CONTINUOUS AND CATEGORICAL TREND ESTIMATORS: SIMULATION RESULTS AND AN APPLICATION TO RESIDENTIAL RADON

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1 CONTINUOUS AND CATEGORICAL TREND ESTIMATORS: SIMULATION RESULTS AND AN APPLICATION TO RESIDENTIAL RADON A Schaffrath Rosario 1,2*, J Wellmann 1,3, IM Heid 1 and HE Wichmann 1,2 1 Institute of Epidemiology, GSF National Research Center for Environment and Health, Neuherberg, Germany 2 Chair for Epidemiology, Ludwig-Maximilians-University, Munich, Germany 3 Institute of Epidemiology and Social Medicine, University of Münster, Germany ABSTRACT Exposure variables in epidemiological studies often have a skewed distribution that can be approximated by a lognormal distribution. A typical example are indoor radon gas concentrations. In this situation, the standard continuous estimator for trend can be strongly influenced by single outlying observations. In a simulation study, we found two types of estimators that serve as useful sensitivity analyses against outliers: trimmed estimators and categorical estimators based on statistically optimal categories using expected values within categories as scores. In the West German study on indoor radon and lung cancer, the alternative estimators are used to assess the influence of two outliers among the controls. INDEX TERMS Statistical methods, Categorization, Skewed distribution, Epidemiology, Indoor radon, Lung cancer INTRODUCTION We consider the estimation of a linear trend in an epidemiological study, say a case-control study. Typically, the trend is estimated by entering the exposure variable as a continuous variable in a logistic regression model. This approach is the most efficient one (Lagakos, 1988; Zhao and Kolonel, 1992; Greenland, 1995a), but it can be sensitive to single outlying observations. When testing for trend, it is quite common to use trend tests based on categories instead of tests based on the continuous variable. Sometimes, categories are also employed when estimating a trend, for example in the Iowa Radon Lung Cancer Study (Field et al., 2000). However, several questions arise when applying categorical trend estimators: How many categories should be used? How to choose the category boundaries? How to choose the category scores? Should the categories be based on the control distribution, the case distribution, or the marginal distribution (i.e. cases and controls combined)? Whereas questions (i) and (ii) are discussed in the literature, questions (iii) and (iv) usually receive little attention. * Contact author rosario@gsf.de 653

2 We conducted a simulation study to evaluate which categorical trend estimators perform best and how they compare with the continuous trend estimator. We then applied different estimators to the West German case-control study on indoor radon and lung cancer (Wichmann et al., 1998; Kreienbrock et al., 2001) to assess the effect of two outliers among the controls found in this study. Since indoor radon levels typically show a right-skewed distribution that can be approximated by a lognormal distribution (i.e. the logarithm of the radon levels is approximately normally distributed), we based our simulations on the lognormal distribution. METHODS We restrict our discussion to case-control studies and assume a standard logistic regression model (Breslow and Day, 1980), i.e. we assume that the log odds ratio is linear in the exposure x. We define a categorical estimator as follows: The original exposure variable x is categorized into k categories. Each category is assigned a score, e.g. the category mean. We replace the original variable x by a new variable z that assigns to each observation the respective category score. The new variable z is entered in the logistic regression model as a continuous variable. The categorical estimators have also been termed 'discrete estimators' (Field et al., 2000). Furthermore, we investigated the behaviour of trimmed estimators (extreme observations are discarded from the analysis) and winsorized estimators (observations above a certain cutpoint are replaced by the value of the cutpoint). Apart from category means and medians, we employed expected values within categories as scores (Cox, 1957; Connor, 1972). In contrast to the empirical measures mean and median, the expected values are calculated based on the distributional assumption. Formulae for the calculation of the expected value for the lognormal distribution can be found in Schaffrath Rosario et al. (2002). It is common practice in epidemiology to use percentile categories, i.e. categories containing about the same number of observations. Another option for right-skewed variables are categories based on logarithmic cutpoints such as 25, 50, 100 and 200 Bq/m³. Statistically optimal categories (Connor, 1972) minimize the exposure variance within categories. For a lognormally distributed exposure variable and k = 5 categories, the approximately optimal categories contain 40%, 34%, 18%, 7% and 1% of the data, respectively. For k = 8 categories, the percentages are 21.8%, 26.2%, 21.5%, 15%, 9%, 4.5%, 1.7% and 0.3% (Schaffrath Rosario et al., 2002). The logarithmic categories are thus similar to the optimal categories when the exposure is right-skewed, with a large reference category and smaller categories in the upper range of the data. For each scenario, we simulated 3,000 case-control studies using the functions RANNOR and RANTBL of SAS version 6.12 under Unix HP-UX. The true odds ratio (OR) was set at 1.2 or 2 per 100 Bq/m³. We mostly simulated large studies with 1500 cases and 2250 controls or 500 cases and 750 controls. Outliers were produced by multiplying the maximum observed value among controls (or cases) by three. For each simulated study, we calculated the following trend estimators: the estimator based on the continuous exposure variable x (original data); trimmed estimators, cutting off the upper 1%; 654

