Reducing bias through directed acyclic graphs
© Shrier and Platt; licensee BioMed Central Ltd. 2008
Received: 11 March 2008
Accepted: 30 October 2008
Published: 30 October 2008
The objective of most biomedical research is to determine an unbiased estimate of effect for an exposure on an outcome, i.e. to make causal inferences about the exposure. Recent developments in epidemiology have shown that traditional methods of identifying confounding and adjusting for confounding may be inadequate.
The traditional methods of adjusting for "potential confounders" may introduce conditional associations and bias rather than minimize it. Although previous published articles have discussed the role of the causal directed acyclic graph approach (DAGs) with respect to confounding, many clinical problems require complicated DAGs and therefore investigators may continue to use traditional practices because they do not have the tools necessary to properly use the DAG approach. The purpose of this manuscript is to demonstrate a simple 6-step approach to the use of DAGs, and also to explain why the method works from a conceptual point of view.
Using the simple 6-step DAG approach to confounding and selection bias discussed is likely to reduce the degree of bias for the effect estimate in the chosen statistical model.
The objective of most biomedical research, whether experimental or observational, is to predict what will happen to an outcome if the treatment is applied to a group of individuals or if a harmful exposure is removed. In other words, the clinician/policy maker is interested in making causal inferences from the results of a study. The purpose of this manuscript is to demonstrate a simple 6-step algorithm for determining whether a proposed set of covariates would reduce possible sources of bias when assessing the total causal effect of a treatment on an outcome.
There are many nuances to the definition of cause. For the purposes of this manuscript, we define it in counterfactual terms: "Had the exposure differed, the outcome would have differed", where exposure or outcome may be dichotomous (e.g. presence/absence of exposure; occurrence/disappearance of disease) or continuous (e.g. a different value of blood pressure whether blood pressure is exposure or outcome). Further refinements into sufficient, complementary and necessary causes  are important but do not alter the essence of the definition. Although the above causal definition is deterministic at the individual level, in almost all practical settings the outcome under the counterfactual condition is unknown. Therefore, researchers are limited to causal inference at the population level (e.g. comparing average risks) . A straightforward explanation of the use of counterfactuals to define cause can be found in .
The traditional approach to confounding is to 'adjust for it' by including certain covariates in a multiple regression model (or by stratification). One common practice is to consider a covariate to be a confounder (and "adjust" for it) if it is associated with the exposure, associated with outcome, and changes the effect estimate when included in the model. According to standard textbooks, additional criteria also need to be applied and the covariate should not be affected by exposure and needs to be an independent cause of the outcome . However, recent advances in epidemiology have proven that even these additional criteria are insufficient. In fact, the methods described above may introduce conditional associations (sometimes called selection bias [5, 6], collider bias  and confounding bias [6, 7]; this terminology may be confusing and we prefer the terminology suggested by the structural approach to bias as described later) and create bias where none existed, which is in direct contrast to the objective of eliminating an existing bias [2, 8, 9]. Some published examples include the effectiveness of HIV treatment , and why birth weight should not be included as a covariate when examining the causal effects of exposure during pregnancy on perinatal outcomes .
One method to help understand whether bias is potentially reduced or increased when conditioning on covariates is the graphical representation of causal effects between variables. In the causal directed acyclic graph (DAG) approach, an arrow connecting two variables indicates causation; variables with no direct causal association are left unconnected. Therefore the bi-directional arrows in figure 1a are replaced with unidirectional arrows (figure 1b). There are of course situations where each variable may cause the other – the functional disability created by chronic pain may cause depression, and depression may cause chronic pain through diminished pain thresholds. These more complex situations are simplified by understanding that time is a component in the above relationship. Therefore, there is a variable for depression at time 1, chronic pain at time 1, depression at time 2, and chronic pain at time 2; the same construct measured at different times represents distinct variables and must be treated as such.
Although other articles have previously described the DAG approach to confounding [9, 12, 13], the articles demonstrate relatively simple DAGs. However, many clinical problems require complicated DAGs and little has been published on how to assess whether a particular subset of covariates potentially reduces or increases bias in this context [6, 9]. Therefore, although many investigators now understand the problem, they continue to use traditional practices because they do not have the tools necessary to choose the statistical model that is most likely to yield an unbiased effect estimate. The objective of this manuscript is to demonstrate a simple 6-step approach developed by Pearl  that helps determine when the traditional statistical approaches of regression/stratification on specific covariates is likely to reduce or increase bias, and to provide our explanation as to why the method works from a conceptual point of view. Although this manuscript is limited to the conceptual discussion necessary for clinical researchers to use DAGs, there are many different facets and a complete theoretical development of these materials has been published elsewhere and has been summarised in one source . Readers are also encouraged to learn more about counterfactual random variables, an important complement to the theory of DAGs [3, 16].
