James Lomas’s journal round-up for 21st May 2018

May 21, 2018May 19, 2018jameslomas88 Leave a comment

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.

Decision making for healthcare resource allocation: joint v. separate decisions on interacting interventions. Medical Decision Making [PubMed] Published 23rd April 2018

While it may be uncontroversial that including all of the relevant comparators in an economic evaluation is crucial, a careful examination of this statement raises some interesting questions. Which comparators are relevant? For those that are relevant, how crucial is it that they are not excluded? The answer to the first of these questions may seem obvious, that all feasible mutually exclusive interventions should be compared, but this is in fact deceptive. Dakin and Gray highlight inconsistency between guidelines as to what constitutes interventions that are ‘mutually exclusive’ and so try to re-frame the distinction according to whether interventions are ‘incompatible’ – when it is physically impossible to implement both interventions simultaneously – and, if not, whether interventions are ‘interacting’ – where the costs and effects of the simultaneous implementation of A and B do not equal the sum of these parts. What I really like about this paper is that it has a very pragmatic focus. Inspired by policy arrangements, for example single technology appraisals, and the difficulty in capturing all interactions, Dakin and Gray provide a reader-friendly flow diagram to illustrate cases where excluding interacting interventions from a joint evaluation is likely to have a big impact, and furthermore propose a sequencing approach that avoids the major problems in evaluating separately what should be considered jointly. Essentially when we have interacting interventions at different points of the disease pathway, evaluating separately may not be problematic if we start at the end of the pathway and move backwards, similar to the method of backward induction used in sequence problems in game theory. There are additional related questions that I’d like to see these authors turn to next, such as how to include interaction effects between interventions and, in particular, how to evaluate system-wide policies that may interact with a very large number of interventions. This paper makes a great contribution to answering all of these questions by establishing a framework that clearly distinguishes concepts that had previously been subject to muddied thinking.

When cost-effective interventions are unaffordable: integrating cost-effectiveness and budget impact in priority setting for global health programs. PLoS Medicine [PubMed] Published 2nd October 2017

In my opinion, there are many things that health economists shouldn’t try to include when they conduct cost-effectiveness analysis. Affordability is not one of these. This paper is great, because Bilinski et al shine a light on the worldwide phenomenon of interventions being found to be ‘cost-effective’ but not affordable. A particular quote – that it would be financially impossible to implement all interventions that are found to be ‘very cost-effective’ in many low- and middle-income countries – is quite shocking. Bilinski et al compare and contrast cost-effectiveness analysis and budget impact analysis, and argue that there are four key reasons why something could be ‘cost-effective’ but not affordable: 1) judging cost-effectiveness with reference to an inappropriate cost-effectiveness ‘threshold’, 2) adoption of a societal perspective that includes costs not falling upon the payer’s budget, 3) failing to make explicit consideration of the distribution of costs over time and 4) the use of an inappropriate discount rate that may not accurately reflect the borrowing and investment opportunities facing the payer. They then argue that, because of this, cost-effectiveness analysis should be presented along with budget impact analysis so that the decision-maker can base a decision on both analyses. I don’t disagree with this as a pragmatic interim solution, but – by highlighting these four reasons for divergence of results with such important economic consequences – I think that there will be further reaching implications of this paper. To my mind, Bilinski et al essentially serves as a call to arms for researchers to try to come up with frameworks and estimates so that the conduct of cost-effectiveness analysis can be improved in order that paradoxical results are no longer produced, decisions are more usefully informed by cost-effectiveness analysis, and the opportunity costs of large budget impacts are properly evaluated – especially in the context of low- and middle-income countries where the foregone health from poor decisions can be so significant.

Patient cost-sharing, socioeconomic status, and children’s health care utilization. Journal of Health Economics [PubMed] Published 16th April 2018

