Econometrics Course: Cost As The Dependent Variable (II)

Econometrics Course:Cost as theDependent Variable (II)Paul G. Barnett, PhDApril 10, 2019

POLL QUESTION #1Which method(s) have you usedto evaluate health care costs?(answer all that apply)– None yet– Rank test (non-parametric method)– Ordinary Least Squares regression with rawcost– OLS log transformed cost– GLM model (gamma regression)2

Health care costsdifficult to analyze– Skewed by rare but extremely high costevents– Zero cost incurred by enrollees who don’tuse care– No negative values– Variance can vary with independent variable3

Limitation ofOrdinary Least Squares (OLS) OLS with raw cost– non-normal dependent variable can generatebiased parameters– can predict negative costs OLS with log transformation of cost– Log cost is normally distributed, can use in OLS– Predicted cost is affected by re-transformation bias– Can’t take log of zero– Assumes variance of errors is constant4

Topics for today’s courseWhat is heteroscedasticity, and whatshould be done about it? What should be done when there aremany zero values? How to test differences in groups with noassumptions about distribution? How to determine which model is best? 5

Topics for today’s courseWhat is heteroscedasticity and whatshould be done about it? What should be done when there aremany zero values? How to test differences in groups with noassumptions about distribution? How to determine which model is best? 6

What is heteroscedasticity? Heteroscedasticity– Variance depends on x (or on predicted y)– For example, the variation in incomeincreases with age OLS assumes Homoscedasticity– Identical variance E(εi2) σ27

Homoscedasticity– Errors have identical variance E(εi2) σ2432e10-105,00010,00015,00020,000-2-3-48

Heteroscedasticity– Errors depend on x (or on predicted y)432e10-105,00010,00015,00020,000-2-3-49

Why worry aboutheteroscedasticity? Predictions based on OLS model can be biasedRe-transformation assumes homoscedasticerrorsPredicted cost when the error isheteroscedastic can be “appreciably biased”10

What should be done aboutheteroscedasticity?Use a Generalized Linear Models (GLM) Analyst specifies a link function g( ) Analyst specifies a variance function – Key reading: “Estimating log models: totransform or not to transform,” Mullahy andManning J Health Econ 20:461, 200111

Link function g( ) in GLMg (E (y x) ) α βx Link function can be natural log, squareroot, or other function – E.g. ln ( E ( y x)) α βx– When link function is natural log, then βrepresents percent change in y for a unitchange in x12

GLM vs. OLSOLS of log estimate: E ( ln ( y) x)) GLM estimate: ln (E ( y x)) – Log of expectation of y is not the same asexpectation of log y!13

GLM advantagesDependent variable can be zero No retransformation bias when predicting – Smearing estimator is not used Does not assume homoscedastic errors14

GLM variance functionGLM does not assume constant variance GLM assumes there is function thatexplains the relationship between thevariance and mean – var (y x)15

Variance assumptions for GLM costmodels Gamma Distribution (most common)– Variance is proportional to the square of themean Poisson Distribution– Variance is proportional to the mean16

Estimation methods How to specify log link and gammadistribution with dependent variableCOST and independent variables X1, X2,X317

GLM with log link and gammadistribution in StataGLM COST X1 X2 X3, FAM(GAM)LINK(LOG)18

GLM with log link and gammadistribution in SAS Basic syntax (drops zero cost observations)PROC GENMOD;MODEL COST X1 X2 X3 / DIST GAMMA LINK LOG; Refined syntax (keeps zero cost observations)PROC GENMOD;A MEAN ;B RESP ;D B/A LOG(A)VARIANCE VAR A**2DEVIANCE DEV D;MODEL COST X1 X2 X3 / LINK LOG;19

Choice between GLMand OLS of log cost GLM advantages:– Handles heteroscedasticity– Predicted cost is not subject toretransformation error OLS of log transform advantages– OLS is more efficient (standard errors aresmaller than with GLM)20

