Kharkiv, statistics, and causal inference

As news comes in (14 May 2022) that Ukraine has won the battle of Kharkiv* and Russian troops are withdrawing, it may be of interest to know that a major figure in statistics and causal inference, Jerzy Neyman (1894-1981), trained as a mathematician there 1912-16. If you have ever used a confidence interval or conceptualised causal inference in terms of potential outcomes, then you owe him a debt of gratitude.

“[Neyman] was educated as a mathematician at the University of Kharkov*, 1912-16. After this he became a Lecturer at the Kharkov Institute of Technology with the title of Candidate. When speaking of these years he always stressed his debt to Sergei Bernstein, and his friendship with Otto Struve (later to meet him again in Berkeley). His thesis was entitled ‘Integral of Lebesgue’.” (Kendall et al., 1982)

* Харків (transliterated to Kharkiv) in Ukrainian, Харькoв (transliterated to Kharkov) in Russian.

Efficacy RCTs as survey twins

Surveys attempt to estimate a quantity of a finite population using a probability sample from that population. How people ended up in the population is somebody else’s problem – demographers, perhaps.

Survey participants are sampled at random from this finite population without replacement. Part a of the figure below illustrates. Green blocks denote people who are surveyed and from whom we collect data. Grey blocks denote people we have not surveyed; we would like to infer what their responses would have been, if they had they been surveyed too.

RCTs randomly assign participants to treatment or control conditions. This is illustrated in part b of the figure above: green cells denote treatment and purple cells denote control. There are no grey cells since we have gathered information from everyone in the finite population. But in a way, we haven’t really.

An alternative way to view efficacy RCTs that aim to estimate a sample average treatment effect (SATE) is as a kind of survey. This illustrated in part c. Now the grey cells return.

There is a finite population of people who present for a trial, often with little known about how they ended up in that population – not dissimilarly to the situation for a survey. (But who studies how they end up in a trial – trial demographers?)

Randomly assigning people to conditions generates two finite populations of theoretical twins, identical except for treatment assignment and the consequences thereafter. One theoretical twin receives treatment and the other receives control. But we only obtain the response from one of the twins, i.e., either the treatment or the control twin. (You could also think of these theoretical twins’ outcomes as potential outcomes.)

Looking individually at one of the two theoretical populations, the random assignment to conditions has generated a random sample from that population. We really want to know what the outcome would have been for everyone in the treatment condition, if everyone had been assigned treatment. Similarly for control. Alas, we have to make do with a pair of surveys that sample from these two populations.

Viewing the Table 1 fallacy through the survey twin lens

There is a common practice of testing for differences in covariates between treatment and control. This is the Table 1 fallacy (see also Dean Eckles’s take on whether it really is a fallacy). Let’s see how it can be explained using survey twins.

Firstly, we have a census of covariates for the whole finite population at baseline, so we know with perfect precision what the means are. Treatment and control groups are surveys of the same population, so clearly no statistical test is needed. The sample means in both groups are likely to be different from each other and from the finite population mean of both groups combined. No surprises there: we wouldn’t expect a survey mean to be identical to the population mean. That’s why we use confidence intervals or large samples so that the confidence intervals are very narrow.

What’s the correct analysis of an RCT?

It’s common to analyse RCT data using a linear regression model. The outcome variable is the endpoint, predictors are treatment group and covariates. This is also known as an ANCOVA. This analysis is easy to understand if the trial participants are a simple random sample from some infinite population. But this is not what we have in efficacy trials as modelled by survey twins above. If the total number of participants in the trial is 1000, then we have a finite population of 1000 in the treatment group and a finite population of 1000 in the control group – together, 2000. In total we have 1000 observations, though, split in some proportion between treatment and control.

Following through on this reasoning, it sounds like the correct analysis uses a stratified independent sampling design with two strata, coinciding with treatment and control groups. The strata populations are both 1000, and a finite population correction should be applied accordingly.

It’s a little more complicated, as I discovered in a paper by Reichardt and Gollob (1999), who independently derived results found by Neyman (1923/1990). Their results highlight a wrinkle in the argument when conducting a t-test on two groups for finite populations as described above. This has general implications for analyses with covariates too. The wrinkle is, the two theoretical populations are not independent of each other.

