Sam Watson’s journal round-up for 15th January 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.

Cost-effectiveness of publicly funded treatment of opioid use disorder in California. Annals of Internal Medicine [PubMed] Published 2nd January 2018

Deaths from opiate overdose have soared in the United States in recent years. In 2016, 64,000 people died this way, up from 16,000 in 2010 and 4,000 in 1999. The causes of public health crises like this are multifaceted, but we can identify two key issues that have contributed more than any other. Firstly, medical practitioners have been prescribing opiates irresponsibly for years. For the last ten years, well over 200,000,000 opiate prescriptions were issued per year in the US – enough for seven in every ten people. Once prescribed, opiate use is often not well managed. Prescriptions can be stopped abruptly, for example, leaving people with unexpected withdrawal syndromes and rebound pain. It is estimated that 75% of heroin users in the US began by using legal, prescription opiates. Secondly, drug suppliers have started cutting heroin with its far stronger but cheaper cousin, fentanyl. Given fentanyl’s strength, only a tiny amount is required to achieve the same effects as heroin, but the lack of pharmaceutical knowledge and equipment means it is often not measured or mixed appropriately into what is sold as ‘heroin’. There are two clear routes to alleviating the epidemic of opiate overdose: prevention, by ensuring responsible medical use of opiates, and ‘cure’, either by ensuring the quality and strength of heroin, or providing a means to stop opiate use. The former ‘cure’ is politically infeasible so it falls on the latter to help those already habitually using opiates. However, the availability of opiate treatment programs, such as opiate agonist treatment (OAT), is lacklustre in the US. OAT provides non-narcotic opiates, such as methadone or buprenorphine, to prevent withdrawal syndromes in users, from which they can slowly be weaned. This article looks at the cost-effectiveness of providing OAT for all persons seeking treatment for opiate use in California for an unlimited period versus standard care, which only provides OAT to those who have failed supervised withdrawal twice, and only for 21 days. The paper adopts a previously developed semi-Markov cohort model that includes states for treatment, relapse, incarceration, and abstinence. Transition probabilities for the new OAT treatment were determined from treatment data for current OAT patients (as far as I understand it). Although this does raise the question about the generalisability of this population to the whole population of opiate users – given the need to have already been through two supervised withdrawals, this population may have a greater motivation to quit, for example. In any case, the article estimates that the OAT program would be cost-saving, through reductions in crime and incarceration, and improve population health, by reducing the risk of death. Taken at face value these results seem highly plausible. But, as we’ve discussed before, drug policy rarely seems to be evidence-based.

The impact of aid on health outcomes in Uganda. Health Economics [PubMed] Published 22nd December 2017

Examining the response of population health outcomes to changes in health care expenditure has been the subject of a large and growing number of studies. One reason is to estimate a supply-side cost-effectiveness threshold: the health returns the health service achieves in response to budget expansions or contractions. Similarly, we might want to know the returns to particular types of health care expenditure. For example, there remains a debate about the effectiveness of aid spending in low and middle-income country (LMIC) settings. Aid spending may fail to be effective for reasons such as resource leakage, failure to target the right population, poor design and implementation, and crowding out of other public sector investment. Looking at these questions at an aggregate level can be tricky; the link between expenditure or expenditure decisions and health outcomes is long and causality flows in multiple directions. Effects are likely to therefore be small and noisy and require strong theoretical foundations to interpret. This article takes a different, and innovative, approach to looking at this question. In essence, the analysis boils down to a longitudinal comparison of those who live near large, aid funded health projects with those who don’t. The expectation is that the benefit of any aid spending will be felt most acutely by those who live nearest to actual health care facilities that come about as a result of it. Indeed, this is shown by the results – proximity to an aid project reduced disease prevalence and work days lost to ill health with greater effects observed closer to the project. However, one way of considering the ‘usefulness’ of this evidence is how it can be used to improve policymaking. One way is in understanding the returns to investment or over what area these projects have an impact. The latter is covered in the paper to some extent, but the former is hard to infer. A useful next step may be to try to quantify what kind of benefit aid dollars produce and its heterogeneity thereof.

