Figure 7. Environmental substitutes outperform elevation for breast but not lung cancer.

(A) Environmental variables were strongly collinear. Correlation with elevation was high, hence many environmental variables also covaried with lung and breast cancer. (B) To test whether elevation-association with lung and breast cancer was direct or indirect, we substituted each environmental correlate in place of elevation during best subset selection for each cancer. The optimal model for each elevation-replacement was compared to the unreplaced model by approximating a Bayes factor (K) from the change in BIC. The Bayes factor indicates the odds that the replacement is superior, thus K > 1 favors the substitution while K < 1 provides evidence against. Since the elevation model was compared to itself, ΔBIC = 0 and K = 1 (log₁₀K = 0). The standardized coefficient for each environmental predictor is represented by a triangle, where size is scaled to the magnitude and orientation indicates the sign (upwards for positive). For breast cancer (red), three substitutions increased likelihood suggesting that any association observed with elevation was indirect. For lung cancer (blue), substituting elevation produced models that were many orders of magnitude less likely, suggesting that the elevation association was direct.