Conference on Applied Statistics in Agriculture

ESTIMATING THE SUBJECT BY TREATMENT INTERACTION IN NON-REPLICATED CROSSOVER DIET STUDIES

Researchers in human nutrition commonly refer to the ‘consistent’ diet effect (i.e. the main effect of diet) and an ‘inconsistent’ diet effect (i.e. a subject by diet interaction). However, due to the non-replicated designs of most studies, one can only estimate the first part using ANOVA; the latter (interaction) is confounded with the residual noise. In many diet studies, it appears that subjects do respond differently to the same diet, so the subject by diet interaction may be large. In a search of over 40,000 published human nutrition studies, most using a crossover design, we found that in none was a subject by diet interaction effect estimated. For this paper, we examined LDL-cholesterol data from a non-replicated crossover study with four diets, the typical American diet, with and without added plant sterols, and a cholesterol-lowering Step-1 diet, with and without sterols. We also examined LDL-cholesterol data from a second crossover study with some replications with three diets, representing the daily supplement of 0, 1 or 2 servings of pistachio nuts. These two data sets were chosen because experience suggested that LDLcholesterol responses to diet tend to be subject-specific. The second data set, with some replication, allowed us to estimate the subject by diet interaction term in a traditional ANOVA framework. One approach to estimating an interaction effect in non-replicated studies is through the use of a multiplicative decomposition of the interaction (sometimes called AMMI―additive main effects, multiplicative interaction). In this type of analysis, residuals, formed after estimated main effects are subtracted from the data, are arrayed in a matrix with diets as columns and subjects as rows. A singular value decomposition of the matrix is performed and the first, or first and second, principal component(s) are used as estimates of the interaction, and can be tested for significance using approximate F-tests. Using the R gnm package, we found large and significant subject by diet interaction effects in both data sets; estimates of the interaction in the second data set were similar to interaction estimates from traditional ANOVA. Of an additional 26 dependent variables from the first and a third data set (the latter investigating the effect of mild alcohol consumption on blood variables), 19 had significant subject by diet interactions, based on the AMMI methodology. These results suggest that the subject by diet interaction is often important and should not be ignored when analyzing data obtained from non-replicated crossover designs―the AMMI methodology works well and is readily available in statistical software packages.