The authors derive the standard error of the mean difference between X and Y as

$$\displaystyle \sqrt{\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y}-\left[ \frac{(\sigma_X-\sigma_Y)^2}{N} + \frac{2(1-\rho) \sigma_X \sigma_{Y}}{N} \right]}$$,

where $$\sigma_X^2$$ and $$\sigma_Y^2$$ are the variances of the two groups, $$n_X$$ and $$n_Y$$ are the observed group sample sizes, and $$N$$ is the total sample (the finite population) size. Finally, $$\rho$$ is the unobservable correlation between treat and control outcomes for each participant – unobservable because we only get either the treatment outcome or control outcome for each participant and not both. The terms in square brackets correct for the finite population.

If the variances are equal ($$\sigma_X = \sigma_Y$$) and the correlation $$\rho = 1$$, then the correction vanishes (glance back at numerators in the square brackets to see). This is great news if you are willing to assume that treatments have constant effects on all participants (an assumption known as unit-treatment additivity): the same regression analysis that you would use assuming a simple random sample from an infinite population applies.

If the variances are equal and the correlation is 0, then this is the same standard error as in the stratified independent sampling design with two strata described above. Or at least it was for the few examples I tried.

If the variances can be different and the correlation is one, then this is the same standard error as per Welch’s two-sample t-test.

So, which correlation should we use? Reichardt and Gollob (1999) suggest using the reliability of the outcome measure to calculate an upper bound on the correlation. More recently, Aronow, Green, and Lee (2014) proved a result that puts bounds on the correlation based on the observed marginal distribution of outcomes, and provide R code to copy and paste to calculate it. It’s interesting that a problem highlighted a century ago on something so basic – what standard error we should use for an RCT – is still being investigated now.

References

Aronow, P. M., Green, D. P., & Lee, D. K. K. (2014). Sharp bounds on the variance in randomized experiments. Annals of Statistics, 42, 850–871.

Neyman, J. (1923/1990). On the application of probability theory to agricultural experiments. Essay on principles. Section 9. Statistical Science, 5, 465-472.

Reichardt, C. S., & Gollob, H. F. (1999). Justifying the Use and Increasing the Power of a t Test for a Randomized Experiment With a Convenience Sample. Psychological Methods, 4, 117–128.

Standard errors of marginal means in an RCT

Randomised controlled trials (RCTs) typically use a convenience sample to estimate the mean effect of a treatment for study participants. Participants are randomly assigned to one of (say) two conditions, and an unbiased estimate of the sample mean treatment effect is obtained by taking the difference of the two conditions’ mean outcomes. The estimand in such an RCT is sometimes called the sample average treatment effect (SATE).

Some papers report a standard error for the marginal mean outcomes in treatment and control groups using the textbook formula

$$\displaystyle \frac{\mathit{SD_g}}{\sqrt{n_g}}$$,

where $$\mathit{SD_g}$$ is the standard deviation of outcomes in group $$g$$ and $$n_g$$ the number of observations in that group.

This formula assumes a simple random sample with replacement from an infinite population, so does not work for a convenience sample (see Stephen Senn, A Standard Error). I am convinced, but curious what standard error for each group’s mean would be appropriate, if any. (You could stop here and argue that the marginal group means mean nothing anyway. The whole point of running a trial is to subtract off non-treatment explanations of change such as regression to the mean.)

Let’s consider a two-arm RCT with no covariates and a coin toss determining who receives treatment or control. What standard error would be appropriate for the mean treatment outcome? Let the total sample size be $$N$$ and quantities for treatment and control use subscripts $$t$$ and $$c$$, respectively.

Treatment outcome mean of those who received treatment

If we focus on the mean for the $$n_t$$ participants who were assigned to treatment, we have all observations for that group, so the standard error of the mean is 0. This feels like cheating.

Treatment outcome mean of everyone in the sample

Suppose we want to say something about the treatment outcome mean for all $$N$$ participants in the trial, not only the $$n_t$$ who were assigned to treatment.

To see how to think about this, consider a service evaluation of $$N$$ patients mimicking everything about an RCT except that it assigns everyone to treatment and uses a coin toss to determine whether someone is included in the evaluation. This is now a survey of $$n$$ participants, rather than a trial. We want to generalise results to the finite $$N$$ from which we sampled.

