A Structural Approach to Selection Bias
Hernán, Miguel A.*; Hernández-Díaz, Sonia†; Robins, James M.*
From the *Department of Epidemiology, Harvard School of Public Health, Boston, Massachusetts; and the †Slone Epidemiology Center, Boston University School of Public Health, Brookline, Massachusetts.
Submitted 21 March 2003; final version accepted 24 May 2004.
Miguel Hernán was supported by NIH grant K08-AI-49392 and James Robins by NIH grant R01-AI-32475.
Correspondence: Miguel Hernán, Department of Epidemiology, Harvard School of Public Health, 677 Huntington Avenue, Boston, MA 02115. E-mail: email@example.com
The term “selection bias” encompasses various biases in epidemiology. We describe examples of selection bias in case-control studies (eg, inappropriate selection of controls) and cohort studies (eg, informative censoring). We argue that the causal structure underlying the bias in each example is essentially the same: conditioning on a common effect of 2 variables, one of which is either exposure or a cause of exposure and the other is either the outcome or a cause of the outcome. This structure is shared by other biases (eg, adjustment for variables affected by prior exposure). A structural classification of bias distinguishes between biases resulting from conditioning on common effects (“selection bias”) and those resulting from the existence of common causes of exposure and outcome (“confounding”). This classification also leads to a unified approach to adjust for selection bias.
Click on the links below to access all the ArticlePlus for this article.
Please note that ArticlePlus files may launch a viewer application outside of your web browser.
Epidemiologists apply the term “selection bias” to many biases, including bias resulting from inappropriate selection of controls in case-control studies, bias resulting from differential loss-to-follow up, incidence–prevalence bias, volunteer bias, healthy-worker bias, and nonresponse bias.
As discussed in numerous textbooks,1–5 the common consequence of selection bias is that the association between exposure and outcome among those selected for analysis differs from the association among those eligible. In this article, we consider whether all these seemingly heterogeneous types of selection bias share a common underlying causal structure that justifies classifying them together. We use causal diagrams to propose a common structure and show how this structure leads to a unified statistical approach to adjust for selection bias. We also show that causal diagrams can be used to differentiate selection bias from what epidemiologists generally consider confounding.
CAUSAL DIAGRAMS AND ASSOCIATION
Directed acyclic graphs (DAGs) are useful for depicting causal structure in epidemiologic settings.6–12 In fact, the structure of bias resulting from selection was first described in the DAG literature by Pearl13 and by Spirtes et al.14 A DAG is composed of variables (nodes), both measured and unmeasured, and arrows (directed edges). A causal DAG is one in which 1) the arrows can be interpreted as direct causal effects (as defined in Appendix A.1), and 2) all common causes of any pair of variables are included on the graph. Causal DAGs are acyclic because a variable cannot cause itself, either directly or through other variables. The causal DAG in Figure 1 represents the dichotomous variables L (being a smoker), E (carrying matches in the pocket), and D (diagnosis of lung cancer). The lack of an arrow between E and D indicates that carrying matches does not have a causal effect (causative or preventive) on lung cancer, ie, the risk of D would be the same if one intervened to change the value of E.
Besides representing causal relations, causal DAGs also encode the causal determinants of statistical associations. In fact, the theory of causal DAGs specifies that an association between an exposure and an outcome can be produced by the following 3 causal structures13,14:
1. Cause and effect: If the exposure E causes the outcome D, or vice versa, then they will in general be associated. Figure 2 represents a randomized trial in which E (antiretroviral treatment) prevents D (AIDS) among HIV-infected subjects. The (associational) risk ratio ARRED differs from 1.0, and this association is entirely attributable to the causal effect of E on D.
2. Common causes: If the exposure and the outcome share a common cause, then they will in general be associated even if neither is a cause of the other. In Figure 1, the common cause L (smoking) results in E (carrying matches) and D (lung cancer) being associated, ie, again, ARRED ≠1.0.
3. Common effects: An exposure E and an outcome D that have a common effect C will be conditionally associated if the association measure is computed within levels of the common effect C, ie, the stratum-specific ARRED|C will differ from 1.0, regardless of whether the crude (equivalently, marginal, or unconditional) ARRED is 1.0. More generally, a conditional association between E and D will occur within strata of a common effect C of 2 other variables, one of which is either exposure or a cause of exposure and the other is either the outcome or a cause of the outcome. Note that E and D need not be unconditionally associated simply because they have a common effect. In the Appendix we describe additional, more complex, structural causes of statistical associations.
That causal structures (1) and (2) imply a crude association accords with the intuition of most epidemiologists. We now provide intuition for why structure (3) induces a conditional association. (For a formal justification, see references 13 and 14.) In Figure 3, the genetic haplotype E and smoking D both cause coronary heart disease C. Nonetheless, E and D are marginally unassociated (ARRED = 1.0) because neither causes the other and they share no common cause. We now argue heuristically that, in general, they will be conditionally associated within levels of their common effect C.
Suppose that the investigators, who are interested in estimating the effect of haplotype E on smoking status D, restricted the study population to subjects with heart disease (C = 1). The square around C in Figure 3 indicates that they are conditioning on a particular value of C. Knowing that a subject with heart disease lacks haplotype E provides some information about her smoking status because, in the absence of E, it is more likely that another cause of C such as D is present. That is, among people with heart disease, the proportion of smokers is increased among those without the haplotype E. Therefore, E and D are inversely associated conditionally on C = 1, and the conditional risk ratio ARRED|C=1 is less than 1.0. In the extreme, if E and D were the only causes of C, then among people with heart disease, the absence of one of them would perfectly predict the presence of the other.
As another example, the DAG in Figure 4 adds to the DAG in Figure 3 a diuretic medication M whose use is a consequence of a diagnosis of heart disease. E and D are also associated within levels of M because M is a common effect of E and D.
There is another possible source of association between 2 variables that we have not discussed yet. As a result of sampling variability, 2 variables could be associated by chance even in the absence of structures (1), (2), or (3). Chance is not a structural source of association because chance associations become smaller with increased sample size. In contrast, structural associations remain unchanged. To focus our discussion on structural rather than chance associations, we assume we have recorded data in every subject in a very large (perhaps hypothetical) population of interest. We also assume that all variables are perfectly measured.
A CLASSIFICATION OF BIASES ACCORDING TO THEIR STRUCTURE
We will say that bias is present when the association between exposure and outcome is not in its entirety the result of the causal effect of exposure on outcome, or more precisely when the causal risk ratio (CRRED), defined in Appendix A.1, differs from the associational risk ratio (ARRED). In an ideal randomized trial (ie, no confounding, full adherence to treatment, perfect blinding, no losses to follow up) such as the one represented in Figure 2, there is no bias and the association measure equals the causal effect measure.
Because nonchance associations are generated by structures (1), (2), and (3), it follows that biases could be classified on the basis of these structures:
1. Cause and effect could create bias as a result of reverse causation. For example, in many case-control studies, the outcome precedes the exposure measurement. Thus, the association of the outcome with measured exposure could in part reflect bias attributable to the outcome's effect on measured exposure.7,8 Examples of reverse causation bias include not only recall bias in case-control studies, but also more general forms of information bias like, for example, when a blood parameter affected by the presence of cancer is measured after the cancer is present.
2. Common causes: In general, when the exposure and outcome share a common cause, the association measure differs from the effect measure. Epidemiologists tend to use the term confounding to refer to this bias.
3. Conditioning on common effects: We propose that this structure is the source of those biases that epidemiologists refer to as selection bias. We argue by way of example.
EXAMPLES OF SELECTION BIAS
Inappropriate Selection of Controls in a Case-Control Study
Figure 5 represents a case-control study of the effect of postmenopausal estrogens (E) on the risk of myocardial infarction (D). The variable C indicates whether a woman in the population cohort is selected for the case-control study (yes = 1, no = 0). The arrow from disease status D to selection C indicates that cases in the cohort are more likely to be selected than noncases, which is the defining feature of a case-control study. In this particular case-control study, investigators selected controls preferentially among women with a hip fracture (F), which is represented by an arrow from F to C. There is an arrow from E to F to represent the protective effect of estrogens on hip fracture. Note Figure 5 is essentially the same as Figure 3, except we have now elaborated the causal pathway from E to C.
In a case-control study, the associational exposure–disease odds ratio (AORED|C = 1) is by definition conditional on having been selected into the study (C = 1). If subjects with hip fracture F are oversampled as controls, then the probability of control selection depends on a consequence F of the exposure (as represented by the path from E to C through F) and “inappropriate control selection” bias will occur (eg, AORED|C = 1 will differ from 1.0, even when like in Figure 5 the exposure has no effect on the disease). This bias arises because we are conditioning on a common effect C of exposure and disease. A heuristic explanation of this bias follows. Among subjects selected for the study, controls are more likely than cases to have had a hip fracture. Therefore, because estrogens lower the incidence of hip fractures, a control is less likely to be on estrogens than a case, and hence AORED|C = 1 is greater than 1.0, even though the exposure does not cause the outcome. Identical reasoning would explain that the expected AORED|C = 1 would be greater than the causal ORED even had the causal ORED differed from 1.0.
