Econometrics Course: 

Cost as the 

Dependent Variable (I) 

Paul G. Barnett, PhD 

November 20, 2013 
2 

What is health care cost? 

 Cost of an intermediate product, e.g., 

– chest x-ray

– a day of stay

– minute in the operating room

– a dispensed prescription

 Cost of a bundle of products 

– Outpatient visit 

– Hospital stay
3 

What is health care cost (cont.)? 

 Cost of a treatment episode 

– visits and stays over a time period

 Annual cost

– All care received in the year
4 

Annual per person VHA costs FY10

(5% random sample) 

0.00

0.10

0.20

0.30

0.40

no cost

$1K$5K

$10K$15K$20K$25K$30K

$30K+

Medical Only Medical+Rx
5 

Descriptive statistics: VHA costs FY10

(5% sample, includes outpatient pharmacy) 

Cost

Mean 5,768

Median 1,750

Standard Deviation 18,874

Skewness 13.98

Kurtosis 336.3
6 

Skewness and kurtosis

 Skewness (3rd moment)

– Degree of symmetry

– Skewness of normal distribution =0

– Positive skew: more observations in right tail

 Kurtosis (4th moment)

– Peakness of distribution and thickness of tails

– Kurtosis of normal distribution=3
7 

Distribution of cost: skewness

– Rare but extremely high cost events

 E.g. only some individuals hospitalized

 Some individuals with expensive chronic illness

– Positive skewness (skewed to the right) 
8 

Comparing the cost incurred by 

members of two groups 

 Do we care about the mean 

or the median? 
9 

Annual per person VHA costs FY09 

among those who used VHA in FY10

0.00

0.10

0.20

0.30

0.40

no cost

$1K$5K

$10K$15K$20K$25K$30K

$30K+

Medical Only Medical+Rx
10 

Distribution of cost: zero value 

records 

 Enrollees who don’t use care 

– Zero values

– Truncation of the distribution
11

What hypotheses involving cost 

do you want to test?
12 

What hypotheses involving cost do 

you want to test?

 I would like to learn how cost is affected 

by: 

– Type of treatment

– Quantity of treatment

– Characteristics of patient

– Characteristics of provider

– Other 
13 

Review of Ordinarily Least 

Squares (OLS) 

 Also known as: Classic linear model 

 We assume the dependent variable can be 

expressed as a linear function of the 

chosen independent variables, e.g.:

 Yi = α + β Xi + εi
14 

Ordinarily Least Squares (OLS)

 Estimates parameters (coefficients) α, β

 Minimizes the sum of squared errors 

– (the distance between data points and the 

regression line)
15 

Linear model

 Regression with cost as a linear dependent 

variable (Y)

– Yi = α + β Xi + εi 

 β is interpretable in raw dollars

– Represents the change of cost (Y) for each unit 

change in X

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

increase in X
16 

Expected value of a random variable

 E(random variable) 

 E(W) = Σ Wi p(Wi)

– For each i, the value of Wi times probability 

that Wi occurs

– Probability is between 0 and 1

– A weighted average, with weights by 

probability
17 

Review of OLS assumptions

 Expected value of error is zero E(εi)=0

 Errors are independent E(εiεj)=0

 Errors have identical variance E(εi

2)=σ2

 Errors are normally distributed 

 Errors are not correlated with 

independent variables E(Xi

εi)=0
18 

When cost is the dependent 

variable

 Which of the assumptions of the classical model are 

likely to be violated by cost data?

– Expected error is zero

– Errors are independent

– Errors have identical variance

– Errors are normally distributed

– Error are not correlated with independent variables
19 

Compare costs incurred by 

members of two groups 

 Regression with one dichotomous explanatory 

variable

 Y = α + β X + ε

 Y cost 

 X group membership

– 1 if experimental group

– 0 if control group
20 

Predicted difference in cost of 

care for two group 

Predicted value of Y conditional on X=0 

 (Estimated mean cost of control group)

 Predicted Y when X=1

(Estimated mean cost experimental group)

XY ++= εβα

ˆ

0)(X|Y == α

ˆ

1)(X|Y +== βα a 
21 

Other statistical tests are special 

cases 

 Analysis of Variance (ANOVA) is a 

regression with one dichotomous 

independent variable

 Relies on OLS assumptions
22 

Compare groups controlling for 

case mix 

 Include case-mix variable, Z

1 2ZXY +++= εββα
23 

Compare groups controlling for case 

mix (cont). 

