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1 Department of Biostatistics and Computational Biology, University of Rochester, Rochester, New York; 2 Occupational Health Program, Department of Environmental Health, Harvard School of Public Health; and 3 Massachusetts General Hospital, Harvard Medical School, Boston, Massachusetts
Requests for reprints: Sally W. Thurston, Department of Biostatistics and Computational Biology, University of Rochester, 601 Elmwood Avenue, Box 630, Rochester, NY 14642. Phone: 585-275-2406; Fax: 585-273-1031. E-mail: thurston{at}bst.rochester.edu
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Abstract |
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Introduction |
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Models that use categories assume that, conditional on other model covariates, the risk of lung cancer within a category is constant. Categorizing a continuous variable does not make full use of all the available data (1) and the choice of cutpoints between categories may influence the estimated smoking-lung cancer risk relationship (2). Furthermore, if the underlying variable used to define the categories is measured with error, then the categorization may create nondifferential measurement error because a value close to the cutpoint is more likely to be misclassified than a value in the mid-range of the category (3).
Continuous smoking metrics used in the literature include pack-years, the square-root of pack-years (4-7), or including smoking duration and intensity as separate variables (8-10). Previous reports have found nonlinear relationships between pack-years and lung cancer (11, 12). In our own lung cancer case-control sample, an approximately linear relationship between square-root pack-years and lung cancer risk was found, but indicator variables to distinguish between current, former, and never smokers were necessary for improved model performance (4). The multistage model of carcinogenesis (13-15) has motivated several authors to separate smoking duration and intensity in modeling lung cancer risk. However, when never smokers are included in the model, relative risks associated with duration for a fixed intensity and vice versa are difficult to interpret (16), because duration and intensity are always zero for never smokers. Other continuous variables that may be important include age of smoking initiation and years since smoking cessation; however, these variables, together with smoking duration, are highly collinear with age, another variable commonly included in cancer risk models.
In addition to these issues, some studies appropriately limit their populations to current smokers (8), ever smokers (17), or use separate models for current and former smokers (10). Such analyses require defining cutpoints in smoking duration, timing, and/or intensity to define these samples. Not only do the choice of cutpoints determine which subjects are excluded, but studies differ in their choice of cutpoints, which can ultimately affect results (12).
In many circumstances, different smoking metrics may provide reasonably similar results such that the choice of metric is not critical. However, when smoking becomes integral to the study hypothesis, as is the case with gene-smoking analyses (4-7, 10, 18-20), it may be important to compare how different metrics perform within the study population. The primary aim of this article is to provide a general approach for evaluating the performance of different metrics through the use of a concrete example of how this approach can be applied in a specific study. Our comparison includes a new metric that we call "logcig-years", which we define to be log(cigarettes smoked per day + 1) x years of smoking. We compare the performance of four different metrics using data from a large lung cancer case-control study, and also explore how the performance of these metrics compare to results from Doll and Peto's model of smoking and cancer (14). Throughout this article, we use "log" to mean the natural logarithm. We define cigpday as cigarettes smoked per day, logcigp as log(cigpday + 1), cigtime as years of smoking, yrsquit as years since smoking cessation, and agestart as age of smoking initiation.
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Materials and Methods |
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As we have previously in this article, we included data from all Caucasians with complete data on age, gender, smoking status, cigpday and cigtime (for ever smokers), and yrsquit (for former smokers).
Motivation of the Logcig-Years Metric
The logcig-years metric was derived in part from a model relating smoking to DNA adducts. The formation of DNA adducts from polycyclic aromatic hydrocarbons (such as benzo[a]pyrene) in tobacco smoke is widely believed to be on the causal pathway from smoking to lung cancer (21-24). Given certain assumptions, it follows from the solution to a set of differential equations relating adducts to smoking, that the logarithm of the number of DNA adducts can be modeled as an additive function of the logarithm of smoking intensity. The logarithmic transformation of adduct numbers, whereas not universal, is fairly standard both in models relating smoking and adducts (25, 26) and in models relating adducts to lung cancer risk (27). Because adduct formation is believed to be on the causal pathway to lung cancer, one could model the probability of cancer initiation as a function of the number of adducts, on some scale. If the logarithmic transformation of smoking intensity is useful for a model of the logarithm of DNA adducts, and if the cumulative log(adduct) burden is directly related to cancer risk, this suggests that cumulative log(smoking intensity) may be a useful smoking metric. Pack-years, which is cumulative smoking intensity on the untransformed scale, is widely used but does not necessarily represent the best way to combine smoking intensity and duration into a single cumulative metric. The logcig-years metric is one alternative to pack-years, and is also a cumulative smoking metric.
