GENERAL RELATIVE RISK REGRESSION MODELS FOR ... - CiteSeerX

AMERICAN JOURNAL OP EPIDEMIOLOGY

Copyright © 1987 by The Johns Hopkins University School of Hygiene and Public Health All rights reserved

VoL 126, No. 5 Printed in U.S.A.

GENERAL RELATIVE RISK REGRESSION MODELS FOR EPIDEMIOLOGIC STUDIES 1 SURESH H. MOOLGAVKAR AND DAVID J. VENZON Moolgavfcar S. H. (The Fred Hutchinson Cancer Research Center, Seattle, WA 98104) and D. J. Venzon. General relative risk regression models for epidemiotogic studies. Am J Epidemiol 1987;126:949-61. Three parametric families of relative risk functions for the analysis of casecontrol data are discussed. A desirable feature for any general relative risk function is that inference based on it be independent of the coding of a binary covariate. Only one of the three families considered has this property. Additionally, when the relative risk is not multiplicative, methods of inference based on the asymptotic covariance matrix are likely to be seriously misleading unless the sample size is very large, as has been noted previously in other papers. This is illustrated by means of examples. Likelihood-based procedures should routinely be employed when nonmuttiplicative relative risk functions are used for analysis of case-control data. biometry; retrospective studies; statistics

During the past decade, relative risk regression models have become indispensable tools for the analysis of both clinical and epidemiologic studies. Recent theoretical developments, on the one hand, and the ready availability of high-speed computing, on the other, assure that these models will assume an even more prominent role in the foreseeable future. The standard forms of these regression models, the Cox model for failure time data and the logistic regression model for casecontrol data, postulate that the relative risk associated with a set of risk factors is a product of the relative risks associated with individual factors and, possibly, interaction Received for publication July 15, 1986, and in final form February 19, 1987. 1 The Fred Hutchinson Cancer Research Center, Division of Public Health Sciences, 1124 Columbia Street, Seattle, WA 98104. (Reprint requests to Dr. Suresh H. Moolgavkar.) This work was supported by USPHS Grants CA39949 and GM-24472 from the National Institutes of Health. The authors thank Dr. Sander Greenland for a careful reading of an earlier draft and for useful suggestions. 949

terms. Some investigators have, however, noted that multiplicativity of relative risk is unduly restrictive, and that other models may provide more parsimonious and better descriptions of the data. A linear or additive relative risk function has been suggested as an attractive alternative (1, 2). Furthermore, some authors have argued that the additive scale is the natural scale on which to measure interactions for public health purposes (3). Recently, several mixture models have been proposed (1, 4, 5). These models introduce a family of relative risk functions parametrized by a mixture parameter. The relative risk functions in these mixture families extend from the subadditive to the supramultiplicative, with the additive and multiplicative models corresponding to distinct values of the mixture parameter. Several authors have noted that there are statistical problems in working with nonmultiplicative relative risk models (6-10). The purpose of this paper is to describe some of these problems. For simplicity, we restrict our discussion to matched case-control

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studies with \:m matching. The results are valid quite generally, however. GENERAL RELATIVE RISK MODELS

described above provide the flexibility to describe a large class of dose-response functions. For particular examples, see the paper by Breslow and Storer (5). Whatever the form of the relative risk function, with \:m matching, the appropriate likelihood to be maximized for parameter estimation is a conditional likelihood (1). Specifically, suppose that there are N cases, and that there are m, controls for the ith case. Let xffl be the vector of exposure variables associated with the ith case, and xv be the vector of exposure variables associated with the yth control for the ith case. Then, the conditional likelihood is given by

We follow the notation in the paper by Breslow and Storer (5). Denote by x* = (xu x?, ..., x*) a vector of k regression or explanatory variables which describe the exposures of interest. The exposure variables may include continuous and discrete variables. Denote the relative risk associated with exposure variable x by i?(x,/S), i.e., R(x,f3) is the incidence rate ratio for exposure x relative to absence of exposure (x = 0), and /3 is a vector of parameters. In the standard, i.e., logistic, regression model, R(x,P) = exp(x'/3), and for the additive relative risk model, R(x,@) = 1 + x*ft 1-1 Three families of mixture models, which 7-0 include the multiplicative and additive models as special cases, have been proThe likelihood X may be maximized over posed. The model proposed by Thomas (1) the parameters (X,/3) by the Newtonassumes that Raphson procedure or by using Fisher's method of scoring (11). In practice, the logi?(x, ft X) maximization is greatly facilitated by fixing = Xx'/9 + (1 - X) log(l + x*/J), different values of X and then maximizing where X is the mixture parameter, and log the likelihood over /3 (5). An example in the refers to the natural logarithm. Note that next section will clarify the procedure. X = 0 corresponds to the additive model, Breslow and Storer (5) provide the "maand X = 1 corresponds to the multiplicative cros" for estimating the parameters of their model. The model proposed by Breslow and model using the software package GLIM (12). Storer (5) assumes that