Berkson15 pointed out that 2 diseases (E and D) that are unassociated in the population could be associated among hospitalized patients when both diseases affect the probability of hospital admission. By taking C in Figure 3 to be the indicator variable for hospitalization, we recognize that Berkson's bias comes from conditioning on the common effect C of diseases E and D. As a consequence, in a case-control study in which the cases were hospitalized patients with disease D and controls were hospitalized patients with disease E, an exposure R that causes disease E would appear to be a risk factor for disease D (ie, Fig. 3 is modified by adding factor R and an arrow from R to E). That is, AORRD|C = 1 would differ from 1.0 even if R does not cause D.
Differential Loss to Follow Up in Longitudinal Studies
Figure 6a represents a follow-up study of the effect of antiretroviral therapy (E) on AIDS (D) risk among HIV-infected patients. The greater the true level of immunosuppression (U), the greater the risk of AIDS. U is unmeasured. If a patient drops out from the study, his AIDS status cannot be assessed and we say that he is censored (C = 1). Patients with greater values of U are more likely to be lost to follow up because the severity of their disease prevents them from attending future study visits. The effect of U on censoring is mediated by presence of symptoms (fever, weight loss, diarrhea, and so on), CD4 count, and viral load in plasma, all summarized in the (vector) variable L, which could or could not be measured. The role of L, when measured, in data analysis is discussed in the next section; in this section, we take L to be unmeasured. Patients receiving treatment are at a greater risk of experiencing side effects, which could lead them to dropout, as represented by the arrow from E to C. For simplicity, assume that treatment E does not cause D and so there is no arrow from E to D (CRRED = 1.0). The square around C indicates that the analysis is restricted to those patients who did not drop out (C = 0). The associational risk (or rate) ratio ARRED|C = 0 differs from 1.0. This “differential loss to follow-up” bias is an example of bias resulting from structure (3) because it arises from conditioning on the censoring variable C, which is a common effect of exposure E and a cause U of the outcome.
An intuitive explanation of the bias follows. If a treated subject with treatment-induced side effects (and thereby at a greater risk of dropping out) did in fact not drop out (C = 0), then it is generally less likely that a second cause of dropping out (eg, a large value of U) was present. Therefore, an inverse association between E and U would be expected. However, U is positively associated with the outcome D. Therefore, restricting the analysis to subjects who did not drop out of this study induces an inverse association (mediated by U) between exposure and outcome, ie, ARRED|C = 0 is not equal to 1.0.
Figure 6a is a simple transformation of Figure 3 that also represents bias resulting from structure (3): the association between D and C resulting from a direct effect of D on C in Figure 3 is now the result of U, a common cause of D and C. We now present 3 additional structures, (Figs. 6b–d), which could lead to selection bias by differential loss to follow up.
Figure 6b is a variation of Figure 6a. If prior treatment has a direct effect on symptoms, then restricting the study to the uncensored individuals again implies conditioning on the common effect C of the exposure and U thereby introducing a spurious association between treatment and outcome. Figures 6a and 6b could depict either an observational study or an experiment in which treatment E is randomly assigned, because there are no common causes of E and any other variable. Thus, our results demonstrate that randomized trials are not free of selection bias as a result of differential loss to follow up because such selection occurs after the randomization.
Figures 6c and d are variations of Figures 6a and b, respectively, in which there is a common cause U* of E and another measured variable. U* indicates unmeasured lifestyle/personality/educational variables that determine both treatment (through the arrow from U* to E) and either attitudes toward attending study visits (through the arrow from U* to C in Fig. 6c) or threshold for reporting symptoms (through the arrow from U* to L in Fig. 6d). Again, these 2 are examples of bias resulting from structure (3) because the bias arises from conditioning on the common effect C of both a cause U* of E and a cause U of D. This particular bias has been referred to as M bias.12 The bias caused by differential loss to follow up in Figures 6a–d is also referred to as bias due to informative censoring.
Nonresponse Bias/Missing Data Bias
The variable C in Figures 6a–d can represent missing data on the outcome for any reason, not just as a result of loss to follow up. For example, subjects could have missing data because they are reluctant to provide information or because they miss study visits. Regardless of the reasons why data on D are missing, standard analyses restricted to subjects with complete data (C = 0) will be biased.
Volunteer Bias/Self-selection Bias
Figures 6a–d can also represent a study in which C is agreement to participate (yes = 1, no = 0), E is cigarette smoking, D is coronary heart disease, U is family history of heart disease, and U* is healthy lifestyle. (L is any mediator between U and C such as heart disease awareness.) Under any of these structures, there would be no bias if the study population was a representative (ie, random) sample of the target population. However, bias will be present if the study is restricted to those who volunteered or elected to participate (C = 1). Volunteer bias cannot occur in a randomized study in which subjects are randomized (ie, exposed) only after agreeing to participate, because none of Figures 6a–d can represent such a trial. Figures 6a and b are eliminated because exposure cannot cause C. Figures 6c and d are eliminated because, as a result of the random exposure assignment, there cannot exist a common cause of exposure and any another variable.
Healthy Worker Bias
Figures 6a–d can also describe a bias that could arise when estimating the effect of a chemical E (an occupational exposure) on mortality D in a cohort of factory workers. The underlying unmeasured true health status U is a determinant of both death (D) and of being at work (C). The study is restricted to individuals who are at work (C = 1) at the time of outcome ascertainment. (L could be the result of blood tests and a physical examination.) Being exposed to the chemical is a predictor of being at work in the near future, either directly (eg, exposure can cause disabling asthma), like in Figures 6a and b, or through a common cause U* (eg, certain exposed jobs are eliminated for economic reasons and the workers laid off) like in Figures 6c and d.
This “healthy worker” bias is an example of bias resulting from structure (3) because it arises from conditioning on the censoring variable C, which is a common effect of (a cause of) exposure and (a cause of) the outcome. However, the term “healthy worker” bias is also used to describe the bias that occurs when comparing the risk in certain group of workers with that in a group of subjects from the general population. This second bias can be depicted by the DAG in Figure 1 in which L represents health status, E represents membership in the group of workers, and D represents the outcome of interest. There are arrows from L to E and D because being healthy affects job type and risk of subsequent outcome, respectively. In this case, the bias is caused by structure (1) and would therefore generally be considered to be the result of confounding.
These examples lead us to propose that the term selection bias in causal inference settings be used to refer to any bias that arises from conditioning on a common effect as in Figure 3 or its variations (Figs. 4–6).
In addition to the examples given here, DAGs have been used to characterize various other selection biases. For example, Robins7 explained how certain attempts to eliminate ascertainment bias in studies of estrogens and endometrial cancer could themselves induce bias16; Hernán et al.8 discussed incidence–prevalence bias in case-control studies of birth defects; and Cole and Hernán9 discussed the bias that could be introduced by standard methods to estimate direct effects.17,18 In Appendix A.2, we provide a final example: the bias that results from the use of the hazard ratio as an effect measure. We deferred this example to the appendix because of its greater technical complexity. (Note that standard DAGs do not represent “effect modification” or “interactions” between variables, but this does not affect their ability to represent the causal structures that produce bias, as more fully explained in Appendix A.3).
To demonstrate the generality of our approach to selection bias, we now show that a bias that arises in longitudinal studies with time-varying exposures19 can also be understood as a form of selection bias.
Adjustment for Variables Affected by Previous Exposure (or its causes)
Consider a follow-up study of the effect of antiretroviral therapy (E) on viral load at the end of follow up (D = 1 if detectable, D = 0 otherwise) in HIV-infected subjects. The greater a subject's unmeasured true immunosuppression level (U), the greater her viral load D and the lower the CD4 count L (low = 1, high = 0). Treatment increases CD4 count, and the presence of low CD4 count (a proxy for the true level of immunosuppression) increases the probability of receiving treatment. We assume that, in truth but unknown to the data analyst, treatment has no causal effect on the outcome D. The DAGs in Figures 7a and b represent the first 2 time points of the study. At time 1, treatment E1 is decided after observing the subject's risk factor profile L1. (E0 could be decided after observing L0, but the inclusion of L0 in the DAG would not essentially alter our main point.) Let E be the sum of E0 and E1. The cumulative exposure variable E can therefore take 3 values: 0 (if the subject is not treated at any time), 1 (if the subject is treated at time one only or at time 2 only), and 2 (if the subject is treated at both times). Suppose the analyst's interest lies in comparing the risk had all subjects been always treated (E = 2) with that had all subjects never been treated (E = 0), and that the causal risk ratio is 1.0 (CRRED = 1, when comparing E = 2 vs. E = 0).
To estimate the effect of E without bias, the analyst needs to be able to estimate the effect of each of its components E0 and E1 simultaneously and without bias.17 As we will see, this is not possible using standard methods, even when data on L1 are available, because lack of adjustment for L1 precludes unbiased estimation of the causal effect of E1 whereas adjustment for L1 by stratification (or, equivalently, by conditioning, matching, or regression adjustment) precludes unbiased estimation of the causal effect of E0.