Like other simple smoking metrics, the logcig-years metric does not take into account all of the many steps that occur between cancer initiation and tumor detection. These steps may depend on factors such as the age at which an individual started smoking and the age at which the individual stopped smoking (if ever). In this article, we do not attempt to model the process of carcinogenesis or to better understand the true complexity of how smoking leads to cancer development. For this, we refer the interested reader to articles on the multistage model of carcinogenesis (13-15) and related articles (28-32). Instead, our goal in this report is simply to compare the performance of the logcig-year metric with more standard smoking metrics.
Statistical Analyses
We examined the relationship between the logit probability of cancer and each of four continuous smoking metrics separately, using first generalized additive models (GAM; ref. 33), and then logistic regression. The smoking metrics we considered were pack-years, square-root pack-years, logcig-years, and the "two-metrics" model in which smoking duration and intensity were separate metrics in the same model. In the two-metrics model, we used cigtime as the duration variable, and logcigp as the intensity variable. This transformation of smoking intensity was chosen in part because of the nonlinearity between the logit probability of lung cancer and the untransformed smoking intensity observed here (data not shown) and in a previous article using data from this study (10). This nonlinearity has also been noted by Rachet et al. (9), who used GAM to develop models relating smoking to lung cancer risk in a case-control study that used duration of smoking and smoking intensity as separate variables.
GAM is a powerful statistical tool that extends the generalized linear models framework to allow the shape of the relationship between the outcome and each continuous variable to be an arbitrary smooth function with the shape determined by the data. GAM was used to examine the nature of the relationship between cancer risk and each smoking metric separately, in a model that adjusted for age, years since quitting smoking (defined here and in other reports to be zero for never smokers; refs. 4-7), smoking status (as two indicator variables to distinguish between never, former, and current smokers), and gender. Each continuous variable was allowed to have a possibly nonlinear effect on cancer risk. Specifically, the GAM models we fit to ever and never smokers together are of the form
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We also used GAM to examine similar adjusted relationships among smokers only. In the smokers-only model, we can potentially adjust for age of smoking initiation, a variable that is meaningless for never smokers. However, due to the collinearity between this variable, years of smoking, age, and years since smoking cessation, it is not possible to adjust for all these variables in the two-metrics model. Instead, in all smokers-only models, we categorized age of smoking initiation and included an indicator variable for whether or not the smoker started smoking prior to age 18. The value of 18 was chosen to represent the approximate age at which lung development is nearing completion. In the smokers-only models, we did not include the current smoking indicator because the former smoker indicator was sufficient to distinguish between current and former smokers.
All GAM models were fit using the S-Plus software (34, 35). In addition to examining the GAM plots, we tested for nonlinearity between the outcome and each continuous variable using the approximate 
2 test for the nonlinear contribution of the nonparametric terms (36), supplied by S-Plus.
Any smoking metric that did not have a significant departure from a linear relationship with the logit probability of cancer in the adjusted model was then considered further in logistic regression models, also fit in S-Plus. Any covariate other than the smoking metric that had a nonlinear relationship with cancer risk was transformed such that the relationship using the transformed variable was approximately linear. The transformed covariate was then used in the logistic regression models. Two logistic regression models were fit using these smoking metrics. In the first logistic regression model (the "full model"), in addition to adjusting for the covariates as described above, we also included an interaction term between smoking status and the smoking metric, to allow the slope relating the smoking metric and cancer risk to differ for current versus former smokers. For the two-metrics model, this meant that we included a pair of interaction terms, one for smoking intensity and one for duration. The second logistic regression model (the "all covariates" model) included all covariates described, but did not include the interaction term(s). The necessity of considering these interactions is motivated by our earlier work (4, 6).
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Results |
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Results from Assessing Linearity Between the Smoking Metrics and Risk, Using GAM
In our sample, the adjusted relationship between pack-years and the logit probability of cancer was significantly nonlinear (P < 0.001 for the nonlinear contribution) both in a model fit using all individuals (i.e., both ever and never smokers; Fig. 1), and in a model fit using only smokers. This indicates that in our sample, pack-years is not appropriate to use as a continuous variable in logistic regression models. In separate models, square-root pack-years and logcig-years were linearly related to the logit probability of cancer, after adjusting for other model covariates (P > 0.10 for the nonlinear contribution), when all individuals were included (Fig. 1), and when only ever smokers were included. The corresponding plots for the smokers-only models were very similar (data not shown).