This paper evaluates a policy using a combination of regression discontinuity design and difference-in-difference methods. Not only does it do that, but it tackles an important policy question using a detailed population-wide dataset (a set of linked datasets, more accurately). As if that weren’t enough, one of the policy reforms was actually implemented as a result of a vote where two politicians ‘accidentally pressed the wrong button’, reducing concerns that the policy may have in some way not been exogenous. Needless to say I found the method employed in this paper to be a pretty convincing identification strategy. The policy question at hand is about whether demand for GP visits for children in the Swedish county of Scania (Skåne) is affected by cost-sharing. Cost-sharing for GP visits has occurred for different age groups over different periods of time, providing the basis for regression discontinuities around the age threshold and treated and control groups over time. Nilsson and Paul find results suggesting that when health care is free of charge doctor visits by children increase by 5-10%. In this context, doctor visits happened subject to telephone triage by a nurse and so in this sense it can be argued that all of these visits would be ‘needed’. Further, Nilsson and Paul find that the sensitivity to price is concentrated in low-income households, and is greater among sickly children. The authors contextualise their results very well and, in addition to that context, I can’t deny that it also particularly resonated with me to read this approaching the 70th birthday of the NHS – a system where cost-sharing has never been implemented for GP visits by children. This paper is clearly also highly relevant to that debate that has surfaced again and again in the UK.

Credits

Antony Theobald (CC BY-NC-ND 2.0)

Method of the month: custom likelihoods with Stan

May 18, 2018May 17, 2018Sam Watson Leave a comment

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 custom likelihoods with Stan.

Principles

Regular readers of this blog will know that I am a fan of Bayesian methods. The exponential growth in personal computing power has opened up a whole new range of Bayesian models at home. WinBUGS and JAGS were the go-to pieces of software for estimating Bayesian models, both using Markov Chain Monte Carlo (MCMC) methods. Theoretically, an MCMC chain will explore the posterior distribution. But MCMC has flaws. For example, if the target distribution has a high degree of curvature, such as many hierarchical models might exhibit, then MCMC chains can have trouble exploring it. To compensate, the chains stay in the ‘difficult’ bit of the space for longer before leaving to go elsewhere so its average oscillates around the true value. Asymptotically, these oscillations balance out, but in real, finite time, they ultimately lead to bias. And further, MCMC chains are very slow to converge to the target distribution, and for complex models can take a literal lifetime. An alternative, Hamiltonian Monte Carlo (HMC), provides a solution to these issues. Michael Betancourt’s introduction to HCM is great for anyone interested in the topic.

Stan is a ‘probabilistic programming language’ that implements HMC. A huge range of probability distributions are already implemented in the software, check out the manual for more information. And there is an R package, rstanarm, that estimates a number of standard models using normal R code that even means you can use these tools without learning the code. However, Stan may not have the necessary distributions for more complex econometric or statistical models. It used to be the case that you would have to build your own MCMC sampler – but given the problems with MCMC, this is now strongly advised against in lieu of HMC. Fortunately, we can implement our own probability density functions in Stan. So, if you can write down the (log) likelihood for your model, you can estimate it in Stan!

The aim of this post is to provide an example of implementing a custom probability function in Stan from the likelihood of our model. We will look at the nested logit model. These models have been widely used for multinomial choice problems. An area of interest among health economists is the choice of hospital provider. A basic multinomial choice model, such as a multinomial logit, requires an independence of irrelevant alternatives (IIA) assumption that says the odds of choosing one option over another is independent of any other alternative. For example, it would assume that the odds of me choosing the pharmacy in town over the hospital in town would be unaffected by a pharmacy opening on my own road. This is likely too strong. There are many ways to relax this assumption, the nested logit being one. The nested logit is useful when choices can be ‘nested’ in groups and assumes there is a correlation among choices with each group. For example, we can group health care providers into pharmacies, primary care clinics, hospitals, etc. such as this:

Implementation

Econometric model

Firstly, we need a nesting structure for our choices, like that described above. We’ll consider a 2-level nesting structure, with T branches and R total choices, with R_t choices in each branch t. Like with most choice models we start from an additive random utility model, which is, for individual i=1,…,N, and with choice over branch and option:

$U_{itr} = V_{itr} + \epsilon_{itr}$

Then the chosen option is the one with the highest utility. The motivating feature of the nested logit is that the hierarchical structure allows us to factorise the joint probability of choosing branch t and option r into a conditional and marginal model:

$p_{itr} = p_{it} \times p_{ir|t}$

Multinomial choice models arise when the errors are assumed to have a generalised extreme value (GEV) distribution, which gives use the multinomial logit model. We will model the deterministic part of the equation with branch-varying and option-varying variables:

$V_{itr} = Z_{it}'\alpha + X_{itr}'\beta_t$

Then the model can be written as:

$p_{itr} = p_{it} \times p_{ir|t} = \frac{exp(Z_{it}'\alpha + \rho_t I_{it})}{\sum_{k \in T} exp(Z_{ik}'\alpha + \rho_k I_{ik})} \times \frac{exp(X_{itr}'\beta_t/\rho_t)}{\sum_{m \in R_t} exp( X_{itm}'\beta_t/\rho_t) }$

where $\rho_t$ is variously called a scale parameter, correlation parameter, etc. and defines the within branch correlation (arising from the GEV distribution). We also have the log-sum, which is also called the inclusive value:

$I_{it} = log \left( \sum_{m \in R_t} exp( X_{itm}'\beta_t/\rho_t) \right)$ .