Unlike previous structures, Figures 7a and 7b contain a common cause of the (component E1 of) exposure E and the outcome D, so one needs to adjust for L1 to eliminate confounding. The standard approach to confounder control is stratification: the associational risk ratio is computed in each level of the variable L1. The square around the node L1 denotes that the associational risk ratios (ARRED|L = 0 and ARRED|L = 1) are conditional on L1. Examples of stratification-based methods are a Mantel-Haenzsel stratified analysis or regression models (linear, logistic, Poisson, Cox, and so on) that include the covariate L1. (Not including interaction terms between L1 and the exposure in a regression model is equivalent to assuming homogeneity of ARRED|L = 0 and ARRED|L = 1.) To calculate ARRED|L = l, the data analyst has to select (ie, condition on) the subset of the population with value L1 = l. However, in this example, the process of choosing this subset results in selection on a variable L1 affected by (a component E0 of) exposure E and thus can result in bias as we now describe.
Although stratification is commonly used to adjust for confounding, it can have unintended effects when the association measure is computed within levels of L1 and in addition L1 is caused by or shares causes with a component E0 of E. Among those with low CD4 count (L1 = 1), being on treatment (E0 = 1) makes it more likely that the person is severely immunodepressed; among those with a high level of CD4 (L1 = 0), being off treatment (E0 = 0) makes it more likely that the person is not severely immunodepressed. Thus, the side effect of stratification is to induce an association between prior exposure E0 and U, and therefore between E0 and the outcome D. Stratification eliminates confounding for E1 at the cost of introducing selection bias for E0. The net bias for any particular summary of the time-varying exposure that is used in the analysis (cumulative exposure, average exposure, and so on) depends on the relative magnitude of the confounding that is eliminated and the selection bias that is created. In summary, the associational (conditional) risk ratio ARRED|L1, could be different from 1.0 even if the exposure history has no effect on the outcome of any subjects.
Conditioning on confounders L1 which are affected by previous exposure can create selection bias even if the confounder is not on a causal pathway between exposure and outcome. In fact, no such causal pathway exists in Figures 7a and 7b. On the other hand, in Figure 7C the confounder L1 for subsequent exposure E1 lies on a causal pathway from earlier exposure E0 to an outcome D. Nonetheless, conditioning on L1 still results in selection bias. Were the potential for selection bias not present in Figure 7C (e.g., were U not a common cause of L1 and D), the association of cumulative exposure E with the outcome D within strata of L1 could be an unbiased estimate of the direct effect18 of E not through L1 but still would not be an unbiased estimate of the overall effect of E on D, because the effect of E0 mediated through L1 is not included.
ADJUSTING FOR SELECTION BIAS
Selection bias can sometimes be avoided by an adequate design such as by sampling controls in a manner to ensure that they will represent the exposure distribution in the population. Other times, selection bias can be avoided by appropriately adjusting for confounding by using alternatives to stratification-based methods (see subsequently) in the presence of time-dependent confounders affected by previous exposure.
However, appropriate design and confounding adjustment cannot immunize studies against selection bias. For example, loss to follow up, self-selection, and, in general, missing data leading to bias can occur no matter how careful the investigator. In those cases, the selection bias needs to be explicitly corrected in the analysis, when possible.
Selection bias correction, as we briefly describe, could sometimes be accomplished by a generalization of inverse probability weighting20–23 estimators for longitudinal studies. Consider again Figures 6a–d and assume that L is measured. Inverse probability weighting is based on assigning a weight to each selected subject so that she accounts in the analysis not only for herself, but also for those with similar characteristics (ie, those with the same vales of L and E) who were not selected. The weight is the inverse of the probability of her selection. For example, if there are 4 untreated women, age 40–45 years, with CD4 count >500, in our cohort study, and 3 of them are lost to follow up, then these 3 subjects do not contribute to the analysis (ie, they receive a zero weight), whereas the remaining woman receives a weight of 4. In other words, the (estimated) conditional probability of remaining uncensored is 1/4 = 0.25, and therefore the (estimated) weight for the uncensored subject is 1/0.25 = 4. Inverse probability weighting creates a pseudopopulation in which the 4 subjects of the original population are replaced by 4 copies of the uncensored subject.
The effect measure based on the pseudopulation, in contrast to that based on the original population, is unaffected by selection bias provided that the outcome in the uncensored subjects truly represents the unobserved outcomes of the censored subjects (with the same values of E and L). This provision will be satisfied if the probability of selection (the denominator of the weight) is calculated conditional on E and on all additional factors that independently predict both selection and the outcome. Unfortunately, one can never be sure that these additional factors were identified and recorded in L, and thus the causal interpretation of the resulting adjustment for selection bias depends on this untestable assumption.
One might attempt to remove selection bias by stratification (ie, by estimating the effect measure conditional on the L variables) rather than by weighting. Stratification could yield unbiased conditional effect measures within levels of L under the assumptions that all relevant L variables were measured and that the exposure does not cause or share a common cause with any variable in L. Thus, stratification would work (ie, it would provide an unbiased conditional effect measure) under the causal structures depicted in Figures 6a and c, but not under those in Figures 6b and d. Inverse probability weighting appropriately adjusts for selection bias under all these situations because this approach is not based on estimating effect measures conditional on the covariates L, but rather on estimating unconditional effect measures after reweighting the subjects according to their exposure and their values of L.
Inverse probability weighting can also be used to adjust for the confounding of later exposure E1 by L1, even when exposure E0 either causes L1 or shares a common cause with L1 (Figs. 7a–7c), a situation in which stratification fails. When using inverse probability weighting to adjust for confounding, we model the probability of exposure or treatment given past exposure and past L so that the denominator of a subject's weight is, informally, the subject's conditional probability of receiving her treatment history. We therefore refer to this method as inverse-probability-of-treatment weighting.22
One limitation of inverse probability weighting is that all conditional probabilities (of receiving certain treatment or censoring history) must be different from zero. This would not be true, for example, in occupational studies in which the probability of being exposed to a chemical is zero for those not working. In these cases, g-estimation19 rather than inverse probability weighting can often be used to adjust for selection bias and confounding.
The use of inverse probability weighting can provide unbiased estimates of causal effects even in the presence of selection bias because the method works by creating a pseudopopulation in which censoring (or missing data) has been abolished and in which the effect of the exposure is the same as in the original population. Thus, the pseudopopulation effect measure is equal to the effect measure had nobody been censored. For example, Figure 8 represents the pseudopulation corresponding to the population of Figure 6a when the weights were estimated conditional on L and E. The censoring node is now lower-case because it does not correspond to a random variable but to a constant (everybody is uncensored in the pseudopopulation). This interpretation is desirable when censoring is the result of loss to follow up or nonresponse, but questionably helpful when censoring is the result of competing risks. For example, in a study aimed at estimating the effect of certain exposure on the risk of Alzheimer's disease, we might not wish to base our effect estimates on a pseudopopulation in which all other causes of death (cancer, heart disease, stroke, and so on) have been removed, because it is unclear even conceptually what sort of medical intervention would produce such a population. Another more pragmatic reason is that no feasible intervention could possibly remove just one cause of death without affecting the others as well.24
The terms “confounding” and “selection bias” are used in multiple ways. For instance, the same phenomenon is sometimes named “confounding by indication” by epidemiologists and “selection bias” by statisticians/econometricians. Others use the term “selection bias” when “confounders” are unmeasured. Sometimes the distinction between confounding and selection bias is blurred in the term “selection confounding.”
We elected to refer to the presence of common causes as “confounding” and to refer to conditioning on common effects as “selection bias.” This structural definition provides a clearcut classification of confounding and selection bias, even though it might not coincide perfectly with the traditional, often discipline-specific, terminologies. Our goal, however, was not to be normative about terminology, but rather to emphasize that, regardless of the particular terms chosen, there are 2 distinct causal structures that lead to these biases. The magnitude of both biases depends on the strength of the causal arrows involved.12,25 (When 2 or more common effects have been conditioned on, an even more general formulation of selection bias is useful. For a brief discussion, see Appendix A.4.)
The end result of both structures is the same: noncomparability (also referred to as lack of exchangeability) between the exposed and the unexposed. For example, consider a cohort study restricted to firefighters that aims to estimate the effect of being physically active (E) on the risk of heart disease (D) (as represented in Fig. 9). For simplicity, we have assumed that, although unknown to the data analyst, E does not cause D. Parental socioeconomic status (L) affects the risk of becoming a firefighter (C) and, through childhood diet, of heart disease (D). Attraction toward activities that involve physical activity (an unmeasured variable U) affects the risk of becoming a firefighter and of being physically active (E). U does not affect D, and L does not affect E. According to our terminology, there is no confounding because there are no common causes of E and D. Thus, if our study population had been a random sample of the target population, the crude associational risk ratio ARRED would have been equal to the causal risk ratio CRRED of 1.0.
However, in a study restricted to firefighters, the crude ARRED and CRRED would differ because conditioning on a common effect C of causes of exposure and outcome induces selection bias resulting in noncomparability of the exposed and unexposed firefighters. To the study investigators, the distinction between confounding and selection bias is moot because, regardless of nomenclature, they must stratify on L to make the exposed and the unexposed firefighters comparable. This example demonstrates that a structural classification of bias does not always have consequences for either the analysis or interpretation of a study. Indeed, for this reason, many epidemiologists use the term “confounder” for any variable L on which one has to stratify to create comparability, regardless of whether the (crude) noncomparability was the result of conditioning on a common effect or the result of a common cause of exposure and disease.
