Choosing Models for Cost Analy

Issues of Nonlinearity Nonlinearity and

Endogeneity

Melissa M. Garrido,1,2 Partha Deb,

,

James F. Burgess, Jr.,4,5 Joan D. Penro

1. GRECC/REAP, James J Peters VA Medical Center, Bronx, NY

2. Brookdale Department of Geriatrics and Palliative Medicine, Mount Sinai School of Medicine

3. Department Department of Economics, Economics, Hunter College College and the Graduate Graduate Center, Center, City University University of New York4. Center for Organization, Leadership and Management Research, VA Boston Healthcare System

5. Department of Health Policy and Management, Boston University School of Public Health

Supported by Department Of Veteran Affairs, Health Services Research and Development Service (# 
Cost Analyses and Comparati

Effectiveness Research

“

...The use and costs of health care are likely to 

important outcomes of interventions for som

patients, whether or not the results are used effectiveness analysis.”

Garber A, Sox H. Health Affairs 2010; 29(1

Health spending accounted for 17.6% of the GD
Question #1 (Poll): How comfortable do yo

perf i orm ng h lh ea t care cost anal 

yses

• Very comfortable• Somewhat comfortable• Nei ht er comf bl orta e nor uncomf bl orta e

• Somewhat uncomfortable

• Very uncomfortable• N/A – Do not perform perform health care cost

analyses
Issues in Healthcare Healthcare Cost Analy

Analy

• Skewed data• Non‐negative outcomes• C i ensor ng

• Endogeneity

• Distribution of marginal marginal effects effects
Question #2 (Whiteboard): Which method

use to analyze costs?
How to Analyze Costs with a

Endogenous Regressor?

• Models that use instrumental variables

– Two‐stage least squares on cost (2SLS)– Two‐stage least squares on the natural log of costs (log‐2SLS)– Control function models ( ps ecial case: two‐stage residual incl– Full information maximum simulated likelihood (FIMSL)

• Propensity score models
Example using Veterans Health Administratio

Effect of an Inpatient Palliative Care Consultation 

• 3,321 inpatients hospitalized in five Vetera

Affairs acute care facilities in 2004‐2007 w

or more life‐limiting limiting diseases diseases

• Data from VHA Medical Medical SAS Inpatient Inpatient Data

Data

VA Decision Support System National Data 

• Outcome: Total direct costs per day during 

inpatient admission
Sample

Variable N (%) or Mean (SD)

Had a palliative care consultation 606 (18%)

65 or older 2235 (67%)Advanced disease diagnosis*

Cancer 802 (24%)

HIV/AIDS 54 (2%)Congestive heart failure 1541 (46%)

Chronic obstructive pulmonary 1699 (51%)

disease

Number of comorbidities at initial

hospitalization

2.1 (1.3)

Died during study period 1659 (50%)

Cost per day $1235.27 ($749.36), IQR $813.74

Natural Natural log of costs per day 7 01 . (0 42) . , IQR 6 70 . ‐7 27 .

* Could have more than one 
Skewed Distribution of Costs

08 .001 Mean = $

SD = 

Median = $

06 .000IQR = $813.74 - $004 .000

Density

002 .00 0 .0

0 5000 10000 15000 

Total Direct Costs Per Day
2

Distribution of Natural Log of Co

1 1.2

Mea

S

Media

IQR 6 

.8

= 6

.6

Density

.40 .2

0 2 4 6 8 10 

ln (Total Direct Costs per Day)
Instrumental variables

for equations equations with a continuous continuous outcome outcome and endo

endo

binary treatment

Treatment Latent/Unobserved characterist

Distribution of error te

Treatment

Observed characteristics

Latent/Unobserved characterist

Distribution of error term

Outcome

(Cost)

Observed characteristics
Instrumental variables

for equations equations with a continuous continuous outcome outcome and endo

endo

binary treatment

Treatment probability is endogenous due to common unobsecharacteristics in each step (if neither δ or γ = 0)
Instrumental variables

for equations equations with a continuous continuous outcome outcome and endo

endo

binary treatment

•2SLS, Control Function, and Full Information Maximum Simulated 

models are all based on these structural forms

• z must include at least one variable (the instrumental variable) th

•The instrumental variable must only be correlated with likelihood 

(d) and not with the likelihood of outcome (Y)

• Each uses a different set of assumptions about f and g
Instrumental Variable

• Treatment: Palliative Palliative care (PC) consultation consultation• Outcome: Healthcare costs• Instrumental Instrumental variable: variable: Propensity Propensity for request

request

consultation by an admitting physician

– First‐stage F statistic = 19.46, p < .001– Chi‐square value for the Anderson canonical correlation lik

ratio test p < .001
2SLS

Step 1: Model treatment likelihood, include instrumentaStep 2: Model outcome likelihood, include treatment like

from Step 1

• Often used because:

•Simple

•Mii l n ma assumptions on di ib i distribution of error term

•Problems:

•Grossly inefficient

•Misleading estimates when used with skewed outco
Log‐2SLS

• 2SLS, but with natural log transformation of c• Problem: Retransformation of outcome back 

to calculate marginal effects

– Taking antilog of predicted value will lead to biase

estimates

– Smearing estimators work only in the case of

homoskedastic error terms or when there are fewidentifiable sources of heteroskedasticity heteroskedasticity

