Method of the month: constrained randomisation

Once a month we discuss a particular research method that may be of interest to people working in health economics. We’ll consider widely used key methodologies, as well as more novel approaches. Our reviews are not designed to be comprehensive but provide an introduction to the method, its underlying principles, some applied examples, and where to find out more. If you’d like to write a post for this series, get in touch. This month’s method is constrained randomisation.


Randomised experimental studies are one of the best ways of estimating the causal effects of an intervention. They have become more and more widely used in economics; Banerjee and Duflo are often credited with popularising them among economists. When done well, randomly assigning a treatment ensures both observable and unobservable factors are independent of treatment status and likely to be balanced between treatment and control units.

Many of the interventions economists are interested in are at a ‘cluster’ level, be it a school, hospital, village, or otherwise. So the appropriate experimental design would be a cluster randomised controlled trial (cRCT), in which the clusters are randomised to treatment or control and individuals within each cluster are observed either cross-sectionally or longitudinally. But, except in cases of large budgets, the number of clusters participating can be fairly small. When randomising a relatively small number of clusters we could by chance end up with a quite severe imbalance in key covariates between trial arms. This presents a problem if we suspect a priori that these covariates have an influence on key outcomes.

One solution to the problem of potential imbalance is covariate-based constrained randomisation. The principle here is to conduct a large number of randomisations, assess the balance of covariates in each one using some balance metric, and then to randomly choose one of the most balanced according to this metric. This method preserves the important random treatment assignment while ensuring covariate balance. Stratified randomisation also has a similar goal, but in many cases may not be possible if there are continuous covariates of interest or too few clusters to distribute among many strata.


Conducting covariate constrained randomisation is straightforward and involves the following steps:

  1. Specifying the important baseline covariates to balance the clusters on. For each cluster j we have L covariates x_{il}; l=1,...L.
  2. Characterising each cluster in terms of these covariates, i.e. creating the x_{il}.
  3. Enumerating all potential randomisation schemes or simulating a large number of them. For each one, we will need to measure the balance of the x_{il} between trial arms.
  4. Selecting a candidate set of randomisation schemes that are sufficiently balanced according to some pre-specified criterion from which we can randomly choose our treatment allocation.

Balance scores

A key ingredient in the above steps is the balance score. This score needs to be some univariate measure of potentially multivariate imbalance between two (or more) groups. A commonly used score is that proposed by Raab and Butcher:

\sum_{l=1}^{L} \omega_l (\bar{x}_{1l}-\bar{x}_{0l})^2

where \bar{x}_{1l} and \bar{x}_{0l} are the mean values of covariate l in the treatment and control groups respectively, and \omega_l is some weight, which is often the inverse standard deviation of the covariate. Conceptually the score is a sum of standardised differences in means, so lower values indicate greater balance. But other scores would also work. Indeed, any statistic that measures the distance between the distributions of two variables would work and could be summed up over the covariates. This could include the maximum distance:

max_l |x_{1l} - x_{0l}|

the Manhattan distance:

\sum_{l=1}^{L} |x_{1l}-x_{0l}|

or even the Symmetrised Bayesian Kullback-Leibler divergence (I can’t be bothered to type this one out). Grischott has developed a Shiny application to estimate all these distances in a constrained randomisation framework, detailed in this paper.

Things become more complex if there are more than two trial arms. All of the above scores are only able to compare two groups. However, there already exist a number of univariate measures of multivariate balance in the form of MANOVA (multivariate analysis of variance) test statistics. For example, if we have G trial arms and let X_{jg} = \left[ x_{jg1},...,x_{jgL} \right]' then the between group covariance matrix is:

B = \sum_{g=1}^G N_g(\bar{X}_{.g} - \bar{X}_{..})(\bar{X}_{.g} - \bar{X}_{..})'

and the within group covariance matrix is:

W = \sum_{g=1}^G \sum_{j=1}^{N_g} (X_{jg}-\bar{X}_{.g})(X_{jg}-\bar{X}_{.g})'

which we can use in a variety of statistics including Wilks’ Lambda, for example:

\Lambda = \frac{det(W)}{det(W+B)}

No trial has previously used covariate constrained randomisation with multiple groups, as far as I am aware, but this is the subject of an ongoing paper investigating these scores – so watch this space!

Once the scores have been calculated for all possible schemes or a very large number of possible schemes, we select from among those which are most balanced. The most balanced are defined according to some quantile of the balance score, say the top 15%.