There are, however, advantages of adopting a structural or causal approach to the classification of biases. First, the structure of the problem frequently guides the choice of analytical methods to reduce or avoid the bias. For example, in longitudinal studies with time-dependent confounding, identifying the structure allows us to detect situations in which stratification-based methods would adjust for confounding at the expense of introducing selection bias. In those cases, inverse probability weighting or g-estimation are better alternatives. Second, even when understanding the structure of bias does not have implications for data analysis (like in the firefighters’ study), it could still help study design. For example, investigators running a study restricted to firefighters should make sure that they collect information on joint risk factors for the outcome and for becoming a firefighter. Third, selection bias resulting from conditioning on preexposure variables (eg, being a firefighter) could explain why certain variables behave as “confounders” in some studies but not others. In our example, parental socioeconomic status would not necessarily need to be adjusted for in studies not restricted to firefighters. Finally, causal diagrams enhance communication among investigators because they can be used to provide a rigorous, formal definition of terms such as “selection bias.”
We thank Stephen Cole and Sander Greenland for their helpful comments.
1. Rothman KJ, Greenland S. Modern Epidemiology
, 2nd ed. Philadelphia: Lippincott-Raven; 1998.
2. Szklo M0, Nieto FJ. Epidemiology. Beyond the Basics
. Gaithersburg, MD: Aspen; 2000.
3. MacMahon B, Trichopoulos D. Epidemiology. Principles & Methods
, 2nd ed. Boston: Little, Brown and Co; 1996.
4. Hennekens CH, Buring JE. Epidemiology in Medicine
. Boston: Little, Brown and Co; 1987.
5. Gordis L. Epidemiology
. Philadelphia: WB Saunders Co; 1996.
6. Greenland S, Pearl J, Robins JM. Causal diagrams for epidemiologic research. Epidemiology
7. Robins JM. Data, design, and background knowledge in etiologic inference. Epidemiology
8. Hernán MA, Hernández-Diaz S, Werler MM, et al. Causal knowledge as a prerequisite for confounding evaluation: an application to birth defects epidemiology. Am J Epidemiol
9. Cole SR, Hernán MA. Fallibility in the estimation of direct effects. Int J Epidemiol
10. Maclure M, Schneeweiss S. Causation of bias: the episcope. Epidemiology
11. Greenland S, Brumback BA. An overview of relations among causal modeling methods. Int J Epidemiol
12. Greenland S. Quantifying biases in causal models: classical confounding versus collider-stratification bias. Epidemiology
13. Pearl J. Causal diagrams for empirical research. Biometrika
14. Spirtes P, Glymour C, Scheines R. Causation, Prediction, and Search. Lecture Notes in Statistics 81
. New York: Springer-Verlag; 1993.
15. Berkson J. Limitations of the application of fourfold table analysis to hospital data. Biometrics
16. Greenland S, Neutra RR. An analysis of detection bias and proposed corrections in the study of estrogens and endometrial cancer. J Chronic Dis
17. Robins JM. A new approach to causal inference in mortality studies with a sustained exposure period—application to the healthy worker survivor effect [published errata appear in Mathematical Modelling.
1987;14:917–921]. Mathematical Modelling
18. Robins JM, Greenland S. Identifiability and exchangeability for direct and indirect effects. Epidemiology
19. Robins JM. Causal inference from complex longitudinal data. In: Berkane M, ed. Latent Variable Modeling and Applications to Causality. Lecture Notes in Statistics 120
. New York: Springer-Verlag; 1997:69–117.
20. Horvitz DG, Thompson DJ. A generalization of sampling without replacement from a finite universe. J Am Stat Assoc
21. Robins JM, Finkelstein DM. Correcting for noncompliance and dependent censoring in an AIDS clinical trial with inverse probability of censoring weighted (IPCW) log-rank tests. Biometrics
22. Hernán MA, Brumback B, Robins JM. Marginal structural models to estimate the causal effect of zidovudine on the survival of HIV-positive men. Epidemiology
23. Robins JM, Hernán MA, Brumback B. Marginal structural models and causal inference in epidemiology. Epidemiology
24. Greenland S. Causality theory for policy uses of epidemiologic measures. In: Murray CJL, Salomon JA, Mathers CD, et al., eds. Summary Measures of Population Health
. Cambridge, MA: Harvard University Press/WHO; 2002.
25. Walker AM. Observation and Inference: An introduction to the Methods of Epidemiology
. Newton Lower Falls: Epidemiology Resources Inc; 1991.
26. Greenland S. Absence of confounding does not correspond to collapsibility of the rate ratio or rate difference. Epidemiology
A.1. Causal and Associational Risk Ratio
For a given subject, E has a causal effect on D if the subject's value of D had she been exposed differs from the value of D had she remained unexposed. Formally, letting Di, e = 1 and Di,e = 0 be subject's i (counterfactual or potential) outcomes when exposed and unexposed, respectively, we say there is a causal effect for subject i if Di,e = 1 ≠ Di, e = 0. Only one of the counterfactual outcomes can be observed for each subject (the one corresponding to his observed exposure), ie, Di, e = Di if Ei = e, where Di and Ei represent subject i's observed outcome and exposure. For a population, we say that there is no average causal effect (preventive or causative) of E on D if the average of D would remain unchanged whetherthe whole population had been treated or untreated, ie, when Pr(De = 1 = 1) = Pr(De = 0 = 1) for a dichotomous D. Equivalently, we say that E does not have a causal effect on D if the causal risk ratio is one, ie, CRRED = Pr(De = 1 = 1)/Pr(De = 0 = 1) = 1.0. For an extension of counterfactual theory and methods to complex longitudinal data, see reference 19.
In a DAG, CRRED = 1.0 is represented by the lack of a directed path of arrows originating from E and ending on D as, for example, in Figure 5. We shall refer to a directed path of arrows as a causal path. On the other hand, in Figure 5, CRREC ≠ 1.0 because there is a causal path from E to C through F. The lack of a direct arrow from E to C implies that E does not have a direct effect on C (relative to the other variables on the DAG), ie, the effect is wholly mediated through other variables on the DAG (ie, F).
For a population, we say that there is no association between E and D if the average of D is the same in the subset of the population that was exposed as in the subset that was unexposed, ie, when Pr(D = 1|E = 1) = Pr(D = 1|E = 0) for a dichotomous D. Equivalently, we say that E and D are unassociated if the associational risk ratio is 1.0, ie, ARRED = Pr(D = 1|E = 1) / Pr(D = 1|E = 0) = 1.0. The associational risk ratio can always be estimated from observational data. We say that there is bias when the causal risk ratio in the population differs from the associational risk ratio, ie, CRRED ≠ ARRED.
A.2. Hazard Ratios as Effect Measures
The causal DAG in Appendix Figure 1a describes a randomized study of the effect of surgery E on death at times 1 (D1) and 2 (D2). Suppose the effect of exposure on D1 is protective. Then the lack of an arrow from E to D2 indicates that, although the exposure E has a direct protective effect (decreases the risk of death) at time 1, it has no direct effect on death at time 2. That is, the exposure does not influence the survival status at time D2 of any subject who would survive past time 1 when unexposed (and thus when exposed). Suppose further that U is an unmeasured haplotype that decreases the subject's risk of death at all times. The associational risk ratios ARRED1 and ARRED2 are unbiased measures of the effect of E on death at times 1 and 2, respectively. (Because of the absence of confounding, ARRED1 and ARRED2 equal the causal risk ratios CRRED1 and CRRED2, respectively.) Note that, even though E has no direct effect on D2, ARRED2 (or, equivalently, CRRED2) will be less than 1.0 because it is a measure of the effect of E on total mortality through time 2.
Appendix Figure 1. E...Image Tools
Consider now the time-specific associational hazard (rate) ratio as an effect measure. In discrete time, the hazard of death at time 1 is the probability of dying at time 1 and thus is the same as ARRED1. However, the hazard at time 2 is the probability of dying at time 2 among those who survived past time 1. Thus, the associational hazard ratio at time 2 is then ARRED2|D1 = 0. The square around D1 in Appendix Figure 1a indicates this conditioning. Exposed survivors of time 1 are less likely than unexposed survivors of time 1 to have the protective haplotype U (because exposure can explain their survival) and therefore are more likely to die at time 2. That is, conditional on D1 = 0, exposure is associated with a higher mortality at time 2. Thus, the hazard ratio at time 1 is less than 1.0, whereas the hazard ratio at time 2 is greater than 1.0, ie, the hazards have crossed. We conclude that the hazard ratio at time 2 is a biased estimate of the direct effect of exposure on mortality at time 2. The bias is selection bias arising from conditioning on a common effect D1 of exposure and of U, which is a cause of D2 that opens the noncausal (ie, associational) path E → D1← U→ D2 between E and D2.13 In the survival analysis literature, an unmeasured cause of death that is marginally unassociated with exposure such as U is often referred to as a frailty.
