© 1999 by CRC Press LLC CHAPTER 4 Dose–Response Assessment — Quantitative Methods for the Investigation of Dose–Response Relationships Suresh H. Moolgavkar CONTENTS I. Introduction II. Measures of Disease Frequency and Measures of Effect III. Confounding IV. Empirical Statistical Methods A. Relative Risk Regression Models B. Poisson Regression V. Biologically Based Models VI. Multistage Models A. The Armitage–Doll Multistage Model B. The Two-Mutation Clonal Expansion Model 1. Likelihood Construction and Maximization 2. Examples VII. Other Quantitative Methods A. Physiologically Based Pharmacokinetic (PBPK) Models VIII. Prospects for the Future Bibliography I. INTRODUCTION The estimation of exposure–response or dose–response relationships is a prereq- uisite for a rational approach to the setting of standards for human exposures to © 1999 by CRC Press LLC potentially toxic substances. In many instances when human epidemiologic data are not available, standards are based on assessment of toxic responses in experimental data followed by extrapolation of risks to humans. Additionally, experiments in animals are often carried out at high exposure levels so that the experiments have the requisite statistical power. The resultant issues of interspecies and low-dose extrapolation are among the most contentious scientific issues of the day. In the past few decades, a vast biostatistical literature has appeared on expo- sure–response and dose–response analyses. Summarizing this literature in a single chapter is a formidable task. Although many of the same methods can be used for experimental and epidemiologic studies, this chapter focuses on statistical methods that have been developed for analyses of epidemiologic studies, and on biologically based mathematical models for analyses of data in which the end point of interest is cancer. Physiologically based pharmacokinetic (PBPK) models, developed to investigate the relationship of exposure to dose by consideration of the uptake, distribution, and disposal of agents of interest, will be discussed only briefly. This is not because these models are considered to be unimportant, but because this subject is outside the author’s area of expertise. Interspecies differences in response to exposure to environmental agents can often be explained, at least partially, in terms of differences in uptake and distribution of the agent. Thus, PBPK models have advanced broadly our understanding of differential species toxicology and can be considered important tools in risk assessment. For risk assessment, epidemiologic studies offer two obvious advantages over experimental studies. Firstly, since the studies are done in the species of ultimate interest, the human, the difficult problem of interspecies extrapolation is finessed. Secondly, most epidemiologic studies are done at levels of exposure that are much closer to typical exposures in free-living human populations than is possible with experimental studies. It is true that often epidemiologic studies are conducted in industrial cohorts, which are typically exposed to higher levels of the agent of interest than the general population. Nonetheless, the levels of exposure, even in industrial cohorts, are much closer to those in the general population than the exposures used in experimental studies. Some of what epidemiologic studies gain in the way of relevance over experimental studies is given up in precision, however. It is generally true that both exposures and disease outcomes are measured with less precision in epidemiologic studies than in laboratory studies, leading, possibly, to bias in the estimate of risk and the shape of the dose–response curve. Exposure measurement error is now widely regarded as being an important issue in analyses of epidemiologic data. Another potential problem for risk assessment arises from the fact that human populations, especially industrial cohorts, are rarely exposed to single agents. When exposure to multiple agents is involved, the effect of the single agent of interest is often difficult to investigate. This fact is of particular relevance for air pollution because it is generally a complex mixture of toxic agents. The role of any single component of the mixture can be difficult to study. Epidemiologic studies that can be used to investigate dose–response relationships are classified into three broad categories. The cohort study is, at least conceptually, close to the traditional experimental study in that groups of exposed and unexposed © 1999 by CRC Press LLC individuals are followed in time and the occurrence of disease in the two groups compared. In the case-control study, relative risks are estimated from cases of the disease under investigation and suitably chosen controls. In these two types of study, information on exposures and disease is available on an individual basis for all subjects enrolled in the study. In a third type of study, the ecological study, infor- mation is available only on a group basis. Ecological studies have generally been looked upon with disfavor by epidemiologists for reasons that have been extensively discussed elsewhere (Greenland and Morgenstern 1989; Greenland and Robins 1994). Nonetheless, they can provide useful information and, particularly, in air pollution epidemiology, they have played a central role in recent times. Another type of study, in which disease outcome and some confounders are known on an individual basis and others together with exposure to air pollutants are known only on a group basis, has recently played an important role in air pollution epidemiology. There is currently no generally accepted term for such studies, which share attributes of the cohort study and the ecological study. Such studies are called here hybrid studies. Because epidemiologic studies are observational (i.e., groups of subjects cannot randomly be assigned to