The impact of social expenditure on health inequalities in Europe. Social Science & Medicine Published 11th January 2018

Let us consider for a moment how we might explore empirically whether social expenditure (e.g. unemployment support, child support, housing support, etc) affects health inequalities. First, we establish a measure of health inequality. We need a proxy measure of health – this study uses self-rated health and self-rated difficulty in daily living – and then compare these outcomes along some relevant measure of socioeconomic status (SES) – in this study they use level of education and a compound measure of occupation, income, and education (the ISEI). So far, so good. Data on levels of social expenditure are available in Europe and are used here, but oddly these data are converted to a percentage of GDP. The trouble with doing this is that this variable can change if social expenditure changes or if GDP changes. During the financial crisis, for example, social expenditure shot up as a proportion of GDP, which likely had very different effects on health and inequality than when social expenditure increased as a proportion of GDP due to a policy change under the Labour government. This variable also likely has little relationship to the level of support received per eligible person. Anyway, at the crudest level, we can then consider how the relationship between SES and health is affected by social spending. A more nuanced approach might consider who the recipients of social expenditure are and how they stand on our measure of SES, but I digress. In the article, the baseline category for education is those with only primary education or less, which seems like an odd category to compare to since in Europe I would imagine this is a very small proportion of people given compulsory schooling ages unless, of course, they are children. But including children in the sample would be an odd choice here since they don’t personally receive social assistance and are difficult to compare to adults. However, there are no descriptive statistics in the paper so we don’t know and no comparisons are made between other groups. Indeed, the estimates of the intercepts in the models are very noisy and variable for no obvious reason other than perhaps the reference group is very small. Despite the problems outlined so far though, there is a potentially more serious one. The article uses a logistic regression model, which is perfectly justifiable given the binary or ordinal nature of the outcomes. However, the authors justify the conclusion that “Results show that health inequalities measured by education are lower in countries where social expenditure is higher” by demonstrating that the odds ratio for reporting a poor health outcome in the groups with greater than primary education, compared to primary education or less, is smaller in magnitude when social expenditure as a proportion of GDP is higher. But the conclusion does not follow from the premise. It is entirely possible for these odds ratios to change without any change in the variance of the underlying distribution of health, the relative ordering of people, or the absolute difference in health between categories, simply by shifting the whole distribution up or down. For example, if the proportions of people in two groups reporting a negative outcome are 0.3 and 0.4, which then change to 0.2 and 0.3 respectively, then the odds ratio comparing the two groups changes from 0.64 to 0.58. The difference between them remains 0.1. No calculations are made regarding absolute effects in the paper though. GDP is also shown to have a positive effect on health outcomes. All that might have been shown is that the relative difference in health outcomes between those with primary education or less and others changes as GDP changes because everyone is getting healthier. The question of the article is interesting, it’s a shame about the execution.

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Method of the month: Semiparametric models with penalised splines

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 semiparametric models with penalised splines.

Principles

A common assumption of regression models is that effects are linear and additive. However, nothing is ever really that simple. One might respond that all models are wrong, but some are useful, as George Box once said. And the linear, additive regression model has coefficients that can be interpreted as average treatment effects under the right assumptions. Sometimes though we are interested in conditional average treatment effects and how the impact of an intervention varies according to the value of some variable of interest. Often this relationship is not linear and we don’t know its functional form. Splines provide a way of estimating curves (or surfaces) of unknown functional form and are a widely used tool for semiparametric regression models. The term ‘spline’ was derived from the tool shipbuilders and drafters used to construct smooth edges: a bendable piece of material that when fixed at a number of points would relax into the desired shape.

Implementation

Our interest lies in estimating the unknown function m:

y_i = m(x_i) + e_i

A ‘spline’ in the mathematical sense is a function constructed piece-wise from polynomial functions. The places where the functions meet are known as knots and the spline has order equal to one more than the degree of the underlying polynomial terms. Basis-splines or B-splines are the typical starting point for spline functions. These are curves that are defined recursively as a sum of ‘basis functions’, which depend only on the polynomial degree and the knots. A spline function can be represented as a linear combination of B-splines, the parameters dictating this combination can be estimated using standard regression model estimation techniques. If we have N B-splines then our regression function can be estimated as:

y_i = \sum_{j=1}^N ( \alpha_j B_j(x_i) ) + e_i

by minimising \sum_{i=1}^N \{ y_i - \sum_{j=1}^N ( \alpha_j B_j(x_i) ) \} ^2. Where the B_j are the B-splines and the \alpha_j are coefficients to be estimated.