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0.07 for the contribution of the nonlinear terms). In all models just described, the adjusted relationship between the logit probability of cancer and years since quitting smoking was approximately linear. However, in our sample the adjusted relationship between the logit probability of cancer and age was significantly nonlinear in all models (P < 0.001). The corresponding GAM plots indicated that the relationship with age was approximately linear up to about age 70, and approximately linear thereafter, but with a change in slope at about age 70 (see Fig. 2). This observed age effect is partly due to the difference in age distribution among cases and controls in this sample.
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Due to the nonlinearity associated with the age effect, in all logistic regression models, we adjusted for age using a piecewise linear model, in which we allowed one slope for age <70, and a different slope for age >70, with the constraint that the slopes join at age 70. In all cases, the slopes before and after age 70 were significantly different from each other (P < 0.001). Gender was not significant in any of the models. We started by considering the "full model", which includes the interaction between smoking status and the smoking metric, for each of the three remaining smoking metrics. In models using all individuals and in the smokers only models, the interactions between smoking status and the smoking metric were significantly different from zero for the square-root pack-years models, and for the two-metrics models (in which the interactions were only significant for logcigp but not for cigtime), but not for the logcig-years models.
Next we considered the "all covariates" models that did not include the interactions mentioned above, but adjusted for all remaining covariates. In models fit using all individuals, and ever smokers only, yrsquit was a significant predictor (P < 0.01) in models with square-root pack-years and in the two-metrics model, but was of borderline significance or not significant in the models with logcig-years (P 0.06 for all individuals, P 0.64 for smokers only). In the model fit using all individuals, the smoking status indicator variables were significant predictors in the model using square-root pack-years and the two-metrics model, but not in the model using logcig-years. For models fit using smokers only, smoking status was not significant for any of the three metrics. The indicator variable for starting smoking before age 18, only included in smoker-only models, was significantly different from zero only in the model using logcig-years as the smoking metric. The logistic regression models are summarized in Table 1.
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Sensitivity of the Logcig-Years Metric
We explored the sensitivity of the logcig-years metric to the scale on which smoking intensity is measured. Specifically, we considered generalized metrics of the form log(
cigpday + 1) x cigtime, for a range of values of . We found that the residual deviance of the unadjusted model is smallest for metrics based on values of between 0.5 and 1.5, but the residual deviance for the adjusted model is smallest for metrics based on values of < 1, suggesting that a metric based on between 0.5 and 1 may be somewhat better than the logcig-years metric which uses = 1. In the adjusted smokers-only model using = 0.5, smoking status and yrsquit remained statistically insignificant, whereas in the model based on all individuals using = 0.5, yrsquit and the current smoking indicator both became borderline significant.
We also investigated the sensitivity of the logcig-year metric to adding the constant of one to cigpday before taking the logarithm. For all individuals and separately for smokers only, we fit three additional logistic regression models (and three analogous GAM models) in which logcig-years was replaced with log(cigpday + k) x cigtime, for k = 2, 3, and 4 in turn. For smokers-only, we also fit a fourth model in which logcig-years was replaced with log(cigpday)x cigtime. Each model adjusted for the same covariates as the logcig-years model. In all cases, the GAM plot was visually indistinguishable from the GAM plot using logcig-years, neither smoking status nor yrsquit were statistically significant, and the coefficient for the alternative metric continued to be 
0.02.
Addressing Possible Confounding by Age
In our sample, the median case age was almost 7 years larger than the median control age. Thus, the observed age effect in this study, as in any case-control study which is not perfectly age-matched, reflects a combination of the direct age effect and the difference in age distribution between cases and controls.
In order to remove the possible confounding with age, we fit the logcig-years model to current, former and never smokers together, separately by four age strata. Following the example of Flanders et al. (8), we fit separate GAM and logistic regression models within age deciles of 40 to 49, 50 to 59, 60 to 69, and 70 to 79. Each model included covariates of logcig-years, current and former smoking indicator variables, yrsquit, age, and gender. The reason for including age was to allow for a possible age effect within age decile. Age was only significant in the 70 to 79 year group. Logcig-years was statistically significant in all four age strata models (P < 0.005), and the coefficient for logcig-years ranged from 0.014 to 0.023 within age strata. This coefficient was smallest (0.014-0.015) for the 40 to 49 and 60 to 69 age groups, and largest (0.022-0.023) for the 50 to 59 and 70 to 70 age groups. Our results imply reasonable robustness of our metric in different age group strata.