The Pragmatic Solution: a Six-Step Process Towards Unbiased Estimates 
Figure 2a is one possible causal diagram for the relationship between warming up prior to exercise and the outcome injury (we will show another possible causal diagram later in the manuscript). The question we want to answer is whether including a measure of neuromuscular fatigue (Z1) and tissue weakness (Z2) (in the design or analysis stage) would minimize bias in the estimate of the effect of warming up on injury if this is the true causal diagram. We will later discuss how to approach the more general problem when multiple causal diagrams are possible. As with any analytic approach to bias in an observational study (including the one below), we must make some assumptions regarding how variables are causally related to each other; we seek to determine whether our analytic approach would succeed under these assumptions. The algorithm we describe below only works if the DAGs are drawn so that they include all variables that cause two or more other variables shown in the DAG . In other words, no common causes can be omitted from the DAG. Finally, the DAG approach does not reduce or eliminate other sources of bias (e.g. measurement bias). Finally, at the end of the manuscript, we have provided a glossary of terms used so that readers unfamiliar with DAG terminology have an easy reference immediately available (genealogical terms are often used to describe relationships between variables).
Step 1 (figure 2a): The covariates chosen to reduce bias [fatigue (Z1) and tissue weakness (Z2) in this case] should not be descendants of X (i.e. they should not be caused by warming up)
This does not occur in this situation and one can proceed to Step 2
Step 1 ensures that the covariates chosen are possible confounders in the traditional sense of the word; if the covariates are descendants of X, then the statistical model adjusting for these variables may yield a biased estimate for the total causal effect of X on the outcome and a different set of covariates needs to be chosen. The step is required because confounding bias (as defined by the structural approach) can only occur if a covariate causes the exposure or is a marker for a cause of exposure (note that other biases are still possible and discussed in Step 4). Although more formal proofs exist , this can be deduced from the following standard criteria for a potential confounder: the covariate must be associated with the exposure and with the outcome, but cannot be affected (i.e. caused) by exposure [4, 18] (for completeness, these standard criteria are in fact insufficient to define confounding , and more complicated scenarios such as time-dependent confounding  are not covered by the standard definitions). Because it is inappropriate to include a variable that lies along a causal pathway between the exposure of interest and the outcome, it is also inappropriate to include a marker for a variable that lies along a causal pathway. For example, if the marker is 100% correlated with the causal pathway variable, there is no mathematical difference in a statistical model between the marker and the causal pathway variable. Thus, if a covariate is associated with an exposure, and the exposure cannot cause or be a marker for a cause of the covariate, then the covariate must cause (or be a marker for a cause of) the exposure. By similar reasoning, one can deduce that confounding only occurs if the covariate also causes, or is a marker for a cause of the outcome. Returning to the DAG, if the covariate is a descendant of X, it means the exposure is a cause of the covariate.
Step 2 (figure 2b): Delete all variables that satisfy all of the following: 1) non-ancestors (an ancestor is a variable that causes another variable either directly or indirectly) of X, 2) non-ancestors of the Outcome and 3) non-ancestors of the covariates that one is including to reduce bias (Z1 and Z2in this example)
In figure 2a, the only covariate that fulfills this criterion is previous injury (Z3) and this is deleted in figure 2b. Note that the exposure, outcome and covariates should not be deleted
Step 2 is essential because after completing the step, all variables left are either conditioned on, or have one of their descendants conditioned on. The importance of this result will become clear in Step 4.
Step 3 (figure 3a): Delete all lines emanating from X
In this setting, warming-up causes a change in proprioception, and therefore we delete this arrow
In Step 3, deleting all lines emanating from X effectively simplifies the DAG because we have already said that X should not be a cause of the covariates in the model. We leave the variables in and eliminate the line because these variables may be responsible for bias through an indirect pathway. This will become clearer in Step 4, and the example where we include a third covariate, which results in the introduction of bias.
