Hierarchical regression analysis applied to a study of multiple dietary exposures and breast cancer

Hierarchical regression attempts to improve standard regression estimates by adding a second-stage "prior" regression to an ordinary model. Here, we use hierarchical regression to analyze case-control data on diet and breast cancer. This regression yields semi-Bayes relative risk estimates for dietary items by using a second-stage model to pull estimates toward each other when the corresponding variables have similar levels of nutrients. Unlike classical Bayesian analysis, however, no use is made of previous studies on nutrient effects. Compared with results obtained with one-stage conditional maximum-likelihood logistic regression, our hierarchical regression model gives more stable and plausible estimates. In particular, certain effects with implausible maximum-likelihood estimates have more reasonable semi-Bayes estimates.

When will nondifferential misclassification of an exposure preserve the direction of a trend?

Dosemeci et al. (Am J Epidemiol 1990; 132:746-8) gave examples in which nondifferential misclassification of exposure reversed the direction of a trend. Gilbert (Am J Epidemiol 1991; 134:440-1) proposed that these examples occurred because the errors in exposure were systematic, and she pointed out that the relation between the measured and the true exposure was not monotonic. Assuming that the mean response either monotonically increases or decreases with the true exposure and that the exposure misclassification is nondifferential, the authors show that if the mean value of the measured exposure increases with the true exposure, then the direction of the trend cannot be reversed. Consequently, Gilbert's intimation that reversal of trend can only occur when errors are systematic is correct. However, the present authors' result is stronger in that even when errors in assessing exposure do include a systematic component, if monotonicity can be assumed, reversal of trend cannot occur. The weaker condition of positive correlation between the measured and true exposure is not sufficient to guarantee nonreversal of trend, as they show by example.

Alternative models for ordinal logistic regression

Armstrong and Sloan have reviewed two types of ordinal logistic models for epidemiologic data: the cumulative-odds model and the continuation-ratio model. I review here certain aspects of these models not emphasized previously, and describe a third type, the stereotype model, which in certain situations offers greater flexibility coupled with interpretational advantages. I illustrate the models in an analysis of pneumoconiosis among coal miners.

Predictors of treatment response in obsessive-compulsive disorder: multivariate analyses from a multicenter trial of clomipramine

There have been many attempts to find predictors of the therapeutic response to the clomipramine treatment of obsessive-compulsive disorder. The majority of studies have failed to identify such predictors. Possible reasons for this failure include the small sample size of most studies, samples homogeneous with respect to the study factors of interest, and the use of statistical procedures that are insensitive to individual differences or that inadequately control for confounding. We have reanalyzed data from Ciba-Geigy's large, multicenter clinical trial of clomipramine for obsessive-compulsive disorder, using stratification and regression techniques to identify multiple prognostic factors and control for confounders. We assessed the relationship between therapeutic response and baseline measures such as severity of symptoms, type of symptoms (obsessions, compulsions, depression), length of illness, age of onset, and other demographic factors (age, race, and sex). We found age of onset to be a strong predictor of response to clomipramine: people who develop obsessive-compulsive disorder later in life have a better chance of responding than do those who become ill earlier, independent of length of illness. We also found that baseline depression is associated with response, but the association appears to be nonlinear.

Invited commentary: a critical look at some popular meta-analytic methods

Meta-analysis is essential for obtaining reproducible summaries of study results and valuable for discovering patterns among study results. A good meta-analysis will highlight and delineate the subjective components of these processes and vigorously search for sources of heterogeneity. Unfortunately, these objective are not always met by common techniques. For example, a scatterplot is an objective summarization if the data are uncensored, but inferred patterns should be regarded as subjective recognitions of the analyst, not objective data properties. Random-effects summaries encourage averaging over important data patterns, divert attention from key sources of heterogeneity, and can amplify distortions produced by publication bias; such summaries should only be used when important heterogeneity remains after a thorough search for the sources of such heterogeneity. Quality scoring adds the analyst's subjective bias to the results, wastes information, and can prevent the recognition of key sources of heterogeneity; it should be completely replaced by meta-regression on quality items (the score components).