Dataset for worked examplesNew primary care episodes of nonspecific low-back pain in 2016 10% Sample N 43,909 VA and community care costs in thefollowing year (excluding residential,nursing home) Cost difference of episodes that started inCBOC? 21

Worked exampleTry gamma regression (GLM log link,gamma distribution) Evaluate link function with Box-Coxregression Evaluate distribution with GLM familytest 22

GLM RegressionLog link, gamma distributionInterceptIndex visit at CBOCBaseline pain scoreAgeNumber of chronic conditionsIndicator of diagnosis for substance useor psychiatric diagnosisFemaleHistory of opiate Rx in prior yearMRI within 6 weeksNumber of visits for physical therapywithin 6 weeksParameter Standard WaldPr t EstimateErrorChi-sq7.94010.0240 109230 .0001-0.27500.0106675.9 .00010.04520.0017726.7 .00010.00260.000445.5 .00010.17850.0036 2397.4 .00010.22460.0128308.2 60.7 .0001 .0001 .00010.09490.012062.8 .000123

Which GLM link function?– Maximum likelihood estimation ofBox-cox parameter (called θ or λ)θCOST 1θ α βx ε24

Link function depends onBox-Cox parameterLink functionBox-CoxparameterInverse (1/cost)-1Log(cost)0Square root (cost).5Cost1Cost Squared225

Box Cox regression Stataboxcox cost {indep. vars} if cost 0 SASproc transreg data {dataset} ;model boxcox(cost/lambda -1 to 2 by .5) identity (&ind vars);26

Box-Cox regression with example dataBox-Cox Transformation Information forCost in year after index stayLambdaR-SquareLog Like-1.00.05-409600-0.50.11-3691220.0 0.15-354795 596029 - Best Lambda* - 95% Confidence Interval - Convenient Lambda27

Which variance structure with GLM?Is it appropriate to assume the gammadistribution? GLM family test – Also called modified Park testRun GLM gamma regression Evaluate residuals with secondregression 28

GLM family test (step 1) Run a gamma regression– Assume log link, gamma variance– Include independent variables Find predicted cost (in log scale)– Xβ from first regression Find residual (in raw scale)– COST - 𝑒𝑒 𝑋𝑋𝛽𝛽Square these residuals29

GLM family test (step 2) Run second gamma regression– Dependent variable is squared residualsin raw scale– Independent variable is predicted cost inlog scale (Xβ from first regression)(COST 𝑒𝑒 𝑋𝑋𝛽𝛽 )2 𝛾𝛾0 𝛾𝛾1 (𝑋𝑋𝛽𝛽) 𝜈𝜈30

GLM family test (step 3) Evaluate the regression coefficient γ1γ1Variance to be used in GLM0Gaussian (Normal)1Poisson2Gamma3Wald (Inverse Normal)31

GLM Family Testwith example dataParameterStandardEstimateErrorWald ChiSquarePr ChiSqγ01.570.184672.1 .0001γ11.980.02138587.6 .0001Result:Since γ1 2, the gamma distribution was appropriate32

Other models for skewed data Generalized gamma models– Estimate link function, distribution, andparameters in single model– STATA ado file “pglm”– See: Basu & Rathouz (2005)33

Questions?34

Topics for today’s courseWhat is heteroscedasticity, and whatshould be done about it? (GLM models) What should be done when there aremany zero values? How to test differences in groups with noassumptions about distribution? How to determine which model is best? 35

What should be done when there aremany zero values? Examples– Members of a health plan without utilization– Clinical trial participants with no cost36

The two-part model Part 1: Dependent variable is indicatorany cost is incurred– 1 if cost is incurred (Y 0)– 0 if no cost is incurred (Y 0) Part 2: Regression of how much cost,among those who incurred any cost38

Annual per person VHA costs FY10among those who used VHA in FY090.40Medical OnlyMedical Rx0.300.20 30K 30K 25K 20K 15K 10K 5K0.00no cost 1K0.1037