Now we have our model setup, the log likelihood over all individuals is

$\sum_{i=1}^N \sum_{k \in T} \sum_{m \in R_t} y_{itr} \left[ Z_{it}'\alpha + \rho_t I_{it} - log \left( \sum_{k \in T} exp(Z_{ik}'\alpha + \rho_k I_{ik}) \right) + X_{itr}'\beta_t/\rho_t - log \left( \sum_{m \in R_t} exp( X_{itm}'\beta_t/\rho_t) \right) \right]$

As a further note, for the model to be compatible with an ARUM specification, a number of conditions need to be satisfied. One of these is satisfied is $0<\rho_t \leq 1$ , so we will make that restriction. We have also only included alternative-varying variables, but we are often interested in individual varying variables and allowing parameters to vary over alternatives, which can be simply added to this specification, but we will leave them out for now to keep things “simple”. We will also use basic weakly informative priors and leave prior specification as a separate issue we won’t consider further:

$\alpha \sim normal(0,5), \beta_t \sim normal(0,5), \rho_t \sim Uniform(0,1)$

Software

DISCLAIMER: This code is unlikely to be the most efficient, nor can I guarantee it is 100% correct – use at your peril!

The following assumes a familiarity with Stan and R.

Stan programs are divided into blocks including data, parameters, and model. The functions block allows us to define custom (log) probability density functions. These take a form something like:

real xxx_lpdf(real y, ...){}

which says that the function outputs a real valued variable and take a real valued variable, y, as one of its arguments. The _lpdf suffix allows the function to act as a density function in the program (and equivalently _lpmf for log probability mass functions for discrete variables). Now we just have to convert the log likelihood above into a function. But first, let’s just consider what data we will be passing to the program:

N, the number of observations;
T, the number of branches;
P, the number of branch-varying variables;
Q, the number of choice-varying variables;
R, a T x 1 vector with the number of choices in each branch, from which we can also derive the total number of options as sum(R). We will call the total number of options Rk for now;
Y, a N x Rk vector, where Y[i,j] = 1 if individual i=1,…,N chose choice j=1,…,Rk;
Z, a N x T x P array of branch-varying variables;
X, a N x Rk x Q array of choice-varying variables.

And the parameters:

$\rho$ , a T x 1 vector of correlation parameters;
$\alpha$ , a P x 1 vector of branch-level covariates;
$\beta$ , a P x T matrix of choice-varying covariates.

Now, to develop the code, we will specify the function for individual observations of Y, rather than the whole matrix, and then perform the sum over all the individuals in the model block. So we only need to feed in each individual’s observations into the function rather than the whole data set. The model is specified in blocks as follows (with all the data and parameter as arguments to the function):

functions{
 real nlogit_lpdf(real[] y, real[,] Z, real[,] X, int[] R, 
   vector alpha, matrix beta, vector tau){
//first define our additional local variables
 real lprob; //variable to hold log prob
 int count1; //keep track of which option in the loops
 int count2; //keep track of which option in the loops
 vector[size(R)] I_val; //inclusive values
 real denom; //sum denominator of marginal model
//for the variables appearing in sum loops, set them to zero
 lprob = 0;
 count1 = 0;
 count2 = 0;
 denom = 0;
 
 // determine the log-sum for each conditional model, p_ir|t, 
 //i.e. inclusive value
 for(k in 1:size(R)){
    I_val[k] = 0;
    for(m in 1:R[k]){
       count1 = count1 + 1;
       I_val[k] = I_val[k] + exp(to_row_vector(X[count1,])*
          beta[,k] /tau[k]);
       }
    I_val[k] = log(I_val[k]);
 }
 
 //determine the sum for the marginal model, p_it, denomininator
 for(k in 1:size(R)){
    denom = denom + exp(to_row_vector(Z[k,])*alpha + tau[k]*I_val[k]);
 }
 
