Quantifying the impact of association of environmental mixture in a type 1 and type 2 error balanced framework

In environmental epidemiology, analysis of environmental mixture in association to health effects is gaining popularity. Such models mostly focus on inferences of hypotheses or summarizing strength of association through regression coefficients and corresponding estimates of precision. Nonetheless, when a decision is made against alternative hypothesis, it becomes increasingly difficult to tease apart whether the decision is influenced by sample size or represents genuine absence of association and whether the result warrants further investigation. Similarly, in case of a decision made in favour of alternative hypothesis, a significant association may indicate influence of large sample and not a strong effect. Moreover, the disparate type 1 and type 2 errors, might render these inferences unreliable. Using Cohen's f2 to evaluate the strength of explanatory associations in a more fundamental way, we herein propose a new concept, optimal impact, to quantify the maximum explanatory association solely contributed by an environmental mixture after controlling for confounders and covariates such that the type 2 error remains at its minimum. Optimal impact is built upon a novel hypothesis testing procedure in which the rejection region is determined in a way that type 1 and type 2 errors are balanced. Even when an association does not achieve statistical significance, its optimal impact might deem it meaningful and strong enough for further investigation. This idea was naturally extended to estimate sample size in designing studies by striking a balance between explanatory precision and utility. The properties of this framework are carefully studied and detailed results are established. A straightforward application of this procedure is illustrated using an exposure-mixture analysis of per and poly fluoroalkyl substances and metals with serum cholesterols using data from 2017 & 2018 US National Health and Nutrition Examination Survey.