Causal Inference

Theme Co-ordinators: Simon Cousens, Karla Diaz-Ordaz, Richard Silverwood, Ruth Keogh, Stijn Vansteelandt

Please see here for slides and audio recordings of previous seminars relating to this theme.

This field was recently criticised in a paper by Vandenbroucke et al. Members of this theme have written a response to these criticisms, which is available here.


As part of the CSM’s activities, seminars on causal inference are often organised. Past speakers include Philip DawidVanessa DidelezRichard EmsleyMiguel HernánErica Moodie and Anders Skrondal.

Details of upcoming seminars can be found here.

UK Causal Inference Meeting 2016

We were delighted to host – together with the London School of Economics and Political Science – the 4th annual UK Causal Inference Meeting, which took place at LSHTM from 13-15 April 2016. More details can be found here.

Other events

As well as regular research seminars, we occasionally organise one-off events on topics of particular interest. For example, we are currently planning a half-day meeting on recent controversies in propensity scores. Details of upcoming events can be found here.

A brief overview of a vast and rapidly-expanding subject

Causal inference is a central aim of many empirical investigations, and arguably most studies in the fields of medicine, epidemiology and public health. We would like to know ‘does this treatment work?’, ‘how harmful is this exposure?’, or ‘what would be the impact of this policy change?’.

The gold standard approach to answering such questions is to conduct a controlled experiment in which treatments/exposures are allocated at random, all participants adhere perfectly to the treatment assigned, and all the relevant data are collected and measured without error. Provided that we can then discount ‘chance’ alone as an explanation, any observed differences between treatment groups can be given a causal interpretation (albeit in a population that may differ from the one in which we are interested).

In the real world, however, such experiments rarely attain this ideal status, and for many important questions, such an experiment would not even be ethically, practically, or economically feasible, and our investigations must be based instead on observational data. In reality, therefore, causal inference is a very ambitious goal. However, since it undeniably is the only useful goal in so many contexts, we must try our best. This involves carefully formulating the causal question to be tackled, explicitly stating the assumptions under which the answers may be trusted, often considering novel analysis methods that may require weaker assumptions than would be required by traditional approaches, and finally using sensitivity analyses to explore how robust our conclusions are to violations of the assumptions.

Historically, even when attempting causal inference, the role of statistics was seen to be to quantify the extent to which ‘chance’ could explain the results, with concerns over systematic biases due to the non-ideal nature of the data relegated to the qualitative discussion of the results. The field known as causal inference has changed this state of affairs, setting causal questions within a coherent framework which facilitates explicit statement of all the assumptions underlying the analysis and allows extensive exploration of potential biases. In the paragraphs that follow, we will attempt a brief overview.

 A language for causal inference (potential outcomes and counterfactuals)

Over the last thirty years, a formal statistical language has been developed in which causal effects can be unambiguously defined, and the assumptions needed for their identification clearly stated. Although alternative frameworks have been suggested (see, for example, Dawid, 2000) and developed, the language which has gained most traction in the health sciences is that of potential outcomes, also called counterfactuals (Rubin, 1978).

Suppose X is a binary exposure, Y a binary outcome, and C a collection of potential confounders, measured before X. We write Y0 and Y1 for the two potential outcomes; the first is the outcome that would be seen if X were set (possibly counter to fact) to 0, and the second is what would be seen if X were set to 1. Causal effects can then be expressed as contrasts of aspects of the distribution of these potential outcomes. For example:

  1. E(Y1) –E(Y0)
  2. E(Y1|X=1)/E(Y0|X=1)
  3. log[E(Y1|C)/{1–E(Y1|C)}] –log[E(Y0|C)/{1–E(Y0|C)}]

The first is the average causal effect (ACE) of X on Y expressed as a marginal risk difference and the second is the average causal effect in the exposed (also called the average treatment effect in the treated, or ATT) expressed as a risk ratio (marginal wrt confounders C). The third is a conditional causal log odds ratio, given C.

Sufficient conditions for these and other similar parameters to be identified can also be expressed in terms of potential outcomes. For the ACE, for example, these are:

  1. Consistency: For x=0,1, if X=x then Yx=Y
  2. Conditional exchangeability: For x=0,1, Yx ╨ X | C

The latter formalises the notion of “no unmeasured confounders”.

The increased clarity afforded by this language has led to increased awareness of causal pitfalls (such as the ‘birthweight paradox’ – see Hernández-Díaz et al, 2006) and the building of a new and extensive toolbox of statistical methods especially designed for making causal inferences from non-ideal data under transparent, less restrictive and more plausible assumptions than were hitherto required.