In contrast to this, the conditional hazard ratio ARRED2|D1 = 0,U at D2 given U is equal to 1.0 within each stratum of U because the path E → D1← U→ D2 between E and D2 is now blocked by conditioning on the noncollider U. Thus, the conditional hazard ratio correctly indicates the absence of a direct effect of E on D2. The fact that the unconditional hazard ratio ARRED2|D1 = 0 differs from the common-stratum specific hazard ratios of 1.0 even though U is independent of E, shows the noncollapsibility of the hazard ratio.26
Unfortunately, the unbiased measure ARRED2|D1 = 0,U of the direct effect of E on D2 cannot be computed because U is unobserved. In the absence of data on U, it is impossible to know whether exposure has a direct effect on D2. That is, the data cannot determine whether the true causal DAG generating the data was that in Appendix Figure 1a versus that in Appendix Figure 1b.
A.3. Effect Modification and Common Effects in DAGs
Although an arrow on a causal DAG represents a direct effect, a standard causal DAG does not distinguish a harmful effect from a protective effect. Similarly, a standard DAG does not indicate the presence of effect modification. For example, although Appendix Figure 1a implies that both E and U affect death D1, the DAG does not distinguish among the following 3 qualitatively distinct ways that U could modify the effect of E on D1:
1. The causal effect of exposure E on mortality D1 is in the same direction (ie, harmful or beneficial) in both stratum U = 1 and stratum U = 0.
2. The direction of the causal effect of exposure E on mortality D1 in stratum U = 1 is the opposite of that in stratum U = 0 (ie, there is a qualitative interaction between U and E).
3. Exposure E has a causal effect on D1 in one stratum of U but no causal effect in the other stratum, eg, E only kills subjects with U = 0.
Because standard DAGs do not represent interaction, it follows that it is not possible to infer from a DAG the direction of the conditional association between 2 marginally independent causes (E and U) within strata of their common effect D1. For example, suppose that, in the presence of an undiscovered background factor V that is unassociated with E or U, having either E = 1 or U = 1 is sufficient and necessary to cause death (an “or” mechanism), but that neither E nor U causes death in the absence of V. Then among those who died by time 1 (D1 = 1), E and U will be negatively associated, because it is more likely that an unexposed subject (E = 0) had U = 1 because the absence of exposure increases the chance that U was the cause of death. (Indeed, the logarithm of the conditional odds ratio ORUE|D1 = 1 will approach minus infinity as the population prevalence of V approaches 1.0.) Although this “or” mechanism was the only explanation given in the main text for the conditional association of independent causes within strata of a common effect; nonetheless, other possibilities exist. For example, suppose that in the presence of the undiscovered background factor V, having both E = 1 and U = 1 is sufficient and necessary to cause death (an “and” mechanism) and that neither E nor U causes death in the absence of V. Then, among those who die by time 1, those who had been exposed (E = 1) are more likely to have the haplotype (U = 1), ie, E and U are positively correlated. A standard DAG such as that in Appendix Figure 1a fails to distinguish between the case of E and U interacting through an “or” mechanism from the case of an “and” mechanism.
Although conditioning on common effect D1 always induces a conditional association between independent causes E and U in at least one of the 2 strata of D1 (say, D1 = 1), there is a special situation under which E and U remain conditionally independent within the other stratum (say, D1 = 0). This situation occurs when the data follow a multiplicative survival model. That is, when the probability, Pr[D1 = 0| U = u, E = e], of survival (ie, D1 = 0) given E and U is equal to a product g(u) h(e) of functions of u and e. The multiplicative model Pr[D1 = 0| U = u, E = e] = g(u) h(e) is equivalent to the model that assumes the survival ratio Pr[D1 = 0| U = u, E = e]/Pr[D1 = 0| U = 0, E = 0] does not depend on u and is equal to h(e). (Note that if Pr[D1 = 0| U = u, E = e] = g(u) h(e), then Pr[D1 = 1| U = u, E = e] = 1 – [g(u) h(e)] does not follow a multiplicative mortality model. Hence, when E and U are conditionally independent given D1 = 0, they will be conditionally dependent given D1 = 1.)
Biologically, this multiplicative survival model will hold when E and U affect survival through totally independent mechanisms in such a way that U cannot possibly modify the effect of E on D1, and vice versa. For example, suppose that the surgery E affects survival through the removal of a tumor, whereas the haplotype U affects survival through increasing levels of low-density lipoprotein-cholesterol levels resulting in an increased risk of heart attack (whether or not a tumor is present), and that death by tumor and death by heart attack are independent in the sense that they do not share a common cause. In this scenario, we can consider 2 cause-specific mortality variables: death from tumor D1A and death from heart attack D1B. The observed mortality variable D1 is equal to 1 (death)when either D1A or D1B is equal to 1, and D1 is equal to 0 (survival) when both D1A and D1B equal 0. We assume the measured variables are those in Appendix Figure 1a so data on underlying cause of death is not recorded. Appendix Figure 2 is an expansion of Appendix Figure 1a that represents this scenario (variable D2 is not represented because it is not essential to the current discussion). Because D1 = 0 implies both D1A = 0 and D1B = 0, conditioning on observed survival (D1 = 0) is equivalent to simultaneously conditioning on D1A = 0 and D1B = 0 as well. As a consequence, we find by applying d-separation13 to Appendix Figure 2 that E and U are conditionally independent given D1 = 0, ie, the path, between E and U through the conditioned on collider D1 is blocked by conditioning on the noncolliders D1A and D1B.8 On the other hand, conditioning on D1 = 1 does not imply conditioning on any specific values of D1A and D1B as the event D1 = 1 is compatible with 3 possible unmeasured events D1A = 1 and D1B = 1, D1A = 1 and D1B = 0, and D1A = 0 and D1B = 1. Thus, the path between E and U through the conditioned on collider D1 is not blocked, and thus E and U are associated given D1 = 1.
Appendix Figure 2. M...Image Tools
What is interesting about Appendix Figure 2 is that by adding the unmeasured variables D1A and D1B, which functionally determine the observed variable D1, we have created an annotated DAG that succeeds in representing both the conditional independence between E and U given D1 = 0 and the their conditional dependence given D1 = 1. As far as we are aware, this is the first time such a conditional independence structure has been represented on a DAG.
If E and U affect survival through a common mechanism, then there will exist an arrow either from E to D1B or from U to D1A, as shown in Appendix Figure 3a. In that case, the multiplicative survival model will not hold, and E and U will be dependent within both strata of D1. Similarly, if the causes D1A and D1B are not independent because of a common cause V as shown in Appendix Figure 3b, the multiplicative survival model will not hold, and E and U will be dependent within both strata of D1.
Appendix Figure 3. M...Image Tools
In summary, conditioning on a common effect always induces an association between its causes, but this association could be restricted to certain levels of the common effect.
A.4. Generalizations of Structure (3)
Consider Appendix Figure 4a representing a study restricted to firefighters (F = 1). E and D are unassociated among firefighters because the path EFACD is blocked by C. If we then stratify on the covariate C like in Appendix Figure 4b, E and D are conditionally associated among firefighters in a given stratum of C; yet C is neither caused by E nor by a cause of E. This example demonstrates that our previous formulation of structure (3) is insufficiently general to cover examples in which we have already conditioned on another variable F before conditioning on C. Note that one could try to argue that our previous formulation works by insisting that the set (F,C) of all variables conditioned be regarded as a single supervariable and then apply our previous formulation with this supervariable in place of C. This fix-up fails because it would require E and D to be conditionally associated within joint levels of the super variable (C, F) in Appendix Figure 4c as well, which is not the case.
Appendix Figure 4. C...Image Tools
However, a general formulation that works in all settings is the following. A conditional association between E and D will occur within strata of a common effect C of 2 other variables, one of which is either the exposure or statistically associated with the exposure and the other is either the outcome or statistically associated with the outcome.
Clearly, our earlier formulation is implied by the new formulation and, furthermore, the new formulation gives the correct results for both Appendix Figures 4b and 4c. A drawback of this new formulation is that it is not stated purely in terms of causal structures, because it makes reference to (possibly noncausal) statistical associations. Now it actually is possible to provide a fully general formulation in terms of causal structures but it is not simple, and so we will not give it here, but see references 13 and 14. Cited Here...
This article has been cited 210 time(s).
