Estimating causal effects

Although one goal of aetiologic epidemiology is to estimate ‘the true effect’ of an exposure on disease occurrence, epidemiologists usually do not precisely specify what ‘true effect’ they want to estimate. We describe how the counterfactual theory of causation, originally developed in philosophy and statistics, can be adapted to epidemiological studies to provide precise answers to the questions ‘What is a cause?’, ‘How should we measure effects?’ and ‘What effect measure should epidemiologists estimate in aetiologic studies?’ We also show that the theory of counterfactuals (1) provides a general framework for designing and analysing aetiologic studies; (2) shows that we must always depend on a substitution step when estimating effects, and therefore the validity of our estimate will always depend on the validity of the substitution; (3) leads to precise definitions of effect measure, confounding, confounder, and effect-measure modification; and (4) shows why effect measures should be expected to vary across populations whenever the distribution of causal factors varies across the populations. Imagine that the creator of the universe appears to you in a dream and grants you the answer to one public-health question. The conversation might go as follows: You: What is the true effect of (your exposure here, denoted by E) on the occurrence of (your disease here, denoted by D)? Creator: What do you mean by ‘the true effect’? The true value of what parameter? You: The true relative risk. Creator: Epidemiologists use the term relative risk for several different parameters. Which do you mean? You: The ratio of average risk with and without exposure—what some call the risk ratio1 and others call the incidence proportion ratio.2 Creator: Which incidence proportion ratio? You: Pardon? Creator: Do you want a ratio of average disease risk in two different groups of people with different exposure levels? You: Yes. Creator: So you want a descriptive incidence proportion ratio? You: No, not descriptive. Causal. An incidence proportion ratio that isolates the effect of E on D from all other causal factors. Creator: By ‘isolate’, you mean a measure that applies to a single population under different possible exposure scenarios? You: Yes, that's what I mean. Creator: OK. Which causal incidence proportion ratio? You: Pardon? Creator: For what population, and for what time period? The true value of a causal incidence proportion ratio can be different for different groups of people and for different time periods. It's not necessarily a biological constant, you know. You: Yes, of course. For population (your population here, denoted by P) between the years (your study time period here, denoted by t0 to t1). Creator: By population P, do you mean: (1) everyone in population P, or (2) the people in population P who have a specific set of characteristics? You: Pardon? Creator: As I just said, the true value of a causal incidence proportion ratio is not necessarily a biological constant. It can be different for subgroups of a population. You: Of course. Everyone in population P. Creator: OK. Comparing what two exposure levels? You: Exposed and unexposed. Creator: What do you mean by exposed and unexposed? Exposed for how long, to how much, and during what time period? There are many different ways you could define exposed and unexposed, and each of the corresponding possible ratios can have a different true value, you know. You: Of course. Ever exposed to any amount of E versus never exposed to E. Creator: The incidence proportion ratio for the causal effect on D of ever E compared to never E in population P during the study time period t0 to t1 is (your causal incidence-proportion-ratio parameter value here). The point of the above is that, while one goal of etiologic epidemiology is to estimate ‘the true effect’ of an exposure on disease frequency, we usually do not precisely specify what ‘true effect’ we want to estimate. We may not be able to do so. For example, before reading this paper would you have required less prompting than in the dialog above? How many published papers explicitly state what the authors mean by ‘true’ relative risk or odds ratio, or whether the estimated measure of association is intended to have a descriptive or causal interpretation? How many papers explicitly define the population or time period of interest? How many etiologic papers over-emphasize results that cannot be given a causal interpretation, such as significance tests, P-values, correlation coefficients, or proportion of variance ‘explained’? In this paper we discuss the questions ‘What is a cause?’, ‘How should we measure effects?’ and ‘What effect measure should epidemiologists estimate in etiologic studies?’ We begin by adapting the counterfactual approach to causation, originally developed in philosophy and in statistics,3,4 to epidemiological studies. In the process, we give precise answers to these questions, and we describe how these answers have important implications for etiologic research: (1) Under the counterfactual approach, the measure we term a ‘causal contrast’ is the only meaningful effect measure for etiologic studies. (2) The counterfactual approach provides a general framework for designing and analysing epidemiological studies. (3) The counterfactual definition of causal effect shows why direct measurement of an effect size is impossible: We must always depend on a substitution step when estimating effects, and the validity of our estimate will thus always depend on the validity of the substitution.3,5–7 (4) The counterfactual approach makes clear that a critical step in study interpretation is