The interpretation of multiplicative-model parameters as standardized parameters

Under current conventions, relative-risk estimates obtained from multiplicative models are interpreted as estimates of a homogeneous effect. Such interpretations condition on the unverifiable assumption that the relative risk under study is homogeneous, an assumption that is not likely to be correct even if the model fits well. We propose that such estimates are better interpreted as estimates of standardized relative risks, with a bias component that depends on the degree of model misspecification and on the study design. To evaluate our proposal, we present a study of the maximum-likelihood estimators from Poisson and logistic regression compared to the population-standardized rate ratio. The results indicate that our proposed interpretation would in practice be more cautious and accurate than the homogeneous-effect interpretation.

A case-control study of cancer mortality at a transformer-assembly facility

To address earlier reports of excess cancer mortality associated with employment at a large transformer manufacturing plant, each plant operation was rated for seven exposures: Pyranol (a mixture of polychlorinated biphenyls and trichlorobenzene), trichloroethylene, benzene, mixed solvents, asbestos, synthetic resins, and machining fluids. Site-specific cancer deaths among active or retired employees were cases; controls were selected from deaths (primarily cardiovascular deaths) presumed to be unassociated with any of the study exposures. Using job records, we then computed person-years of exposure for each subject. All subjects were white males. The only unequivocal association was that of resin systems with lung cancer (odds ratio = 2.2 at 16.6 years of exposure, P = 0.001, in a multiple logistic regression including asbestos, age, year of death, and year of hire). Certain other odds ratios appeared larger, but no other association was so robust and remained as distinct after considering the multiplicity of comparisons. Study power was very limited for most associations, and several biases may have affected our results. Nevertheless, further investigation of synthetic resin systems of the type used in the study plant appears warranted.

Invited commentary: ecologic studies--biases, misconceptions, and counterexamples

Many authors have pointed out that relative-risk estimates derived from ecologic data are vulnerable to biases not found in estimates derived from individual-level data. Nevertheless, biases in ecologic studies still are often dealt with in the same manner as biases in other observational studies, and so are not given adequate treatment. This commentary reviews and illustrates some of the more recent findings about bias in ecologic estimates. Special attention is given to problems of ecologic confounder control when individual risks follow a nonlinear model, and to misconceptions about ecologic bias that have appeared in the literature.

A comparison of the performance of model-based confidence intervals when the correct model form is unknown: coverage of asymptotic means

We conducted a simulation study to examine the performance of confidence intervals when multiplicative and additive rate (Poisson regression) models are fit to follow-up data, but the model form may be misspecified. Data were generated from over 129,000 different population structures that ranged from sub-additive to supra-multiplicative. When a multiplicative model was fit, all of the confidence intervals that we examined performed well as interval estimators of the asymptotic means of the point estimators, even when the correct model form was not multiplicative. When an additive model was fit, (1) only the likelihood ratio interval and the score interval with expected information consistently performed well, and they consistently performed better than any of the Wald intervals that we examined; (2) Wald intervals performed better when calculated with observed information rather than with expected information; and (3) Wald intervals with expected information performed better when the information matrix was evaluated at the restricted maximum likelihood estimate rather than the unrestricted maximum likelihood estimate.

Empirical-Bayes and semi-Bayes approaches to occupational and environmental hazard surveillance

Empirical-Bayes methods offer potentially dramatic improvements in statistical accuracy over conventional statistical methods. We provide an elementary introduction to empirical-Bayes analysis of occupational and environmental hazard surveillance data. Such analyses are especially well suited to situations in which many associations must be examined, but few or none can be estimated precisely. Statistical issues in hazard surveillance are reviewed, followed by a discussion of the rationale and methods for empirical-Bayes analyses, using a study of occupational exposures and cancer mortality to illustrate key concepts. Finally, the assumptions underlying empirical-Bayes analyses are discussed critically, with special attention to the "exchangeability" assumptions that distinguish empirical-Bayes from conventional methods.

Simulation study of confounder-selection strategies

In the absence of prior knowledge about population relations, investigators frequently employ a strategy that uses the data to help them decide whether to adjust for a variable. The authors compared the performance of several such strategies for fitting multiplicative Poisson regression models to cohort data: 1) the "change-in-estimate" strategy, in which a variable is controlled if the adjusted and unadjusted estimates differ by some important amount; 2) the "significance-test-of-the-covariate" strategy, in which a variable is controlled if its coefficient is significantly different from zero at some predetermined significance level; 3) the "significance-test-of-the-difference" strategy, which tests the difference between the adjusted and unadjusted exposure coefficients; 4) the "equivalence-test-of-the-difference" strategy, which significance-tests the equivalence of the adjusted and unadjusted exposure coefficients; and 5) a hybrid strategy that takes a weighted average of adjusted and unadjusted estimates. Data were generated from 8,100 population structures at each of several sample sizes. The performance of the different strategies was evaluated by computing bias, mean squared error, and coverage rates of confidence intervals. At least one variation of each strategy that was examined performed acceptably. The change-in-estimate and equivalence-test-of-the-difference strategies performed best when the cut-point for deciding whether crude and adjusted estimates differed by an important amount was set to a low value (10%). The significance test strategies performed best when the alpha level was set to much higher than conventional levels (0.20).

