Econometrics Course: Econometrics Course:

Cost as the Cost as the

Dependent Variable (I

Paul G. Barnett, 

April 25, 201
What is health care cos

What is health care cos

 Cost of an intermediate product e 

Cost of an intermediate product, 

e

– chest x-ray– a day of stay– minute in the operating room minute in the operating room– a dispensed prescription

 Cost of a bundle of products 

– O t ti t i it O

u

tpatien

t v

i

sit

– Hos

pital sta

y

p y
What is health care cost (co

 Cost of a treatment episode

–

vii d i id i

s

its an

d stays over a t

ime per

i

o

d

 Annual cost Annual cost

– All care received in the year
A l VHA t 

Annua

l per person VHA cos

ts 

(5% random sample)

0.40

Medical Only M

0.300 100.200.000.10

no cost

$1K

$5K

$10K

$15K

$20K
D i ti t ti ti Descr

i

ptive s

t

ati

stics: VHA 

VHA cos

(5% sample, includes outpatient pharmacy)

Cost

Mean 5,768M di e an 1 750 ,Standard Deviation Standard Deviation 18 874 ,Skewness 13.98Kurtosis 336.3
Skewness and kurtosis

 Skewness (3rd moment)

– Degree of symmetry Degree of symmetry– Skewness of normal distribution =0– Positive skew: more observations in righ

 Kurtosis (4 Kurtosis (4th moment)

– Peakness of distribution and thickness of – K t i f l di t ib ti 3 Kur

tos

is o

f norma

l di

s

t

rib

ution=

3
Distribution of cost: skew

– Rare but extremely high cost events

 E g only some individuals hospitalize

E.g. only some individuals hospitalize

 Some individuals with expensive chro

– Positive skewness (skewed to the ri
Comparing the cost incurre

Comparing the cost incurre

members of two groups

 Do we care about the mean 

or th di ? e median

?
Annual per person VHA costs 

Annual per person VHA costs 

among those who used VHA in 

0.40

Medical Only Medical+

0.300.200.100.00

o cost

$1K

$5K

$10K

$15K

$20K

$25K

no
Distribution of cost: zero v

Distribution of cost: zero v

records

 Enrollees who don’t use care

– Z l ero va

lues

– Tru c o o e d s bu o n

cati

on 

of th

e

d

i

stributi

o

n
Wh t h th i l i 

Wh

a

t

hypotheses 

invo

l

v

ing 

do you want to test? do you want to test?
What hypotheses involving co

What hypotheses involving co

you want to test?

 I would like to learn how cost is a

by:

–

Type of treatment

– Quantity of treatment– Ch i i f i Character

ist

ics o

f pat

ient

–

Characteristics h

a

racterist

ics

of 

provider rov

ider

– Other
Review of Ordinarily Le

Review of Ordinarily Le

S

q () uares 

(OLS

)

 Also known as: Classic linear mod We assume the dependent variable 

expressed as a linear function of thexpressed as

a linear function of th

chosen independent variables, e.g



Y

i = 

α

+

β

X

i

+

ε

i
Ordinarily Least Squares (

 Estimates parameters (coefficients Minimizes the sum of squared err

– (the distance between data points an

(the distance between data points an

regression line)
Linear model

 Regression with cost as a linear depen

variable (Y)

– Yi = α + β Xi + εi

 β is interpretable in raw dollars

– Represents the change of cost (Y) for eac

change in X

– E.g. if β=10, then cost increases $10 for e

increase in X
Expected value of a random va

 E(random variable) E(W) = 

Σ

W

i p(W

i

)

– For each i the value of W

i For each

i, the value of

W

i times pro

times pro

that W

i occurs

– Probability is between 0 and 1– A weighted average, with weights b

A weighted average, with weights b

probability
Review of OLS assumpti

 Expected value of error is zero E( Errors are independent E(

ε

i

εj)=0

 Errors have identical variance E(

ε

 Errors have identical variance E(

ε

 Errors are normall

y distributed

 Errors are not correlated with 

i d d i bl E(X i

n

depen

dent var

i

ables E(X

i

ε

i) 0=

0
When cost is the depend

When cost is the depend

variable

 Which of the assumptions of the classical m

lik

e y to be v o ated by cost data? l

y to be

v

i

o

lated by cost data?

