Estimating Health Care Cost Among Fragile & Conflict Affected States: An Elastic Net Approach

Kevin Wunderlich (Graduate Student)

8 April 2016

Outline

Introduction
Research Objectives
Models
Data & Simulation
Simulated Results
Concluding Remarks
References

Introduction

What Are Fragile and Conflict Affected States (FCAS)?

“[A] fundamental failure of the state to perform functions necessary to meet citizens' basic needs and expectations.” McIntosh, K. & Buckley, J. (2015)

“[Fragile states are] those where the government cannot or will not deliver core functions to the majority of its people, including the poor.” DFID (2005)

“[S]tates are fragile when states lack political will and/or capacity to provide the basic functions needed for poverty reduction, development and to safeguard the security and human rights of their populations.” OECD (2007)

Introduction (Cont'd)

Locations of HIPC

“The growing concentration of the poor in impoverished, conflict-prone countries could make it impossible to reach the goal of eradicating poverty” World Bank (2014/2015)

Locations of FCAS

Daily wage in 2014 World Bank data:
- FCAS: $4.25
- Heavily indebted poor countries(HIPC) daily wage: $2.36

Introduction (Cont'd)

“Conflict and population displacement on a large scale poses significant risks for infectious disease, combined with livelihood disruption and food and water shortages” Kruk ,Freedman, et al. (2015)
Deadly and infectious diseases result from insufficient health care, predominately in FCAS (recall the 2014 Ebola virus outbreak in West Africa) Kalra, Sarathi, et al. (2014) 2013 Vaccine-Preventable Outbreaks Map

Research Objectives

“Since the mid-1990s, a stronger donor emphasis on rewarding countries with relatively effective governments and stable macroeconomic policies has led to further neglect of fragile states” Department for International Development (2005)
This study seeks to accomplish the following:

Identify the key predictors of health care cost using the Elastic Net (Enet) algorithm

Determine the maximum health care cost among these resource constrained populations using risk measures

Use the preceeding outcomes to contribute to the global discussion regarding improvements for health care in FCAS

Models

Linear Regression Model
- Suppose $ Y \epsilon R $ is a response variable and $ X \epsilon R^p $ is a vector of predictors. The linear regression model is estimated by:

\[ E[Y|X=x]=\beta_{0}+x^{T}\beta\tag{1} \]

where:
- ( $ x_i , y_i $ ) for $ i=1,2,...,N $ are pairs of observations
- $ \beta_0 $ is the intercept
- $ \beta $ is the slope vector
  - Use OLS to estimate the parameters of the model

Models (Cont'd)

Enet Solution:

\[ argmin_{(\beta_{0},\beta) \epsilon R^{p+1}} \bigg[\frac{1}{2N}\sum\limits_{i=1}^N \bigg(y_{i}-\beta_{0}-x_{i}^T \beta \bigg)^2+\lambda P_{\alpha}(\beta)\bigg]\tag{2} \] where: \[ P_{\alpha}(\beta)=\sum\limits_{j=1}^k\bigg[\frac{1}{2}(1-\alpha)\beta_{j}^2+\alpha |\beta_{j}| \bigg]\tag{3} \]

Enet is a compromise between Ridge regression and LASSO (Zou and Hastie (2005))
Parameters are estimated using the Coordinate Descent algorithm (Friedman et al. (2007))

Ridge
- $ \alpha=0 $

LASSO
- $ \alpha=1 $

Enet
- $ 0<\alpha<1 $

Data and Simulation

Data
- Response variable: Health Expenditure per capita
- Predictors: Currently obtaining data, but will include income (GDP)
- Data sources: The World Bank $ \& $ UNDP
- Collection is still on-going
Simulation
- To see how the model works in practice, we provide the following simulation:
- To obtain the data set, we simulate an $ n\times k $ matrix of standard normal random variables where some of the variables share correlation ($ \rho_1=0.6 $ and $ \rho_2=0.5 $)
- $ k $ represents the number of predictors $ (k=10) $
- $ n $ represents the number of observations $ (n=25) $
- For the response variable, we simulate an $ n\times1 $ standard normal random vector $ (n=25) $