Step 4 (figure 3b): Connect any two parents (direct causes of a variable) sharing a common child (this step appears simple but it requires practice not to miss any)
For example, team motivation and poor proprioception can both cause an individual to warm-up more than someone without these factors – these two variables are joined because they share a common effect
Step 4 is essential for the following reason. If two covariates both cause a third covariate, then adjustment for the third covariate (or an effect of the third covariate) creates a conditional association between the first two covariates (i.e. if one conditions on the child or descendant of the child, there is a conditional association between the parents), and could introduce bias . For example, both rain and sprinklers can cause a football field to be wet. If one knows the grass is wet, then knowing the sprinklers were off improves your assessment of the probability that it rained; rain and sprinklers become associated when the common effect of "field wetness" is known. Consider a second example from the health sciences: both a thrombus and a haemorrhage can cause a stroke. If we condition on the patient having symptoms of a stroke and learn that there was no haemorrhage, the probability that a thrombotic event occurred is increased. By connecting the two parents of a common child in the figure after Steps 1–3 are completed, we are explicitly stating that we understand that these variables are associated because we have either conditioned on the value of the child or one of the child's descendants (otherwise the variable would have been removed in Step 2). As we shall later see, it is this conditional association that can cause the introduction of bias when traditional rules of confounding adjustment are applied without reference to a DAG. In DAG terminology, the child is called a "collider" because two directed arrows collide at the covariate (node).
Step 5 (figure 4a): Strip all arrowheads from lines
In Step 5, we strip all the arrowheads from the lines. This is because the arrowheads (causal direction) were only necessary to note the conditional associations created between two parents of a collider. Once this is done, we can simplify the diagram as we have now completed all the steps related to causation.
Step 6 (figure 4b): Delete all lines between the covariates in the model and any other variables
All lines into and out of Neuromuscular fatigue (Z1) and tissue weakness (Z2) are deleted
Step 6 is simply the graphical equivalent of standard regression techniques. When a covariate is included, the estimate of the effect represents the relationship between the exposure and the outcome independent of any causal pathway going through that covariate; including the covariate "blocks" all associations occurring through this pathway. Therefore, we can delete all lines between the covariates included in the model and any other covariates.
Interpretation: If X is dissociated from the outcome after Step 6, then the statistical model chosen (i.e. one that includes only the chosen covariates) will minimize the bias of the estimate of X on the chosen outcome
If this causal model is correct, then a statistical model that includes a measure of tissue weakness and neuromuscular fatigue minimizes the bias in the estimate of the effect of warming up on the risk of injury
We have now deleted all the direct causal pathways between the exposure of interest and the outcome, and between the covariates and the outcome, and explicitly noted the conditional associations created by including specific covariates with two different causes as explained in step 4. If there is no uninterrupted series of lines through nodes from X to the outcome after completing the six steps (figure 4b), then within this specific causal DAG, there is no non-causal structural association between X and the outcome. In other words, any measured association between the exposure and outcome that exists conditional on the covariates in the model minimizes the bias in the estimate of the causal relationship.
When including covariates creates a conditional association and introduces bias
Understanding the conditional associations naturally leads to what is sometimes known as the structural approach to bias [5, 15]. Using this approach, epidemiologic biases can be categorized as either lack of conditioning on a common cause (known as confounding bias), or conditioning on a common effect of two parents (or a descendant of the common effect; known as selection bias). The typical selection bias described in observational studies is due to conditioning on a common effect (one conditions on willingness to participate), as are Berkson's bias (conditioning on admission to hospital), loss to follow-up or missing data (conditioning on presence of data; occurs in both observational or randomized trials), some forms of Simpson's Paradox, etc [5, 12]. Indeed, we believe it is possible to represent all epidemiologic biases in DAGs; therefore, the restrictions we set out at the beginning of this article concerning an ideal study were used only as a pedagogical tool and are not necessary for this approach.
Selecting a subset of covariates that minimizes the bias in the estimate of the effect requires trial and error and a sound foundation of the theoretical model. At the present time, there is no algorithm and the six-step process should be repeated until a subset of covariates is found such that X is dissociated from the outcome after the 6-step process is completed.
There are two other potential advantages to the DAG approach. First, only a subset of covariates that are associated with both exposure and outcome are necessary to yield an unbiased estimate of effect. Second, because one may require fewer covariates in the model, the statistical efficiency of the analysis is increased (i.e. there are more degrees of freedom if one uses fewer covariates).