 //put everything together in the log likelihood
 for(k in 1:size(R)){
    for(m in 1:R[k]){
       count2 = count2 + 1;
       lprob = lprob + y[count2]*(to_row_vector(Z[k,])*alpha + 
         tau[k]*I_val[k] - log(denom) + 
         to_row_vector(X[count2,])*beta[,k] - I_val[k]);
       }
    }
// return the log likelihood value
 return lprob;
 }
}
data{
 int N; //number of observations
 int T; //number of branches
 int R[T]; //number of options per branch
 int P; //dim of Z
 int Q; //dim of X
 real y[N,sum(R)]; //outcomes array
 real Z[N,T,P]; //branch-varying variables array
 real X[N,sum(R),Q]; //option-varying variables array
}
parameters{
 vector<lower=0, upper=1>[T] rho; //scale-parameters
 vector[P] alpha; //branch-varying parameters
 matrix[Q,T] beta; //option-varying parameters
}
model{
//specify priors
 for(p in 1:P) alpha[p] ~ normal(0,5); 
 for(q in 1:Q) for(t in 1:T) beta[q,t] ~ normal(0,5);

//loop over all observations with the data 
 for(i in 1:N){
    y[i] ~ nlogit(Z[i,,],X[i,,],R,alpha,beta,rho);
 }
}

Simulation model

To see whether our model is doing what we’re hoping it’s doing, we can run a simple test with simulated data. It may be useful to compare the result we get to those from other estimators; the nested logit is most frequently estimated using the FIML estimator. But, neither Stata nor R provide packages that estimate a model with branch-varying variables – another reason why we sometimes need to program our own models.

The code we’ll use to simulate the data is:

#### simulate 2-level nested logit data ###

N <- 300 #number of people
P <- 2 #number of branch variant variables
Q <- 2 #number of option variant variables
R <- c(2,2,2) #vector with number of options per branch
T <- length(R) #number of branches
Rk <- sum(R) #number of options

#simulate data

Z <- array(rnorm(N*T*P,0,0.5),dim = c(N,T,P))
X <- array(rnorm(N*Rk*Q,0,0.5), dim = c(N,Rk,Q))

#parameters
rho <- runif(3,0.5,1)
beta <- matrix(rnorm(T*Q,0,1),c(Q,T))
alpha <- rnorm(P,0,1)

#option models #change beta indexing as required
vals_opt <- cbind(exp(X[,1,]%*%beta[,1]/rho[1]),exp(X[,2,]%*%beta[,1]/rho[1]),exp(X[,3,]%*%beta[,2]/rho[2]),
 exp(X[,4,]%*%beta[,2]/rho[2]),exp(X[,5,]%*%beta[,3]/rho[3]),exp(X[,6,]%*%beta[,3]/rho[3]))

incl_val <- cbind(vals_opt[,1]+vals_opt[,2],vals_opt[,3]+vals_opt[,4],vals_opt[,5]+vals_opt[,6])

vals_branch <- cbind(exp(Z[,1,]%*%alpha + rho[1]*log(incl_val[,1])),
 exp(Z[,2,]%*%alpha + rho[2]*log(incl_val[,2])),
 exp(Z[,3,]%*%alpha + rho[3]*log(incl_val[,3])))

sum_branch <- rowSums(vals_branch)

probs <- cbind((vals_opt[,1]/incl_val[,1])*(vals_branch[,1]/sum_branch),
 (vals_opt[,2]/incl_val[,1])*(vals_branch[,1]/sum_branch),
 (vals_opt[,3]/incl_val[,2])*(vals_branch[,2]/sum_branch),
 (vals_opt[,4]/incl_val[,2])*(vals_branch[,2]/sum_branch),
 (vals_opt[,5]/incl_val[,3])*(vals_branch[,3]/sum_branch),
 (vals_opt[,6]/incl_val[,3])*(vals_branch[,3]/sum_branch))

Y = t(apply(probs, 1, rmultinom, n = 1, size = 1))

Then we’ll put the data into a list and run the Stan program with 500 iterations and 3 chains:

data <- list(
 y = Y,
 X = X,
 Z = Z,
 R = R,
 T = T,
 N = N,
 P = P,
 Q = Q
)

require(rstan)
rstan_options(auto_write = TRUE)
options(mc.cores = parallel::detectCores())

fit <- stan("C:/Users/Samuel/Dropbox/Code/nlogit.stan",
 data = data,
 chains = 3,
 iter = 500)

Which gives results (with 25t and 75th percentiles dropped to fit on screen):