the formal quantification of bias in study results. (5) The counterfactual approach leads to precise definitions of effect measure, confounding, confounder, and to precise criteria for effect-measure modification. In the discussion that follows, we assume that the study outcome is a disease (e.g. lung cancer); this discussion can be readily extended to any outcome (e.g. a health behaviour such as cigarette smoking). We also assume for simplicity that disease occurrence is deterministic; under a stochastic model, the quantities we discuss are probabilities or expected values.6,7 In 1748, the renowned Scottish philosopher David Hume wrote ‘we may define a cause to be an object followed by another … where, if the first object had not been, the second never had existed.’3,8 A key innovation of this definition was that it pivoted on a clause of the form ‘if C had not occurred, D would not have either’, where C and D are what actually occurred. Such a clause, which hypothesizes what would have happened under conditions contrary to actual conditions, is called a counterfactual conditional. Despite its early appearance, this counterfactual concept of causation received no formal basis until 1923 when the statistician Jerzy Neyman presented a quantitative conceptual model for causal analysis.9 This model was originally known as the ‘randomization model’10 and was later called the ‘potential-outcomes model’ (or, inaccurately, ‘Rubin's model’) when extended to observational studies.11 The model has since been widely (though not universally12) adopted by statisticians and others seeking a logical foundation for statistical analysis of causation.4,10,13–16 These developments were paralleled by more extensive analysis of counterfactual reasoning by philosophers.17–20 A comprehensive review of causality theory is provided by Pearl,15 who shows how structural-equation models and graphical causal models (causal diagrams) translate directly to counterfactual models, shedding light on all three approaches. A brief review of these connections is given by Greenland,21 and Greenland et al.22 provide a more extensive review of graphical causal modelling for epidemiological research. We will use the term target population for the group of people about which our scientific or public-health question asks, and therefore for which we want to estimate the causal effect of an exposure. The target population could be composed of one group of people (as in most epidemiological studies), several groups of people (as in an intervention study in several communities), or one person. For simplicity, in the rest of this discussion we assume that the target population is one group of people. Let the etiologic time period be the time period about which our scientific or public-health question asks. The beginning and end of this time period is specified by the study question. For example, in a study of the effectiveness of a back-injury prevention programme in a workplace, the etiologic time period could be any time period after the implementation of the programme. Note that this period may vary among individuals (e.g. the etiologic time period for a study of weight gain during pregnancy and pre-eclampsia spans only pregnancy), and not all of the period need be time at risk. For example, the etiologic time period for a study of intrauterine diethylstilbesterol exposure and subsequent fertility problems could include childhood, a time at no risk of such problems but during which etiologically relevant events (e.g. puberty) occur. Let target refer to the target population during the etiologic time period. Consider one target population during one etiologic time period, but under two different exposure distributions, as illustrated below. Let the subscript 1 denote one exposure distribution, and let the subscript 0 denote the other. These distributions represent different possible mixtures of individual exposure conditions. With, say, smoking as the exposure, distribution 0 could represent conditions under which 20% of the target population regularly smoked cigarettes during a given time period, whereas distribution 1 could represent conditions under which 40% (instead of 20%) of the population did so during that period. Then R1= A1/B1 is disease frequency if the target population had experienced exposure distribution 1, and R0= A0/B0 is disease frequency if the same group of people during the same time period had instead experienced exposure distribution 0. Synonyms for causal contrast are effect measure and causal parameter2. A causal contrast compares disease frequency under two exposure distributions, but in one target population during one etiologic time period. This type of contrast has two important consequences. First, the only possible reason for a difference between R1 and R0 is the exposure difference. A causal contrast, therefore, measures the causal effect of the difference between exposure distributions 1 and 0 in the target population during the etiologic time period.2,4–7,23 Second, a causal contrast cannot be observed directly, as we explain below, and therefore a different type of measure must be used as a substitute for it. Why is it not possible to directly observe a causal contrast? Because at least one of the disease-frequency parameters needed for a causal contrast, R1 and R0, must be counterfactual and therefore unobservable. A parameter (such as a disease frequency) that describes events under actual conditions is said to be actual (or factual); in contrast, a parameter that describes events under a hypothetical alternative to actual conditions is said to be counterfactual.2,3,7,17 Counterfactual parameters cannot be observed