It is all too common for a box and arrow diagram to be cobbled together in an afternoon and christened a “theory of change”. One formalised version of such a diagram is a structural equation model (SEM), the arrows of which are annotated with coefficients estimated using data. Here is John Fox (2002) on SEM and informal boxology:

“A cynical view of SEMs is that their popularity in the social sciences reflects the legitimacy that the models appear to lend to causal interpretation of observational data, when in fact such interpretation is no less problematic than for other kinds of regression models applied to observational data. A more charitable interpretation is that SEMs are close to the kind of informal thinking about causal relationships that is common in social-science theorizing, and that, therefore, these models facilitate translating such theories into data analysis.”

This is an amusing opening to a paper on face validity, by Mosier (1947):

“Face validity is a term that is bandied about in the field of test construction until it seems about to become a part of accepted terminology. The frequency of its use and the emotional reaction which it arouses-ranging almost from contempt to highest approbation-make it desirable to examine its meaning more closely. When a single term variously conveys high praise or strong condemnation, one suspects either ambiguity of meaning or contradictory postulates among those using the term. The tendency has been, I believe, to assume unaccepted premises rather than ambiguity, and beautiful friendships have been jeopardized when a chance remark about face validity has classed the speaker among the infidels.”

I think dozens of beautiful friendships have been jeopardized by loose talk about randomised controlled trials, theory-based evaluation, realism, and positivism, among many others. I’ve just seen yet another piece arguing that you wouldn’t evaluate a parachute with an RCT and I can’t even.

(1) Alex’s train left 2 minutes before they arrived at the platform.

(2) If Alex had arrived at the platform 10 minutes earlier, then they probably would have caught their train.

Is the counterfactual in sentence 2 true or false, or can’t you tell because you didn’t run an RCT?

I reckoned that the counterfactual is true. I reasoned that Alex probably missed the train because they were late, so turning up earlier would have fixed that.

I could think of other possible outcomes, but they became increasingly contrived and far from the (albeit minimal) evidence provided. For instance, it is conceivable that if Alex arrived earlier, they would have believed they had time to pop to Pret for a coffee – and missed the train again.

Process tracing is an application of Bayes’ theorem to test hypotheses using qualitative evidence.¹ Application areas tend to be complex, e.g., evaluating the outcomes of international aid or determining the causes of a war by interpreting testimony and documents. This post explores what happens if we apply process tracing to a simple hypothetical quantitative study: an RCT that includes a mediation analysis.

Process tracing is often conducted without probabilities, using heuristics such as the “hoop test” or “smoking gun test” that make its Bayesian foundations digestible. Alternatively, probabilities may be made somewhat easier to digest by viewing them through verbal descriptors such as those provided by the PHIA Probability Yardstick. Given the simple example we will tackle, I will apply Bayes’ rule directly to point probabilities.

I will assume that there are three mutually exclusive hypotheses:

Null: the intervention has no effect.

Out: the intervention improves outcomes; however, not through the hypothesised mediator (it works but we have no idea how).

Med: the intervention improves the outcome and it does so through the hypothesised mediator.

Other hypotheses I might have included are that the intervention causes harm or that the mediator operates in the opposite direction to that hypothesised. We might also be interested in whether the intervention pushes the mediator in the desired direction without shifting the outcome. But let’s not overcomplicate things.

There are two sources of evidence, estimates of:

Average treatment effect (ATE): I will treat this evidence source as binary: whether there is a statistically significant difference between treat and control or not (alternative versus null hypothesis). Let’s suppose that the Type I error rate is 5% and power is 80%. This means that if either Out or Med holds, then there is an 80% chance of obtaining a statistically significant effect. If neither holds, then there is a 5% chance of obtaining a statistically significant effect (in error).

Average causal mediation effect (ACME): I will again treat this as binary: is ACME statistically significantly different to zero or not (alternative versus null hypothesis). I will assume that if ATE is significant and Med holds, then there is a 70% chance that ACME will be significant. Otherwise, I will assume a 5% chance (by Type I error).