{

(1 + x'ft)x - 1 X '

BINARY COVARIATES AND GENERAL RELATIVE RISK MODELS

Binary covariates (presence or absence of risk factors) are ubiquitous in epidemiologic research. A desirable feature of any data analytic procedure is that the results be independent of the way in which these binary covariates are coded. For example, a binary covariate is often coded as a "zerol o g ( l + Xx'jS) one" covariate. The choice of whether preslogff(x,ftA) = ence or absence of risk factor is designated X = 0. 1 is, however, entirely arbitrary, and one Thus, in this model, X = 0 corresponds to would feel uncomfortable with a procedure multiplicative relative risk, and X = 1 cor- the results of which depended upon the coding. responds to additive relative risk. The three general relative risk models From now on, "original coding" will refer

log(l + x'/S), X = 0. Note that also in this model, X = 0 corresponds to additive relative risk, whereas X = 1 corresponds to multiplicative relative risk. The third model, proposed by Guerrero and Johnson (4), assumes that

GENERAL RELATIVE RISK MODELS

to the situation in which presence of a binary covariate is labeled 1, and "reversed coding" will refer to the situation in which absence of the covariate is labeled 1. Now, for any relative risk model, there are two possibilities when the coding of a binary covariate is changed. The recoding may induce a (linear or nonlinear) change of parametrization while maintaining the same functional form for the relative risk. The general structure of the argument leading to explicit expressions for the parameter transformations is illustrated for the Guerrero-Johnson (4) model in the appendix. For example, suppose that there are two covariates, one binary and the other continuous. Suppose that Xi is the binary covariate coded so that 1 represents presence of the risk factor and 0 represents absence of the risk factor. Note that a change in coding of Xi leads to a "new" covariate yi = 1 — xi. If the relative risk function is multiplicative, i.e., #(x,0) = exp(x*0) = exp(xi0i + *202), a change in coding of Xi leads to a reparametrization of the relative risk function, fl(y,co) = exp(y!O)i + y ^ ) , with yx = (1 - x^, y2 = xz, ui = -ft, and W2 = 02- In this case, the change in parametrization is linear. When the relative risk function is additive, R(x,fi) = 1 + Xi0i + X2/J2, the situation is somewhat more complicated. A change in coding of Xi leads now to a nonlinear transformation of the parameters. The relative risk is now R(y,u) = 1 + yi«i + y ^ , where yi = (1 - Xi), y2 = X2, «i = - f t / ( l + ft), and o>2 = 0 2 /(l + ft). Note that now the parameter associated with x2 is also transformed. With the Guerrero-Johnson family of relative risks, ft(x,0,X) = (1 + Xx0)1/x = (1 + Xxxft + AX202)1A. X 9^ 0, a change in coding again leads to a nonlinear transformation of parameters. The relative risk is now R(y,w,a) = (1 + oyxcoi + ayvu^)11", a 5* 0, where yx = (1 - Xi), y2 = xj., «i = - 0 i / ( l + Xft), o>2 = 02/(1 + *A), and a = X. Note that both 0i and 02 are transformed, but that the mixture parameter X is not. That is, the functional form of the relative risk is main-

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tained for any fixed value X. Note also that when X = 1, the transformation is that for the additive model, and when X = 0, the transformation is that for the multiplicative model given above. The transformation formulae given here can be generalized to an arbitrary number of covariates of which an arbitrary subset may be binary. The interested reader is directed to the appendix. On the other hand, with the Thomas (1) and Breslow-Storer (5) models, the coding of a binary covariate may affect the results of the analysis in a rather dramatic way. In fact, a change in coding in a binary covariate leads to completely different models. What this means precisely is that a change in coding leads to a different likelihood function (see appendix). This is best illustrated by means of an example. We analyzed the endometrial cancer data from appendix 3 of Breslow and Day (13). The data consist of 49 cases of endometrial cancer with four controls each, and eight cases of endometrial cancer with three controls each, after eliminating individuals with missing values for the covariates considered. Two risk factors were considered, one binary (presence or absence of gallbladder disease, xx) and the other continuous (length of estrogen use in months, x2). The Guerrero-Johnson, the Breslow-Storer, and the Thomas mixture models were used to analyze the data. Figure 1 shows the maximized likelihoods plotted against values of the mixture parameter with the 1 - 0 (original) coding and the 0 — 1 (reversed) coding for gallbladder disease. As expected, the same curve is obtained for the GuerreroJohnson model with both codings. With the Breslow-Storer and Thomas models, however, different curves are obtained with the different codings. Since the additive and multiplicative models are invariant under coding changes, the two curves intersect at X = 0 and X = 1. Figure 2 shows the relative risks associated with estrogen use plotted against length of estrogen use. In this figure, for each of the mixture models, the parameters