Useful technical explainers of splines and B-splines can be found here [PDF] and here [PDF].

One issue with fitting splines to data is that we run the risk of ‘overfitting’. Outliers might distort the curve we fit, damaging the external validity of conclusions we might make. To deal with this, we can enforce a certain level of smoothness using so-called penalty functions. The smoothness (or conversely the ‘roughness’) of a curve is often defined by the integral of the square of the second derivative of the curve function. Penalised-splines, or P-splines, were therefore proposed which added on this smoothness term multiplied by a smoothing parameter \lambda. In this case, we look to minimising:

\sum_{i=1}^N \{ y_i - \sum_{j=1}^N ( \alpha_j B_j(x_i) ) \}^2 + \lambda\int m''(x_i)^2 dx

to estimate our parameters. Many other different variations on this penalty have been proposed. This article provides a good explanation of P-splines.

An attractive type of spline has become the ‘low rank thin plate spline‘. This type of spline is defined by its penalty, which has a physical analogy with the resistance that a thin sheet of metal puts up when it is bent. This type of spline removes the problem associated with thin plate splines of having too many parameters to estimate by taking a ‘low rank’ approximation, and it is generally insensitive to the choice of knots, which other penalised spline regression models are not.

Crainiceanu and colleagues show how the low rank thin plate smooth splines can be represented as a generalised linear mixed model. In particular, our model can be represented as:

m(x_i) = \beta_0 + \beta_1x_i  + \sum_{k=1}^K u_k |x_i - \kappa_k|^3

where \kappa_k, k=1,...,K, are the knots. The parameters, \theta = (\beta_0,\beta_1,u_k)', can be estimated by minimising

\sum_{i=1}^N \{ y_i - m(x_i) \} ^2 + \frac{1}{\lambda} \theta ^T D \theta .

This is shown to give the mixed model

y_i = \beta_0 + \beta_1 + Z'b + u_i

where each random coefficient in the vector b is distributed as N(0,\sigma^2_b) and Z and D are given in the paper cited above.

As a final note, we have discussed splines in one dimension, but they can be extended to more dimensions. A two-dimensional spline can be generated by taking the tensor product of the two one dimensional spline functions. I leave this as an exercise for the reader.

Software

R

  • The package gamm4 provides the tools necessary for a frequentist analysis along the lines described in this post. It uses restricted maximum likelihood estimation with the package lme4 to estimate the parameters of the thin plate spline model.
  • A Bayesian version of this functionality is implemented in the package rstanarm, which uses gamm4 to produce the matrices for thin plate spline models and Stan for the estimation through the stan_gamm4 function.

If you wanted to implement these models for yourself from scratch, Crainiceanu and colleagues provide the R code to generate the matrices necessary to estimate the spline function:

n<-length(covariate)
X<-cbind(rep(1,n),covariate)
knots<-quantile(unique(covariate),
 seq(0,1,length=(num.knots+2))[-c(1,(num.knots+2))])
Z_K<-(abs(outer(covariate,knots,"-")))^3
OMEGA_all<-(abs(outer(knots,knots,"-")))^3
svd.OMEGA_all<-svd(OMEGA_all)
sqrt.OMEGA_all<-t(svd.OMEGA_all$v %*%
 (t(svd.OMEGA_all$u)*sqrt(svd.OMEGA_all$d)))
Z<-t(solve(sqrt.OMEGA_all,t(Z_K)))

Stata

I will temper this advice by cautioning that I have never estimated a spline-based semi-parametric model in Stata, so what follows may be hopelessly incorrect. The only implementation of penalised splines in Stata is the package and associated function psplineHowever, I cannot find any information about the penalty function used, so I would advise some caution when implementing. An alternative is to program the model yourself, through conversion of the above R code in Mata to generate the matrix Z and then the parameters could be estimated with xtmixed. 

