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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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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Review: Health Econometrics Using Stata (Partha Deb et al)

Health Econometrics Using Stata

Partha Deb, Edward C. Norton, Willard G. Manning

Paperback, 264 pages, ISBN: 978-1-59718-228-7, published 31 August 2017

Amazon / Google Books / Stata Press

This book is the perfect guide to understanding the various econometric methods available for modelling of costs and counts data for the individual who understands econometrics best after applying it to a dataset (like myself). Pre-requisites include a decent knowledge of Stata and a desire to apply econometric methods to a cost or count outcome variable

It’s important to say that this book does not cover all aspects of econometrics within health economics, but instead focuses on ‘modelling health care costs and counts’ (the title of the short course from which the book evolved). As expected from this range of texts, the vast majority of the book comes with detailed example Stata code for all of the methods described, with illustrations either using a publicly available sample of MEPS data or simulated data.

Like many papers in this field, the focus of the book revolves around the non-normal characteristics of health care resource use distributions. These are the mass point at zero, right-hand skew and inherent heteroskedasticity. As such the book covers the broad suite of models that have been developed in order to account for these features, ranging from two-part models, transformation of the data (and the problematic re-transformation of estimated effects) to non-linear modelling methods such as generalised linear models (GLMs). Unlike many papers in this field, the authors emphasise the need – and provide guidance on how – to delve deep into the underlying data in order to appreciate the most appropriate methods (there is even a chapter on design effects) and encourage rigorous testing of model specification. In addition, Health Econometrics Using Stata considers the important issue of endogeneity and is not solely fixated on distributional issues, providing important insight and code for estimation of non-linear models that control for potential endogeneity (interested readers may wish to heed the published cautionary notes for some of these methods, e.g. Chapman and Brooks). Finally, the book describes more advanced methods for estimating heterogeneous effects, although code is not provided for all of these methods, which is a bit of a shame (but perhaps understandable given the complexity).

This could be a very dry text, but it is not – emphatically! The personality of the authors comes through very strongly from the writing. Reading it brought back many pleasant memories from the course ‘modelling health care costs and counts’ that I sat in 2012. The book also features a dedication to Willard Manning, which is a fitting tribute to a man who was both a great academic and an outstanding mentor. One particular highlight, with which past course attendants will be familiar, is the section ‘top 10 myths in health econometrics’. This straightforward and punchy presentation, backed up by rigorous methodological research, is a great way to get these key messages across in an accessible format. Other great features of this book include the use of simulations to illustrate important features of the econometric models (with code provided to recreate) and a personal highlight (granted, a niche interest…) was the code to generate appropriate standard errors when using the poisson family within GLMs for costs.

Of course, Health Econometrics Using Stata cannot be comprehensive and there are developments in this field that are not covered. Most notably, there is no discussion of how to model these data in a panel/longitudinal setting, which is crucially important for estimating parameters for decision models, for example. Potential issues around missing data and censoring are also not discussed. Also, this text does not cover advances in flexible parametric modelling, which enable modelling of data that are both highly skewed and leptokurtic (see Jones 2017 for an excellent summary of this literature along with a primer on data visualisation using Stata).

I heartily recommend Health Econometrics Using Stata to interested colleagues who want practical advice – on model selection and specification testing with cost and count outcome data – from some of the top specialists in our field, in their own words.

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