Power of Permutation Tests Using Generalized Additive Models with Bivariate Smoothers

In spatial epidemiology, when applying Generalized Additive Models (GAMs) with a bivariate locally weighted regression smooth over longitude and latitude, a natural hypothesis is whether location is associated with an outcome. An approximate chi-square test (ACST) is available but has an inflated type I error rate. Permutation tests provide alternatives. This research evaluated powers of ACST and four permutation tests: the conditional (CPT), fixed span (FSPT), fixed multiple span (FMSPT), and unconditional (UPT) permutation tests. For CPT, the span size was determined by minimizing the Akaike Information Criterion (AIC) and was held constant for models applied to permuted datasets. For FSPT, a single span was selected a priori. For FMSPT, GAMs were applied using 3-5 different spans selected a priori and the significance cutoff was reduced to account for multiple testing. For UPT, the span was selected by minimizing the AIC for observed and for permuted datasets. Data with a cluster of increased/decreased risk centered in a study region were simulated. ACST and CPT had high power estimates when applied with reduced significance cutoffs to adjust for inflated type I errors. FSPT power depended on the span size; FMSPT power estimates were slightly lower. Overall, UPT had low power.