– Expected error is zero– E id d t Errors are 

i

n

depen

den

t

– Errors have identical variance– Errors are normally distributed– Error are not correlated with inde

penden
Compare costs incurred 

Compare costs incurred 

members of two 

g p rou

p

 Regression with one dichotomous exp

variable

 Y = 

α + 

β X + 

ε

 Y t cos

t

 X group membership

– 1 if experimental group–

0 co t o g oup if co

n

t

r

ol 

g

roup
Predicted difference in co

Predicted difference in co

care for two 

g p rou

p

Y

=

α

+

β

X

+

ε

Predicted value of Y conditional on X=0 

(Estimated mean cost of control group

Estimated mean cost of control group

Yˆ |(X

=0)

=

α

 Predicted Y when X=1

(Estimated mean cost experimental group) (Estimated mean cost experimental group)

Y | (

X

=

1

)

=

α

+

β

a

ˆ

| (

)

β
Other statistical tests are sp

Other statistical tests are sp

cases

 Analysis of Variance (ANOVA) i

regressi ith di h t ion with one di

c

h

o

tomous 

inde

pendent variable

 Relies on OLS assumptions
Compare groups controllin

Compare groups controllin

case mix

 Include case Include case

-mix variable Z mix variable, 

Z

Y

=

α

+

β

X

+

β

Z

+

ε

1

2
Compare groups controlling fo

Compare groups controlling fo

mix (cont).

 Estimated mean cost of control group 

controlling for case mix (evaluated at controlling for case mix (evaluated at 

value for case-mix variable)

Yˆ | (

X

=

0

)

=

α

+

β 2

Z

where

Z is mean

of

Z
Compare groups controlling fo

Compare groups controlling fo

mix (cont).

 Estimated mean cost of experimental g

controlling for case mix (evaluated at controlling for case mix (evaluated at 

value for case-mix variable)

Yˆ | (

X

=

1

)

=

α

+

β1

+

β 2

Z

where

Z is mean of

Z
Assumptions are about er

Assumptions are about er

term

 Formally, the OLS assumptions ar

th t e error 

term

 The residuals (estimated errors) of

The residuals (estimated errors) of

have a similar distribution to the 

d d t i bl depen

den

t var

i

abl

e
Why worry about using OLS 

Why worry about using OLS 

skewed (non-normal) data

 “In small and moderate sized samples

case can have tremendous influence on estimate”

– Wil

– El

gar Companion to Health Economics AM Jones, E

 There are no values skewed to left to b

this influence this influence

 In Rand Health Insurance Experiment

ob ti t d f 17% f th 

bservation accoun

t

e

d

for 17% o

f the 

particular health plan
The influence of a single ou

The influence of

a single ou

observation

350

Y = 0.72 + 0.88 X

2503001 0200

Y

1001

5

00

50

0 100 200 

X
The influence of a single ou

The influence of

a single ou

observation

350

Y = 22.9 + 0.42 X

250300200

Y

100150

0

50

0 100 200 3

X
Log Transformation of C

 Take natural log (log with base e) of c Examples of log transformation: Examples of log transformation:

COST LN(COST)

$10 2.30

$1,000 6.91

$100,000 11.51
7

Same data- outlier is less influe

Ln Y = 2.87 + 0.011 X 

56

Y4

3

Ln 

120

0 100 200 

X 
7

Same data- outlier is less influe

Ln Y = 2.99 + 0.008 X

6

45

Y

3

Ln 

120

0 100 200 

X 
Annual per person VHA costs 

0.40

Medical Only M

0.300 100.200.000.10

no cost

$1K

$5K

$10K

$15K

$20K
Effect of log transformati

Effect of log transformati

Annual pegr person VHA costs FY

0.30 M

M

0.200.100.00

$1

$2

$3

$4

$5

$6

$7

$8

$9

$10

$11
D i ti t ti ti Descr

i

ptive s

t

ati

stics: VHA 

VHA cos

(5% sample, includes outpatient pharmacy)

Cost Ln Co

Mean 5,768 7.68M di e an 1 750 , 7 67 .Standard 18 874 1 50

Deviation

18

,874

1.50

Skewness 13.98 -0.18Kurtosis 336 3 1 12

336.3

1.12
Log linear model

 Regression with log dependent 

Ln 

Y

=

α

+

β

X

+

μ
Log linear model

 Ln (Y) = 

α + 

β X + 

μ

 P t ( ffi i t ) t Parame

ters 

(coeffi

c

ien

t

s

) are no

t

interpretable in raw dollars

– Parameter represents the relative ch

cost (Y) for each nit change in X cost (Y) for each unit change in

X

– E.

g. if 

β=0.10, then cost increases 1

each unit increase in X
What is the mean cost of t

experimental group controllin

case

-mix?