M i ann ng WG. J l ourna of H lh ea t E i conom cs. 1998
Control Functions

Step 1: Model treatment likelihood, include instrumental vaStep 2: Model outcome likelihood, include a function of the r

the treatment likelihood equation

• Uses principles of instrumental variable regression

• Flexible function of the residuals can produce the correct 

for endogeneity in the outcome equation

• No need to transform dependent variable

Heckman JJ, Robb R. Journal of Econometric

Newey WK, Powell JL, Vella F. Econometrica 1
Control Functions: Choice of Resid

Types of Residuals

‐ Response

‐ Difference Difference between between predicted predicted and observed observed treatment treatment likelihood likelihood

‐ Anscombe

‐ Include transformations of the predicted and observed treatment 

that are aimed at achieving achieving normality normality

‐ Deviance

‐ Obtained from a function of the log likelihood ratio

• These are equivalent in linear but not nonlinear sett• Anscombe Anscombe and deviance deviance residuals residuals better approximat

approximat

normal distribution in nonlinear settings

Gill, J. 2000. Generalized Linear Models: A Unified Approach. Thousan
Special Case of Control Function Appro

2SRI

Step 1: Model treatment likelihood, include instrumental varStep 2: Model outcome likelihood, include the response resid

treatment likelihood equation

Lee S. J l ourna of E ti conometrics 2007; 

Terza JV, Basu A, Rathouz PJ. Journal of Health Economics 20

Often a misapplication misapplication of CF: No reason to believe believe that 

just the response residual eliminates endogeneity bia

More research research needed to determine determine which functional functional fo

specification of the residuals is needed for CF to behoptimally in nonlinear studies

Basu A, Manning Manning WG. Medical Medical Care 200
Full Information Maximum Simulated Like

(FIMSL)

• Derive the joint distribution of the treatment and outco

variables cond l itional on the common latent variables (I 

• Define and maximize a simulated likelihood function• Estimate treatment effect with draws from a pseudo‐ra

Halton sequence and average these effects

Deb P, Trivedi PK. Econometrics Journa

Deb P, Trivedi PK. The Stata Journal. 20
Propensity Scores

• Pros:

• Accounts Accounts for selection selection into treatment treatment based o

observable characteristics

• Cons:

• Little is known about the performance of pro

scores in nonlinear models

– May not be able to balance distributions of covaria

between those who did and did not receive treatm

Basu A, Manning WG. Medical Care 2009; 47 (7 Su

Basu A, Polsky D, Manning WG. Health, Econometrics and Data Group Working Pa
100

120

Mean days until death

406080

Days PCC = 1

0

2040

1 2 3 4 5 6 7 8 9 10

PCC = 0

25 SD days until dea

Deciles of propensity scores

1520

Days

05

10

PCC = 1PCC = 0

0

1 2 3 4 5 6 7 

Deciles of propensity 
Model Median Mean Range

Two Stage Least Squares

Cost ‐3185.83 ‐3185.83 (‐3185.Natural log of cost (retransformed) ‐1347.77 ‐1514.07 (‐3866.

Propensity Score

Nearest neighbor ‐129.89

Stratification ‐95.97

Control Function Approach

Response

1st Response

–

1 degree (2SRI) 152 33 153 68 ( 254 7

st degree (2SRI) ‐152.33 ‐153.68 (‐254.7

Response – 3rd degree ‐217.64 ‐220.06 (‐388.6

Anscombe – 1st degree ‐75.29 ‐75.94 (‐125.4

Anscombe – 3rd degree ‐319.64 ‐324.62 (‐625.0

Deviance – 1st degree ‐133.34 ‐134.40 (‐221.1

Deviance

–

3rd de

gree ‐439.39 ‐450.02 (‐972.3

Full Information MSL ‐422.92 ‐429.38 (‐733.0
Importance of Distribution of

Margi l na Effects

• Partial Partial effects effects differ across observations observations 

nonlinear models

• Where and for whom the marginal effe

call d cu ated d d epen s on the research que
Marginal Effects of PC Consultation on Direct Costs 

015. .01

nsity .005

De

0

-4000 -3000 -2000 -1000 

Cost ($)

Log-2SLSCF, 3rd degree Anscombe residuals CF, 1st degree response re

2SLS FIMSL
Marginal Effects of PC Consultation on Direct Costs p

15 .0 .01

sity

.005

Den

0

-1000 -800 -600 -400 -200 

Cost ($)

CF, 1st degree response residuals (2SRI)

CF, 3rd degree Anscombe residuals

CF, 3rd degree devian

FIMSL PS, nea
Summary

• Estimates from the FIMSL model were the most simi

those of the CF 3rd degree Anscombe and deviance r

• Clustering of these three estimate distributions sugg

we have robust estimates of the effect of PC consulta

costs

• FIMSL provided similar results to CF without the nee

several specification tests and with lower variance fo

estimates

• Further testing with other datasets is needed to dete

how often FIMSL results mirror those from CF mode
Recommendations for

H lh ea t Care Cost A l na yses

• Obtain estimates estimates from several several models to che

robustness of results

• Evaluate robustness by examining distributio

marginal effects

• Account for nonlinearity and endogeneity at 

same time
Questions?

For list of references references or other questions, questions, p

e‐mail me at melissa.garrido@va.gov

The views expressed in this presentation are those of the authors and do not necessarily reflect 

policy of the Department of Veterans Aff i a rs or the Uni d te States government