Applications

Applications of these semi-parametric models in the world of health economics have tended to appear more in technical or statistical journals than health economics journals or economics more generally. For example, recent examples include Li et al who use penalised splines to estimate the relationship between disease duration and health care costs. Wunder and co look at how reported well-being varies over the course of the lifespan. And finally, we have Stollenwerk and colleagues who use splines to estimate flexible predictive models for cost-of-illness studies with ‘big data’.

Credit

 

Method of the month: Synthetic control

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 synthetic control.

Principles

Health researchers are often interested in estimating the effect of a policy of change at the aggregate level. This might include a change in admissions policies at a particular hospital, or a new public health policy applied to a state or city. A common approach to inference in these settings is difference in differences (DiD) methods. Pre- and post-intervention outcomes in a treated unit are compared with outcomes in the same periods for a control unit. The aim is to estimate a counterfactual outcome for the treated unit in the post-intervention period. To do this, DiD assumes that the trend over time in the outcome is the same for both treated and control units.

It is often the case in practice that we have multiple possible control units and multiple time periods of data. To predict the post-intervention counterfactual outcomes, we can note that there are three sources of information: i) the outcomes in the treated unit prior to the intervention, ii) the behaviour of other time series predictive of that in the treated unit, including outcomes in similar but untreated units and exogenous predictors, and iii) prior knowledge of the effect of the intervention. The latter of these only really comes into play in Bayesian set-ups of this method. With longitudinal data we could just throw all this into a regression model and estimate the parameters. However, generally, this doesn’t allow for unobserved confounders to vary over time. The synthetic control method does.

Implementation

Abadie, Diamond, and Haimueller motivate the synthetic control method using the following model:

y_{it} = \delta_t + \theta_t Z_i + \lambda_t \mu_i + \epsilon_{it}

where y_{it} is the outcome for unit i at time t, \delta_t are common time effects, Z_i are observed covariates with time-varying parameters \theta_t, \lambda_t are unobserved common factors with \mu_i as unobserved factor loadings, and \epsilon_{it} is an error term. Abadie et al show in this paper that one can derive a set of weights for the outcomes of control units that can be used to estimate the post-intervention counterfactual outcomes in the treated unit. The weights are estimated as those that would minimise the distance between the outcome and covariates in the treated unit and the weighted outcomes and covariates in the control units. Kreif et al (2016) extended this idea to multiple treated units.

Inference is difficult in this framework. So to produce confidence intervals, ‘placebo’ methods are proposed. The essence of this is to re-estimate the models, but using a non-intervention point in time as the intervention date to determine the frequency with which differences of a given order of magnitude are observed.

Brodersen et al take a different approach to motivating these models. They begin with a structural time-series model, which is a form of state-space model:

y_t = Z'_t \alpha_t + \epsilon_t

\alpha_{t+1} = T_t \alpha_t + R_t \eta_t

where in this case, y_t is the outcome at time t, \alpha_t is the state vector and Z_t is an output vector with \epsilon_t as an error term. The second equation is the state equation that governs the evolution of the state vector over time where T_t is a transition matrix, R_t is a diffusion matrix, and \eta_t is the system error.

From this setup, Brodersen et al expand the model to allow for control time series (e.g. Z_t = X'_t \beta), local linear time trends, seasonal components, and allowing for dynamic effects of covariates. In this sense the model is perhaps more flexible than that of Abadie et al. Not all of the large number of covariates may be necessary, so they propose a ‘slab and spike’ prior, which combines a point mass at zero with a weakly informative distribution over the non-zero values. This lets the data select the coefficients, as it were.

Inference in this framework is simpler than above. The posterior predictive distribution can be ‘simply’ estimated for the counterfactual time series to give posterior probabilities of differences of various magnitudes.

Software

Stata

  • Synth Implements the method of Abadie et al.

R

  • Synth Implements the method of Abadie et al.
  • CausalImpact Implements the method of Brodersen et al.

Applications

Kreif et al (2016) estimate the effect of pay for performance schemes in hospitals in England and compare the synthetic control method to DiD. Pieters et al (2016) estimate the effects of democratic reform on under-five mortality. We previously covered this paper in a journal round-up and a subsequent post, for which we also used the Brodersen et al method described above. We recently featured a paper by Lépine et al (2017) in a discussion of user fees. The synthetic control method was used to estimate the impact that the removal of user fees had in various districts of Zambia on use of health care.

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