It is common practice in science to take a weighted average of estimators of a single parameter. If the original estimators are unbiased, any weighted average will be an unbiased estimator as well. The best estimator ...It is common practice in science to take a weighted average of estimators of a single parameter. If the original estimators are unbiased, any weighted average will be an unbiased estimator as well. The best estimator among the weighted averages can be obtained by choosing weights that minimize the variance of the weighted average. If the variances of the individual estimators are given, the ideal weights have long been known to be the inverse of the variance. Nonetheless, I have not found a formal proof of this result in the literature. In this article, I provide three different proofs of the ideal weights.展开更多
Estimating causal effects is a principal goal in epidemiology and other branches of science. Nonetheless, what constitutes an effect and which measure of effect is pre-ferred are unsettled questions. I argue that, und...Estimating causal effects is a principal goal in epidemiology and other branches of science. Nonetheless, what constitutes an effect and which measure of effect is pre-ferred are unsettled questions. I argue that, under indeterminism, an effect is a change in the tendency of the outcome variable to take each of its values, and then present a critical analysis of commonly used measures of effect and the measures of frequency from which they are calculated. I conclude that all causal effects should be quantified using a unifying measure of effect called the log likelihood ratio (which is the log probability ratio when the outcome is a discrete variable). Furthermore, I suggest that effects should be estimated for all causal contrasts of the causal variable (i.e., expo-sure), on all values of the outcome variable, and for all time intervals between the cause and the outcome. This goal should be kept in mind in practical approximations.展开更多
文摘It is common practice in science to take a weighted average of estimators of a single parameter. If the original estimators are unbiased, any weighted average will be an unbiased estimator as well. The best estimator among the weighted averages can be obtained by choosing weights that minimize the variance of the weighted average. If the variances of the individual estimators are given, the ideal weights have long been known to be the inverse of the variance. Nonetheless, I have not found a formal proof of this result in the literature. In this article, I provide three different proofs of the ideal weights.
文摘Estimating causal effects is a principal goal in epidemiology and other branches of science. Nonetheless, what constitutes an effect and which measure of effect is pre-ferred are unsettled questions. I argue that, under indeterminism, an effect is a change in the tendency of the outcome variable to take each of its values, and then present a critical analysis of commonly used measures of effect and the measures of frequency from which they are calculated. I conclude that all causal effects should be quantified using a unifying measure of effect called the log likelihood ratio (which is the log probability ratio when the outcome is a discrete variable). Furthermore, I suggest that effects should be estimated for all causal contrasts of the causal variable (i.e., expo-sure), on all values of the outcome variable, and for all time intervals between the cause and the outcome. This goal should be kept in mind in practical approximations.