Note where I obtained the probabilities above. I got the 5% and 80% for free, following conventions for Type I error and power in the social sciences. I arrived at the 70% using finger-in-the-wind: it should be possible to choose a decent mediator based on the prior literature, I reasoned; however, I have seen examples where a reasonable choice of mediator still fails to operate as expected in a highly powered study.

Finally, I need to choose prior probabilities for Null, Out, and Med. Under clinical equipoise, I feel that there should be a 50-50 chance of the intervention having an effect or not (findings from prior studies of the same intervention notwithstanding). Now suppose it does have an effect. I am going to assume there is a 50% chance of that effect operating through the mediator.

This means that

P(Null) = 50%
P(Out) = 25%
P(Med) = 25%

So, P(Out or Med) = 50%, i.e., the prior probabilities are setup to reflect my belief that there is a 50% chance the intervention works somehow.

I’m going to use a Bayesian network to do the sums for me (I used GeNIe Modeler). Here’s the setup:

The lefthand node shows the prior probabilities, as chosen. The righthand nodes show the inferred probabilities of observing the different patterns of evidence.

Let’s now pretend we have concluded the study and observed evidence. Firstly, we are delighted to discover that there is a statistically significant effect of the intervention on outcomes. Let’s update our Bayesian network (note how the Alternative outcome on ATE has been underlined and emboldened):

P(Null) has now dropped to 6% and P(ACME > 0) has risen to 36%. We do not yet have sufficient evidence to distinguish between Out or Med: their probabilities are both 47%.²

Next, let’s run the mediation analysis. Amazingly, it is also statistically significant:

So, given our initial probability assignments and the pretend evidence observed, we can be 93% sure that the intervention works and does so through the mediator.

If the mediation test had not been significant, then P(Out) would have risen to 69% and P(Med) would have dropped to 22%. If the ATE had been indistinguishable from zero, then P(Null) would have been 83%.

¹ Okay, I accept that it is controversial to say that process tracing is necessarily an application of Bayes, particularly when no sums are involved. However, to me Bayes’ rule explains in the simplest possible terms why the four tests attributed to Van Evera (1997) [Guide to Methods for Students of Political Science. New York, NY: Cornell University Press.] work. It’s clear why there are so many references to Bayes in the process tracing literature.

² These are all actually conditional probabilities. I have made this implicit in the notation for ease of reading. Hopefully it’s clear given the prose.

For example, P(Med | ATE = Alternative) = 47%; in other words, the probability of Med given a statistically significant ATE estimate is 47%.

‘I am honoured to introduce this special issue dedicated to John Mayne, a “thought leader,” “practical thinker,” “bridge builder,” and “scholar practitioner” in the field of evaluation. Guest editors Steffen Bohni Nielsen, Sebastian Lemire, and Steve Montague bring together 14 colleagues whose articles document, analyze, and expand on John’s contributions to evaluation in the Canadian public service as well as his contributions to evaluation theory.’ –Jill A. Chouinard

“Donaldson and Lipsey (2006), Leeuw and Donaldson (2015), and Weiss (1997) noted that there is a great deal of confusion today about what is meant by theory-based or theory-driven evaluation, and the differences between using program theory and social science theory to guide evaluation efforts. For example, the newcomer to evaluation typically has a very difficult time sorting through a number of closely related or sometimes interchangeable terms such as theory-oriented evaluation, theory-based evaluation, theory-driven evaluation, program theory evaluation, intervening mechanism evaluation, theoretically relevant evaluation research, program theory, program logic, logic modeling, logframes, systems maps, and the like. Rather than trying to sort out this confusion, or attempt to define all of these terms and develop a new nomenclature, a rather broad definition is offered in this book in an attempt to be inclusive.

“Program Theory–Driven Evaluation Science is the systematic use of substantive knowledge about the phenomena under investigation and scientific methods to improve, to produce knowledge and feedback about, and to determine the merit, worth, and significance of evaluands such as social, educational, health, community, and organizational programs.”

– Donaldson, S. I. (2022, p. 9). Introduction to Theory-Driven Program Evaluation (2nd ed.). Routledge.