 We want to find the fitted value o

We want to find the fitted value o

 Conditional on X=1 With covariates held at the mean

Ln (Y)

=

α

+

β1

X

+

β 2

Z

+

?

Yˆ What is
Can we retransform by tak

Can we retransform by tak

antilog of fitted values?

With the model

:

Ln 

( ) Y

)

=

α

+

β1

X

+

β 2

Z

Does

β

β

?

ˆ

1

X

2

Z

Y

e

oes

α

+

β

+

β

Y

=

e

?
What is fitted value of Y

(

)

(

)

1

2

=

+

+

+ i E

Y

E

e

α

β

X

β

Z

μ

(

)

1

2

=

+

+ i e

E

e

α

β

X

β

Z

μ

1

2

=

+

+

e

α

β

X

β

Z

only

if we can assume

:

(

)

=

1 i E

e

μ
Retransformation bias

Since

E

(

μi

)

=

0

(

)

1

?

(

)

i does

E

e

i

μ

μ

=

(

)

?

(

)

1

?

(

)

i

i Does

e

E

e

does

E

e

E

μ

μ

=

(

)

?

(

)

i

i Does

e

E

e

μ

μ

=
Retransformation bias

(

)

(

) Example of why

E

e

≠

e

μi E

μi

1

1: when

1

= and

2

=

−

i

μ

μ

2

2

0

3

1

1

1

(

)

1

1

0

=

=

=

+

−

e

e

e i E

μ

2

2.7

2

0.3

7

2

(

)

1

1

+

=

+

=

−

e

e

E

e

μi

2

2
Retransformation bias

 The expected value of the antilog of th

residuals

does not equal

 The antilog of the expected value of th

residuals

(

)

(

)

i

E

i

E

e

e

μ

μ

E

(

e

)

≠

e
One way to eliminate 

retransformation bias: the sm

estimator

+

X

+

Z

+

i

E

(

Y

)

E

(

α

β

1

β

2

μ

(

)

+

+

+

+

+

=

X

Z

X

Z

i

E

Y

E

e

(

)

(

)

(

1

2

α

β

β

μ

α

β

β

μ

(

)

+

+

=

X

Z

i

e

E

(

e

)

α

β

1

β

2

μ

(

)

∑

+

+

=

n

X

Z

i

e

(

e

)

1

α

β

1

β

2

μ

(

)

∑

n

i

=

1

(

)
Smearing Estimator

∑

n

i

(

)

1

μ

∑i

i

e

n

1

(

)

μ

n

i

=

1
Smearing estimator

 This is the mean of the anti-log of 

resid l ua

l

s

 Most statistical programs allow yo

Most statistical programs allow yo

save the residuals from the regress

– Find their antilog – Find the mean of this antilog Find the mean of this antilog

 The estimator is often greater than 
Correcting retransformatio

 See Duan J Am Stat Assn 78:605 Smearing estimator assumes ident

variance of errors (homoscedasticvariance of errors (homoscedastic

 Other methods when this assumpt

can’t be made
Retransformation

 Log models can be useful when da

sk d ewe

d

 Fitted values must correct for Fitted values must correct for

retransformation bias
Zero values in cost dat

 The other problem: left edge of distrib

is truncated by observations where no is truncated by observations where no 

is incurred

 How can we find Ln(Y) when Y = 0? Recall that Ln (0) is undefined Recall that Ln (0) is undefined
Log transformation

 Can we substitute a small positive 

numb f t d d 

ber 

for zero cos

t recor

ds, an

d 

take the lo

g of cost?

