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%.
Is this process tracing or simply putting Bayes’ rule to work as usual? Does this example show that RCTs can be theory-based evaluations, since process tracing is a theory-based method, or does the inclusion of a control group rule out that possibility, as Figure 3.1 of the Magenta Book would suggest? I will leave the reader to assign probabilities to each possible conclusion. Let me know what you think.
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%.