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’.

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Sam Watson’s journal round-up for 11th December 2017

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.

Can incentives improve survey data quality in developing countries?: results from a field experiment in India. Journal of the Royal Statistical Society: Series A Published 17th November 2017

I must admit a keen interest in the topic of this paper. As part of a large project looking at the availability of health services in slums and informal settlements around the world, we are designing a household survey. Much like the Demographic and Health Surveys, which are perhaps the Gold standard of household surveys in low-income countries, interviewers will go door to door to sampled households to complete surveys. One of the issues with household surveys is that they take a long time, and so non-response can be an issue. A potential solution is to offer respondents incentives, cash or otherwise, either before the survey or conditionally on completing it. But any change in survey response as a result of an incentive might create suspicion around data quality. Work in high-income countries suggests incentives to participate have little or no effect on data quality. But there is little evidence about these effects in low-income countries. We might suspect the consequences of survey incentives to differ in poorer settings. For a start, many surveys are conducted on behalf of the government or an NGO, and respondents may misrepresent themselves if they believe further investment in their area might be forthcoming if they are sufficiently badly-off. There may also be larger differences between the interviewer and interviewee in terms of education or cultural background. And finally, incentives can affect the balance between a respondent’s so-called intrinsic and extrinsic motivations for doing something. This study presents the results of a randomised trial where the ‘treatment’ was a small conditional payment for completing a survey, and the ‘control’ was no incentive. In both arms, the response rate was very high (>96%), but it was higher in the treatment arm. More importantly, the authors compare responses to a broad range of socioeconomic and demographic questions between the study arms. Aside from the frequent criticism that statistical significance is interpreted here as the existence of a difference, there are some interesting results. The key observed difference is that in the incentive arm respondents reported having lower wealth consistently across a number of categories. This may result from any of the aforementioned effects of incentives, but may be evidence that incentives can affect data quality and should be used with caution.

Association of US state implementation of newborn screening policies for critical congenital heart disease with early infant cardiac deaths. JAMA [PubMedPublished 5th December 2017

Writing these journal round-ups obviously requires reading the papers that you choose. This can be quite an undertaking for papers published in economics journals, which are often very long, but they provide substantial detail allowing for a thorough appraisal. The opposite is true for articles in medical journals. They are pleasingly concise, but often at the expense of including detail or additional analyses. This paper falls into the latter camp. Using detailed panel data on infant deaths by cause by year and by state in the US, it estimates the effect of mandated screening policies for infant congenital heart defects on deaths from this condition. Given these data and more space, one might expect to see more flexible models than the differences in differences type analysis presented here, such as allowing for state-level correlated time trends. The results seem clear and robust – the policies were associated with a reduction in death from congenital heart conditions by around a third. Given this, one might ask: if it’s so effective, why weren’t doctors doing it anyway? Additional analyses reveal little to no association of the policies with death from other conditions, which may suggest that doctors didn’t have to reallocate their time from other beneficial functions. Perhaps then the screening bore other costs. In the discussion, the authors mention that a previous economic evaluation showed that universal screening was relatively costly (approximately $40,000 per life year saved), but that this may be an overestimate in light of these new results. Certainly then an updated economic evaluation is warranted. However, the models used in the paper may lead one to be cautious about causal interpretations and hence using the estimates in an evaluation. Given some more space the authors may have added additional analyses, but then I might not have read it…

Subsidies and structure: the lasting impact of the Hill-Burton program on the hospital industry. Review of Economics and Statistics [RePEcPublished 29th November 2017

As part of the Hospital Survey and Construction Act of 1946 in the United States, the Hill-Burton program was enacted. As a reaction to the perceived lack of health care services for workers during World War 2, the program provided subsidies of up to a third for building nonprofit and local hospitals. Poorer areas were prioritised. This article examines the consequences of this subsidy program on the structure of the hospital market and health care utilisation. The main result is that the program had the consequence of increasing hospital beds per capita and that this increase was lasting. More specific analyses are presented. Firstly, the increase in beds took a number of years to materialise and showed a dose-response; higher-funded counties had bigger increases. Secondly, the funding reduced private hospital bed capacity. The net effect on overall hospital beds was positive, so the program affected the composition of the hospital sector. Although this would be expected given that it substantially affected the relative costs of different types of hospital bed. And thirdly, hospital utilisation increased in line with the increases in capacity, indicating a previously unmet need for health care. Again, this was expected given the motivation for the program in the first place. It isn’t often that results turn out as neatly as this – the effects are exactly as one would expect and are large in magnitude. If only all research projects turned out this way.

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