Limitations to the 6-Step approach
The immediate question that always arises is how can one know the true underlying causal structure in order to draw the DAG (i.e. Step 0) – if we knew it, we wouldn't have to study the disease. Although it can be a challenging exercise, the fact remains that understanding the causal structure is an essential step when one wants to know if including a covariate is likely to reduce or increase bias in the effect estimate. In other words, the DAG representing the true causal structure exists even if we do not know what it is, and all causal inferences based on statistical models are implicitly based on a causal structure – the DAG approach simply makes the assumptions explicit.
Drawing causal DAGs can be challenging. Causal DAGs represent theory, and theory needs to be developed within the context of all the evidence (basic science, observational and clinical trials) available. Because of this, generating a causal DAG necessarily requires the collaboration of methodological experts, clinicians, physiologists, and others (e.g. psychologists, sociologists) depending on the particular question. The inclusion of latent (unmeasured) variables poses additional problems [21, 22]. For many conditions, it is likely that even after reviewing all the evidence, we still won't have enough information to determine if one particular DAG is more appropriate than another DAG. Under these conditions, it is necessary to draw each of the possible DAGs and determine if the same choice of covariates yields an unbiased estimate for each. If not, then one should present each of the interpretations and future research will determine which causal diagram, and which interpretation is correct. Not using the causal approach because of uncertainty on which is the correct DAG simply means that one is allowing chance rather than rational deliberation to make the choice among the different causal diagrams. A further corollary of the structural approach to bias is that an understanding of biological mechanisms and basic science is necessary for appropriate epidemiological studies, and that cross-discipline collaborations should be encouraged.
The DAG approach requires a "node" for each of the covariates. Effect modifiers or covariates that interact with other covariates in a synergistic or antagonistic manner are not currently indicated as such in a DAG. Although there is some theoretical work currently being done in this area (e.g. [23, 24]), one can conceptually think of two binary variables that interact as a single variable with multiple levels, and include them as a single node in the DAG. This is somewhat analogous to treating socio-economic status as one variable in a model even though it represents the two distinct constructs of sociological and economical influences. As is known, when two variables both cause a third variable, there is interaction on either the multiplicative scale, additive scale, or both. Therefore, if a DAG were to model synergism or antagonism, one would need different DAGs for different measures of effect (e.g. risk ratio versus risk difference). Finally, issues related to sufficient and component causes have also recently been addressed elsewhere .
The DAG approach is not a statistical technique that yields an estimate of effect. However, it will allow users of traditional stratification and regression techniques to reduce the magnitude of the bias in the estimate. Although researchers should generally not adjust for a covariate (or a marker for a covariate) that lies along a causal pathway when assessing the total causal effect, this may not be the case for researchers interested in decomposing total causal effects into direct and indirect effects. In these cases, one may sometimes need to include covariates that lie along the causal path, but this is a process that needs to be carefully thought out or incorrect inferences may occur [26, 27]. We also think it is important to highlight the effect of newer statistical techniques to assess total causal effect like marginal structural models  that are often necessary in special situations, such as when the covariate is affected by exposure or when a covariate is both a "collider" and a "confounder" at the same time [29, 30].
The traditional approach to confounding bias by determining only associations and avoiding discussions related to causation is problematic and has led to inappropriate data analysis and interpretation [10, 13]. The DAG approach can be used to help choose which covariates should be included in traditional statistical approaches in order to minimize the magnitude of the bias in the estimate produced. Investigators should become aware of the other statistical causal approaches available so that the appropriate technique is used to answer the appropriate question.
A short summary of the Six-Step Process Towards Unbiased Estimates
Step 1. The covariates chosen to reduce bias should not be descendants of X
Step 2. Delete all variables that satisfy all the following criteria: 1) non-ancestors of X, 2) non-ancestors of the outcome and 3) non-ancestors of the covariates that one is including in the model to reduce bias.
Step 3. Delete all lines emanating from X.
Step 4. Connect any two parents sharing a common child.
Step 5. Strip all arrowheads from lines.
Step 6. Delete all lines between the covariates in the model and any other covariates
Interpretation: If X is dissociated from the outcome after Step 6, then the statistical model chosen (i.e. one that includes only the chosen covariates) minimizes the bias of the estimate of X on the chosen outcome.
Glossary of Terms
Genealogy: The DAG approach often uses terms familiar in genealogy
1. Parent: A parent is a direct cause of a particular variable.
2. Ancestor: An ancestor is a direct cause (i.e. parent) or indirect cause (e.g. grandparent) of a particular variable.
3. Child: A child is the direct effect of a particular variable, i.e. the child is a direct effect of the parent.
4. Descendant: A descendant is a direct effect (i.e. child) or indirect effect (e.g. grandchild) of a particular variable.