Since the population is finite and the sampling is done without replacement, the standard error of the mean should be multiplied by a finite population correction,

$$\displaystyle \mathit{FPC} = \sqrt{\frac{N-n}{N-1}}$$.

This setup for a survey is equivalent to what we observe in the treatment group of an RCT. Randomly assigning participants to treatment gives us a random sample from a finite population, the sample frame of which we get by the end of the trial: all treatment and control participants. So we can estimate the SEM around the mean treatment outcome as:

$$\displaystyle \mathit{SEM_t} = \frac{\mathit{SD_t}}{\sqrt{n_t}} \sqrt{\frac{N-n_t}{N-1}}$$.

If, by chance (probability $$1/2^N$$), the coin delivers everyone to treatment, then $$N = n_t$$ and the FPC reduces to zero, as does the standard error.

Conclusion

If the marginal outcome means mean anything, then there are a couple of standard errors you could use, even with a convenience sample. But the marginal means seem irrelevant when the main reason for running an RCT is to subtract off non-treatment explanations of change following treatment.

If you enjoyed this, you may now be wondering what standard error to use when estimating a sample average treatment effect. Try Efficacy RCTs as survey twins.

“… the following are two incontrovertible facts about a randomized clinical trial:

1. over all randomizations the groups are balanced;

2. for a particular randomization they are unbalanced.

Now, no [statistically] ‘significant imbalance’ can cause 1 to be untrue and no lack of a significant balance can make 2 untrue. Therefore the only reason to employ such a test must be to examine the process of randomization itself. Thus a significant result should lead to the decision that the treatment groups have not been randomized…”

– Senn (1994,  p. 1716)

Senn, S. (1994). Testing for baseline balance in clinical trials. Statistics in Medicine, 13, 1715–1726.

Book review: High-quality psychotherapy research, by Areán and Kraemer (2013)

Randomised controlled trials (RCTs) are great, the gold standard of empirical research. The only thing better than RCTs are systematic reviews of lots and lots of RCTs. (So the story goes.) The reader may have noticed that RCTs evaluating CBT for psychosis have been vigorously debated for many months after a review was published in the British Journal of Psychiatry (Jauhar et al., 2014). Maybe not everyone agrees that RCTs are great (disclosure: I have analysed a couple), but I think it’s fair to say they are unavoidable whether you are trying to design or demolish them.

High-quality psychotherapy research by Patricia Arean and Helena Chmura Kraemer sets out to be a “practical, step-by-step guide” to designing and running RCTs. So why bother with an RCT? Observational trials, the authors explain, might involve studying participants who choose one of two or more interventions of interest by simply observing how they get on. This is problematic as differences in outcomes might be due to whatever factors led to them ending up receiving an intervention rather than the effect it had. RCTs use randomisation to overcome this problem so that people differ only in terms of the intervention received. That’s about it for the “why”: don’t expect debate on the epistemology.

The book’s strengths emerge as it develops: it catalogues issues that should worry study investigators and the authors draw on their own experience to offer hints. The Delphi consensus-building approach is introduced to solve the problem of developing an intervention manual and examples are given of how to word a letter asking for feedback on the proposed result. Randomisation techniques are introduced including horror stories of how they have gone wrong and invalidated RCTs. Ideas are provided for control groups, e.g., waiting list, usual care, and “gold standard” controls, and their strengths and drawbacks. The importance of not using pilot study results to determine sample size choices is explained. Guidance is provided on the people required; for example you need three or more therapists, at least two research assistants in case one takes ill, and a good statistician amongst other people. The Appendix includes a sample budget justification. All practical advice.

The text runs to under 200 pages so this could never be a comprehensive guide to all aspects of RCTs. What this book does do well is provide a systematic menu of options and ideas for things to consider. It might possibly give some ideas of what to demolish too, should you be so inclined, but this book is really only for those who are already sold on RCTs and want to get on with the seemingly painful task of designing and running one.

References

Areán, P. A., & Kraemer, H. C. (2013). High-quality psychotherapy research: from conception to piloting to national trials. Oxford University Press.

Jauhar, S., McKenna, P. J., Radua, J., Fung, E., Salvador, R., & Laws, K. R. (2014). Cognitive-behavioural therapy for the symptoms of schizophrenia: systematic review and meta-analysis with examination of potential bias. The British Journal of Psychiatry, 204, 20–29. doi:10.1192/bjp.bp.112.116285