because, by their very definition, they describe consequences of conditions that did not exist—they describe events following hypothetical alternatives to actual conditions, not actual conditions. The entire collection of outcome parameters for the target, actual and counterfactual (here, R1, R0 and all Ri under all other exposure conditions), is sometimes called the set of potential outcomes, to note that each is a possibility before the exposure distribution becomes fixed.11,23 For a given R1 and R0, one and only one of the following three scenarios may occur. R1 is an actual disease frequency; it occurs, and therefore it can be observed. R0, then, must be counterfactual; as a hypothetical alternative to R1 it does not occur, and therefore it cannot be observed (as illustrated below). R0 is an actual disease frequency; it occurs, and therefore it can be observed. R1, then, must be counterfactual; as a hypothetical alternative to R0 it does not occur, and therefore it cannot be observed. Both R1 and R0 are counterfactual disease frequencies—both are hypothetical alternatives to the actual disease frequency that occurs under the actual exposure distribution (which is neither exposure distribution 1 nor 0), and therefore neither R1 nor R0 can occur and be observed. A causal contrast requires two quantities, at least one of which must be counterfactual and therefore unobservable. How, then, can we estimate a causal contrast? By using substitutes (illustrated below) for the counterfactual disease frequencies in the target. As before, the subscript indicates the exposure distribution. In a substitute under exposure distribution 1, let C1 be the numerator of a disease-frequency measure, and let D1 be the denominator (number of people or amount of person-time at risk). Likewise, in a substitute under exposure distribution 0, let E0 be the numerator, and let F0 be the denominator. In epidemiological practice, a substitute will usually be a population other than the target population during the etiologic time period. It may be the target population observed at a time other than the etiologic time period, or a population other than the target population. In theory, however, a substitute can be any source of information about a counterfactual parameter. Below we describe how these quantities are used to predict5–7 or impute11 the counterfactual quantities in the causal contrast of interest. In this scenario, the exposed substitute is often the exposed subset of the target population, and the unexposed substitute is often the unexposed subset of the target population. That is, C1, D1, E0, and F0 are often subsets of A1, B1, A0, and B0, respectively. They may, however, have no overlap at all with the target population; this is the case whenever we generalize from a study to an external target population (see below). The counterfactual approach and the concept of a causal contrast have important implications for designing, analysing, and interpreting aetiologic studies. Under the counterfactual approach, causal contrasts are the only meaningful effect measures for aetiologic studies. Note that many measures are not causal contrasts; for example, the following are not, because they cannot be expressed as contrasts of a target under two exposure distributions of intrinsic interest: correlation coefficients, percent of variance explained (R2), P-values, χ2 statistics, and standardized regression coefficients.24 Causal contrasts can be given precise interpretations. RRcausal can be interpreted as the net proportionate change in disease frequency caused by the difference in exposure distributions 1 and 0 in the target population during the aetiologic time period. RDcausal can be similarly interpreted as the net absolute change in disease frequency.2,5–7 Because a causal contrast is a contrast in a target—not in a substitute or a ‘study base’—it should be interpreted as a measure of causal effect in the target. We caution that not all population causal contrasts can be interpreted as averages of individual causal effects of exposure, or as averages of effects on subpopulations. This limitation arises when the denominators (Bi) of the disease-frequency measures are affected by exposure, as when the Bi represent non-cases or person-years, so that the Ri represent odds or incidence rates.2,6,25 For example, if the Bi represent person-years, so that RRcausal and RDcausal are the causal rate ratio and rate difference, and exposure, age, and sex all affect the rate, then RRcausal will not equal the average rate ratio across age and sex groups and RDcausal will not equal the average rate difference across these groups. For explanations of such problems, see refs2,6,7,25. The counterfactual approach leads to a general framework for designing and analysing aetiologic studies. Because all aetiologic designs should estimate causal contrasts, different designs can be viewed simply as different ways of (1) choosing a target that corresponds to the study question, and (2) choosing substitutes and sampling subjects from target and substitutes into the study to balance tradeoffs among bias, variance, study costs and study time. This approach works for all etiologic studies.26 Beginning with Fisher and Neyman's work on permutation tests,16 careful study of counterfactual models has also led to the invention of new analysis methods and new study designs.11,27–29 The above framework applies to randomized trials as well as observational studies. In fact, it was invented for the analysis of randomized trials, and only later extended to non-experimental studies.3,4,16,17 A typical randomized trial is an example of the scenario discussed above