Clinical Psychology and Epidemiology, Max Planck Institute of Psychiatry, Munich, Germany

Abstract

Background

The counterfactual or potential outcome model has become increasingly standard for causal inference in epidemiological and medical studies.

Discussion

This paper provides an overview on the counterfactual and related approaches. A variety of conceptual as well as practical issues when estimating causal effects are reviewed. These include causal interactions, imperfect experiments, adjustment for confounding, time-varying exposures, competing risks and the probability of causation. It is argued that the counterfactual model of causal effects captures the main aspects of causality in health sciences and relates to many statistical procedures.

Summary

Counterfactuals are the basis of causal inference in medicine and epidemiology. Nevertheless, the estimation of counterfactual differences pose several difficulties, primarily in observational studies. These problems, however, reflect fundamental barriers only when learning from observations, and this does not invalidate the counterfactual concept.

Background

Almost every empirical research question is causal. Scientists conducting studies in medicine and epidemiology investigate questions like "Which factors cause a certain disease?" or "How does a certain therapy affect the duration and course of disease?" Clearly, not every association is temporarily directed, and not every temporarily directed association involves a causal component but might be due to measurement error, shared prior factors or other bias only. The only sine qua non condition for a causal effect in an individual is the precedence of the factor to its effect, and 100% evidence for causality is impossible. This insight dates back at least to the 18^{th }century Scottish philosopher David Hume [

The history of causal thinking, especially in philosophy, is a history of controversies and misunderstandings. For a detailed description of these controversies, see [

The article is organized as follows: In the first two sections of the Discussion part, the counterfactual model of causal effects is defined, and some general aspects on statistical inference are discussed. The next chapters provide an overview on causal interactions and causal inference in randomised and nonrandomised studies. In the last two sections, several special topics and related approaches for assessing causal effects are reviewed.

Discussion

1. The counterfactual model of causal effects

Statistics cannot contribute to causal inference unless the factor of interest

To define a causal effect in an individual _{i}. The outcome can be binary or quantitative (e.g. the amount of segregation of a hormone or a psychological score). According to Greenland and Brumback

(a) at the fixed time point of assignment, the individual _{i }= _{i }=

(b) the outcome _{i }exists under both _{i }= _{i,t}) and _{i }= _{i,c}).

Counterfactuals and potential outcomes

Obviously, the outcome can be observed only (or more precisely, at most) under one, and not under both conditions. If individual _{i,c }is unobservable; likewise, if individual _{i,t }is unobservable. The treatment that individual

A meaningful counterfactual constitutes a principally possible condition for individual

In general, counterfactuals are quite natural, and, although sometimes claimed

Definition of causal effect

There is a

_{i,t }≠ _{i,c}.

The

_{i,t }- _{i,c}.

If the outcome is strictly positive, one may also use the ratio. The choice of a measure, however, affects the interpretability of a summary of individual effects as the population average effect, and the interpretability of heterogeneity of individual effect magnitudes as causal interaction (see sections 2 and 3).

To imagine a causal effect in a binary outcome suppose that an individual _{i }= _{i,t }= 0). The question is whether the treatment was the cause of the remission of the disease – in comparison to another treatment level (e.g. _{i }= _{i,c }= 1. According to Maldonado and Greenland ^{th }century when the 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."

Counterfactual causality was the central idea that stimulated invention of randomised experiments by Ronald A. Fisher and statistical inference on them by Fisher around 1920 and, later, by Jerzey Neyman and Egon Pearson in a somewhat different way

Choosing the reference treatment

The first difficulty in assessing counterfactual causal effects is to choose the reference condition when comparing one treatment level

Multiple causal factors and causal mechanisms

In the counterfactual model, a causal factor is a

Causal graph for an indirect effect of

Causal graph for an indirect effect of

Investigating a causal effect does not require knowing its mechanism. The ability to explain an association, however, often supports the conclusion that it has a causal component (especially if the explanation is given before a researcher looks at the data). The mechanism of an effect is closely related to the terms of effect-modifying and mediating variables. An

2. Statistical inference on counterfactual effects

As already mentioned, one can evaluate a fixed individual _{i }= _{i }= _{i,t }= 1) has been caused by the received treatment or by other factors. One exception is ballistic evidence for a bullet stemming from a particular gun and found in a killed person

Average causal effects

The aim is to estimate the

To be interpreted as an estimate of the population average effect, the difference between the arithmetic mean in

The following discussion is restricted to the more frequent case of a sample consisting of different individuals rather than of different time points (or both).

