but not yet found their way into applied statistics. Type-I error rates with Wald's test-statistic in cluster sampling from 10 actual forest cover-type maps from 5 sites and 81 sample designs are compared to the error rates of 11 alternatives. The effects of site, sampling design, evenness of cover-type class proportions, and intra-cluster correlation on Type-I error rates are quantified with logistic regressions for Wald's statistic and five promising alternatives. Our proposed second-order bias correction of Finney's [Finney, D.J., 1971. Probit Analysis, vol. 3. Cambridge University Press, p. 350] and Brier's [Brier, S.S., 1980. Analysis of contingency tables under cluster sampling. Biometrika 67, 591–596] method of moments correction of Pearson's Chi-squared test statistic emerged as the overall best alternative in this study. It was the least sensitive to design and cluster effects. Test power was investigated for the alternative simple hypothesis of equality of cover-type proportions in two site-specific maps. The proposed alternative test statistic had slightly (3%) less power than Wald's test for designs with a power of 80% or greater, yet a consistently better odds ratio of a correct test decision.