> print(fit)
Inference for Stan model: nlogit.
3 chains, each with iter=500; warmup=250; thin=1; 
post-warmup draws per chain=250, total post-warmup draws=750.

             mean se_mean   sd    2.5%     50%     75%   97.5% n_eff Rhat
rho[1]       1.00    0.00 0.00    0.99    1.00    1.00    1.00   750  1
rho[2]       0.87    0.00 0.10    0.63    0.89    0.95    1.00   750  1
rho[3]       0.95    0.00 0.04    0.84    0.97    0.99    1.00   750  1
alpha[1]    -1.00    0.01 0.17   -1.38   -0.99   -0.88   -0.67   750  1
alpha[2]    -0.56    0.01 0.16   -0.87   -0.56   -0.45   -0.26   750  1
beta[1,1]   -3.65    0.01 0.32   -4.31   -3.65   -3.44   -3.05   750  1
beta[1,2]   -0.28    0.01 0.24   -0.74   -0.27   -0.12    0.15   750  1
beta[1,3]    0.99    0.01 0.25    0.48    0.98    1.15    1.52   750  1
beta[2,1]   -0.15    0.01 0.25   -0.62   -0.16    0.00    0.38   750  1
beta[2,2]    0.28    0.01 0.24   -0.16    0.28    0.44    0.75   750  1
beta[2,3]    0.58    0.01 0.24    0.13    0.58    0.75    1.07   750  1
lp__      -412.84    0.14 2.53 -418.56 -412.43 -411.05 -409.06   326  1

Samples were drawn using NUTS(diag_e) at Sun May 06 14:16:43 2018.
For each parameter, n_eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor on split chains (at 
convergence, Rhat=1).

Which we can compare to the original parameters:

> beta
           [,1]       [,2]      [,3]
[1,] -3.9381389 -0.3476054 0.7191652
[2,] -0.1182806  0.2736159 0.5237470
> alpha
[1] -0.9654045 -0.6505002
> rho
[1] 0.9503473 0.9950653 0.5801372

You can see that the posterior means and quantiles of the distribution provide pretty good estimates of the original parameters. Convergence diagnostics such as Rhat and traceplots (not reproduced here) show good convergence of the chains. But, of course, this is not enough for us to rely on it completely – you would want to investigate further to ensure that the chains were actually exploring the posterior of interest.

Applications

I am not aware of any examples in health economics of using custom likelihoods in Stan. There are not even many examples of Bayesian nested logit models, one exception being a paper by Lahiri and Gao, who ‘analyse’ the nested logit using MCMC. But given the limitations of MCMC discussed above, one should prefer this implementation in the post rather than the MCMC samplers of that paper. It’s also a testament to computing advances and Stan that in 2001 an MCMC sampler and analysis could fill a paper in a decent econometrics journal and now we can knock one out for a blog post.

In terms of nested logit models in general in health economics, there are many examples going back 30 years (e.g. this article from 1987). More recent papers have preferred “mixed” or “random parameters” logit or probit specifications, which are much more flexible than the nested logit. We would advise these sorts of models for this reason. The nested logit was used as an illustrative example of estimating custom likelihoods for this post.

Credit

Popular Science Monthly

Thesis Thursday: James Oswald

May 17, 2018May 16, 2018Chris Sampson Leave a comment

On the third Thursday of every month, we speak to a recent graduate about their thesis and their studies. This month’s guest is Dr James Oswald who has a PhD from the University of Sheffield. If you would like to suggest a candidate for an upcoming Thesis Thursday, get in touch.

Title
Essays on well-being and mental health: determinants and consequences
Supervisors
Sarah Brown, Jenny Roberts, Bert Van Landeghem
Repository link
http://etheses.whiterose.ac.uk/18915/

What measures of health did you use in your research and how did these complement broader measures of well-being?

I didn’t use any measures of physical health. I used a few measures of subjective well-being (SWB) and mental health which vary across the chapters. In Chapter 2 I used life satisfaction and the Rutter Malaise Inventory. Life satisfaction is a global retrospective judgement of one’s life and is a measure of evaluative well-being (see Dolan and Metcalfe, 2012). The Rutter Malaise Inventory, a measure of affective well-being, is an index that is composed of 9 items that measure the respondent’s symptoms of psychological distress or depression. The measure of the mental health problems of adolescents that is used in Chapter 3 is the Strengths and Difficulties Questionnaire (SDQ). The SDQ is made up of four 5-item subscales: emotional problems, peer relationship problems, conduct problems, and hyperactivity/ inattention problems. Chapter 3 utilises the General Health Questionnaire (GHQ) as a measure of the mental health of parents. The GHQ is a screening instrument that was initially developed to diagnose psychiatric disorders. Chapter 4 utilises one measure of subjective well-being, which relates to the number of days of poor self-reported mental health (stress, depression, and problems with emotions) in the past 30 days.