Occupational and Environmental MedicineMarginal structural models to control for time-varying confounding in occupational and environmental epidemiologyOccupational and Environmental Medicine
Occupational and Environmental MedicineWork related asthma. A causal analysis controlling the healthy worker effectOccupational and Environmental Medicine
European Journal of EpidemiologyTime-dependent propensity score and collider-stratification bias: an example of beta(2)-agonist use and the risk of coronary heart diseaseEuropean Journal of Epidemiology
American Journal of SociologyExposure to Classroom Poverty and Test Score Achievement: Contextual Effects or Selection?American Journal of Sociology
Statistics in MedicineMethods for dealing with time-dependent confoundingStatistics in Medicine
Pediatric Blood & CancerAssessment of potential bias from non-participation in a dynamic clinical cohort of long-term childhood cancer survivors: Results from the St. Jude lifetime cohort studyPediatric Blood & Cancer
Paediatric Respiratory ReviewsOpportunities and pitfalls of registry data for clinical researchPaediatric Respiratory Reviews
Journal of Clinical EpidemiologyIndex event bias-a numerical exampleJournal of Clinical Epidemiology
Archives of Internal MedicineComparative Risk for Angioedema Associated With the Use of Drugs That Target the Renin-Angiotensin-Aldosterone SystemArchives of Internal Medicine
Preventive MedicineSelection bias in cohorts of casesPreventive Medicine
Bmc PsychiatryPreventing the onset of major depression based on the level and profile of risk of primary care attendees: protocol of a cluster randomised trial (the predictD-CCRT study)Bmc Psychiatry
Science of the Total EnvironmentLong-term exposure to traffic-related air pollution and the risk of death from hemorrhagic stroke and lung cancer in Shizuoka, JapanScience of the Total Environment
American Journal of EpidemiologyThe Healthy Worker Effect in Cancer Incidence StudiesAmerican Journal of Epidemiology
Children and Youth Services ReviewChild welfare and the challenge of causal inferenceChildren and Youth Services Review
Annual Review of Public Health, Vol 34Causal Inference in Public HealthAnnual Review of Public Health, Vol 34
Preventive MedicineChanges in combined lifestyle risks and disability transition in older adults: Korean Longitudinal Study of Aging, 2006-2008Preventive Medicine
Annals of EpidemiologyAssessing the component associations of the healthy worker survivor bias: occupational asbestos exposure and lung cancer mortalityAnnals of Epidemiology
Tropical Medicine & International HealthBlood lead concentrations in sub-Saharan African children below 6 years: systematic reviewTropical Medicine & International Health
Circulation-Cardiovascular ImagingUse of Cardiac Magnetic Resonance and Echocardiography in Population-Based Studies Why, Where, and When?Circulation-Cardiovascular Imaging
American Journal of BioethicsDoes Consent Bias Research?American Journal of Bioethics
American Journal of BioethicsHow and When Does Consent Bias Research?American Journal of Bioethics
Bmc CancerEpidemiology and outcomes of previously undiagnosed diabetes in older women with breast cancer: an observational cohort study based on SEER-MedicareBmc Cancer
Transfusion Medicine ReviewsAssociation Vs Causality in Transfusion Medicine: Understanding Multivariable Analysis in Prediction Vs Etiologic ResearchTransfusion Medicine Reviews
International Journal of ObesityBirth cohort effects on abdominal obesity in the United States: the Silent Generation, Baby Boomers and Generation XInternational Journal of Obesity
Diabetes CareTiming and Duration of Obesity in Relation to Diabetes Findings from an ethnically diverse, nationally representative sampleDiabetes Care
Statistics in MedicineSimulating from marginal structural models with time-dependent confoundingStatistics in Medicine
Clinical Infectious DiseasesThe Effect of a "Universal Antiretroviral Therapy" Recommendation on HIV RNA Levels Among HIV-Infected Patients Entering Care With a CD4 Count Greater Than 500/mu L in a Public Health SettingClinical Infectious Diseases
American Journal of GastroenterologyGI-REASONS: A Novel 6-Month, Prospective, Randomized, Open-Label, Blinded Endpoint (PROBE) TrialAmerican Journal of Gastroenterology
Plos OneHarnessing the Medicaid Analytic eXtract (MAX) to Evaluate Medications in Pregnancy: Design ConsiderationsPlos One
Statistical Methods in Medical ResearchObservational data for comparative effectiveness research: An emulation of randomised trials of statins and primary prevention of coronary heart diseaseStatistical Methods in Medical Research
American Journal of EpidemiologyEnvironmental Confounding in Gene-Environment Interaction StudiesAmerican Journal of Epidemiology
Paediatric and Perinatal EpidemiologyAccuracy Loss Due to Selection Bias in Cohort Studies with Left TruncationPaediatric and Perinatal Epidemiology
American Journal of EpidemiologyMarginal structural models for estimating the effect of highly active antiretroviral therapy initiation on CD4 cell countAmerican Journal of Epidemiology
Fertility and SterilityGestational hypertension in pregancies supported by infertility treatments: role of infertility, treatments, and multiple gestationsFertility and Sterility
Occupational and Environmental MedicineReducing healthy worker survivor bias by restricting date of hire in a cohort study of Vermont granite workersOccupational and Environmental Medicine
Statistics & Probability LettersA note on distortions induced by truncation with applications to linear regression systemsStatistics & Probability Letters
Clinical Infectious DiseasesLong-term effects of highly active antiretroviral therapy on CD4(+) cell evolution among children and adolescents infected with HIV: 5 years and countingClinical Infectious Diseases
BiostatisticsAdjusting for selection bias in retrospective, case-control studiesBiostatistics
American Journal of EpidemiologyConstructing inverse probability weights for marginal structural modelsAmerican Journal of Epidemiology
Annals of EpidemiologyTime-Dependent Confounding in the Study of the Effects of Regular Physical Activity in Chronic Obstructive Pulmonary Disease: An Application of the Marginal Structural ModelAnnals of Epidemiology
Clinical Gastroenterology and HepatologyCurrent and Past Smoking Significantly Increase Risk for Barrett's EsophagusClinical Gastroenterology and Hepatology
American Journal of EpidemiologyInvited Commentary: Causal Diagrams and Measurement BiasAmerican Journal of Epidemiology
American Journal of EpidemiologyControl Selection and Participation in an Ongoing, Population-based, Case-Control Study of Birth DefectsAmerican Journal of Epidemiology
Long-term effectiveness of potent antiretroviral therapy in preventing AIDS and death: a prospective cohort study
Statistical issues arising in the Women's Health Initiative - Discussion
American Journal of EpidemiologyShould meta-analyses of interventions include observational studies in addition to randomized controlled trials? A critical examination of underlying principlesAmerican Journal of Epidemiology
Pharmacoepidemiology and Drug SafetyValidating the effects of drug treatment on blood pressure in the General Practice Research DatabasePharmacoepidemiology and Drug Safety
Journal of the Royal Statistical Society Series B-Statistical Methodology
Sampling bias and logistic models - Discussion
Journal of the Royal Statistical Society Series B-Statistical Methodology, 70():
Acta OncologicaA hierarchical step-model for causation of bias-evaluating cancer treatment with epidemiological methodsActa Oncologica
Environmental Health PerspectivesMaternal serum polychlorinated biphenyl concentrations across critical windows of human develoomentEnvironmental Health Perspectives
Journal of Epidemiology and Community HealthUsing directed acyclic graphs to consider adjustment for socioeconomic status in occupational cancer studiesJournal of Epidemiology and Community Health
Journal of Epidemiology and Community HealthUsing directed acyclic graphs to guide analyses of neighbourhood health effects: an introductionJournal of Epidemiology and Community Health
Journal of Evaluation in Clinical PracticeEvaluating the effect of change on change: a different viewpointJournal of Evaluation in Clinical Practice
American Journal of EpidemiologyThe birth weight "paradox" uncovered?American Journal of Epidemiology
Journal of Traumatic StressMaximizing follow-up in longitudinal studies of traumatized populationsJournal of Traumatic Stress
Journal of Clinical EpidemiologyQuantitative assessment of unobserved confounding is mandatory in nonrandomized intervention studiesJournal of Clinical Epidemiology
NeuroepidemiologySelection Bias Importance of Identification and AdjustmentNeuroepidemiology
Perspectives on Psychological ScienceProceeding From Observed Correlation to Causal Inference The Use of Natural ExperimentsPerspectives on Psychological Science
Kidney InternationalThe effect of epoetin dose on hematocritKidney International
Lifetime Data AnalysisSensitivity analysis for unmeasured confounding in a marginal structural Cox proportional hazards modelLifetime Data Analysis
Journal of the Royal Statistical Society Series B-Statistical MethodologyA general dynamical statistical model with causal interpretationJournal of the Royal Statistical Society Series B-Statistical Methodology
Cancer Causes & ControlConsumption of sweet foods and breast cancer risk: a case-control study of women on Long Island, New YorkCancer Causes & Control
Journal of Evaluation in Clinical PracticeEvaluating health management programmes over time: application of propensity score-based weighting to longitudinal dataJournal of Evaluation in Clinical Practice
International Journal of Public HealthEffects of time-varying exposures adjusting for time-varying confounders: the case of alcohol consumption and risk of incident human immunodeficiency virus infectionInternational Journal of Public Health