As a simple simulated example of how this might be coded in R, let’s consider a trial of 8 clusters with two standard-normally distributed covariates. We’ll use the Raab and Butcher score from above:

#simulate the covariates
n <- 8
x1 <- rnorm(n)
x2 <- rnorm(n)
x <- matrix(c(x1,x2),ncol=2)
#enumerate all possible schemes - you'll need the partitions package here
schemes <- partitions::setparts(c(n/2,n/2))
#write a function that will estimate the score
#for each scheme which we can apply over our
#set of schemes
balance_score <- function(scheme,covs){
treat.idx <- I(scheme==2)
control.idx <- I(scheme==1)
treat.means <- apply(covs[treat.idx,],2,mean)
control.means <- apply(covs[control.idx,],2,mean)
cov.sds <- apply(covs,2,sd)
#Raab-butcher score
score <- sum((treat.means - control.means)^2/cov.sds)
#apply the function
scores <- apply(schemes,2,function(i)balance_score(i,x))
#find top 15% of schemes (lowest scores)
scheme.set <- which(scores <= quantile(scores,0.15))
#choose one at random
scheme.number <- sample(scheme.set,1)
scheme.chosen <- schemes[,scheme.number]


A commonly used method of cluster trial analysis is by estimating a mixed-model, i.e. a hierarchical model with cluster-level random effects. Two key questions are whether to control for the covariates used in the randomisation, and which test to use for treatment effects. Fan Li has two great papers answering these questions for linear models and binomial models. One key conclusion is that the appropriate type I error rates are only achieved in models adjusted for the covariates used in the randomisation. For non-linear models type I error rates can be way off for many estimators especially with small numbers of clusters, which is often the reason for doing constrained randomisation in the first place, so a careful choice is needed here. I would recommend adjusted permutation tests if in doubt to ensure the appropriate type I error rates. Of course, one could take a Bayesian approach to analysis, although there is no analysis that I’m aware of, of the performance of these models for these analyses (another case of “watch this space!”).


There are many trials that used this procedure and listing even a fraction would be a daunting task. But I would be remiss for not noting a trial of my own that uses covariate constrained randomisation. It is investigating the effect of providing an incentive to small and medium sized enterprises to adhere to a workplace well-being programme. There are good applications used as examples in Fan Li’s papers mentioned above. A trial that featured in a journal round-up in February used covariate constrained randomisation to balance a very small number of clusters in a trial of a medicines access programme in Kenya.


Sam Watson’s journal round-up for 25th February 2019

Every Monday our authors provide a round-up of some of the most recently published peer reviewed articles from the field. We don’t cover everything, or even what’s most important – just a few papers that have interested the author. Visit our Resources page for links to more journals or follow the HealthEconBot. If you’d like to write one of our weekly journal round-ups, get in touch.

Democracy does cause growth. Journal of Political Economy [RePEc] Published January 2019

Citizens of a country with a democratic system of government are able to affect change in its governors and influence policy. This threat of voting out the poorly performing from power provides an incentive for the government to legislate in a way that benefits the population. However, democracy is certainly no guarantee of good governance, economic growth, or population health as many events in the last ten years will testify. Similarly, non-democracies can also enact policy that benefits the people. A benevolent dictator is not faced with the same need to satisfy voters and can enact politically challenging but beneficial policies. People often point to China as a key example of this. So there remains the question as to whether democracy per se has any tangible economic or health benefits.

In a past discussion of an article on democratic reform and child health, I concluded that “Democratic reform is neither a sufficient nor necessary condition for improvements in child mortality.” Nevertheless democracy may still be beneficial, on average, given the in-built safeguards against poor leaders. This paper, which has been doing the rounds for years as a working paper, is another examination of the question of the impact of becoming democratic. Principally the article is focused on economic growth, but health and education outcomes feature (very) briefly. The concern I have with the article mentioned at the beginning of this paragraph and with this newly published article are that they do not consider in great detail why democratisation occurred. As much political science work points out, democratic reform can be demanded in poor economic conditions due to poor governance. For these endogenous changes economic growth causes democracy. Whereas in other countries democracy could come about in a more exogenous manner. Lumping them all in together may be misleading.

While the authors of this paper provide pages after pages of different regression specifications, including auto-regressive models and instrumental variables models, I remain unconvinced. For example, the instrument relies on ‘waves’ of transitions: a country is more likely to shift politically if its regional neighbours do, like the Arab Spring. But neither economic nor political conditions in a given country are independent of its neighbours. In somewhat of a rebuttal, Ruiz Pozuelo and other authors conducted a survey to try to identify and separate out those countries which transitioned to democracy endogenously and exogenously (from economic conditions). Their work suggests that the countries that transitioned exogenously did not experience growth benefits. Taken together this shows the importance of theory to guide empirical work, and not the other way round.

Effect of Novartis Access on availability and price of non-communicable disease medicines in Kenya: a cluster-randomised controlled trial. Lancet: Global Health Published February 2019

Access to medicines is one of the key barriers to achieving universal health care. The cost-effectiveness threshold for many low income countries rules out many potentially beneficial medicines. This is in part driven though by the high prices charged by pharmaceutical countries to purchase medicine, which often do not discriminate between purchasers with high and low abilities to pay. Novartis launched a scheme – Novartis Access – to provide access to medicines to low and middle income countries at a price of US$1 per treatment per month. This article presents a cluster randomised trial of this scheme in eight counties of Kenya.