one exposure group or another), careful attention must be paid to controlling factors that may bias estimates of risk. Thus, controlling for what epidemiologists call “confounding” is of paramount importance both in the design and analyses of epidemiologic studies. Within the last two decades sophisticated statistical tools have been developed for the analyses of epidemiologic data. Many of these methods fall under the rubric of the so-called relative risk regression models. Additionally, recent research in air pollution epidemiology has exploited regression methods for analyses of time-series of counts. Both parametric and semiparametric Poisson regression models have been developed for analyses of these data. Special methods are required when multiple observations are made on the same individual, as is done in panel studies, or in the same geographic location, as is done with Poisson regression analyses of time-series of counts. Account must then be taken of serial correlations in the observations. Various statistical methods are used to address this issue. Finally, when the health effect of interest is cancer, stochastic models based on biological considerations can be used for data analyses. These models provide a useful complement to the more empirical statistical approaches to data analyses. Each of these approaches will be discussed briefly in this chapter. II. MEASURES OF DISEASE FREQUENCY AND MEASURES OF EFFECT When discussing dose– or exposure–response relationships it is important to define clearly what response one is talking about. Often the term dose– or expo- sure–response is used with no indication of what response means. In order to define response precisely it is important to have a clear idea of the various commonly used measures of disease frequency and of effect. Perhaps the most fundamental measure of disease frequency is the incidence rate, also called the hazard rate in the statistical © 1999 by CRC Press LLC literature. The incidence or hazard rate measures the rate (per person per unit time) at which new cases of a disease appear in the population under study. Because the incidence rates of many chronic diseases, including cancer, vary strongly with age, a commonly used measure of frequency is the age-specific incidence rate, usually reported in five-year age categories. For example, the age-specific incidence rate per year in the five-year age group 35–39 may be estimated as the ratio of the number of new cases of cancer occurring in that age group in a single year to the number of individuals in that age group who are cancer free at the beginning of the year. Strictly speaking, the denominator should be not the total number of individuals who are cancer free at the beginning of the year but the person-years at risk during the year. This is because some individuals contribute less than a full year of expe- rience to the denominator, either because they enter the relevant population after the year has begun (for example, an individual may reach age 35 sometime during the year) or because they may leave the population before the year is over (for example, an individual may reach age 40, die, or migrate during the year). Mathematically, the concept of incidence rate is an instantaneous concept, and is most precisely defined in terms of the differential calculus. A precise definition of the concept is given in the next section, and the reader is referred to texts on survival analysis (e.g., Kalbfleisch and Prentice 1980; Cox and Oakes 1984) for further details. Another commonly used measure of disease frequency is the probability that an individual will develop disease in a specified period of time. For risk assessment, interest is most often focused on the lifetime probability, often called lifetime risk of developing disease. Here, lifetime is arbitrarily defined in the U.S. usually as 70 years. The incidence (or hazard) rate and probability of developing disease are related by a simple formula. This relationship is expressed by the following equation: where P(t) is the probability of developing the disease of interest by age t, and I(s) is the incidence or hazard rate at age s. Note that although the probability of disease, P(t), is called cumulative incidence in some epidemiology textbooks (Rothman 1986), the integral is actually the cumulative incidence. When the incidence rate is small, as is true for most chronic diseases, the probability of disease by time t, P(t), is approximately equal to the cumulative incidence, Pt Is ds t () =− − ()         ∫ 1 0 exp Isds t () ∫ 0 Pt Is ds t () ≈ () ∫ 0 © 1999 by CRC Press LLC The impact of an environmental agent on the risk of disease can be measured on either the absolute or the relative scale. The last two decades have seen an explosion of statistical literature on relative measures of risk, which can be estimated in both case-control and cohort studies. Let I e be the incidence rate in the exposed population and I u be the incidence rate in the unexposed population. Then the relative incidence (relative risk) is defined by RR = I e /I u A closely related measure is excess relative risk, which is defined as ERR = (I e – I u )/I u = RR – 1 Yet another measure of risk is the attributable or etiologic fraction, which is defined as AF = (I e – I u )/I e = (RR – 1)/RR The AF is the fraction of incident cases in the exposed population that would not have occurred in the absence of exposure, and “can be interpreted as the pro- portion of exposed cases for whom the disease is attributable to the exposure” (Rothman 1986). In most regression analyses of epidemiologic data, RR is modeled either as a “multiplicative” or an “additive” function of the covariates of interest. Since RR is readily estimated from both case-control and cohort studies, the various measures of effect discussed above which are functions of RR alone can be estimated. On