3 winsorized estimators, winsorizing the upper 1%; categorical estimators with k = 5, 10, 20 percentile categories with mean scores and median scores; categorical estimators with k = 5 categories based on logarithmic cutpoints (25, 50, 100, 200 Bq/m³; for distributions with high exposure range: 50, 100, 200, 400 Bq/m³) with mean scores and expected value scores; categorical estimators with k = 5 or 8 categories based on optimal cutpoints with expected value scores. For each estimator, four versions using different reference distributions were determined: a) cutpoints and scores determined separately for cases and controls (status-specific); b) cutpoints and scores based on the distribution of observed values among the controls, with the same cutpoints and scores applied to the cases; c) cutpoints and scores based on the case distribution; d) cutpoints and scores based on the marginal distribution, i.e. cases and controls combined. RESULTS Simulation results We present here a summary of the simulation results. Details can be found in Schaffrath Rosario et al. (2002). When there was no outlier in the data, the continuous estimator was unbiased and had smallest standard error, both for an OR of 1.2 and 2. A strong overestimation of the OR occured when status-specific cutpoints and scores were employed. Categorical estimators based on the control distribution somewhat overestimated the true OR, while those based on the case distribution had a small downward bias. The bias diminished with an increasing number of categories. For trimmed estimators, the direction of bias as related to the choice of the reference distribution was reversed. Trimmed and categorical estimators based on the marginal distribution, however, were nearly unbiased. Some degree of overestimation was present in the winsorized estimators regardless of the choice of the reference distribution. We also observed an overestimation of the effect when category medians were used as scores, almost comparable in size to that of status-specific procedures. Using quintiles with category medians under a lognormal distribution with low exposure range, the true OR of 1.2 was on average estimated at 1.24, while the true OR of 2 was estimated at In general, the bias was small for the low OR of 1.2, but tended to be substantial for OR = 2. Categorical estimators based on percentile categories with mean scores were more biased and had larger standard errors than those based on logarithmic or optimal cutpoints with expected value scores. When there was an outlier among controls in the data, the continuous estimator showed a strong downward bias, but it was hardly influenced at all by an outlier among cases. Categorical estimators based on mean scores, however, were biased downward both by an outlier among controls and by an outlier among cases. Categorical estimators using expected value scores, on the other hand, remained uninfluenced by the outlier. The same holds true for trimmed estimators. 655

4 In summary, the following alternative estimators provided unbiased results in nearly all situations investigated: the trimmed estimator based on the marginal distribution; the grouped estimator using optimal categories with expected value scores, again based on the marginal distribution. Application to the West German case-control study on indoor radon and lung cancer In the West German case-control study on indoor radon and lung cancer (Wichmann et al., 1998; Kreienbrock et al., 2001), we are facing an outlier problem. There are two controls with high radon concentrations (899 and 827 Bq/m³), while the maximum radon concentration found among cases is 682 Bq/m³. Table 1. Sensitivity analysis in the West German radon study OR per 100 Bq/m³ 95% confidence interval Continuous estimator 0.98 (0.82, 1.17) 1% Trimmed estimator 1.18 (0.89, 1.58) Categorical estimators 1 : 5 optimal categories 1.05 (0.83, 1.34) 8 optimal categories 1.00 (0.80, 1.25) 1 with expected value scores, based on the marginal distribution In a conditional logistic regression stratified by age, sex and region, and adjusting for smoking as log (packyears+1), ex-smoking and occupational asbestos exposure, the continuous trend estimator indicates no increased lung cancer risk due to indoor radon (OR = 0.98 per 100 Bq/m³, see Table 1). The alternative estimators yield higher OR estimates, ranging from 1.00 over 1.05 to However, they are associated with wider confidence intervals. A similar increase in risk estimates is apparent for the subgroup of subjects living in regions with higher radon levels (see Schaffrath Rosario et al., 2002). DISCUSSION Right-skewed exposure variables such as indoor radon levels are especially sensitive to outliers, as there are few observations in the upper range of the data. Our simulations have confirmed the sensitivity of the continuous trend estimator to single outlying observations. But categorical estimators also turned out to be sensitive to outliers when they were based on mean scores. This finding may be unexpected, but it has already been noted (Greenland, 1995a-c). However, categorical estimators with expected value scores were not sensitive to outliers. This can be explained by the fact that the outlier strongly influences the mean in the category where the outlier occurs, whereas the expected value scores are calculated based on the whole data set. Apart from the sensitivity to outliers, the continuous estimator fared best in terms of bias and precision in all our simulations. This supports the statements that caution against the use of categorical analysis because of the associated loss of efficiency (Lagakos, 1988; Zhao and Kolonel, 1992; Greenland, 1995a). In particular, the commonly employed 4 or 5 percentile categories are not appropriate for right-skewed data. Instead, optimal categories or categories with logarithmic cutpoints should be used. Moreover, categorical estimators with statusspecific cutpoints and scores and estimators based on median scores turned out to be biased upwards, as did winsorized estimators. The first finding is parallel to the result that status- 656

