A Generalized Linear Model for an Estimation of Drug Expenditures

1. A Generalized Linear Modelfor an Estimation ofDrug Expenditures Supon Limwattananon, BSc (Pharm), MPHM, PhD Faculty of Pharmaceutical Sciences, Khon Kaen University, THAILAND

2. Objective To demonstrate a generalized linear model (GLM), a statistical approach appropriate for an estimation of non-normally distributed drug expenditures as explained by policy change and other explanatory variables Data source Longitudinal data on drug use in a panel of 6,794 adults with chronic asthma in 17 hospitals for 3 years (2000 – 2002) Research question What is the magnitude of effect of 2002-Universal Coverage (UC) Policy on expenditures of drug use in patients with chronic asthma ?

3. Drug Utilization Study Research Issues • For a descriptive research • To explain variation in patterns of drug use •  variation in drug expenditures • Such variation is probably conditional on policy change • or interventions given • For statistical inference of the policy/intervention effect • A need for multivariate models, controlled for other explanatory variables or covariates

4. Year as an Indicator for Policy Change Use of expensive inhaled corticosteroids Patient demographics Annualized Expenditure for antiasthmatics Health insurance schemes Years of drug use UC Policy Hospital indicators

5. Statistical Model for an Estimation of Expenditure for Antiasthmatics Left hand side, dependent variable (Yi) Expenditure for antiasthmatics per year in a given patient Right hand side, explanatory variables(Xi) ICSi: Receiving inhaled corticosteroids (ICS) in a year (1=Yes, 0=No) AGEi: Age groups (AGE36: 36-50 yr., AGE51: >50 yr., vs. 18-35 yr.) SEXi: Male vs. Female SCHEMEi: Health insurance schemes (CSMB, UCLIC, UCROP, ROP vs. SS) YEARi: Years of antiasthmatic use (YR2001, YR2002 vs. YR2000) HOSPi: Indicator variables for 17 study hospitals i: Unexplained portion of the expenditures in the specified model

6. Longitudinal Data for Drug Use Study Methodological Issues Longitudinal data on use of drugs For a given patient, there are multiple Rx visits or drug use in a year Patient-year as a unit of analysis Aggregation  Annualized expenditures per patient • Behavior of expenditure data • Such annualized expenditures tend to vary a lot across patients • Such expenditures vary a lot more in groups with high expenditures • Variation in the expenditures is reflected by variance of i

7. Classical Linear Regression Ordinary Least Squares (OLS) Method To estimate beta coefficients (), using a CLR model ( = magnitude of the effect on Y of each X) Required assumption: well-behaved distribution of data 1. Normality (E[i] = 0) 2. Homoscedasticity (uniform variance of i) 3. Independence of i 4. Linearity * i = Unexplained portion of the expenditures in the specified model

8. Distribution of Drug Expenditures (N = 6,794 in Year 2002) Mean: 2,493 Baht Median: 1,540 Baht Skewness: 2.57 (P < 0.001)

9. Distribution of Log-transformed Expenditures (N = 6,794 in Year 2002)

10. Results Regression for Log-transformed Expenditure . regressLnBaht ICS age* male CSMB - ROP YR* HOSP* ------------------------------------------------------------------------------ LnBaht | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- ICS | 1.434188 .019684572.860.0001.3956051.472771 age36 | .1495537 .02892915.170.000 .0928502 .2062571 age51 | .3315848 .028162211.770.000 .2763845 .3867851 male | .3524408 .017066320.650.000 .3189895 .3858921 CSMB | -.0160937 .0318193-0.510.613 -.0784622 .0462748 UCLIC | -.093156 .0302468-3.080.002 -.1524422 -.0338699 UCROP | -.0935148 .0348599-2.680.007 -.1618431 -.0251865 ROP | -.3409096 .041128-8.290.000 -.4215239 -.2602952 YR2001 | .3201426 .020359715.720.000 .2802358 .3600493 YR2002 | .091071 .02038064.470.000 .0511234 .1310186