Experimental and quasi-experimental evaluations usually define a programme effect as the difference between (a) the actual outcome following a social programme and (b) an estimate of what the outcome would have been without the programme – the counterfactual outcome. (The latter might be a competing programme or some genre of “business as usual”.)

It is also usually argued that qualitative or so-called “theory-based” approaches to evaluation are not counterfactual evaluations. Reichardt (2022) adds to a slowly accumulating body of work that challenges this and argues that any approach to evaluation can be understood in counterfactual terms.

Reichardt provides seven examples of evaluation approaches, quantitative and qualitative, and explains how a counterfactual analysis is relevant:

Comparisons Across Participants. RCTs and friends. The comparison group is used to estimate the counterfactual. (Note: the comparison group is not the counterfactual. A comparison group is factual.)

Before-After Comparisons. The baseline score is often treated as counterfactual outcome (though it’s probably not, thanks, e.g., due to regression to the mean).

What-If Assessments. Asking participants to reflect on a counterfactual like, “How would you have felt without the programme?” Participants provide the estimate of the counterfactual, the evaluators use it to estimate the effect.

Just-Tell-Me Assessments. Cites Copestake (2014): “If we are interested in finding out whether particular men, women or children are less hungry as a result of some action it seems common-sense just to ask them.” In this case participants may be construed as carrying out the “What-If” assessment of the previous point and using this to work out the programme effect themselves.

Direct Observation. Simply seeing the causal effect rather than inferring. An example given is of tapping a car brake and seeing the effect. Not sure I buy this one and neither does Reichardt. Whatever it is, I agree a counterfactual of some sort is needed (and inferred): you need to have a theory to explain what would have happened had you not tapped the brake.

Theories-of-Change Assessments. Contribution analysis and realist evaluation are offered as examples. The gist is, despite what proponents of these approaches claim, to use a theory of change to work out whether the programme is responsible for or “contributes to” outcomes, you need to use the theory of change to think about the counterfactual. I’ve blogged about realist evaluation and contribution analysis elsewhere and their definitions of a causal effect.

The Modus Operandi (MO) Method. The evaluator looks for evidence of traces or tell-tales that the programme worked. Not sure I quite get how this differs from theory-of-change assessments. Maybe it doesn’t. It sounds like potentially another way to evidence the causal chains in a theory of change.

The conclusion:

“I suspect there is no viable alternative to the counterfactual definition of an effect and that when the counterfactual definition is not given explicitly, it is being used implicitly. […] Of course, evaluators are free to use an alternative to the counterfactual definition of a program effect, if an adequate alternative can be found. But if an alternative definition is used, evaluators should explicitly describe that alternative definition and forthrightly demonstrate how their definition undergirds their methodology […].”

I like four of the seven, as kinds of evidence used to infer the counterfactual outcome. I also propose a fifth: evaluator opinion.

Comparisons Across Participants.

Before-After Comparisons.

What-If Assessments.

Just-Tell-Me Assessments.

Evaluator opinion.

The What-If and Just-Tell-Me assessments could involve subject experts rather than only beneficiaries of a programme, which would have an impact on how those assessments are interpreted, particularly if the experts have a vested interest. To me, the Theory of Change Assessment in Reichardt’s original could be carried out with the help of one or more of these five. They are all ways to justify causal links (mediating variables or intermediate variables), not just evaluate outcomes, and help assess the validity of a theory of change. Though readers may not find them all equally compelling, particularly the last.

References

Copestake, J. (2014). Credible impact evaluation in complex contexts: Confirmatory and exploratory approaches. Evaluation, 20(4), 412–427.

It is a cliché that randomised controlled trials (RCTs) are the gold standard if you want to evaluate a social policy or intervention and quasi-experimental designs (QEDs) are presumably the silver standard. But often it is not possible to use either, especially for complex policies. Theory-Based Evaluation is an alternative that has been around for a few decades, but what exactly is it?

In this post I will sketch out what some key texts say about Theory-Based Evaluation; explore one approach, contribution analysis; and conclude with discussion of an approach to assessing evidence in contribution analyses (and a range of other approaches) using Bayes’ rule.

theory (lowercase)

Let’s get the obvious out of the way. All research, evaluation included, is “theory-based” by necessity, even if an RCT is involved. Outcome measures and interviews alone cannot tell us what is going on; some sort of theory (or story, account, narrative, …) – however flimsy or implicit – is needed to design an evaluation and interpret what the data means.