the absolute scale, the impact of an agent can be measured simply by the difference of incidence rates (or probabilities) among exposed and nonexposed subjects. Absolute measures of risk cannot be estimated from case-control studies without ancillary information (Rothman and Greenland 1998). The impact of an environmental agent on the risk of disease on a population will depend not only on the strength of its effect in the exposed subpopulation, but also on how large this subpopulation is. Even if the agent is a very potent carcinogen, its impact on the cancer burden of the entire population will be small if only a small fraction of the population is exposed. On the other hand, if exposure to a weak carcinogen is widespread, the population impact could be substantial. A measure of risk that attempts to quantify the population burden of disease due to a specific exposure is the population attributable fraction, PAF, which is defined as the fraction of all cases in the population that can be attributed to the exposure, and is given by the expression PAF = (I T – I u )/I T where I T is the incidence in the total population. In addition to the RR, estimation of the PAF requires information on the fraction of the population exposed to the agent of interest (see Rothman 1986). The PAF can be estimated directly from © 1999 by CRC Press LLC case-control data only if the controls are a random sample from the population (Rothman and Greenland 1998). When the RR associated with exposure to an agent is high and the exposure is widespread, a major fraction of disease in the population can be attributed to the agent. For example, it has been estimated that approximately 84% of all lung cancers and 43% of all bladder cancers in Australian men in 1992 could be attributed to cigarette smoking (English et al. 1995). The calculation of the PAF can be extended to situations where there are multiple levels of exposure (by considering each level in turn and adding up the PAFs) or where the exposure is a continuous variable rather than a categorical one (by creating discrete categories such as quartiles or quintiles of exposure, or by using regression models). Joint effects of several exposures may be considered similarly. In the case of two or more exposures, the separate PAFs may be calculated for each exposure while ignoring the other exposures, or a combined PAF may be calculated by considering all possible combinations of exposures, calculating the PAF for each and adding up. When two or more exposures are involved, the sum of the separate PAFs will frequently exceed the combined PAF calculated in this way and may actually exceed 100%. The reason for this is clear: cases that occur in the joint exposure categories are counted multiple times when PAFs for single exposures are computed, once for each exposure in the joint exposure category. Attribution of causation in the case of joint exposures is best done by considering all possible combinations of exposures. For example, with two exposures, attribution of causation may be summarized by subdividing the cases into those that can be considered as being caused by the combination of the two agents, each agent exclusively, or neither agent (Enterline 1983). For a more advanced treatment of PAFs, see Bruzzi et al. (1985), Wahrendorf (1987), Benichou (1991), and Greenland and Drescher (1993). III. CONFOUNDING A detailed discussion of confounding, a concept of central importance in epide- miology, is outside the scope of this chapter. Confounding arises in epidemiologic studies as a consequence of the fact that these are observational (not randomized). Suppose one is interested in alcohol as a possible cause of oral cancer. Suppose that an epidemiologic study shows an association between alcohol consumption and oral cancer. That is, suppose the incidence of oral cancer in the subpopulation of individ- uals that imbibes alcohol is higher than the incidence of oral cancer in the subpop- ulation of teetotalers. The crucial question then is the following: Could the association between alcohol consumption and oral cancer be “spurious” in the sense that it is due to another agent that is itself a cause of oral cancer, and more likely to be found in the subpopulation of alcohol imbibers than in the subpopulation of teetotalers? One example of such an agent is tobacco smoke. Individuals who imbibe alcohol are more likely than teetotalers to be smokers. Moreover, smoking is strong risk factor for oral cancer. Thus the observed association between alcohol consumption and oral cancer may actually be due to the association between smoking and alcohol con- sumption. In a study of oral cancer and alcohol, tobacco smoke is a confounder. © 1999 by CRC Press LLC As shown by this example, confounding is the distortion of the effect of the agent of interest by an extraneous factor. To be a confounder, a factor must satisfy two conditions. First, the putative confounder must be a risk factor for the disease in the absence of the agent of interest. Second, the putative confounder must be associated with the exposure of interest in the population in which the study is conducted. Sometimes a third condition (Rothman and Greenland 1998) is added—the putative confounder must not be an intermediate step in the pathway between exposure and disease. While these three criteria define a confounder for most epidemiologists, other definitions which are close but not identical to the definition given here have been given by biostatisticians. These are usually couched in terms of collapsibility of contingency tables. For a more detailed discussion, the reader is directed to Greenland and Robins (1986) and Rothman and Greenland (1998). Confounding in epidemiologic studies can be addressed in one of two ways—it can be prevented by appropriate study design or controlled by appropriate analyses. The specific methods