11. Families of Data Distribution (Variance Var.[Y] and Mean E[Y]) Family Variance function Binomial Var.[Y] = (E[Y]) (1 - E[Y]) Gaussian (normal) Var.[Y] = Constant Poisson Var.[Y] = E[Y] Gamma Var.[Y] = (E[Y])2 Expenditures in the high-cost groups tend to vary a lot more than in the lower ones

12. Cook Book for Diagnosis Approach Regress Y on explanatory variables Save i(studentized residuals) Save Yi hat (predicted Yi) = Mean Squared i= Variance ln(i2) = Log variance ln(Yi hat) = Log mean Regress log variance on log mean

13. Results Variance Function of the Mean . regress LnSqrRes LnYhat ------------------------------------------------------------------------------ LnSqrRes | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- LnYhat | 1.290466 .022588257.130.0001.2461921.334741 _cons | -11.63317 .1727979-67.320.000-11.97187-11.29447 ------------------------------------------------------------------------------ Ln(Var. [Y]) = 1.3*Ln(E[Y]) Var. [Y] = E[Y]1.3

14. Variance Function of the Mean Other Drug Classes Drug class Power of Mean  Variance ACE inhibitors 1.8 ACE inhibitors & A2 receptor antagonists 1.6 Calcium channel blockers 1.5 Statins & fibrates 1.3 NSAIDs & COX2 inhibitors 1.5 H2 antagonists & proton pump inhibitors 1.5 Antiretrovirals 1.7 Antiepileptics 1.6 Source: Limwattananon S, et al. Cost and Utilization Patterns of Drugs Prescribed to Hospital-Visited Patients: an Impact of Universal Health Coverage Policy, 2003

15. Estimation Approach Generalized Linear Model . glmBaht ICS age* male CSMB - ROP YR* HOSP*, family(gamma) link(log) ------------------------------------------------------------------------------ Baht | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- ICS | 1.07386 .016359965.640.0001.0417961.105925 age36 | .1386217 .02366365.860.000 .0922419 .1850015 age51 | .2623115 .023235811.290.000 .2167701 .3078528 male | .2951559 .014128520.890.000 .2674646 .3228471 CSMB | .0410909 .02628911.560.118 -.0104348 .0926165 UCLIC | -.0687261 .0248604-2.760.006 -.1174515 -.0200006 UCROP | -.0936884 .0287291-3.260.001 -.1499963 -.0373804 ROP | -.3036286 .034236-8.870.000 -.3707299 -.2365274 YR2001 | .1986592 .016703111.890.000 .1659218 .2313966 YR2002 | .0566952 .01683553.370.001 .0236982 .0896921

16. Interpretation of Beta-coefficient (Semi-logarithmic Functional Form: ln Y =  X + ) For a continuous variable X :  = Percentage effect on Y per unit change in X For an indicator variable X: exp() – 1 = Percentage effect on Y of a change in X from 0 to 1 status Ref: Halvorsen R and Palmquist R. The interpretation of dummy variables in semilogarithmic equations. American Economic Review 1980; 70: 474-475. Kennedy P. Estimation with correctly interpreted dummy variables in semilogarithmic equations. American Economic Review 1981; 71: 802.

17. Explanatory variable % difference in expenditure for antiasthmatics Point estimate Lower 95% CI Upper 95% CI

18. Comparison between GLM and CLR GLM CLR (gamma, log link) (OLS on log expenditure)

19. Useful Readings 1. Blough DK, Ramsey SD. Using generalized linear models to assess medical care costs. Health Services and Outcomes Research Methodology. 2000; 1: 185-202. 2. Blough DK, Madden CW, Hornbrook MC. Modeling risk using generalized linear models. Journal of Health Economics 1999; 18: 153-171. 3. Manning WG. The logged dependent variable, heteroscedasticity, and the retransformation problem. Journal of Health Economics1998; 17: 283-295. 4. Manning WG, Mullahy J. Estimating log models: to transform or not to transform? Journal of Health Economics2001; 17: 461-494.