A practical guide to statistical methods for comparing means from two-stage sampling

Two-staged sampling is the method of sampling populations that occur naturally in groups and is common in ecological field studies. This sampling method requires special statistical analyses that account for this sample structure. We present and compare several analytical methods for comparing means from two-stage sampling: (1) simple ANOVA ignoring sample structure, (2) unit means ANOVA, (3) Nested Mixed ANOVA, (4) restricted maximum likelihood (REML) Nested Mixed analysis, and (5) REML Nested Mixed analysis with heteroscedasticity.We consider a fisheries survey example where the independent sampling units are subsampled (i.e., hauls are the sampling unit and fish are subsampled from hauls). To evaluate the five analytical methods, we simulated 1000 samples of fish lengths subsampled from hauls in two regions with various levels of: (1) differences between the region means, (2) unbalance among numbers of hauls within regions and numbers of fish within hauls, and (3) heteroscedasticity. For each simulated sample, we tested for a difference in mean lengths between regions using each of the five methods. The inappropriate, simple ANOVA that ignored the sample structure resulted in grossly inflated Type I errors (rejecting a true null hypothesis of no difference in the means). We labeled this analysis the Pseudoreplication ANOVA based on the term “pseudoreplication” that describes the error of using a statistical analysis that assumes independence among observations when in fact the measurements are correlated. The result of this error is artificially inflated degrees of freedom, giving the illusion of having a more powerful test than the data support.The other four analyses performed well when the data were balanced and homoscedastic. When there were unequal numbers of fish per haul, the REML Nested Mixed analyses and the Unit Means ANOVA performed best. The Unequal-Variance REML Nested Mixed analysis showed clear benefit in the presence of heteroscedasticity and unbalance in hauls. For the REML Nested Mixed analysis, we compared three software packages, S-PLUS, SAS, and SYSTAT.A second simulation that compared samples with varying ratios of among-haul to among-fish variance components showed that the Pseudoreplication ANOVA was only appropriate when the haul effect yielded a p-value >0.50.