Causes, Effects and Associations
1. Common Cause: A common cause is covariate that is an ancestor of two other covariates.
2. Common Effect (also known as collider): A common effect is a covariate that is a descendant of two other covariates. The term collider is used because the two arrows from the parents "collide" at the node of the descendant.
3. Conditioning: Conditioning on a variable means that one has used either sample restriction or stratification/regression (stratification/regression being two forms of the same mathematical approach) to examine the association of exposure and outcome within levels of the conditioned variable. Other terms often used such as "adjusting for" or "controlling for" suggest an interpretation of the statistical model that is sometimes misleading and therefore we prefer the word conditioning.
4. Unconditional Association: If knowing the value of one covariate provides information on the value of the other covariate without conditioning on any other variable, the two variables are said to be unconditionally associated. This is also known as marginal statistical dependence and its absence as marginal statistical independence.
5. Conditional Association: If knowing the value of one covariate provides information on the value of the other covariate after conditioning on one or more covariates (i.e. within any level of the conditioned covariate(s)), the two variables are said to be conditionally associated. This is also known as conditional statistical dependence and its absence as conditional statistical independence.
1. Confounding bias: occurs when there is a common cause of the exposure and outcome that is not "blocked" by conditioning on other specific covariates.
2. Selection bias: occurs when one conditions on a common effect (e.g. Berkson's Bias, loss to follow-up, missing data, healthy worker bias, etc) such that there is now a conditional association between the exposure and the outcome.
Dr. Shrier is a recipient of a Clinical Investigator Award from the Fonds de la Recherche en Santé du Québec (FRSQ). Dr. Platt is a recipient of an Investigator Award from the FRSQ, and is a member of the the Research Institute of the McGill University Health Centre, which is supported in part by the FRSQ.
- Rothman KJ, Greenland S: Causation and causal inference. Modern Epidemiology. Edited by: Rothman KJ, Greenland S. 1998, Philadelphia: Lippencott-Raven Publishers, 2: 7-28.Google Scholar
- Hernan MA: A definition of causal effect for epidemiological research. J Epidemiol Community Health. 2004, 58 (4): 265-271. 10.1136/jech.2002.006361.View ArticlePubMedPubMed CentralGoogle Scholar
- Greenland S, Morgenstern H: Confounding in health research. Annu Rev Public Health. 2001, 22: 189-212. 10.1146/annurev.publhealth.22.1.189.View ArticlePubMedGoogle Scholar
- Rothman KJ, Greenland S: Precision and validity in epidemiologic studies. Modern Epidemiology. Edited by: Rothman KJ, Greenland S. 1998, Philadelphia: Lippencott-Raven Publishers, 2: 115-134.Google Scholar
- Hernan MA, Hernandez-Diaz S, Robins JM: A structural approach to selection bias. Epidemiology. 2004, 15 (5): 615-625. 10.1097/01.ede.0000135174.63482.43.View ArticlePubMedGoogle Scholar
- Glymour MM, Greenland S: Causal Diagrams. Modern Epidemiology. Edited by: Rothman KJ, Greenland S. 2008, Philadelphia: Lippencott-Raven Publishers, 3: 183-209.Google Scholar
- Greenland S: Quantifying biases in causal models: classical confounding vs collider-stratification bias. Epidemiology. 2003, 14 (3): 300-306. 10.1097/00001648-200305000-00009.PubMedGoogle Scholar
- Weinberg CR: Toward a clearer definition of confounding. Am J Epidemiol. 1993, 137 (1): 1-8.PubMedGoogle Scholar
- Greenland S, Pearl J, Robins JM: Causal diagrams for epidemiologic research. Epidemiology. 1999, 10 (1): 37-48. 10.1097/00001648-199901000-00008.View ArticlePubMedGoogle Scholar
- Hernan MA, Brumback B, Robins JM: Marginal structural models to estimate the causal effect of zidovudine on the survival of HIV-positive men. Epidemiology. 2000, 11 (5): 561-570. 10.1097/00001648-200009000-00012.View ArticlePubMedGoogle Scholar