5 specific imputation of missing radon levels overestimates the relative risk (Weinberg et al., 1996). It should be noted that our findings are restricted to skewed exposure variables that can be approximated by a lognormal distribution. Furthermore, we take the assumption of a linear exposure-response relationship for granted. Two types of estimators emerged as appropriate tools for a sensitivity analysis: the trimmed estimator and the categorical estimator with optimal categories and expected value scores, both based on the marginal distribution. We suggest that such sensitivity analyses should routinely be applied to skewed exposure variables. A SAS macro for these analyses is available upon request from the first author. The application of these estimators to the West German case-control study on indoor radon and lung cancer has shown that the estimated trend in this study is strongly influenced by two outliers among the controls. This instability in risk estimates calls for caution in the interpretation of the results of this study. In the future, we plan to supplement the data with short-term measurements, after appropriate seasonal correction. This should increase the number of observations at the upper end of the exposure range, thus enabling more stable risk estimates. CONCLUSION AND IMPLICATIONS When the exposure variable has a right-skewed distribution, as is the case with indoor radon gas concentrations, well-chosen categorical and trimmed estimators are a useful tool for sensitivity analyses that can easily be applied in routine practice. The West German radon study has turned out to be influenced by two outliers among controls. Risk estimates from other studies on indoor radon and lung cancer might be similarly sensitive. ACKNOWLEDGEMENTS We thank all colleagues at the GSF and outside who have helped to collect and prepare the data for the German radon studies, especially Dr. Michael Gerken. Financial support by BMU (Federal Ministry for the Environment, Nature Conservation and Nuclear Safety), grant number StSch REFERENCES Breslow NE, and Day NE Statistical methods in cancer research, Vol. 1: The analysis of case-control studies. Lyon: International Agency for Research on Cancer. Connor RJ Grouping for testing trends in categorical data. J Am Stat Assoc. Vol. 67, pp Cox DR A note on grouping. J Am Stat Assoc. Vol. 19, pp Field RW, Stack DJ, Smith BJ, et al Residential radon gas exposure and lung cancer: The Iowa Radon Lung Cancer Study. Am J Epidemiol. Vol. 151 (11), pp Greenland S. 1995a. Avoiding power loss associated with categorization and ordinal scores in dose-response and trend analysis. (Commentary). Epidemiology. Vol. 6 (4), pp Greenland S. 1995b. Problems in the average-risk interpretation of categorical dose-response analyses. Epidemiology. Vol. 6 (5), pp Greenland S. 1995c. Previous research on power loss associated with categorization in doseresponse and trend analysis. (Letter). Epidemiology. Vol. 6 (6), pp Kreienbrock L, Kreuzer M, Gerken M, et al Case-control study on lung cancer and residential radon in West Germany. Am J Epidemiol. Vol. 153 (1), pp

6 Lagakos SW Effects of mismodelling and mismeasuring explanatory variables on tests of their association with a response variable. Statistics in Medicine. Vol. 7 (1-2), pp Schaffrath Rosario A, Wellmann J, Heid IM, et al Radon epidemiology: Continuous and categorical trend estimators when the exposure distribution is skewed and outliers may be present. J Toxicol Environmental Health. Submitted (Special issue on residential radon and lung cancer). Wichmann HE, Kreienbrock L, Kreuzer M, et al Lungenkrebsrisiko durch Radon in der Bundesrepublik Deutschland (West) [Lung cancer risk due to radon in the Federal Republic of Germany (West)]. Fortschritte in der Umweltmedizin [Advances in environmental medicine]. Landsberg am Lech, Germany: ecomed Verlagsgesellschaft. [In German.] Weinberg CR, Moledor ES, Umbach DM, et al Imputation for exposure histories with gaps, under an excess relative risk model. Epidemiology. Vol. 7 (5), pp Zhao LP, and Kolonel LN Efficiency loss from categorizing quantitative exposures into qualitative exposures in case-control studies. Am J Epidemiol. Vol. 136 (4), pp