Journal of Evaluation in Clinical PracticeCausal diagrams for encoding and evaluation of information biasJournal of Evaluation in Clinical Practice
Journal of Clinical OncologyUltimate Fate of Oncology Drugs Approved by the US Food and Drug Administration Without a Randomized TrialJournal of Clinical Oncology
American Journal of EpidemiologyNeighborhood Poverty and Injection Cessation in a Sample of Injection Drug UsersAmerican Journal of Epidemiology
Genetic EpidemiologyCase-Only Gene-Environment Interaction Studies: When Does Association Imply Mechanistic Interaction?Genetic Epidemiology
European Journal of EpidemiologyRepresentativeness, losses to follow-up and validity in cohort studiesEuropean Journal of Epidemiology
European Journal of EpidemiologyFeasibility of recruiting a birth cohort through the Internet: the experience of the NINFEA cohortEuropean Journal of Epidemiology
Human ReproductionShould we adjust for gestational age when analysing birth weights? The use of z-scores revisitedHuman Reproduction
International Journal of ObesityCausal models for estimating the effects of weight gain on mortalityInternational Journal of Obesity
Preventive Veterinary MedicineLinking causal concepts, study design, analysis and inference in support of one epidemiology for population healthPreventive Veterinary Medicine
Cancer Causes & ControlA summary measure of pro- and anti-oxidant exposures and risk of incident, sporadic, colorectal adenomasCancer Causes & Control
Environmental ResearchChanges in maternal serum chlorinated pesticide concentrations across critical windows of human reproduction and developmentEnvironmental Research
Journal of RheumatologyPrednisone, Lupus Activity, and Permanent Organ DamageJournal of Rheumatology
American Journal of EpidemiologyInvited commentary: Hypothetical interventions to define causal effects - Afterthought or prerequisite?American Journal of Epidemiology
Social Science & MedicineSome models just can't be fixed. A commentary on MortensenSocial Science & Medicine
Bmc Musculoskeletal DisordersThe Knee Clinical Assessment Study - CAS(K). A prospective study of knee pain and knee osteoarthritis in the general population: baseline recruitment and retention at 18 monthsBmc Musculoskeletal Disorders
NeurotoxicologyHershey Medical Center Technical Workshop Report: Optimizing the design and interpretation of epiderniologic studies for assessing neurodevelopmental effects from in utero chemical exposureNeurotoxicology
American Journal of EpidemiologyAdherence to lipid-lowering therapy and the use of preventive health services: An investigation of the healthy user effectAmerican Journal of Epidemiology
Environmental Health PerspectivesEstimated effects of disinfection by-products on preterm birth in a population served by a single water utilityEnvironmental Health Perspectives
Statistical ScienceDefining and estimating intervention effects for groups that will develop an auxiliary outcomeStatistical Science
Human HeredityImproper Adjustment for Baseline in Genetic Association Studies of Change in PhenotypeHuman Heredity
Journal of Clinical EpidemiologyMethodological considerations, such as directed acyclic graphs, for studying "acute on chronic" disease epidemiology: Chronic obstructive pulmonary disease exampleJournal of Clinical Epidemiology
International Journal of EpidemiologyIllustrating bias due to conditioning on a colliderInternational Journal of Epidemiology
Environmental Health PerspectivesAssociation of Exposure to Phthalates with Endometriosis and Uterine Leiomyomata: Findings from NHANES, 1999-2004Environmental Health Perspectives
International Journal of EpidemiologySelection bias and its implications for case-control studies: a case study of magnetic field exposure and childhood leukaemiaInternational Journal of Epidemiology
Scandinavian Journal of GastroenterologyLong-term clinical outcome and effect of glycyrrhizin in 1093 chronic hepatitis C patients with non-response or relapse to interferonScandinavian Journal of Gastroenterology
Journal of the American Society of NephrologyAssociation between serum lipids and survival in hemodialysis patients and impact of raceJournal of the American Society of Nephrology
American Journal of EpidemiologyInvited commentary: Effect modification by time-varying covariatesAmerican Journal of Epidemiology
Annual Review of Public HealthUS disparities in health: Descriptions, causes, and mechanismsAnnual Review of Public Health
BiometricsConditional generalized estimating equations for the analysis of clustered and longitudinal dataBiometrics
NeuroepidemiologyAdjustment for Selection Bias in Observational Studies with Application to the Analysis of Autopsy DataNeuroepidemiology
Obesity ReviewsWalking for prevention of cardiovascular disease in men and women: a systematic review of observational studiesObesity Reviews
Journal of the American Geriatrics SocietyBlood Pressure and Brain Injury in Older Adults: Findings from a Community-Based Autopsy StudyJournal of the American Geriatrics Society
European Journal of EpidemiologyPredictors of follow-up and assessment of selection bias from dropouts using inverse probability weighting in a cohort of university graduatesEuropean Journal of Epidemiology
American Journal of EpidemiologyDetermining the effect of highly active antiretroviral therapy on changes in human immunodeficiency virus type 1 RNA viral load using a marginal structural left-censored mean modelAmerican Journal of Epidemiology
Journal of Biopharmaceutical StatisticsAnalysis of treatment effectiveness in longitudinal observational dataJournal of Biopharmaceutical Statistics
Paediatric and Perinatal EpidemiologyRelationships between birthweight and biomarkers of chronic disease in childhood: Aboriginal Birth Cohort Study 1987-2001Paediatric and Perinatal Epidemiology
Paediatric and Perinatal EpidemiologyBronchopulmonary dysplasia and brain white matter damage in the preterm infant: a complex relationshipPaediatric and Perinatal Epidemiology
Family PracticeThe course of newly presented unexplained complaints in general practice patients: a prospective cohort studyFamily Practice
Cancer EpidemiologyAcute myeloid leukemia incidence following radiation therapy for localized or locally advanced prostate adenocarcinomaCancer Epidemiology
On recovering a population covariance matrix in the presence of selection bias
Nut consumption and weight gain in a Mediterranean cohort: The SUN study
American Journal of EpidemiologyDirected acyclic graphs, sufficient causes, and the properties of conditioning on a common effectAmerican Journal of Epidemiology
International Journal of ObesityAdjusting for reverse causality in the relationship between obesity and mortalityInternational Journal of Obesity
American Journal of Health-System PharmacyBias: Considerations for research practiceAmerican Journal of Health-System Pharmacy
Developmental PsychologyNew Approaches to Studying Problem Behaviors: A Comparison of Methods for Modeling Longitudinal, Categorical Adolescent Drinking DataDevelopmental Psychology
Paediatric and Perinatal EpidemiologyQuantification of collider-stratification bias and the birthweight paradoxPaediatric and Perinatal Epidemiology
American Journal of Public HealthLife-Course Socioeconomic Position and Incidence of Diabetes Mellitus Among Blacks and Whites: The Alameda County Study, 1965-1999American Journal of Public Health
Paediatric and Perinatal EpidemiologyRepresentativeness of child controls recruited by random digit diallingPaediatric and Perinatal Epidemiology
Pharmacoepidemiology and Drug SafetyStructural accelerated failure time models for survival analysis in studies with time-varying treatmentsPharmacoepidemiology and Drug Safety
European Journal of EpidemiologyExposure to famine during gestation, size at birth, and blood pressure at age 59 y: evidence from the dutch famineEuropean Journal of Epidemiology
International Journal of EpidemiologyCommentary: Selected samples and nebulous measures: some methodological difficulties in life-course epidemiologyInternational Journal of Epidemiology
Nephrology Dialysis Transplantation
Association between number of months below K/DOQI haemoglobin target and risk of hospitalization and death
Nephrology Dialysis Transplantation, 23(5):
Disability and RehabilitationAdiposity and tendinopathyDisability and Rehabilitation
Clinical Journal of the American Society of NephrologyEstimated Effect of Epoetin Dosage on Survival among Elderly Hemodialysis Patients in the United StatesClinical Journal of the American Society of Nephrology
The Role of Disability in Explaining Long-Term Care Utilization
Medical Care, 47():
Lifetime Data AnalysisAdjusting for time-varying confounding in the subdistribution analysis of a competing riskLifetime Data Analysis
Journal of Clinical LipidologyVitamin D is associated with atheroprotective high-density lipoprotein profile in postmenopausal womenJournal of Clinical Lipidology
American Journal of EpidemiologyHernandez-Diaz et al. Respond to "The perils of birth weight"American Journal of Epidemiology
European Journal of EpidemiologyOn the relationship of sufficient component cause models with potential outcome (counterfactual) modelsEuropean Journal of Epidemiology
Journal of Medical ScreeningValidity of self-reported Pap smear history in Norwegian womenJournal of Medical Screening
Paediatric and Perinatal EpidemiologyZ-scores and the birthweight paradoxPaediatric and Perinatal Epidemiology
Lifetime Data AnalysisOn Granger causality and the effect of interventions in time seriesLifetime Data Analysis
European Journal of EpidemiologySelection by socioeconomic factors into the Danish National Birth CohortEuropean Journal of Epidemiology
International Journal of EpidemiologyIntervening on risk factors for coronary heart disease: an application of the parametric g-formulaInternational Journal of Epidemiology
American Journal of EpidemiologyUsing Marginal Structural Measurement-Error Models to Estimate the Long-term Effect of Antiretroviral Therapy on Incident AIDS or DeathAmerican Journal of Epidemiology
