బయోమెట్రిక్స్ & బయోస్టాటిస్టిక్స్ జర్నల్

A possibly multimodal log density is modeled via orthogonal polynomials. An efficient Fisher scoring algorithm for maximum likelihood density estimation is described. Statistical hypothesis testing is emphasized such as the test for normality and density multimodality. The density estimation is illustrated with two biomedical examples: brain oxygen distribution and toenail arsenic distribution among New Hampshire residents.

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, i.e. whether the smoothing term is necessary. An approximate chi-square test (ACST) is available but has an inflated type I error rate. Permutation tests can provide appropriately sized 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 for observed data 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 were simulated with a single, circular cluster of increased or decreased risk that was centered in a circular study region. Previous research found CPT to have an inflated type I error when applied with the nominal cutoff. ACST and CPT had high power estimates when applied with reduced significance cutoffs to adjust for the respective inflated type I error rates. FSPT power depended on the span size, while FMSPT power estimates were slightly lower than those of FSPT. Overall, UPT had low power estimates when compared to the other methods.