The two-part model Expected value of Y conditional on XE (Y X ) P (Y 0) X ) E (Y Y 0, X )Part 1.The probability thatY is greater than zero,conditional on X}}Is the product of:Part 2.Expected value of Y,conditional on Y beinggreater than zero,conditional on X39

Predicted cost in two-part model Predicted value of YPart 1.Probability of any costbeing incurred}Is the product of:}E (Y X ) P (Y 0) X ) E (Y Y 0, X )Part 2.Predicted costconditional onincurring any cost40

Question for classP(Y 0) X ) Part one estimates probability Y 0– Y 0 is dichotomous indicator– 1 if cost is incurred (Y 0)– 0 if no cost is incurred (Y 0)41

POLL QUESTION #2Which regression method(s) are used fora dichotomous (zero/one) dependentvariable? (check all that apply)– Ordinary Least Squares– Logistic Regression– Probit Regression– Survival (Cox) regression42

First part of modelRegression with dichotomous variable Logistic regression or probitLogistic regression uses maximumlikelihood function to estimate log oddsratio:Pi α β1 Xlog1 Pi43

Logistic regression syntax in SASProc Logistic;Model HASCOST X1 X2 X3 / Descending;Output out {dataset} prob {variable name}; HASCOST an indicator variableOutput statement saves the predicted probability thatthe dependent variable equals one (cost was incurred)Descending option in model statement is required,otherwise SAS estimates the probability that thedependent variable equals zero44

Logistic regression syntax inStataLogit HASCOST X1 X2 X3Predict {variable name}, pr Predict statement generates the predictedprobability that the dependent variableequals one (cost was incurred)45

Second part of modelConditional quantityRegression involves only observationswith non-zero cost (conditional costregression) Use GLM or OLS with log cost 46

Two-part models Separate parameters for participation andconditional quantity– How independent variables predict participation in care quantity of cost conditional on participation– each parameter may have its policyrelevance47

Stata TPM command Fits two part regressions– First part: binary choice (Prob depvar 0)– Second part: distribution of depvarconditional on depvar 0 User developed ADO file– must be installed from web Federico Belotti & Partha Deb (2012)48

Stata TPM command First part options– Logit or Probit Second part options– OLS of raw value, OLS of log, or GLM Example syntaxTPM COST X1 X2 X3, f(logit) s(glm, fam(gamma) link(log))49

Stata TPM command Post-estimation commands– Predict values of depvar– Allows out of sample predictions– Corrects for retransformation bias in OLSmodels50

Alternatives to two-part modelOLS with untransformed cost OLS with log cost, using small positivevalues in place of zero – not recommended GLM models, e.g. gamma regression– Cannot have “too many” values with zero51

Topics for today’s courseWhat is heteroscedasticity, and whatshould be done about it? (GLM models) What should be done when there aremany zero values? (Two-part models) How to test differences in groups with noassumptions about distribution? How to determine which model is best? 52

Non-parametric statistical testsMake no assumptions about distribution,variance Wilcoxon rank-sum test Assigns rank to every observation Compares ranks of groups Calculates the probability that the rankorder occurred by chance alone 53

Extension to more than twogroupsGroup variable with more than twomutually exclusive values Kruskall-Wallis test – is there any difference between any pairs ofthe mutually exclusive groups? If KW is significant, then a series ofWilcoxon tests allows comparison ofpairs of groups54

Limits of non-parametric test It is too conservative– Compares ranks, not means– Ignores influence of outliers– E.g. all other ranks being equal, Wilcoxon willgive same result regardless of whether Top ranked observation is 1 million more costly thansecond observation, or Top ranked observation just 1 more costly Doesn’t allow for additional explanatoryvariables55

Topics for today’s courseWhat is heteroscedasticity, and whatshould be done about it? (GLM models) What should be done when there aremany zero values? (Two-part models) How to test differences in groups with noassumptions about distribution? (Nonparametric statistical tests) How to determine which model is best? 56