Bootstrap Confidence Intervals

Standard parametric confidence intervals can provide a measure of significance for regression coefficients. Yet they require acceptance of Gaussian assumptions regarding estimates of coefficients for their validity. Diagnostic analysis did not support these assumptions, especially given the limited data available to estimate the variability from the multitude of sources. Alternatives to the standard parametric confidence intervals are the semiparametric or nonparametric methods using bootstrap estimates of the variability of the coefficient estimates [4,3]. Our analysis used nonparametric bootstrap percentile confidence intervals to infer the observed significance level of the effects. The multiple linear regression was performed with 1000 bootstrap replications, by fixing the design matrix and resampling from the possible responses conditional on each treatment combination. The bootstrap distribution of each regression coefficient was compiled, and the 5th and 95th percentiles of the empirical distribution formed the limits for the 95% bootstrap percentile confidence interval. Plots for the effects of interest are included below. If the confidence interval failed to include 0, then the p-value was deemed to be less than or equal to 0.05, and the effect was said to be significant. Figure 5 shows the 95% percentile intervals for the interactions terms of interest.

Figure 5: Nonparametric bootstrap intervals