Of course this does not mean that all causal questions can be answered, but at least they can be formally formulated and the plausibility of the required assumptions assessed.

Considerations of causality are not new. Neyman used potential outcomes in his PhD thesis in the 1920s, and who could forget Bradford Hill’s much-cited guidelines published in 1965? The last few decades, however, have seen the focus move towards developing solutions, as well as acknowledging limitations.

Traditional methods

Not all reliable causal inference requires novel methodology. A carefully-considered regression model, with an appropriate set of potential confounders (possibly identified using a causal diagram – see below) measured and appropriately included as covariates, is a reasonable approach in some settings.

Causal diagrams

An ubiquitous feature of methods for estimating causal effects from non-ideal data is the need for untestable assumptions regarding the causal structure of the variables being analysed (from which conditions such as conditional exchangeability can be deduced). Such assumptions are often represented in a causal diagram or graph, with variables identified by nodes and the relationships between them by edges. The simplest and most commonly-used class of causal diagram is the (causal) directed acyclic graph (DAG), in which all edges are arrows, and there are no cycles, i.e. no variable explains itself (Greenland et al, 1999). These are used not only to represent assumptions but also to inform the choice of a causally-interpretable analysis, specifically to help decide which variables should be included as confounders.

Fully-parametric approaches to problems involving many variables

Another common feature of causal inference methods is that, as we move further from the ideal experimental setting, more aspects of the joint distribution of the variables must be modelled, which would have been ancillary had the data arisen from a perfect experiment. Structural equation modelling (SEM) (Kline, 2011) is a fully-parametric approach, in which the relationship between each node in the graph and its parents is specified parametrically. This approach offers an elegant (full likelihood) treatment of ignorable missing data and measurement error, when this affects any variable for which validation or replication data are available.

Semiparametric approaches

Concerns over the potential impact of model misspecification in fully-parametric approaches have led to the development of alternative semiparametric approaches to causal inference, in which the number of additional aspects to be modelled is reduced. These include methods based on the propensity score (Rosenbaum and Rubin, 1983), including inverse probability weighting, and g-estimation, and the so-called doubly-robust estimation proposed by Robins, Rotnitzky and others.

Inferring the effects of time-varying exposures

Novel causal inference methods are particularly relevant for studying the causal effect of a time-varying exposure on an outcome, because standard methods fail to give causally-interpretable estimators when there exist time-varying confounders of the exposure and outcome that are themselves affected by previous levels of the exposure. Methods developed to deal with this problem include the fully-parametric g-computation formula (Robins, 1986), and two semiparametric approaches: g-estimation of structural nested models (Robins et al, 1992), and inverse probability weighted estimation of marginal structural models (Robins et al, 2000). For an accessible tutorial on these methods, see Daniel et al (2013). Related to this longitudinal setting is the identification of optimal treatment regimes, for example in HIV/AIDS research where questions such as ‘at what level of CD4 should HAART (highly active antiretroviral therapy) be initiated?’ are often asked. These can be addressed using the methods listed above, and other related methods (see Chakraborty and Moodie, 2013).

Instrumental variables and Mendelian Randomisation

It is important to appreciate that non-ideal experimental data (e.g. suffering from noncompliance, missing data or measurement error) are not on a par with data arising from observational studies (as may be inferred from what is written above). Randomisation can be used as a tool to aid causal inference even when the randomised experiment is ‘broken’, for example as a result of non-compliance to randomised treatment. Such methods make use of randomisation as an instrumental variable (Angrist and Pischke, 2009). Instrumental variables have even been used with observational data, in particular when the instrument is a variable that holds genetic information (in which case it is known as Mendelian randomisation; see Davey Smith and Ebrahim, 2003) with genotype used in place of randomisation. This is motivated by the idea that genes are ‘randomly’ passed down from parents to offspring in the same way that treatment is allocated in double-blind randomised trials. Although this assumption is generally untestable (Hernán and Robins, 2006), there are situations in which it may be deemed more plausible than the other candidate set of untestable assumptions, namely conditional exchangeability.

Mediation analysis

Approaches (such as SEM) amenable to complex causal structures have opened the way to looking beyond the causal effect of an exposure on an outcome as a black box, and to asking ‘how does this exposure act?’. For example, if income has a positive effect on certain health outcomes, does this act simply by increasing access to health care, or are there other important pathways? Addressing such questions is the goal of mediation analysis and the estimation of direct/indirect effects (see Emsley et al, 2010, for a review). This area has seen an explosion of new methodology in recent years, with several semiparametric alternatives to SEM introduced.