Evaluating Health Policy Effect with Generalized Linear Model and Generalized Estimating Equation Model A Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Actuarial Science and Quantitative Risk Management in the Graduate School of The Ohio State University By Chen Zhao, B.S. Graduate Program in Department of Mathematics The Ohio State University 2020 Master's Examination Committee: Dr. Chunsheng Ban, Co-Advisor Dr. Bo Lu, Co-Advisor c Copyright by Chen Zhao 2020 Abstract According to the Affordable Care Act (ACA) in 2010, the scope of Medicaid is expanded to cover the people with annual incomes under the 138% of the Federal Poverty Level (FPL). Each state has the option to determine whether they join the program or not. So far, there are over 30 states have implemented the Medicaid expansion program. Also, in the rural counties, due to the lack of nearby hospitals and low population, there is a general trend leading to loss of medical care workforce supply. In this thesis, I will examine the association between the Medicaid expansion and the change of health care workforce supply in the rural counties. To answer this question, I used the difference-in-differences (DD) model. The effect of the Medicaid expansion program on rural health care workforce supply was explored by analyzing the change of number of Primary Care Physicians and Nurse Practitioners before and after the Medicaid expansion. I used the Generalized Linear Model (GLM) and Generalized Estimating Equation (GEE) to construct DD estimates and standard errors, and compare the fitting results of both. Finally, I also compared the model fitting in Poisson distribution and negative binomial distribution. ii This is dedicated to my family. iii Acknowledgments I would like to express deep gratitude to my advisor, Dr. Chunsheng Ban, for his encouragement and suggestion that always provide me with a sense of direction in pursuit of knowledge. The completion of this project could not have been possible without the participation and assistance of my research mentor, Dr. Bo Lu, I greatly appreciate all of his guidance on my study. Also, it has been an honor to join Dr. Yi Xu's research group, I cannot thank enough for her support and encouragement. I deeply appreciate everyone advice and guidance in support of this project. iv Vita November 14, 1995 . Born - Beijing, China 2018 . .B.S. Statistics, Beijing Technology and Business University. 2018-present . .Graduate Student, The Ohio State University. Fields of Study Major Field: Quantitive Risk Management and Data Science v Table of Contents Page Abstract . ii Dedication . iii Acknowledgments . iv Vita......................................... v List of Tables . viii List of Figures . ix 1. Introduction . 1 1.1 The introduction of Causal Inference . 1 1.1.1 Average Treatment Effect . 2 1.1.2 Randomized Experiment . 3 1.1.3 Estimated Causal Effect in Regression . 3 1.1.4 Causal Inference in Observational Study . 4 1.2 Health Policy Background . 6 1.3 Method . 7 1.3.1 Instrumental Variables . 7 1.3.2 Regression Discontinuity . 8 2. Methods . 10 2.1 Difference-in-Differences . 10 2.1.1 Framework of difference-in-differences . 10 2.1.2 Difference-in-differences with Regression . 14 2.2 Generalized Linear Model . 15 2.2.1 Framework of General Linear Model . 15 vi 2.2.2 Setting of Generalized Linear Model . 15 2.2.3 Robust Variance Estimation . 17 2.3 Generalized Estimating Equation . 19 3. Data Analysis . 21 3.1 Data . 21 3.2 Model . 24 3.3 Results . 27 4. Conclusion . 33 Appendices 35 A. Stata commands and results . 35 Bibliography . 40 vii List of Tables Table Page 3.1 The number of rural counties in each state . 22 3.2 Status of State Action on the Medicaid Expansion Decision . 23 3.3 State-level Characteristics . 25 3.4 Medicaid expansion effect on Primary Care Physician and Nurse Prac- titioner using GLM and GEE . 29 3.5 Medicaid expansion effect on Primary Care Physician and Nurse Prac- titioner using GEE with Negative Binomial distribution . 31 viii List of Figures Figure Page 2.1 Difference-in-differences parallel trend. Note: Adapted from [10] . 12 3.1 the Primary Care Physician in rural county over year . 25 3.2 Distribution of Nurse Practitioner in rural county over year . 26 3.3 Distribution of Primary Care Physician in rural county . 26 3.4 Distribution of Nursing Practitioner in rural county . 27 3.5 Distribution of PCP density by year . 31 3.6 Distribution of NP density by year . 