Stable-unit-treatment assumption

Before treatment assignment, there are two random variables for each individual _{i,c}) and the outcome under treatment _{i,t}). Although the theory can be extended accordingly

Exchangeability

Suppose the average causal effect is defined as the difference in means in the target population between both conditions

a) The distribution of the unobserved outcome _{t }under actual treatment _{t }under actual treatment _{t}.

b) The distribution of the unobserved outcome _{c }under actual treatment _{c }under actual treatment _{c}.

Note that, if individuals actually having received treatment

In the section on causal inference, I will provide an outline on how exchangeability relates to different study designs and what statistical methods can contribute to approach unbiased estimation of causal effects if the optimal design (a perfect randomised experiment) is not feasible.

3. Heterogeneity in causal effects

An important issue is the assessment of differences in causal effects between individuals. Clearly, a necessary condition for a factor

Choice of the effect measure

Whether and, if yes, to what extent the degree of an effect differs according to the values of

Moreover, the risk difference is the only measure for which effect heterogeneity is logically linked with causal co-action in terms of counterfactual effects. To explain this, it is necessary to define the causal synergy of two binary factors, _{i }and _{i }(coded as 0 or 1), on a binary outcome _{i }in an individual

Clearly, if _{i }and _{i }do not act together in causing the event _{i }= 1, then

(a) if _{i }= 1 is caused by _{i }only,

_{i }= 1 if (_{i }= 1 and _{i }= 0) or

(_{i }= 1 and _{i }= 1)

and _{i }= 0 in all other cases. Thus, _{i }= 1 occurs in all cases where _{i }= 1 and in no other cases.

(b) if _{i }= 1 is caused by _{i }only,

_{i }= 1 if (_{i }= 0 and _{i }= 1) or

(_{i }= 1 and _{i }= 1)

and _{i }= 0 in all other cases. Thus, _{i }= 1 occurs in all cases where _{i }= 1 and in no other cases.

Therefore, causal synergy means that 1) _{i }= 1 if either one or both factors are present and 2) _{i }= 0 if neither factor is present. Now, one is often interested in

P(

If superadditivity is present, one can show that there must be causal synergy between

Another crucial point for the choice of effect index is whether the interaction terms in regression models corresponds with so-called

Deterministic versus probabilistic causality

A fundamental question relating to heterogeneity in causal effects is the distinction between deterministic and probabilistic causality [

Within the probabilistic understanding of causality, individual variation exists within the outcome

On the other hand, after incorporating major effect-modifiers into a model, the effect of

4. Causal inference in randomised and non-randomised studies

Randomised experiments

As already mentioned, if the individuals are exchangeable between the treatments and there are no other biases, causal effects can be directly estimated, most simply with the difference in the mean of _{i }= P (_{i }=

In a simple randomised experiment, _{i }is equal for all individuals. For example, in an experiment with balanced groups, the individuals are assigned to each treatment with a probability of 50%: _{i }= 1/2 for all

- Z

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Imperfect experiments

In the discussion above I have implicitly assumed that treatment and control protocols were followed exactly; in that sense, the experiments were supposed to be perfect. In many studies, however, the actual treatment and control conditions do not equal the intended protocols, at least, not for some individuals or measurement points (

Instrumental variables

If one ignores the fact that the treatment conditions were not exactly followed, one estimates the effect of the intended, not of the actual treatment. This is referred to as

Observational studies

Not every interesting factor can be translated into equivalent lab settings or can be manipulated. Factors like social support or peer relationships are difficult to observe outside their natural environment. Other conditions should not be assigned to human beings for ethical reasons (e.g. smoking). In such cases, there is no way but to conduct an observational study. In observational studies, the group assignment is neither manipulated nor randomised. The group status

The propensity score then typically depends on a variety of variables (denoted as vector

- Z

In many practical situations, one should assume substantial residual bias due to unobserved determinants of the exposure _{l }is a candidate for a confounder, the difference between the means under _{l }might be biased more strongly than the unadjusted mean difference. This can happen, for instance, if other, more important factors of group assignment are distributed more unequally across _{l }[

Pearl [

Methods to adjust for unobserved confounding and other biases

There are various approaches to address unobserved confounding, bias due to measurement error, selection, and other biases. The first method is sensitivity analysis, which examines what impact one or several supposed scenarios of bias would have had on the results at hand. The results depend on the presumed values of bias parameters like misclassification probabilities, the distribution of a confounder, and the magnitude of it's effects on

This drawback is solved with Monte Carlo sensitivity analysis. Here, distributions are assigned to the unknown bias parameters, which reflect a researcher's knowledge or assumptions about their true values. Bias-corrected point and interval estimates can then be calculated. The results from these methods have approximatively a Bayesian interpretation if additional uncertainty is added (as would be the case if one drew random numbers from the posterior distribution of the unknown effect), the estimator of the causal effect is approximatively efficient, and the data provide no information on the bias parameters [