The trial provided access to four treatment counties and used four counties as controls. Individuals selected at random within the counties with non-communicable diseases and pharmacies were the principal units within the counties at which outcomes were analysed. Given the small number of clusters, a covariate-constrained randomisation procedure was used, which generates randomisation that ensures a decent balance of covariates between arms. However, the analysis does not control for the covariates used in the constrained randomisation, which can lead to lower power and incorrect type one error rates. This problem is emphasized by the use of statistical significance to decide on what was and was not affected by the Novartis Access program. While practically all the drugs investigated show an improved availability, only the two with p<0.05 are reported to have improved. Given the very small sample of clusters, this is a tricky distinction to make! Significance aside, the programme appears to have had some success in improving access to diabetes and asthma medication, but not quite as much as hoped. Introductory microeconomics though would show how savings are not all passed on to the consumer.


Sam Watson’s journal round-up for 26th November 2018

Every Monday our authors provide a round-up of some of the most recently published peer reviewed articles from the field. We don’t cover everything, or even what’s most important – just a few papers that have interested the author. Visit our Resources page for links to more journals or follow the HealthEconBot. If you’d like to write one of our weekly journal round-ups, get in touch.

Alcohol and self-control: a field experiment in India. American Economic Review Forthcoming

Addiction is complex. For many people it is characterised by a need or compulsion to take something, often to prevent withdrawal, often in conflict with a desire to not take it. This conflicts with Gary Becker’s much-maligned rational theory of addiction, which views the addiction as a choice to maximise utility in the long term. Under Becker’s model, one could use market-based mechanisms to end repeated, long-term drug or alcohol use. By making the cost of continuing to use higher then people would choose to stop. This has led to the development of interventions like conditional payment or cost mechanisms: a user would receive a payment on condition of sobriety. Previous studies, however, have found little evidence people would be willing to pay for such sobriety contracts. This article reports a randomised trial among rickshaw drivers in Chennai, India, a group of people with a high prevalence of high alcohol use and dependency. The three trial arms consisted of a control arm who received an unconditional daily payment, a treatment arm who received a small payment plus extra if they passed a breathalyser test, and a third arm who had the choice between either of the two payment mechanisms. Two findings are of much interest. First, the incentive payments significantly increased daytime sobriety, and second, over half the participants preferred the conditional sobriety payments over the unconditional payments when they were weakly dominated, and a third still preferred them even when the unconditional payments were higher than the maximum possible conditional payment. This conflicts with a market-based conception of addiction and its treatment. Indeed, the nature of addiction means it can override all intrinsic motivation to stop, or do anything else frankly. So it makes sense that individuals are willing to pay for extrinsic motivation, which in this case did make a difference.

Heterogeneity in long term health outcomes of migrants within Italy. Journal of Health Economics [PubMed] [RePEc] Published 2nd November 2018

We’ve discussed neighbourhood effects a number of times on this blog (here and here, for example). In the absence of a randomised allocation to different neighbourhoods or areas, it is very difficult to discern why people living there or who have moved there might be better or worse off than elsewhere. This article is another neighbourhood effects analysis, this time framed through the lens of immigration. It looks at those who migrated within Italy in the 1970s during a period of large northward population movements. The authors, in essence, identify the average health and mental health of people who moved to different regions conditional on duration spent in origin destinations and a range of other factors. The analysis is conceptually similar to that of two papers we discussed at length on internal migration in the US and labour market outcomes in that it accounts for the duration of ‘exposure’ to poorer areas and differences between destinations. In the case of the labour market outcomes papers, the analysis couldn’t really differentiate between a causal effect of a neighbourhood increasing human capital, differences in labour market conditions, and unobserved heterogeneity between migrating people and families. Now this article examining Italian migration looks at health outcomes, such as the SF-12, which limit the explanations since one cannot ‘earn’ more health by moving elsewhere. Nevertheless, the labour market can still impact upon health strongly.

The authors carefully discuss the difficulties in identifying causal effects here. A number of model extensions are also estimated to try to deal with some issues discussed. This includes a type of propensity score weighting approach, although I would emphasize that this categorically does not deal with issues of unobserved heterogeneity. A finite mixture model is also estimated. Generally a well-thought-through analysis. However, there is a reliance on statistical significance here. I know I do bang on about statistical significance a lot, but it is widely used inappropriately. A rule of thumb I’ve adopted for reviewing papers for journals is that if the conclusions would change if you changed the statistical significance threshold then there’s probably an issue. This article would fail that test. They use a threshold of p<0.10 which seems inappropriate for an analysis with a sample size in the tens of thousands and they build a concluding narrative around what is and isn’t statistically significant. This is not to detract from the analysis, merely its interpretation. In future, this could be helped by banning asterisks in tables, like the AER has done, or better yet developing submission guidelines around its use.