used depend upon the type of epidemiologic study. The reader is referred to recent texts (Rothman and Greenland 1998) for details. The main statistical tools for exposure– and dose–response analyses of epide- miologic data will now be discussed briefly. Many of these methods can be used for analyses of experimental data as well. IV. EMPIRICAL STATISTICAL METHODS Because most epidemiologic studies are observational, issues of sampling and data analysis are particularly important to assure appropriate interpretation of results in the presence of possible confounding. Some of the main statistical tools developed over the last few decades to address these issues are discussed below. A. Relative Risk Regression Models The development here will follow that in the paper by Prentice et al. (1986). Although this paper was written over a decade ago, it lays out the basic framework for these models. The concept of hazard function was introduced above as being the appropriate statistical concept that captures the epidemiologic idea of an incidence rate. A more precise definition of this concept follows. Consider a large, conceptually infinite, population that is being followed forward in time, and about which one wishes to draw inferences regarding the occurrence of some health related event, generically referred to as a “failure.” Typically one is interested in relating the failure to preceding levels of one or more risk factors, such as genetic and lifestyle factors and exposure to external agents, collectively referred to as covariates. Let z(t) denote the vector of covariates for an individual at time t. Time may be the age of the individual, or, in some settings, it may be more natural to consider other specifica- tions, such as time from a certain calendar date, or duration of employment in a specific occupation. Let T denote the time of failure for a subject, and suppose that © 1999 by CRC Press LLC Z(t) represents the covariate history up to time t. Then the population frequency of failure, which may be thought of as the probability of failure, in a time interval t to t + ∆ with covariate history Z(t), will be denoted by P[t + ∆|Z(t)]. The hazard or incidence function (which, if failure refers to death, is often called the force of mortality) is then defined by In order to simplify notation, the dependence of h, P, etc. on the covariate history Z(t) will be suppressed unless this is not clear from the context. Thus, for example, h[t; Z(t)] will be written as h(t). An intuitive interpretation of the hazard is that it is the rate of failure at time t among those who have not failed up to that time. Now suppose that one is interested in the incidence of failures among individuals with a specific covariate history, Z(t). For example, one may be interested in the incidence among individuals who are exposed to certain environmental agents thought to be associated with the disease under investigation. Let Z 0 (t) represent some standard covariate history; for example, Z 0 (t) could be thought of as the covariate history among those not exposed to the agents of interest. One can then write h[t; Z(t)] = h 0 (t) RR[t; Z(t)] where h 0 (t) = h[t; Z 0 (t)] and RR[t; Z(t)] denotes the relative risk of failure at time t associated with covariate history Z(t). Relative risk regression models attempt to describe risks in populations by focusing on the relative risk function. Various functional forms for RR have been used, the most commonly used being “multiplicative” and “additive” functions of the covariates. The multiplicative model is given by RR[t; Z(t)] = exp(β 1 z 1 + β 2 z 2 + … + β n z n ) and the additive model by RR[t; Z(t)] = 1 + β 1 z 1 + β 2 z 2 + … + β n z n where z 1 through z n are the covariates of interest and the β’s are parameters to be estimated from the data. Note that the additive model posits that the relative risk is a linear function of the exposures of interest and that the effect of joint exposures is additive. The multiplicative model posits that the logarithm of relative risk is a linear function of the exposures and that the effect of joint exposures is multiplicative. Quite often the relative risk cannot be adequately described by either a multiplicative or an additive model. For example, the relative risk associated with joint exposure to radon and cigarette smoke is greater than additive but less than multiplicative (BEIR IV 1988). Various mixture models have been proposed (Thomas 1981; Breslow and Storer 1985; Guerrero and Johnson 1982) to address such situations. htZt Pt Zt T t P t Zt Pt Zt; lim ; () [] =+ () ≥ [] = ′ () [] − () [] () → − ∆ ∆∆ 0 1 1 © 1999 by CRC Press LLC The use of these models presents special statistical problems (Moolgavkar and Venzon 1987; Venzon and Moolgavkar 1988). There is a vast biostatistical literature on the application of relative risk regres- sion models to the analyses of various study designs encountered in epidemiology. It is outside the scope of this chapter to review this literature. The interested reader is referred to the appropriate publications (Breslow and Day 1980; Breslow and Day 1987). In the field of air pollution epidemiology, relative risk regression models were used for analyses of two important studies of the long-term effects of air pollution on health. These are the Harvard Six Cities Study (Dockery et al. 1993) and the ACS II study (Pope et al. 1995). In these studies, cohorts of individuals were assembled from cities with different pollution profiles and information collected on certain life-style factors, such as cigarette smoking. These individuals were then followed and their mortality experience recorded. The authors of these studies refer to them as cohort studies. There is, however, an important element of the ecologic design to these studies. The exposure of interest, namely air pollution, is measured not on the individual level, but