Summarization, smoothing, and inference in epidemiologic analysis. 1991 Ipsen Lecture, Hindsgavl, Denmark

In a recent article (Epidemiology 1990; 1: 421-429) I resurrected some historical criticisms of conventional statistics in non-randomized, non-randomly sampled studies, and suggested some improvements to current practice in response to these criticisms. Here, I propose that some resolution can be achieved by separating data analysis into summarization, smoothing, and inferential phases. Methods of statistical inference are in fact smoothing methods, as are many methods of descriptive statistics, and as such can be viewed as pattern-recognition devices. Scientific inference is not a statistical process, but instead concerns derivation of explanations for patterns detected by statistical methods. Improvements could be made to all three phases simply by keeping the phases distinct.

Basic problems in interaction assessment

This paper reviews problems with the definition and estimation of interactions in epidemiologic studies. Methods for modeling interactions and dose-response also are reviewed, and references to more detailed literature are provided. Concepts are illustrated in the context of evaluating the joint effects of household radon exposure and environmental tobacco smoke.

Maximum likelihood estimation of the attributable fraction from logistic models

Bruzzi et al. (1985, American Journal of Epidemiology 122, 904-914) provided a general logistic-model-based estimator of the attributable fraction for case-control data, and Benichou and Gail (1990, Biometrics 46, 991-1003) gave an implicit-delta-method variance formula for this estimator. The Bruzzi et al. estimator is not, however, the maximum likelihood estimator (MLE) based on the model, as it uses the model only to construct the relative risk estimates, and not the covariate-distribution estimate. We here provide maximum likelihood estimators for the attributable fraction in cohort and case-control studies, and their asymptotic variances. The case-control estimator generalizes the estimator of Drescher and Schill (1991, Biometrics 47, 1247-1256). We also present a limited simulation study which confirms earlier work that better small-sample performance is obtained when the confidence interval is centered on the log-transformed point estimator rather than the original point estimator.

A meta-analysis of coffee, myocardial infarction, and coronary death

This paper presents a meta-analysis of 22 studies of coffee use and myocardial infarction or coronary death. In the eight case-control studies, a fairly homogeneous increased risk was found among coffee users (geometric mean rate ratio of 1.42 for 5 cups per day vs none, with 95% confidence limits of 1.30, 1.55, homogeneity P-value of 0.89). The 14 cohort studies tended to exhibit lower but very heterogeneous rate ratios, with a trend toward larger rate ratios in studies with longer follow-up periods and later publication dates (geometric mean rate ratio of 0.92 for the five cohort studies published up to 1981, 1.27 for the nine cohort studies published in 1986 or later; overall homogeneity P-value of 0.0008). The evidence thus remains ambiguous regarding both the existence and size of a coffee effect, and although a rate ratio of over 1.5 for 5 cups per day appears unlikely, stronger effects for 10-cup-per-day drinkers cannot be ruled out.

Interpreting model coefficients when the true model form is unknown

In this paper, we critically examine mathematical modeling. We outline the major assumptions required by modeling methods used in epidemiology and discuss in detail one fundamental assumption that is usually violated in epidemiologic studies: the assumption that the structural model form is correctly specified. We apply concepts from the econometrics literature to examine how epidemiologic inference may be affected when the structural model form is incorrectly specified. Because the structural model is almost always misspecified in practice, tests and confidence intervals for model coefficients do not refer to "true population parameters" in the ordinary sense. Rather, these statistics concern parameters that depend on features of study design, as well as the effects under study. In cohort studies analyzed with multiplicative rate models, model parameters are interpretable as approximations to log standardized rate ratios; unfortunately, such interpretations are not as accurate for other models and designs. We therefore conclude that model coefficients can serve as reasonable effect summaries in some, but not all, situations.