Journal of Clinical OncologyAspirin Intake and Survival After Breast CancerJournal of Clinical Oncology
Journal of Oral and Maxillofacial SurgeryOral Function After Oncological Intervention in the Oral Cavity: A Retrospective StudyJournal of Oral and Maxillofacial Surgery
Mutation Research-Fundamental and Molecular Mechanisms of MutagenesisMolecular epidemiology: New rules for new tools ?Mutation Research-Fundamental and Molecular Mechanisms of Mutagenesis
Selection bias in cancer risk estimation from A-Bomb survivors
Radiation Research, 167(6):
Statistics in MedicineUntitledStatistics in Medicine
Pediatric Blood & CancerAssessment of Selection Bias in Clinic-Based Populations of Childhood Cancer Survivors: A Report From the Childhood Cancer Survivor StudyPediatric Blood & Cancer
International Journal of EpidemiologyCohort Profile: The Diabetes Study of Northern California (DISTANCE) - objectives and design of a survey follow-up study of social health disparities in a managed care populationInternational Journal of Epidemiology
Archives of Dermatology
Teledermatologic Consultation and Reduction in Referrals to Dermatologists A Cluster Randomized Controlled Trial
Archives of Dermatology, 145(5):
Statistics in MedicineA simple G-computation algorithm to quantify the causal effect of a secondary illness on the progression of a chronic diseaseStatistics in Medicine
American Journal of EpidemiologyGeneralizing Evidence From Randomized Clinical Trials to Target PopulationsAmerican Journal of Epidemiology
American Journal of EpidemiologyCigarette smoking and incidence of first depressive episode: An 11-year, population-based follow-up studyAmerican Journal of Epidemiology
Journal of Epidemiology and Community HealthDo mother's education and foreign born status interact to influence birth outcomes? Clarifying the epidemiological paradox and the healthy migrant effectJournal of Epidemiology and Community Health
Bmc Public HealthEpidemiology of frequent attenders: a 3-year historic cohort study comparing attendance, morbidity and prescriptions of one-year and persistent frequent attendersBmc Public Health
American Journal of Respiratory and Critical Care MedicineAssociation between Tobacco Smoking and Active Tuberculosis in Taiwan Prospective Cohort StudyAmerican Journal of Respiratory and Critical Care Medicine
Journal of the American Society of Nephrology
Activated injectable vitamin D and hemodialysis survival: A historical cohort study
Journal of the American Society of Nephrology, 16(4):
Annals of Internal Medicine
Evaluation of the quality of prognosis studies in systematic reviews
Annals of Internal Medicine, 144(6):
American Journal of EpidemiologyInvited commentary: Beyond the metrics for measuring neighborhood effectsAmerican Journal of Epidemiology
American Journal of EpidemiologyBirth order and sibship size: Evaluation of the role of selection bias in a case-control study of non-Hodgkin's lymphomaAmerican Journal of Epidemiology
American Journal of Kidney DiseasesRandomized and Observational Studies in Nephrology: How Strong Is the Evidence?American Journal of Kidney Diseases
British Journal of General PracticePredictability of persistent frequent attendance: a historic 3-year cohort studyBritish Journal of General Practice
Bmc Medical Research MethodologyReducing bias through directed acyclic graphsBmc Medical Research Methodology
Statistical Methods in Medical ResearchObservation plans in longitudinal studies with time-varying treatmentsStatistical Methods in Medical Research
Lifetime Data AnalysisRelation between three classes of structural models for the effect of a time-varying exposure on survivalLifetime Data Analysis
American Journal of EpidemiologyTime Scale and Adjusted Survival Curves for Marginal Structural Cox ModelsAmerican Journal of Epidemiology
Basic & Clinical Pharmacology & Toxicology
Comparison of dynamic treatment regimes via inverse probability weighting
Basic & Clinical Pharmacology & Toxicology, 98(3):
Cancer Epidemiology Biomarkers & Prevention
Nonsteroidal anti-inflammatory drugs and risk of esophageal and gastric cancer
Cancer Epidemiology Biomarkers & Prevention, 14(2):
International Journal of Environmental Research and Public HealthProbabilistic Approaches to Better Quantifying the Results of Epidemiologic StudiesInternational Journal of Environmental Research and Public Health
American Journal of EpidemiologyInvited commentary: When bad genes look good - APOE*E4, cognitive decline, and diagnostic thresholdsAmerican Journal of Epidemiology
Cancer Epidemiology Biomarkers & PreventionAntibiotic use and the risk of lung cancerCancer Epidemiology Biomarkers & Prevention
Fertility and SterilitySelecting controls is not selecting "normals": Design and analysis issues for studying the etiology of polycystic ovary syndromeFertility and Sterility
European Journal of EpidemiologyFrom causal diagrams to birth weight-specific curves of infant mortalityEuropean Journal of Epidemiology
Marginal structural models application to estimate the effects of antiretroviral therapy in 5 cohorts of HIV seroconverters
Gaceta Sanitaria, 21(1):
Clinical TrialsNonparametric estimator of relative time with application to the Acyclovir Prevention TrialClinical Trials
Statistics in MedicineAdherence to antiretroviral therapy, virological response, and time to resistance in the Dakar cohortStatistics in Medicine
Journal of Evaluation in Clinical PracticeUsing propensity score-based weighting in the evaluation of health management programme effectivenessJournal of Evaluation in Clinical Practice
Annals of EpidemiologyLifecourse Social Conditions and Racial Disparities in Incidence of First StrokeAnnals of Epidemiology
British Journal of PsychiatrySelective drop-out in longitudinal studies and non-biased prediction of behaviour disordersBritish Journal of Psychiatry
Paediatric and Perinatal EpidemiologySelf-selection and bias in a large prospective pregnancy cohort in NorwayPaediatric and Perinatal Epidemiology
Nephron Clinical PracticeSelection Bias and Information Bias in Clinical ResearchNephron Clinical Practice
Scandinavian Journal of Primary Health CareSerum total cholesterol levels and all-cause mortality in a home-dwelling elderly population: a six-year follow-upScandinavian Journal of Primary Health Care
Bmj-British Medical JournalUse of antidepressants near delivery and risk of postpartum hemorrhage: cohort study of low income women in the United StatesBmj-British Medical Journal
Journal of Clinical EpidemiologyDifferent analyses estimate different parameters of the effect of erythropoietin stimulating agents on survival in end stage renal disease: a comparison of payment policy analysis, instrumental variables, and multiple imputation of potential outcomesJournal of Clinical Epidemiology
Journal of Clinical EpidemiologySuper learning to hedge against incorrect inference from arbitrary parametric assumptions in marginal structural modelingJournal of Clinical Epidemiology
International Journal of EpidemiologyMatched designs and causal diagramsInternational Journal of Epidemiology
American Journal of MedicineAngiotensin-converting Enzyme Inhibitors and Outcomes in Heart Failure and Preserved Ejection FractionAmerican Journal of Medicine
Environmental Health PerspectivesMaternal Urinary Bisphenol A during Pregnancy and Maternal and Neonatal Thyroid Function in the CHAMACOS StudyEnvironmental Health Perspectives
Environmental Health PerspectivesUrban Tree Canopy and Asthma, Wheeze, Rhinitis, and Allergic Sensitization to Tree Pollen in a New York City Birth CohortEnvironmental Health Perspectives
Environmental Health PerspectivesPrenatal and Postnatal Bisphenol A Exposure and Body Mass Index in Childhood in the CHAMACOS CohortEnvironmental Health Perspectives
Statistics in MedicineUncovering selection bias in case-control studies using Bayesian post-stratificationStatistics in Medicine
American Journal of CardiologySelf-Reported Snoring and Risk of Cardiovascular Disease Among Postmenopausal Women (from the Women's Health Initiative)American Journal of Cardiology
Administrative Science QuarterlyThe Failure of Private Regulation: Elite Control and Market Crises in the Manhattan Banking IndustryAdministrative Science Quarterly
American Journal of EpidemiologyDisparate Rates of New-Onset Depression During the Menopausal Transition in 2 Community-based Populations: Real, or Really Wrong?American Journal of Epidemiology
Bmc Medical Research MethodologyA counterfactual approach to bias and effect modification in terms of response typesBmc Medical Research Methodology
Journal of NeurotraumaWho Gets Recruited in Mild Traumatic Brain Injury Research?Journal of Neurotrauma
Arthritis and RheumatismPropensity-Adjusted Association of Methotrexate With Overall Survival in Rheumatoid ArthritisArthritis and Rheumatism
Canadian Medical Association JournalUnexpected predictor-outcome associations in clinical prediction research: causes and solutionsCanadian Medical Association Journal
European Journal of EpidemiologyCompeting risk bias to explain the inverse relationship between smoking and malignant melanomaEuropean Journal of Epidemiology
Alzheimer Disease & Associated DisordersSmall Head Circumference is Associated With Less Education in Persons at Risk for Alzheimer Disease in Later LifeAlzheimer Disease & Associated Disorders
Critical Care MedicineAcute lung injury in patients with subarachnoid hemorrhage: Incidence, risk factors, and outcomeCritical Care Medicine
EpidemiologyThe Hazards of Hazard RatiosEpidemiology
EpidemiologyDoes Low Participation in Cohort Studies Induce Bias?Epidemiology
EpidemiologyBias Formulas for Sensitivity Analysis for Direct and Indirect EffectsEpidemiology
Obstetrics & GynecologyMaternal Hypertension and Neonatal Outcome Among Small for Gestational Age InfantsObstetrics & Gynecology
Journal of Developmental & Behavioral PediatricsStress and Emotional Problems During Pregnancy and Excessive Infant CryingJournal of Developmental & Behavioral Pediatrics
Supplemental Digital Content
© 2004 Lippincott Williams & Wilkins, Inc.