Did your research result in any novel findings regarding the social determinants of mental well-being?

The findings of Chapter 2 suggested that bullying victimisation at age 11 has a large, adverse effect on SWB as an adult. Childhood bullying remains prevalent – recent estimates suggest that approximately 20-30% of children are bullied by other children. The evidence provided in Chapter 3 indicated that greater externalising problems of adolescents are positively associated with the likelihood that they engage in antisocial behaviour. Chapter 4 indicated two important findings. Firstly, Hurricane Katrina had a negative effect on the SWB of individuals living in the states that were directly affected by the disaster. Secondly, the analysis suggested that the Indian Ocean tsunami and the Haiti earthquake increased the SWB of Americans living closest to the affected areas.

How can natural disasters affect mental health?

My thesis presents evidence to suggest that their impact depends upon whether you live in the disaster area. I explored the role of geography by exploring the effects of three disasters – hurricane Katrina in 2005 (USA), Indian Ocean tsunami in 2004 (East Asia), and the Haiti earthquake in 2010. Firstly, Hurricane Katrina had a negative effect on the SWB of individuals living in the states that were directly affected by the disaster. As a result, the findings suggest that government intervention in the aftermath of disasters is needed to help mitigate the adverse effects of natural disasters on the SWB of people who live in the directly affected areas. For example, appropriate mental health services and counselling could be offered to people suffering unhappiness or distress. Secondly, the analysis suggested that the Indian Ocean tsunami and the Haiti earthquake increased the SWB of Americans living closest to the affected areas. This surprising finding may be explained by the interdependence of utility functions. Following the disasters, Americans were exposed to widespread coverage of the disasters via social and traditional media sources. Because of the media coverage, they may have thought about the catastrophic repercussions of the disasters for the victims. Consequently, Americans who lived closest to the affected areas may have compared themselves to the disaster victims, leading them to feel thankful that the disaster did not affect them, thus increasing their SWB.

The empirical results support the case that the utility functions of strangers may be interdependent, rather than independent, an assumption generally made in economics. Furthermore, the findings indicated no evidence that the effects of the disasters were more pronounced for individuals of the same ethnicity as the disaster victims. The results therefore suggest that geographical proximity to the affected areas, rather than sharing similar characteristics with the disaster victims, may determine the effects of natural disasters on SWB outside of the areas that were directly affected by the disasters. This issue is discussed in greater detail in Chapter 4 of my thesis.

How did you go about identifying some of the consequences of mental health problems?

My thesis uses a range of econometric methods to explore the determinants and consequences of mental health and subjective well-being. In Chapter 2 – for childhood bullying and adult subjective well-being – I used a range of methods including random effects ordered probit models, Hausman tests, and Heckman models. Chapter 3 investigates how the mental health of adolescents affects their participation in antisocial behaviour. The analysis uses random effects probit, multivariate probit, and conditional logit models. Chapter 4 investigates the effects of three natural disasters on subjective well-being in the USA. The chapter uses difference-in-differences methodology with a count data model called a zero-inflated negative binomial model.

Are there any policy recommendations that you would make in light of your research?

Chapter 2 suggests that being bullied as a child adversely affects subjective well-being as an adult. My analysis supports the case that preventing children bullying in schools may have a positive effect on the SWB of a large percentage of the adult population. Chapter 3 indicated that greater externalising problems of adolescents are positively associated with the likelihood that they engage in antisocial behaviour. Previous research has suggested that adolescents who commit antisocial behaviour have an increased probability of committing crime as adults. Consequently, the findings suggest that mental health interventions to target the externalising problems of adolescents may reduce future crime. The findings also suggest that the money spent on the “Troubled Families” programme may be spent more cost-effectively in reducing antisocial behaviour by expanding access to mental health interventions for adolescents, such as via the Improving Access to Psychological Therapies programme.

The findings of Chapter 4 suggest that government intervention in the aftermath of disasters is needed to help mitigate the adverse effects of natural disasters on the SWB of people who live in the directly affected areas. For example, appropriate mental health services and counselling could be offered to people suffering unhappiness or distress because of natural disasters.