Meta-analysis of epidemiologic dose-response data

We discuss methods for summarizing epidemiologic studies of dose-response. The data from such a study typically appear as a series of dose-specific relative risks, with one category serving as the common reference group. We present methods for estimating the dose-response parameters from single and multiple study reports, for assigning levels to exposure categories when modeling relative risks, and for investigating the effects of study design and subject characteristics on dose-response curves. Finally, we discuss the choice of fixed vs random effects models. We illustrate our points with data from case-control studies of the relation between duration of oral contraceptive use and risk of breast cancer.

Risk and rate advancement periods as measures of exposure impact on the occurrence of chronic diseases

The predominant epidemiologic measures of risk factor impact have been various concepts of the "attributable fraction." For many chronic diseases, however, risk factors merely advance the occurrence of disease, and traditional concepts of the attributable fraction do not convey information on the time dimension of premature disease occurrence. Measures of years of disease-free life lost have been proposed to reflect this time dimension but are not always estimable without special assumptions. In this paper, we examine two alternative measures, the risk and rate advancement periods, which are the time periods by which the risk or rate of disease is advanced among exposed subjects conditional on disease-free survival to some baseline age. These measures are applicable for risk factors that promote progression of chronic diseases whose rates increase with age. Point and interval estimates of these measures are easily derived from the output of standard modeling analyses. The measures are illustrated with examples from the literature. The uses and limitations of risk and rate advancement periods are critically discussed.

Methods for epidemiologic analyses of multiple exposures: a review and comparative study of maximum-likelihood, preliminary-testing, and empirical-Bayes regression

Many epidemiologic investigations are designed to study the effects of multiple exposures. Most of these studies are analysed either by fitting a risk-regression model with all exposures forced in the model, or by using a preliminary-testing algorithm, such as stepwise regression, to produce a smaller model. Research indicates that hierarchical modelling methods can outperform these conventional approaches. I here review these methods and compare two hierarchical methods, empirical-Bayes regression and a variant I call 'semi-Bayes' regression, to full-model maximum likelihood and to model reduction by preliminary testing. I then present a simulation study of logistic-regression analysis of weak exposure effects to illustrate the type of accuracy gains one may expect from hierarchical methods. Finally, I compare the performance of the methods in a problem of predicting neonatal mortality rates. Based on the literature to date, I suggest that hierarchical methods should become part of the standard approaches to multiple-exposure studies.

Cohort and nested case-control studies of hematopoietic cancers and brain cancer among electric utility workers

Recent studies have raised concern about the potential health effects of occupational exposures to power frequency electric and magnetic fields. We evaluated cancer mortality for leukemia, brain cancer, and lymphoma from 1960 to 1988 in a cohort of 36,221 electric utility workers using cohort analyses and three nested case-control studies. From a volunteer sample of the current workforce that represented a variety of different occupations and work locations, we collected 776 days of magnetic field measurements. We derived exposure information from company job history information and developed exposure scores by linking job history data to measured magnetic fields. In job title analyses, we compared "electrical workers" with other field and craft occupations, office, and technical support staff. Age-specific cancer rates for electrical and reference workers were similar. "Electrical workers" had rate ratios or odds ratios ranging from 0.7 to 1.4. Most ratios were close to 1.0. Lymphomas were slightly elevated compared with leukemias and brain cancers (ratios of 0.9-1.4 vs 0.7-1.2, respectively). Odds ratios for magnetic field exposure indices, based on scores for the mean, median, 99th percentile, and fractions exceeding 10 milligauss and 50 milligauss, were all close to or less than 1.0. The interval estimates indicate no strong association but are somewhat limited by imprecision.

The use of restricted controls to prevent recall bias in case-control studies of reproductive outcomes

Recall bias or report bias is said to occur when associations are distorted or created because case informants report events differently from controls. Some investigators have suggested that this bias can be prevented by choosing controls who have conditions similar to those found in the case group. We use the term "restricted-control group" for such a control series. Although using a restricted-control series may eliminate differential misclassification, this approach will usually not eliminate nondifferential misclassification and may create selection bias. In this article, we present a way to algebraically examine the effects of misclassification and selection bias on observed associations. We use this method to compare the impact of recall bias in a study using a population control group with the effects of selection bias and nondifferential misclassification if a restricted-control group is used. Our approach is exemplified using data from a case-control study of sudden infant death syndrome. Our findings show that even when recall bias exists, the observed association can be closer to the true association when a population control series is used than when a restricted-control group is used.