Data and Strategies for Analysis

The data presented several intriguing aspects which complicated an otherwise straightforward analysis of means. To begin, the research question focused on the synergistic effects of the cell treatments (BSO, BCNU, and MB) with selenium supplementation; that is, the effect of the interaction between cell treatment and selenium on activity was more important biologically than the effect of treatment of selenium alone. Secondly, the data varied considerably by day, which could mask or confound assessment of the significance of the effects. Furthermore, inference about ratios is complex; there is no clear estimator for the specific activity response for an experiment. Despite adjustment for the between-days experimental variability, analysis showed that the errors were neither constant over days nor even approximately Gaussian. We lacked confidence in the standard assumptions for parametric multivariate and instead developed nonparametric bootstrap-based estimates of the effects.

Motivated by the cited considerations, we quantified the impacts of MB, BSO, and BCNU, both singly and in combination, upon the rate of GPx synthesis for each cell line by a regression technique we developed. Implementation was with Splus 3.4 [7]. This analysis allowed us to examine first and second order interactions between selenium supplementation and the cell treatments. It also included separate predictors for each experimental day, in order to explain the impacts of the cell treatments beyond that which could be explained by aggregating over days [8,5]. An estimator for the GPx activity ratio was constructed using resampling techniques [4,3], and a Box-Cox transformation of the dependent variable helped to adjust for heterogeneous and non-Gaussian errors [2]. A bootstrap method was also used to assess variability of the linear regression coefficients and estimate their significance in the absence of standard least-squares assumptions [4,3]. A discussion of each of the regression methods is below, followed by the Splus code used in the analysis. The basis of the analysis was a least-squares multiple regression of the rate of GPx synthesis (specific activity) on selenium supplementation and cell treatments of BSO, MB, and BCNU. A rough representation of the basic model is

$$\begin{array}{ll} SpecificActivity_i\, =&\beta_0\,+\, \beta_... ...& +\,\beta_9(Se\cdot BSO\cdot BCNU)_i\,+\,\epsilon_i\end{array}$$

where $Se_i\,=\,1$ if selenium supplementation was applied in the sample $i$, and $0$ otherwise, with the other variables coded similarly, and $\epsilon_i\,\sim N(0,\sigma^2)$.