 Estimated mean cost of control group 

controlling for case mix (evaluated at mean 

value for case-mix variable)

Zwhere is mean Zof

ˆ

XY )0(| +== βα 2 Z
24 

Compare groups controlling for case 

mix (cont). 

 Estimated mean cost of experimental group 

controlling for case mix (evaluated at mean 

value for case-mix variable)

Zwhere is mean Zof

ˆ

XY )1(| ++== ββα 21 Z
25 

Assumptions are about error 

term 

 Formally, the OLS assumptions are about 

the error term 

 The residuals (estimated errors) often 

have a similar distribution to the 

dependent variable
26 

Why worry about using OLS with 

skewed (non-normal) data? 

 “In small and moderate sized samples, a single 

case can have tremendous influence on an 

estimate”

– Will Manning – Elgar Companion to Health Economics AM Jones, Ed. (2006) p. 439 

 There are no values skewed to left to balance 

this influence

 In Rand Health Insurance Experiment, one 

observation accounted for 17% of the cost of a 

particular health plan
27 

Y = 0.72 + 0.88 X

0

50

100

150

200

250

300

350

0 100 200 300

X

Y

The influence of a single outlier 

observation 
28 

Y = 22.9 + 0.42 X

0

50

100

150

200

250

300

350

0 100 200 300

X

Y

The influence of a single outlier 

observation 
29 

Log Transformation of Cost

 Take natural log (log with base e) of cost

 Examples of log transformation:

COST LN(COST) 

$10 2.30 

$1,000 6.91 

$100,000 11.51 
30 

Ln Y = 2.87 + 0.011 X 

0

1

2

3

4

5

6

7

0 100 200 300

X

Ln Y

Same data- outlier is less influential
31 

Ln Y = 2.99 + 0.008 X

0

1

2

3

4

5

6

7

0 100 200 300

X

Ln Y

Same data- outlier is less influential
32 

Annual per person VHA costs FY10

0.00

0.10

0.20

0.30

0.40

no cost

$1K$5K

$10K$15K$20K$25K$30K

$30K+

Medical Only Medical+Rx
33 

Effect of log transformation

Annual per person VHA costs FY10 

0.00

0.10

0.20

0.30

$1$2$3$4$5$6$7$8$9

$10$11$12$13

$13+

Medical Only

Medical+Rx
34 

Descriptive statistics: VHA costs FY10

(5% sample, includes outpatient pharmacy) 

Cost Ln Cost

Mean 5,768 7.68

Median 1,750 7.67

Standard 

Deviation 

18,874 1.50

Skewness 13.98 -0.18

Kurtosis 336.3 1.12
35 

Log linear model

 Regression with log dependent variable

++= X YLn μβα
36 

Log linear model

 Ln (Y) = α + β X + μ

 Parameters (coefficients) are not 

interpretable in raw dollars

– Parameter represents the relative change of 

cost (Y) for each unit change in X

– E.g. if β=0.10, then cost increases 10% for 

each unit increase in X
37 

What is the mean cost of the 

experimental group controlling for 

case-mix?

 We want to find the fitted value of Y 

 Conditional on X=1

 With covariates held at the mean 

?

ˆ

Ln (Y) Z X 1 2

What is Y

+++= μββα
38 

Can we retransform by taking 

antilog of fitted values?

?

ˆ

Ln (Y) Z X 

 With the model:

21 X 

1 2

Z eY

Does

ββα

ββα μ

++

=

+++=
39 

What is fitted value of Y? 