(Monte Carlo) sensitivity analyses and Bayesian methods outperform conventional analyses, which often yield overconfident and biased results because they are based on wrong point priors at zero (e.g. misclassification probabilities) at the parameters determining bias

5. Some more special issues

Time-varying exposures

In many applications, the exposure level

The problem with time-varying systems is that they are subject to feedback mechanisms: The causes at fixed time

Details of statistical models are rather technical and thus beyond the scope of this paper. Briefly, Robins

Competing risks

Suppose that one is interested in the health burden attributable to a variable that is actually an outcome and not a treatment action in the earlier sense. Let me borrow an example from Greenland _{i }= 4 | _{i }= 1). Now the estimation of _{i }under _{i }= 0 is unclear because it depends on how _{i }= 0 was caused, how death from lung cancer was prevented. If death by lung cancer had been prevented through convincing the individual not to smoke at all in his entire lifetime, then the risk of other causes of death (e.g. coronary heart disease, diabetes, or other kinds of cancer) would be lower as well. In this case, _{i }under _{i }= 0 might be considerably higher than 4 years. On the other hand, if _{i }under _{i }= 0 might not have been much higher here than under _{i }= 1. The expected increase in years lived would thus be much smaller if lung cancer was prevented by chemotherapy than it would be if lung cancer was prevented by lifetime absence of smoking.

To conclude, there is no single intervention in this case that would be independent of an individual's history prior to exposure. The evaluation of the effect of removing _{i }= 1 depends on the

The probability of causation

A common problem is how to determine the probability that an event in an individual has been caused by a certain exposure, that is, the

where _{X = 0 }and _{X = 1 }denote the incidence rates in the target population under exposure and under non-exposure, respectively [

where:

- _{1 }is the number of individuals in the population in which exposure has caused an

- _{2 }denotes the number of individuals in whom exposure has caused

- _{T }is the total number of persons exposed to the disease (including also those individuals who have not been affected by the exposure,

Now, one can show that, if the probability of the exposure having an effect in the exposed is low, the rate fraction _{2}/_{T }_{1 }is small as compared to _{2}. This means that the effect is required to have an all-or-none effect in the vast majority of exposed and diseased individuals. Otherwise, the probability of causation is underestimated proportionally to the ratio _{1}/_{2}. A fundamental problem with the estimation of _{1 }– the number of exposed and diseased persons who would have developed the disease later under non-exposure. This estimation would require some biological model (which seems to be rarely available) for the progress of the disease

6. Related approaches to causal inference

The sufficient-component-cause model

Rothman

Structural equation models

Especially in the fields of psychology, social sciences and economics, structural equation models (SEMs) with latent variables are frequently used for causal modelling. These models consist of (a) parameters for the relations among the latent variables, (b) parameters for the relations among latent and observed variables and (c) distributional parameters for the error terms within the equations. Pearl

There are, however, several practical problems with the use of SEMs. First, in an under-determined system of equations, several assumptions are necessary to identify the parameters (i.e. to make the estimates unique). In psychological applications, the assumptions tend to be justified only partially

It is therefore recommended that one should be extremely careful in the application of SEMs. For more sophisticated discussions of the relations among structural equation models, graphical models, the corresponding causal diagrams and counterfactual causality; see

The controversy on counterfactual causality raised by Dawid's article

Dawid _{c }and _{t }for fixed individuals.

Together with Dawid's paper in the Journal of the American Statistical Association, not less than seven commentaries as well as Dawid's rejoinder _{c }and _{t }as not always necessary.

Dawid

Summary

1. The counterfactual concept is the basis of causal thinking in epidemiology and related fields. It provides the framework for many statistical procedures intended to estimate causal effects and demonstrates the limitations of observational data

2. Counterfactual causality has also stimulated the invention of new statistical methods such as g-estimation.

3. The intuitive conception makes the counterfactual approach also quite useful for teaching purposes

4. Counterfactual considerations should replace vague conceptions of "real" versus "spurious" association, which occasionally can still be read. In this context, the Yule-Simpson paradox is often mentioned. This paradox indicates that an association can have a different sign (positive or negative association, resp.) in each of two different subpopulations than it has in the entire population. However, if the temporal direction of the variables is added to this paradox and there is no bias and random error, the paradox is resolved: It is then determinable which association is

5. Causal effects have been treated like a stepchild for a long time, maybe because many researchers shared the opinion that causality would lie outside what could be scientifically assessed or mathematically formalised. Pearl

Competing interests

The author(s) declare that they have no competing interests.

Acknowledgements

I wish to thank Sander Greenland for very helpful comments on a former version of the manuscript and Evelyn Alvarenga for language editing.

Pre-publication history

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