If you are evaluating a psychological therapy, then you probably assume that attending sessions exposes therapy clients to something that is likely to be helpful. You might make assumptions about the importance of the therapeutic relationship to clients’ openness, of any homework activities carried out between sessions, etc. RCTs can include statistical mediation tests to determine whether the various things that happen in therapy actually explain any difference in outcome between a therapy and comparison group (e.g., Freeman et al., 2015).

It is great if a theory makes accurate predictions, but theories are underdetermined by evidence, so this cannot be the only criterion for preferring one theory’s explanation over another (Stanford, 2017) – again, even if you have an effect size from an RCT. Lots of theories will be compatible with any RCT’s results. To see this, try a particular social science RCT and think hard about what might be going on in the intervention group beyond what the intervention developers have explicitly intended.

In addition to accuracy, Kuhn (1977) suggests that a good theory should be consistent with itself and other relevant theories; have broad scope; bring “order to phenomena that in its absence would be individually isolated”; and it should produce novel predictions beyond current observations. There are no obvious formal tests for these properties, especially where theories are expressed in ordinary language and box-and-arrow diagrams.

Theory-Based Evaluation (title case)

Theory-Based Evaluation is a particular genre of evaluation that includes realist evaluation and contribution analysis. According the UK’s government’s Magenta Book (HM Treasury, 2020, p. 43), Theory-Based methods of evaluation

“can be used to investigate net impacts by exploring the causal chains thought to bring about change by an intervention. However, they do not provide precise estimates of effect sizes.”

The Magenta Book acknowledges (p. 43) that “All evaluation methods can be considered and used as part of a [Theory-Based] approach”; however, Figure 3.1 (p. 47) is clear. If you can “compare groups affected and not affected by the intervention”, you should go for experiments or quasi-experiments; otherwise, Theory-Based methods are required.

Theory-Based Evaluation attempts to draw causal conclusions about a programme’s effectiveness in the absence of any comparison group. If a quasi-experimental design (QED) or randomised controlled trial (RCT) were added to an evaluation, it would cease to be Theory-Based Evaluation, as the title case term is used.

Example: Contribution analysis

Contribution analysis is an approach to Theory-Based Evaluation developed by JohnMayne (28 November 1943 – 18 December 2020). Mayne was originally concerned with how to use monitoring data to decide whether social programmes actually worked when quasi-experimental approaches were not feasible (Mayne, 2001), but the approach evolved to have broader scope.

According to a recent summary (Mayne, 2019), contribution analysis consists of six steps (and an optional loop):

Step 1: Set out the specific cause-effect questions to be addressed.

Step 2: Develop robust theories of change for the intervention and its pathways.

Step 3: Gather the existing evidence on the components of the theory of change model of causality: (i) the results achieved and (ii) the causal link assumptions realized.

Step 4: Assemble and assess the resulting contribution claim, and the challenges to it.

Step 5: Seek out additional evidence to strengthen the contribution claim.

Step 6: Revise and strengthen the contribution claim.

Step 7: Return to Step 4 if necessary.

Here is a diagrammatic depiction of the kind of theory of change that could be plugged in at Step 2 (Mayne, 2015, p. 132), which illustrates the cause-effect links an evaluation would aim to evaluate.

In this example, mothers are thought to learn from training sessions and materials, which then persuades them to adopt new feeding practices. This leads to children having more nutritious diets. The theory is surrounded by various contextual factors such as food prices. (See also Mayne, 2017, for a version of this that includes ideas from the COM-B model of behaviour.)

Step 4 is key. It requires evaluators to “Assemble and assess the resulting contribution claim”. How are we to carry out that assessment? Mayne (2001, p. 14) suggests some questions to ask:

“How credible is the story? Do reasonable people agree with the story? Does the pattern of results observed validate the results chain? Where are the main weaknesses in the story?”