in which the target experiences neither exposure distribution 1 nor 0. Here the treatment arms are substitutes for the target under different treatments. For example, when a drug is approved for treatment of a particular disease, a generalization is being made from the clinical trials on which the approval was based to some external (target) population of patients with that disease. In effect, the treatment and placebo arms in those trials serve as substitutes for the target under different treatment scenarios. The concept of a causal contrast facilitates precise and general definitions of confounding and confounder. Confounding is present if our substitute imperfectly represents what our target would have been like under the counterfactual condition. An association measure is confounded (or biased due to confounding) for a causal contrast if it does not equal that causal contrast because of such an imperfect substitution.2,5–7,30 Roughly speaking, a confounder is a variable that at least partly explains why confounding is present. Many authors to define a confounder more precisely as a variable that is a risk for disease and is with exposure but not affected by exposure. This definition has several is that it applies only to the in which is just one variable to That variable may be a of several such as an used for or however, we must several at while as when some have been and others have In that the of a variable as a confounder, as well as the and of confounding, can change to which are is that of a variable that the above definition can at more confounding than it This for example, when is or no confounding to in that case we may many that the above definition, but confounding effects have this of one but not the others can see for an the that must be to confounding are A0/B0 in scenario 1 and A1/B1 in scenario in scenario are needed in the case discussed Both these however, represent and no on particular or their of confounding thus on which these are than on the particular used to the to in of for have been developed using counterfactual and graphical causal This shows that the size of a causal risk ratio not only to vary with the proportion of individuals in the target population outcome is by exposure are in and but also to vary with the proportion of individuals in the target population for disease is by the end of the etiologic time period are in This shows that the size of the causal risk difference will to vary only with the proportion of individuals in the target population outcome is by exposure. It that a that or can the size of a ratio or difference effect measure, and can the size of a ratio effect measure if it only one should not be if an effect measure varies from one population to another or from one time period to another one other causal factors to have distributions across the populations or periods. Because in the distribution of other factors can in effect the of an association measure across populations should not be viewed as a causal if one other causal factors to have distributions across a set of one should in particular in the distribution of across the populations and of confounding in the association of an association measure does not in provide logical for if the distribution of all other factors is across the populations. These from the counterfactual provide a logical basis for about the A question is on what is to be for the same risk ratios to under many is to for which is to In other the has general only as a than a quantitative and then only on the that or is The same to that are as a for of among study than as an in estimating a We must always use measures of association as for causal This to the question, ‘How different are measures of association from causal In other how bias is in the measures of association that we In practice, these questions are usually is, it is a of The of bias, however, is a of many and may be Many authors that formal such as and be used to the of of bias requires that describe the of bias as a of relevant parameters. The counterfactual approach can For example, it can be used to show that in the expected value of a relative risk estimate the causal relative risk a bias for confounding, a bias for to a bias for a bias for a bias for subjects from a bias for information This can be used in a analysis to bias under different scenarios. By their very definition, counterfactuals cannot be observed. people this and counterfactuals as a foundation for causal they may use statistical methods that hypothetical study for reason for their is that the counterfactual definition of effect to the that we can observe This arises because of the to use the for different to an observed outcome such as lung was an effect of however, to an effect measure such as which has at least one counterfactual Although we observe the effects of a we can only the cause of an effect, because our will always depend on that may be called into question. Causal is possible because we can about the that we do not observe following we have how problems of causal can be made precise to logical by into problems of about other of reasoning about cause and effect, models and causal to results to those using This to a among methods for causal The to be and … a As it does in counterfactual analysis can some of the in for it leads to a general framework for designing, analysing, and interpreting etiologic studies. It has led to a of analysis and we have it an We that this paper will in epidemiologists to problems from the counterfactual This was made possible by from from the of are the of the authors and do not necessarily represent the of the We are to and the for their on of this