on the level of the city. That is, because information on concentrations of pollutants is available only from central monitoring stations, exposure to air pollution is assumed to be identical for all study subjects in a city. Because these studies combine elements of the cohort design with ecologic design, the term hybrid studies has been coined by this author for designs of this type. This study design can pose formidable problems in the interpretation of the results of analyses (Moolgavkar and Luebeck 1996). B. Poisson Regression Quite often information is available, not on individual members of a study cohort, but on subgroups that are reasonably homogeneous with respect to important char- acteristics, including exposure, that determine disease incidence. As a concrete example, consider the well-known British doctors’ study of tobacco smoking and lung cancer. For the cohort of individuals in this study, information on the number of lung cancer deaths is cross-tabulated by daily level of smoking (reported in fairly narrow ranges) and five-year age categories. Another well-known example is pro- vided by the incidence and mortality data among the cohort of atomic bomb survi- vors, for which the numbers of cancer cases are reported in cross-tabulated form by (ranges of) age at exposure, total dose received (in narrow ranges) and by five-year attained age categories. When data are presented in this way the method of Poisson regression is often used for analyses. Only a very brief outline of the method is given here. For more details the reader is referred to the standard text by McCullagh and Nelder (1989). For Poisson regression, the number of events of the outcome of interest (death or number of cases of disease) in each cell in the cross-tabulated data is assumed to be distributed as a Poisson random variable with expectation (mean) that is a function of the covariates of interest. The numbers of events in distinct cells of the cross-tabulated data are assumed to be independent. Suppose that the data are © 1999 by CRC Press LLC presented in I distinct cross-tabulated cells, and let E i be the expectation of the number of events in cell I. Suppose that the observed number of events in cell I is O i . Then under the assumption that the number of events is Poisson distributed, the likelihood of the data is L = Π i {E i Oi exp(–E i )}/O i ! where the product is taken over all the cells in the cross-tabulated data. The expec- tations E i are made functions of the covariates of interest. Generally, log E i is modeled as a linear function of the covariates. More elaborate functions have been used, however, for example in the analyses of the atomic bomb survivors data (BEIR V 1990) and the analyses of lung cancer in cohorts of underground miners (BEIR IV 1988; DHHS 1994). The expectation has been modeled as well by the hazard function of biologically based carcinogenesis models (Moolgavkar et al. 1989). Whatever the model form for the expectation, the parameters are estimated by maximizing the likelihood function. Poisson regression models have played a prominent role in recent analyses of associations between indices of air quality in various urban areas and health out- comes such as mortality (Schwartz and Dockery 1992; Schwartz 1993) and hospital admissions (Burnett et al. 1994; Moolgavkar et al. 1997) for specific causes (respi- ratory disease, heart disease). These studies purport to investigate the acute effects of air pollution in contrast to the hybrid studies referred to above, which investigated the long-term effects of air pollution. In these analyses, daily counts of events (deaths or hospital admissions) in a defined geographical area are regressed against levels of air pollution as measured at monitoring stations in that area. Explicitly, the number of events on any given day is assumed to be a Poisson random variable, the expec- tation of which depends upon indices of air quality and weather on the same or previous days. In this type of study, inferences regarding the association of air pollution with the health events of interest depend upon relating fluctuations in daily counts of events to levels of air pollution on the same or previous days. As indicated above, in the simplest form of Poisson regression the logarithm of the expectation is a linear function of the covariates. This restriction on the shape of the expo- sure–response function may not be appropriate, and recently more flexible methods that make no assumptions regarding the shape of this relationship have been intro- duced for analyses of these data (Health Effects Institute 1995). An important difference between Poisson regression analyses of air pollution data and the other examples given above (e.g., analysis of the atomic bomb survivor data) is that in the air pollution data information on exposure is available only from central monitors of air quality. It is not possible to form strata of individuals with like exposures within a narrow range. It is not possible, therefore, to investigate the number of deaths or hospital admissions among individuals similarly exposed. This fact makes this type of study of air pollution an ecological study in that exposures and outcomes are known only on the group level, and it is not clear that the number of events is related to the level of exposure. 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Nonetheless, they can provide useful information and, particularly, in air pollution epidemiology, they. deaths is cross-tabulated by daily level of smoking (reported in fairly narrow ranges) and five-year age categories. Another well-known example is pro- vided by the incidence and mortality data. between indices of air quality in various urban areas and health out- comes such as mortality (Schwartz and Dockery 1992; Schwartz 1993) and hospital admissions (Burnett et al. 19 94; Moolgavkar et