Addressing Possible Confounding by Age of Smoking Initiation
Among ever smokers, we also explored models in which age of smoking initiation was included as a continuous variable (results not shown). In the two-metrics model, this meant that we were not able to adjust for age, and in this model, larger values of age of smoking initiation and larger values of years since quitting smoking were both associated with increased cancer risk. Under the multistage model of carcinogenesis, the effect of a carcinogen will depend on age of smoking initiation, time since initial exposure, or both, depending on the stage(s) in which the carcinogen has an effect (38). The results just described are consistent with cigarette smoke carcinogens acting on both early and late stage transitions (38), as other studies have suggested. However, the implication that years since smoking cessation is positively related to lung cancer risk is neither biologically reasonable nor consistent with other studies. In this data, age of smoking initiation ranged from 6 to 61 years, with 78 smokers starting at age 30 or greater, including 8 who started smoking after age 50. In the two-metrics model, age of smoking initiation as a continuous variable, years of smoking, and years since quitting smoking together comprise the overall age effect, possibly explaining the apparent positive association between cancer risk and years since quitting smoking in this model.
Our decision to dichotomize age of smoking initiation allows us to also adjust for age in models using each smoking metric. It has been suggested that the lung is most sensitive to the effects of smoking during lung development (26, 39). Dichotomizing age of smoking initiation at age 18 is meant to capture whether smoking started before or after lung development was essentially complete. However, this dichotomization does not capture smoking initiation effects which may be important at a later age, such as cancer promotion in intermediate-stage cancer cells. Individuals who started smoking earlier were on average heavier smokers who smoked longer than those who started smoking later. There was no evidence of an interaction between this indicator variable and logcig-years.
Addressing the Definition of Years Since Quitting Smoking for Never Smokers
We defined yrsquit to be zero for never smokers, yet it could be argued that yrsquit, like agestart, is not meaningful for never smokers. For smokers, the variable age is the sum of agestart, cigtime, and yrsquit. For never smokers, this suggests defining yrsquit to be zero and agestart to be age. In a model that includes never smokers and adjusts for yrsquit (defined to be zero for never smokers), whether or not never smokers are influential in determining the coefficient for yrsquit can be visually assessed by examining the GAM plot for yrsquit. In all models discussed here, the adjusted relationship between yrsquit and the logit probability of cancer for never smokers was consistent with the relationship for ever smokers.
Exploring Smoking-Lung Cancer Risk Implications of Each Metric
Here we compare what each smoking metric implies about lung cancer risk predictions over a range of different values of smoking intensity and duration. For pack-years, the increase in predicted cancer risk is the same for a doubling in smoking intensity (for fixed duration) as it is for a doubling in number of years smoked (for fixed intensity). The same is true for square-root pack-years. In contrast, for logcig-years, the predicted increase in cancer risk for a doubling of smoking duration (for fixed intensity) is much larger than it is for a doubling in smoking intensity (for a fixed duration).
In Fig. 3, we give contours of these three smoking metrics, as well as a two-dimensional smoothed estimate of cancer risk as a function of smoking intensity and smoking duration estimated from the lung cancer data from the model
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In Fig. 4, we show the estimated lung cancer relative risk on the logarithmic scale, for ever smokers relative to never smokers using the logcig-years model, including only the significant or borderline significant covariates (logcig-years, age as a piecewise linear term, and years since quitting smoking). The estimated log relative risk was 0.019 x logcig-years – 0.009 x yrsquit. Because smoking status was not needed in this model, the estimated relative risk does not change abruptly when smoking cessation occurs. This feature is not shared by any of the other smoking metrics when fit to data using all individuals.