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Thomas model -0.25

0

0.25

0.75

0.5

Breslow— Storer model -2

-1

-1.5

-0.5

0

0.5

Guerrero—Johnson model 0.5

1

1.5

2

2.5

3

3.5

-69

-74 FIGURE 1. The log of the likelihood (vertical axis) plotted against the value of the mixture parameter A (horizontal axis) for various mixture models.

were fixed at the values that maximized the likelihood, and the relative risk function predicted by the models is plotted in each of the two strata determined by gallbladder disease. With the original coding of gallbladder disease, the Thomas model achieves its maximum likelihood at X = —0.034 (figure 1), indicating that the relative risk is subadditive. With the reversed coding, however, the maximum of the likelihood is achieved at X = 0.92, and is somewhat higher, suggesting that the relative risk is almost multiplicative, although the likeli-

hood ratio test rejects the hypothesis that X = 1. The plots of the relative risk functions in figure 2 suggest quite different interpretations of the data, depending on which coding scheme is used. With the original coding (X = —0.034), the importance of gallbladder disease as a risk factor decreases with increasing estrogen use. With the reversed coding (X = 0.92), gallbladder disease is an important risk factor independent of the length of estrogen use. For the Breslow-Storer model, with the original coding, the likelihood is maximized at X = —0.28. With the reversed coding,

GENERAL RELATIVE RISK MODELS

there is a local maximum at X = 0.47 and a maximum at approximately —1.0, in the neighborhood of which the likelihood is quite flat. Figure 2 indicates that, with the original coding, the Breslow-Storer model concludes that gallbladder disease is unimportant as a risk factor for endometrial cancer once estrogen use has been taken into account. With the reversed coding and X = 0.47, on the other hand, the BreslowStorer model concludes that gallbladder disease is an important risk factor. The

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conclusion from the Guerrero-Johnson model is that the importance of gallbladder disease decreases with increasing use of estrogens. T H E ADDITIVE RELATIVE RISK MODEL

We have seen above that the Thomas and Breslow-Storer general relative risk models are not invariant under coding of binary risk factors. In particular, inference based on these models depends upon an arbitrary choice of coding of binary co-

Thomas model Original.

>-

-0.03

1.5

Reversed,

>= 0.92

0.5 -0.5 -1.5 -2.5

Breslow—Storer model Original, X = -0.2B

1 5

Reversed,

A= 0.47

Guerrero—Johnson model 4T

X = 2.3

0 20 40 60 80 100 Months of estrogen use FIGURE 2. The log of the relative risk function (vertical axis) plotted against length of estrogen use in months (horizontal axis) in each of the gallbladder disease strata. The parameter X is fixed at the value that maximizes the likelihood (see figure 1) for each of the mixture models. At any fixed value of length of estrogen use, the vertical distance between the two curves is the log of the risk in women with gallbladder disease relative to those without gallbladder disease.

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variates. In addition, with nonmultiplicative models, there are problems with parameter estimation, tests of hypotheses, and construction of confidence regions. Problems with parameter estimation have been addressed in recent papers (8,14). We illustrate the problems of inference with the additive relative risk function as an example. Inference We now consider the problems of inference that may arise with additive relative risk functions. Problems may arise both with tests of hypotheses and with the construction of confidence regions. Once the model has been fitted to the data and the parameters estimated, three distinct statistical tests are available to test the hypothesis that the parameters, say 0' = (0u ..., 8k), are different from some null value 8' = (§u ..., dk). In most instances, the null value of interest is 0 = 0. The three available tests are the Wald test, the likelihood ratio test, and the score test. These three tests are asymptotically equivalent. We do not consider the score test further here. For a discussion in the context of case-control studies, see the paper by Lustbader et al. (7). Most software packages for regression analyses yield the maximum likelihood estimates 0 and the asymptotic covariance matrix of these estimates, say V. Then, the Wald test, which is asymptotically chisquared distributed with k degrees of freedom is given by W = (0 - SY V-1 (0 - 0). To test hypotheses regarding a single parameter, say 0i, out of a vector, the appropriate diagonal element, 0n, is picked out from &. This is the asymptotic variance of 0i. Then, (