– $0.01, or $0.10, or $1.00?
Substitute $1 for Zero Cost Rec

13

Ln Y = -.40 + 0.12 X 11

79

Y

35

Ln 

-1

1 Substitute $1

-3

0 20 40 60 80 0 20 40 60 80 

X
Substitute $0.10 for Zero Cost R

13 Ln Y = 2.47 + 0.15 X

11

7935

Ln Y

11

-3-

0 20 40 60 80 

Substitute $0.10

0 20 40 60 80 

X
Substitute small positive for zero 

 Log model assumes parameters are linear in 

Log model assumes parameters are linear in 

 Thus it assumes that change from $0.01 to $

the same as chan

g, , e from $1,000 to $10,000

 Possible to use a small positive in place of z

– if just a few zero cost records are involved– if results are not sensitive to choice of small pos

 There are better methods!

– Transformati h ll ( 

ions t

hat allows zeros 

(squar

– Two-part model– Oth fi er types o

f regress

ions
Is there any use for OLS w

Is there any use for OLS w

untransformed cost?

 OLS with untransformed cost can be u

– When costs are not ver

y skewe

d

– When there aren’t too many zero observa– When there is lar

ge number of observatio

 Parameters are much easier to explain Can estimate in a single regression eve

Can estimate in

a single regression eve

though some observations have zero c

 The reviewers will probably want to b

The reviewers will probably want to b

that you considered alternatives!
Review

 Cost data are not normal

– Th b k d (hi h li 

They can be skewed (high cost outli

– They c be u c ed ( e o v ues) can be truncated (zero values)

 Ordinary Least Squares (classical 

model) assumes error term (hence dependent variable) is normally dependent variable) is normally

distributed
Review

 Applying OLS to data that aren’t n

can result i bi d t ( 

lt in biased parameters (ou

are too influential)p y es ecially in sm

moderate sized samples
Review

 Log transformation can make cost 

normall di t ib t d ti

lly distributed so we can sti

OLS

 Log transformation is not always 

necessary or th l th d f d 

the only method of d

with skewed cost
Review

 Meaning of the parameters depend

th d l e model

– With linear dependent variable: 

 β is the change in absolute units of Y 

change in X change in X

– With logged dependent variable: 

 β is the proportionate change in Y for 

change in X
Review

 To find fitted value a with linear depen

variable

 Find the linear combination of parame

and variables, e.g. 

Y

ˆ

| (X =1, Z = Z) =α + β1 + β 
Review

 To find the fitted value with a logg

d d t i bl dependent variable

 Can t’ simply take anti simply take anti-log of the l

log of the l

combination of parameters and va

 Must correct for retransformation 
Review

 Retransformation bias can be correcte

multiplying the anti multiplying the anti-log of the fitted v

log of the fitted v

by the smearing estimator

 Smearing estimator is the mean of the Smearing estimator is the mean of the

antilog of the residuals
Review

 Cost data have observations with z

val t t d di t ib ti lues, a truncated distribution

 Ln (0) is not defined Ln (0) is not defined It is sometimes possible to substitu

small positive values for zero, but can result in biased parameters can result in biased parameters

 There are better methods
Next session- May 9

 Two-part models Regressions with link functions Non

-parametric statistical tests parametric statistical tests

 How to determine which method i
Reading assignment on c

Reading assignment on c

models

Basic overview of methods of analy

cos

t

s

– P Die

r

, D Yanez

, , A Ash

, M Hornbr

Lin. Methods for analyzing health utilization and costs utilization and costs Ann Rev Publi

Ann Rev Publi

(1999) 20:125-144

 HERC@va.gov
Supplemental reading on 

Supplemental reading on 

Models

 Smearing estimator for retransformatio

models

– Duan N. Smearing estimate: a nonparam

retransformation method. Journal of the A

St ti ti l A i ti St

ati

stica

l

Assoc

i

ation (1983) 78 605 (1983) 78:605

-610

 Alternatives to smearing estimator

– M i WG Th l d d d 

Mann

ing WG. The 

logge

d

depen

dent var

heteroscedasticity, and the retransformati

problem. Journal of Health Economics Journal of Health Economics (1

17(3):283-295.
Appendix: Derivation of the mea

Appendix: Derivation of the mea

the parameter in log mode

Ln 

Y

=

α

+

β

X

+

μ

=

β as dLnY

= dY

dx

dLn

Y

,

/

=

β dY

Y

dx

/

β dx

β is the proportional change in Y for a small ch