For me, the most credible stories would include experimental or quasi-experimental tests, with mediation analysis of key hypothesised mechanisms, and qualitative detective work to get a sense of what’s going on beyond the statistical associations. But the quant part of that would lift us out of the Theory-Based Evaluation wing of the Magenta Book flowchart. In general, plausibility will be determined outside contribution analysis in, e.g., quality criteria for whatever methods for data collection and analysis were used. Contribution analysis says remarkably little on this key step.

Although contribution analysis is intended to fill a gap where no comparison group is available, Mayne (2001, p. 18) suggests that further data might be collected to help rule out alternative explanations of outcomes, e.g., from surveys, field visits, or focus groups. He also suggests reviewing relevant meta-analyses, which could (I presume) include QED and RCT evidence.

It is not clear to me what the underlying theory of causation is in contribution analysis. It is clear what it is not (Mayne, 2019, pp. 173–4):

“In many situations a counterfactual perspective on causality—which is the traditional evaluation perspective—is unlikely to be useful; experimental designs are often neither feasible nor practical…”

“[Contribution analysis] uses a stepwise (generative) not a counterfactual approach to causality.”

(We will explore counterfactuals below.) I can guess what this generative approach could be, but Mayne does not provide precise definitions. It clearly isn’t the idea from generative social science in which causation is defined in terms of computer simulations (Epstein, 1999).

One way to think about it might be in terms of mechanisms: “entities and activities organized in such a way that they are responsible for the phenomenon” (Illari & Williamson, 2011, p. 120). We could make this precise by modelling the mechanisms using causal Bayesian networks such that variables (nodes in a network) represent the probability of activities occurring, conditional on temporally earlier activities having occurred – basically, a chain of probabilistic if-thens.

Why do people get vaccinated for Covid-19? Here is the beginning of a (generative?) if-then theory:

If you learned about vaccines in school and believed what you learned and are exposed to an advert for Covid-19 jab and are invited by text message to book an appointment for one, then (with a certain probability) you use your phone to book an appointment.

If you have booked an appointment, then (with a certain probability) you travel to the vaccine centre in time to attend the appointment.

If you attend the appointment, then (with a certain probability) you are asked to join a queue.

… and so on …

In a picture:

This does not explain how or why the various entities (people, phones, etc.) and activities (doing stuff like getting the bus as a result of beliefs and desires) are organised as they are, just the temporal order in which they are organised and dependencies between them. Maybe this suffices.

What are counterfactual approaches?

Counterfactual impact evaluation usually refers to quantitative approaches to estimate average differences as understood in a potential outcomes framework (or generalisations thereof). The key counterfactual is something like:

“If the beneficiaries had not taken part in programme activities, then they would not have had the outcomes they realised.”

Logicians have long worried how to determine the truth of counterfactuals, “if A had been true, B.” One approach, due to Stalnaker (1968), proposes that you:

Start with a model representing your beliefs about the factual situation where A is false. This model must have enough structure so that tweaking it could lead to different conclusions (causal Bayesian networks have been proposed; Pearl, 2013).

Add A to your belief model.

Modify the belief model in a minimal way to remove contradictions introduced by adding A.

Determine the truth of B in that revised belief model.

This broader conception of counterfactual seems compatible with any kind of evaluation, contribution analysis included. White (2010, p. 157) offered a helpful intervention, using the example of a pre-post design where the same outcome measure is used before and after an intervention:

“… having no comparison group is not the same as having no counterfactual. There is a very simple counterfactual: what would [the outcomes] have been in the absence of the intervention? The counterfactualis that it would have remained […] the same as before the intervention.”

The counterfactual is untested and could be false – regression to the mean would scupper it in many cases. But it can be stated and used in an evaluation. I think Stalnaker’s approach is a handy mental trick for thinking through the implications of evidence and producing alternative explanations.

Cook (2000) offers seven reasons why Theory-Based Evaluation cannot “provide the valid conclusions about a program’s causal effects that have been promised.” I think from those seven, two are key: (i) it is usually too difficult to produce a theory of change that is comprehensive enough for the task and (ii) the counterfactual remains theoretical – in the arm-chair, untested sense of theoretical – so it is too difficult to judge what would have happened in the absence of the programme being evaluated. Instead, Cook proposes including more theory in comparison group evaluations.