- Hernández-Díaz S, Schisterman EF, Hernán MA: The birth weight "paradox" uncovered?. Am J Epidemiol. 2006, 164 (11): 1115-1120. 10.1093/aje/kwj275.View ArticlePubMedGoogle Scholar
- Pearl J: Simpson's paradox, confounding, and collapibility. Causality: models, reasoning and inference. 2000, Cambridge University of Cambridge, 173-200.Google Scholar
- Hernan MA, Hernandez-Diaz S, Werler MM, Mitchell AA: Causal knowledge as a prerequisite for confounding evaluation: an application to birth defects epidemiology. Am J Epidemiol. 2002, 155 (2): 176-184. 10.1093/aje/155.2.176.View ArticlePubMedGoogle Scholar
- Pearl J: The art and science of cause and effect. Causality: models, reasoning and inference. 2000, Cambridge University of Cambridge, 331-358.Google Scholar
- Pearl J: Causality: models, reasoning and inference. 2000, Cambridge University of CambridgeGoogle Scholar
- Holland PW: Statistics and causal inference. J Amer Statist Assoc. 1986, 81: 945-960. 10.2307/2289064.View ArticleGoogle Scholar
- Spirtes P, Glymour C, Scheines R: Causation and prediction: axioms and explications. Causation, prediction and search. 2000, Cambridge: MIT Press, 19-58.Google Scholar
- Greenland S, Rothman KJ: Introduction to stratified analysis. Modern Epidemiology. Edited by: Rothman KJ, Greenland S. 1998, Philadelphia: Lippencott-Raven Publishers, 2: 253-279.Google Scholar
- Robins JM: The control of confounding by intermediate variables. Stats Med. 1989, 8: 679-701. 10.1002/sim.4780080608.View ArticleGoogle Scholar
- Pearl J: Introduction to probabilities, graphs, and causal models. Causality: models, reasoning and inference. 2000, Cambridge University of Cambridge, 1-40.Google Scholar
- Spirtes P, Glymour C, Scheines R: Discovery algorithms for causally sufficient structures. Causation, prediction and search. 2000, Cambridge: MIT Press, 73-122.Google Scholar
- Spirtes P, Glymour C, Scheines R: Discovery algorithms without causal sufficiency. Causation, prediction and search. 2000, Cambridge: MIT Press, 123-155.Google Scholar
- Weinberg CR: Can DAGs clarify effect modification?. Epidemiology. 2007, 18 (5): 569-572. 10.1097/EDE.0b013e318126c11d.View ArticlePubMedPubMed CentralGoogle Scholar
- VanderWeele TJ, Robins JM: Four types of effect modification: a classification based on directed acyclic graphs. Epidemiology. 2007, 18 (5): 561-568. 10.1097/EDE.0b013e318127181b.View ArticlePubMedGoogle Scholar
- Vanderweele TJ, Robins JM: Directed Acyclic Graphs, Sufficient Causes, and the Properties of Conditioning on a Common Effect. Am J Epid. 2007, 166: 1096-1104. 10.1093/aje/kwm179.View ArticleGoogle Scholar
- Kaufman JS, Maclehose RF, Kaufman S: A further critique of the analytic strategy of adjusting for covariates to identify biologic mediation. Epidemiol Perspect Innov. 2004, 1 (1):
- Cole SR, Hernan MA: Fallibility in estimating direct effects. Int J Epidemiol. 2002, 31 (1): 163-165. 10.1093/ije/31.1.163.View ArticlePubMedGoogle Scholar
- Robins JM, Hernan MA, Brumback B: Marginal structural models and causal inference in epidemiology. Epidemiology. 2000, 11 (5): 550-560. 10.1097/00001648-200009000-00011.View ArticlePubMedGoogle Scholar
- Haight T, Tager I, Sternfeld B, Satariano W, Laan van der M: Effects of body composition and leisure-time physical activity on transitions in physical functioning in the elderly. Am J Epidemiol. 2005, 162 (7): 607-617. 10.1093/aje/kwi254.View ArticlePubMedGoogle Scholar
- Witteman JC, D'Agostino RB, Stijnen T, Kannel WB, Cobb JC, de Ridder MA, Hofman A, Robins JM: G-estimation of causal effects: isolated systolic hypertension and cardiovascular death in the Framingham Heart Study. Am J Epidemiol. 1998, 148 (4): 390-401.View ArticlePubMedGoogle Scholar
- The pre-publication history for this paper can be accessed here:http://www.biomedcentral.com/1471-2288/8/70/prepub
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