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Adding one to cigtime before taking the logarithm resulted in a very bimodal distribution for this variable. Among the n = 137 current male smokers meeting Doll and Peto's criteria, the smallest cigtime was 9, so the variable log(cigtime + 1) had n = 177 values of 0, and n = 137 values between 2.30 and 4.11. The GAM plot indicated that the logit probability of cancer was strongly and positively related to log(cigtime + 1) among current smokers, but that the adjusted relationship for never smokers did not fit this pattern. As a result, the inclusion of never smokers caused the overall relationship to be extremely nonlinear. In the adjusted logistic regression model, the coefficient (SE) for log(cigpday + 6) was 2.70 (0.70), consistent with the Doll and Peto model. The coefficient (SE) for log(age) was 1.35 (0.85), whereas the coefficient of –0.35 for log(cigtime + 1) is not meaningful due to the nonlinearity noted above. The age distributions among cases and controls in our overall case-control study and in the subset used for this analysis is not necessarily representative of the corresponding age distributions in the population. The age effect seen here partly reflects this difference, which could explain why our estimated age effect differs from Doll and Peto's estimate. Other reasons why our results did not more closely match Doll and Peto's could include our assumption of the baseline risk among never smokers, and the fact that cigarette smoke exposure characteristics have changed over the past four decades, which may also affect the smoking metric. In fact, Flanders et al. (8) did a more recent cohort analysis, and also found major differences with Doll and Peto. In this data subset, logcig-years continued to be linearly related to the logit probability cancer with a regression coefficient (SE) of 0.02(0.002).
The Doll and Peto sample did not include former smokers in their base population, and this subset comprised over half of the ever smokers in our sample. Differences in the epidemiology of lung cancer in former smokers and current smokers (for example, proportion of adenocarcinomas, peripheral versus central lung cancers, etc.) suggest possible differences in lung carcinogenesis, and this too may affect the smoking metric. Thus, choosing an appropriate metric may be affected by differences in study design and population.
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Discussion |
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Our general approach to evaluating continuous variable smoking metrics can be summed up as follows: (a) evaluate each metric for linearity with disease outcome using the appropriate link function (e.g., the logit probability of cancer, for logistic regression), in one's study population; (b) evaluate the effect on risk estimates by inclusion of other potentially clinically important variables along with your metric(s) of interest. Examples include smoking status (never, current, or former smokers), year since quitting smoking, age of smoking initiation, and/or age; (c) compare the implications of the different smoking metrics for lung cancer risk predictions; and (d) explore possible reasons why the metric that performs best in your study population may be different from other metrics chosen in other studies or for other hypotheses. The best performing continuous smoking metrics seem to have the following three properties: (a) a linear relationship with disease risk using the appropriate link function because this is a model assumption; (b) the ability to include or exclude never smokers from the model without substantial changes in choice of model covariates or estimated disease risk in smokers; and (c) an insensitivity of disease risk estimates to changes in smoking status for fixed values of other model covariates. Models that include smoking status imply a jump in estimated risk at the age of smoking initiation and/or smoking cessation, an assumption that is appropriate for certain types of analyses, but not for others, and one that is somewhat implausible from the biological perspective.
A limitation of this study concerns the derivation of the logcig-years metric, which was based on several simplifying assumptions. Our derivation only considered polycyclic aromatic hydrocarbon formation from smoking, but other substances such as well-done red meat are also sources of polycyclic aromatic hydrocarbons. We did not account for other possible sources of polycyclic aromatic hydrocarbons in this article [but see Cortessis and Thomas (40) who model smoking and well-done red meat consumption jointly]. Although we have stated various limitations of the logcig-years metric, it should be noted that all metrics suffer from an inability to explain or account for many biological premises associated with tobacco carcinogenesis. Although initially motivated by a DNA adducts model, our metric was chosen mainly because it did better than other metrics in this sample data set. It should be understood that in other contexts, other metrics, including those not mentioned in this article, might be most appropriate for analysis. In all circumstances, the derived metric should have at least face validity.
In summary, we recommend that a process such as we outlined here be followed before assuming that a particular smoking metric suitably adjusts for or evaluates smoking in a statistical model. Different studies may use different metrics because the base population and study designs may differ between studies. We do not recommend that this comprehensive approach be used for all studies that incorporate smoking variables, but that the process be adapted to evaluate smoking metrics in studies where smoking is an integral part of the biology of the disease or the study hypothesis.
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Acknowledgments |
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Footnotes |
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The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked advertisement in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Note: This manuscript was presented in part at the 2004 AACR Annual Meeting in Orlando, Florida. S. Thurston, G. Liu, D.P. Miller, D.C. Christiani. "Modeling cancer risk in case-control studies using a new dose metric of smoking based on a DNA-adduct model of carcinogenesis."
Received 5/31/04; revised 5/19/05; accepted 6/13/05.
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