Data and Simulation (Cont'd)

Partitioning of simulated data
- Training Set
- Testing Set

Training set estimates the Enet models ($ \alpha=0.5 $)

Minimize the Cross-Validation (CV) - MSE ($ \lambda\geq 0 $) to optimize Enet model

Determine the prediction accuracy of the model based on the MSE obtained from the testing set

Simulated Results

plot of chunk unnamed-chunk-2

Coefficient Paths

plot of chunk unnamed-chunk-3

CV Curve

Simulated Results (Cont'd)

Variable	Parameter Estimate
x1	-0.344
x2	0.000
x3	-0.443
x4	0.361
x5	-0.224
x6	0.000
x7	0.014
x8	0.125
x9	0.000
x10	0.000

$ \lambda_{optimal}= $ 0.144
- Now that we have chosen an appropriate $ \lambda $, the Enet algorithm performs variable selection and achieves the slope estimates on the left
- With these estimates, we fit the model on the testing set and compute the MSE
  - MSE= 2.001

Concluding Remarks

The Enet procedure will be of great use when fitting an appropriate model for health care in FCAS because we may have more predictors than number of observations (FCAS)
Choosing an appropriate $ \lambda $ for the model, will optimize the Enet algorithm approach
Future Work
- Gathering and collecting data on many variables from the World Bank & UNDP
- Choosing an optimal $ \alpha $ for the model, will improve the Enet method
- Estimate the value at risk (VaR) along with conditional tail expectation (CTE) to obtain maximum health care cost

References

Council of Foreign Relations. (2013). Vaccine Preventable Infectious Diseases.
Department for International Development. (2005). Why We Need To Work More Effectively in Fragile States.
Freedman, Lynn P. (2010). Rebuilding Health Systems to Improve Health and Promote State Building in Post-conflict Countries: A Theoretical Framework and Research Agenda., 2010. Social Science & Medicine, 70(1), 89-97.
Friedman, et al. (2007). Pathwise Coordinate Optimization. The Annals of Applied Statistics, 1(2), 302-32.
Kim, Susanna.(2015) How Much More Starbucks Customers Will Pay Each Year With Today's Price Hike. ABCNews.
McIntosh, K., & Buckley, J. (2015). Economic development in fragile and conflict-affected states: Topic guide. Birmingham, UK: GSDRC, University of Birmingham.
McPake, et al. (2015). Ebola in the Context of Conflict Affected States and Health Systems: Case Studies of Northern Uganda and Sierra Leone, 2015. Conflict and Health, 9(23), 1-9.
Organization for Economic Cooperation and Development. (2015). Governance and Peace.
Sarathi, Karla. (2014). The Emergence of Ebola as a Global Health Security Threat: From 'Lessons Learned' to Coordinated Multilateral Containment Efforts. Journal of Global Infectious Diseases, 6(4), 164-67.
World Bank. (2015). Ending Poverty and Sharing Prosperity.
Zou & Hastie. (2005). Regularization and Variable Selection via the Elastic Net. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 67(5), 768.

Questions/Comments/Suggestions

KGWunderlich1s@semo.edu

Estimating Health Care Cost Among Fragile & Conflict Affected States: An Elastic Net Approach

Kevin Wunderlich (Graduate Student)

8 April 2016

Outline

Introduction

What Are Fragile and Conflict Affected States (FCAS)?

Introduction (Cont'd)

Introduction (Cont'd)

Research Objectives

Models

Models (Cont'd)

Data and Simulation

Data and Simulation (Cont'd)

Simulated Results

Simulated Results (Cont'd)

Concluding Remarks

References

Questions/Comments/Suggestions

Thank You