RE: “INTERPRETING MULTIPLE LOGISTIC REGRESSION COEFFICIENTS IN PROSPECTIVE OBSERVATIONAL STUDIES”
RE: “INTERPRETING MULTIPLE LOGISTIC REGRESSION COEFFICIENTS IN PROSPECTIVE OBSERVATIONAL STUDIES”
复制标题
回复：“解释前瞻性观察研究中的多个逻辑回归系数”
DOI：
10.1093/oxfordjournals.aje.a114108
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发表时间：
1985
影响因子：
5
通讯作者：
J. Fleiss
中科院分区：
文献类型：
--
作者：
B. Levin;J. Fleiss
Abbott and Carroll's paper (1) is motivated by the problem of estimating the effect of a reduction in triglycerides on the probability of developing coronary heart disease. If high density lipoprotein cholesterol (HDL-C) is an intermediary in the causal pathway between triglycerides and coronary heart disease, use of the triglycerides coefficient from a multiple regression model which controls for HDL-C is inappropriate when interest lies not in the effect of a change in triglycerides while HDL-C is held constant, but rather in the full effects of change in triglycerides, including those which might be mediated through changes in HDL-C. On the other hand, use of a regression model which does not include HDL-C might also be inappropriate because failure to control for HDL-C could result in confounding, since the independent effect of HDL-C on coronary heart disease may be wrongly attributed to its correlate, triglycerides. In other words, the problem is that HDL-C can act in one or both of two ways: as an" overmatcher" in Breslow and Day's first sense (2, pp. 104-105), in which case it should be excluded from the model, or as a confounder, in which case it should be included in the model. Abbott and Carroll propose the" projected slope" as a resolution of this dilemma, but they do not explain what the projected slope measures. In fact, it is basically the univariate slope on Xi. The problem which motivated Abbott and Carroll can also occur when the outcome variable is continuous. Applying their proposed solution in a classic multiple regression context reveals its flaw. Replace the left-hand side of their equation 6 with E (V,), the expected value of a continuous variable observed with risk factor xu and covariate ij,. In their notation, 4i= ft+ 7ift> where ii is the projected slope, ft and ft are the respective regression coefficients from the multiple regression of Y on Xi and Xt, and 71 is the slope of the regression of X2 on Xt. It is well known, however, that ft+ 7,/S,=/3* j where/?*, is the slope of the univariate regression of Y on Xt(see, for example, reference 3, pp. 394-398). Since the e/s, the residuals from the regression of Xj on X,, are uncorrelated with the X, I'B, 61 is not affected by inclusion of the 6j* term in their equation 6. So St=(J*, and the projected slope simply equals the univariate coefficient and does not" consider the relationship" between Xi and Xt. The analagous result can easily be obtained when the projected slope is used with several independent variables. The distinction between the interpretation of coefficients from models with the same right-hand side in classic and logistic regression is that the coefficients from the former apply to changes in expected values of continuous variables, while the coefficients from the latter apply to changes in the logits of the probabilities of binary events. The estimates of projected slope and the univariate coefficients in Abbott and Carroll's tables are not identical because of the iterated reweighting of each observation in logistic regression. If the estimates of ft,/J*], and y, were all based on weighted regressions, with the same vector of weights in each regression, the estimates of the projected slope and the univariate coefficients would be equal, just as in classic regression. So in logistic regression, the projected slope is simply an approximation to the univariate coefficient. The projected slope does not solve the problem raised by Abbott and Carroll. However, it may not even be an important issue in their example. Assume that a preventive measure such as a change in diet is expected to have effects on both triglycerides and HDL-C. The effect of the intervention on risk could be estimated as the sum of the estimated effects due to …