32 ix Chapter 1: Introduction 1.1 The introduction of Causal Inference From a very general perspective, causal inference can be thought of as a field related to statistics, computer science, psychology, economics, and many more. One could think of causal inference as a field of quantitative research in understanding the causality. The causality has played a critical role in developing social science research. For example, people were wondering if schooling has a causal connection to the future income. It is quite possible that the causal inference helps us understand the impact of certain variables on the outcomes. i.e. we can infer what will happen to the average income for every additional year in school. Different from the association relation which is computed from the data alone, the causal relation focus on the data-generating process. [16] In causal inference, we denote the treatment A and outcome Y . If we want to understand causality, we have to define the potential outcomes that refer to the values of all outcomes describing experimental units with different treatments [19]. Here, I simplify the treatment at two levels for individual i (Ai=0 is without treatment, Ai= 1 is with treatment). A=1 The potential outcome for subject i getting treatment is Yi , on the other hand, A=0 the potential outcome without the treatment is Yi . The observed outcome Yi for 1 individual i could be expressed in terms of potential outcomes A=1 A=0 Yi = AiYi + (1 − Ai)Yi (1.1) By comparing the difference of the potential outcomes between two treatment A=0 A=1 status, Yi − Yi , we can identify the causal effect of a treatment A on outcomes Y for subject i. 1.1.1 Average Treatment Effect Fundamental problem for causal inference is that we cannot obtain all potential outcomes for the same individual. We can only observe one of potential outcomes that treatment actually assigned, and the others are missing. Therefore, inference for causal effect of the treatment is a missing data problem. [18] In general, we will consider the Average Treatment Effect (ATE) which is the average causal effect of population. People are assigned to treatment groups (YijAi = 1) and control groups (YijAi = 0), we can obtain a causal relationship between the treatment and the outcomes by comparing the average results of the treatment group and the control group. A=1 A=0 E(YijAi = 1) − E(YijAi = 0) = E(Yi jAi = 1) − E(Yi jAi = 1) (1.2) A=0 A=0 + E(Yi jAi = 1) − E(Yi jAi = 0) A=1 A=0 The term E(Yi jAi = 1)−E(Yi jAi = 1), on the right hand side, is the average A=0 A=0 causal effect in the treatment group. The leftover, E(Yi jAi = 1)−E(Yi jAi = 0), is called the selection bias. The selection bias is generated by the pre-treatment covariates difference between two groups. Due to the presence of selection bias, we cannot directly conclude the causal relationship based on the observed outcomes. 2 Thus, many causal inference methods, such as matching, are aimed to eliminate the selection bias. 1.1.2 Randomized Experiment One way to solve the selection bias problem is to use randomization in treatment assignment. In the randomized experiment, since each individual are assigned to different groups randomly, the treatment A is independent of potential outcomes A=0;1 Yi . Thus, the selection bias in (1.2) will disappear A=0 A=0 A=0 A=0 E(Yi jAi = 1) − E(Yi jAi = 0) = E(Yi ) − E(Yi ) = 0 (1.3) The purpose of such experiments is to ensure that the observed results are only dependent on the treatment. Because at this time the treatment group and the experimental group are drawn from one distribution. It is impossible to guarantee that all other variables are identical if we pick up individuals in the treated group. However, on the average, since we randomly pick up from the same distribution, the average differences are arbitrary between the two groups, and systematic differences disappear. Therefore, the difference in average outcomes between the two groups is solely determined by the treatment. 1.1.3 Estimated Causal Effect in Regression We used the conditional expectation function (CEF) to model (1.2) the causal effect in the last sections. According to the Theorem of The regression-CEF [1], the best linear estimated CEF is the linear regression model. Here, we set up our causal effect model as 3 Yi = α + ρAi + βXi + i (1.4) where ρ is the causal effect, Xi denotes the vector of all other variables related to 2 response variables Yi, and i ∼ N(0; σ ) is the random error term. In order to remove selection bias, we need to control the covariates Xi to keep the balance over treatment groups. Thus, one key assumption that can guarantee regres- sion having the causal interpretation is the conditional independence assumption [5] A A E(Yi jXi;Ai = 1) = E(Yi jXi;Ai = 0) = E(YijXi;Ai) (1.5) Xi is the covariates, the expected potential outcomes are independent by the treat- ment, which means that there is no systematic difference of potential outcomes across the treated groups.