Bayesian contribution tracing

Contribution analysis has been supplemented with a Bayesian variant of process tracing (Befani & Mayne, 2014; Befani & Stedman-Bryce, 2017; see also Fairfield & Charman, 2017, for a clear introduction to Bayesian process tracing more generally).

The idea is that you produce (often subjective) probabilities of observing particular (usually qualitative) evidence under your hypothesised causal mechanism and under one or more alternative hypotheses. These probabilities and prior probabilities for your competing hypotheses can then be plugged into Bayes’ rule when evidence is observed.

Suppose you have two competing hypotheses: a particular programme led to change versus pre-existing systems. You may begin by assigning them equal probability, 0.5 and 0.5. If relevant evidence is observed, then Bayes’ rule will shift the probabilities so that one becomes more probable than the other.

Process tracers often cite Van Evera’s (1997) tests such as the hoop test and smoking gun. I find definitions of these challenging to remember so one thing I like about the Bayesian approach is that you can think instead of specificity and sensitivity of evidence, by analogy with (e.g., medical) diagnostic tests. A good test of a causal mechanism is sensitive, in the sense that there is a high probability of observing the relevant evidence if your causal theory is accurate. A good test is also specific, meaning that the evidence is unlikely to be observed if any alternative theory is true. See below for a table (lighted edited from Befani & Mayne, 2014, p. 24) showing the conditional probabilities of evidence for each of Van Evera’s tests given a hypothesis and alternative explanation.

Van Evera test
if Eᵢ is observed

P(Eᵢ | Hyp)

P(Eᵢ | Alt)

Fails hoop test

Low

—

Passes smoking gun

—

Low

Doubly-decisive test

High

Low

Straw-in-the-wind test

High

High

Let’s take the hoop test. This applies to evidence which is unlikely if your preferred hypothesis were true. So if you observe that evidence, the hoop test fails. The test is agnostic about the probability under the alternative hypothesis. Straw-in-the-wind is hopeless for distinguishing between your two hypotheses, but could suggest that neither holds if the test fails. The double-decisive test has high sensitivity and high specificity, so provides strong evidence for your hypothesis if it passes.

The arithmetic is straightforward if you stick to discrete multinomial variables and use software for conditional independence networks. Eliciting the subjective probabilities for each source of evidence, conditional on each hypothesis, may be less straightforward.

Conclusions

I am with Cook (2000) and others who favour a broader conception of “theory-based” and suggest that better theories should be tested in quantitative comparison studies. However, it is clear that it is not always possible to find a comparison group – colleagues and I have had to make do without (e.g., Fugard et al., 2015). Using Theory-Based Evaluation in practice reminds me of jury service: a team are guided through thick folders of evidence, revisiting several key sections that are particularly relevant, and work hard to reach the best conclusion they can with what they know. There is no convenient effect size to consult, just a shared (to some extent) and informal idea of what intuitively feels more or less plausible (and lengthy discussion where there is disagreement). To my mind, when quantitative comparison approaches are not possible, Bayesian approaches to assessing qualitative evidence are the most compelling way to synthesise qualitative evidence of causal impact and make transparent how this synthesis was done.

Finally, it seems to me that the Theory-Based Evaluation category is poorly named. Better might be, Assumption-Based Counterfactual approaches. Then RCTs and QEDs are Comparison-Group Counterfactual approaches. Both are types of theory-based evaluation and both use counterfactuals; it’s just that approaches using comparison groups gather quantitative evidence to test the counterfactual. However, the term doesn’t quite work since RCTs and QEDs rely on assumptions too… Further theorising needed.

Edited to add: Reichardt’s (2022), The Counterfactual Definition of a Program Effect, is a very promising addition to the literature and, I think, offers a clear way out of the theory-based versus non-theory-based and counterfactual versus not-counterfactual false dichotomies. I’ve blogged about it here.

Kuhn, T. S. (1977). Objectivity, Value Judgment, and Theory Choice. In The Essential Tension: Selected Studies in Scientific Tradition and Change (pp. 320–339). The University of Chicago Press.