1)(

:

)(

()( )

21

21

21

=

=

=

=

++

++

+++

i

i

i

eE

only if we assumecan

e

eEe

eEYE

ZX

ZX

ZX

μ

ββα

ββα μ

μββα
40 

Retransformation bias

?)(

?1)(

0)(

)( i i

i

Does eEe

eEdoes

ESince

E

i

μ μ

μ

μ

=

=

=
41 

Retransformation bias

5.1

2

37.072.2

2 )(

1

1 :1

)(

11

011)(

1 2

)(

=

+

=

+

=

===

−==

≠

−

−+

ee eE

eee

when and

Example eeEwhyof

i

i

i i

E

E

μ

μ

μ μ

μμ
42 

Retransformation bias

 The expected value of the antilog of the 

residuals

does not equal

 The antilog of the expected value of the 

residua

!)(

)( i E i eeE μ μ ≠

ls
43 

One way to eliminate 

retransformation bias: the smearing 

estimator 

( )

( ) ∑

=

++

++

+++

=

=

=

n

i

ZX

ZX

ZX

i

i

i

e

n

e

eEe

eEYE

1

)( 1

)(

()( )

21

21

21

ββα μ

ββα μ

μββα
44 

Smearing Estimator

∑

=

n

i

i e

n 1

)( 1 μ
45 

Smearing estimator

 This is the mean of the anti-log of the 

residuals 

 Most statistical programs allow you to 

save the residuals from the regression 

– Find their antilog 

– Find the mean of this antilog

 The estimator is often greater than 1 
46 

Correcting retransformation bias

 See Duan J Am Stat Assn 78:605 

 Smearing estimator assumes identical 

variance of errors (homoscedasticity) 

 Other methods when this assumption 

can’t be made
47 

Retransformation 

 Log models can be useful when data are 

skewed 

 Fitted values must correct for 

retransformation bias
48 

Zero values in cost data

 The other problem: left edge of distribution 

is truncated by observations where no cost 

is incurred

 How can we find Ln(Y) when Y = 0?

 Recall that Ln (0) is undefined
49 

Log transformation

 Can we substitute a small positive 

number for zero cost records, and then 

take the log of cost? 

– $0.01, or $0.10, or $1.00?
50 

Substitute $1 for Zero Cost Records

Ln Y = -.40 + 0.12 X 

-3

-1

1

3

5

7

9

11

13

0 20 40 60 80 100

Ln Y

X

Substitute $1
51 

Substitute $0.10 for Zero Cost Records

Ln Y = 2.47 + 0.15 X

-3

-1

1

3

5

7

9

11

13

0 20 40 60 80 100

X

Ln Y

Substitute $0.10
52 

Substitute small positive for zero cost?

 Log model assumes parameters are linear in logs

 Thus it assumes that change from $0.01 to $0.10 is 

the same as change from $1,000 to $10,000

 Possible to use a small positive in place of zeros

– if just a few zero cost records are involved

– if results are not sensitive to choice of small positive value

 There are better methods!

– Transformations that allows zeros (square root)

– Two-part model

– Other types of regressions
53 

Is there any use for OLS with 

untransformed cost? 

 OLS with untransformed cost can be used:

– When costs are not very skewed

– When there aren’t too many zero observations

– When there is large number of observations

 Parameters are much easier to explain

 Can estimate in a single regression even 

though some observations have zero costs

 The reviewers will probably want to be sure 

that you considered alternatives!
54 

Review 

 Cost data are not normal 

– They can be skewed (high cost outliers)

– They can be truncated (zero values)

 Ordinary Least Squares (classical linear 

model) assumes error term (hence 

dependent variable) is normally 

distributed 
55 

Review 

 Applying OLS to data that aren’t normal 

can result in biased parameters (outliers 

are too influential) especially in small to 

moderate sized samples
56 

Review 

 Log transformation can make cost more 

normally distributed so we can still use 

OLS 

 Log transformation is not always 

necessary or the only method of dealing 

with skewed cost 
57 

Review 

 Meaning of the parameters depends on 

the model 

– With linear dependent variable: 

 β is the change in absolute units of Y for a unit 

change in X

– With logged dependent variable: 

 β is the proportionate change in Y for a unit 

change in X
58 

Review 

 To find fitted value a with linear dependent 

variable

 Find the linear combination of parameters 

and variables, e.g. 

ˆ

ZZXY ),1(| ++=== ββα 21 Z
59 

Review 

 To find the fitted value with a logged 

dependent variable

 Can’t simply take anti-log of the linear 

combination of parameters and variables 

 Must correct for retransformation bias 
60 

Review 

 Retransformation bias can be corrected by

multiplying the anti-log of the fitted value

by the smearing estimator

 Smearing estimator is the mean of the

antilog of the residuals
61 

Review 

 Cost data have observations with zero 

values, a truncated distribution

 Ln (0) is not defined 

 It is sometimes possible to substitute 

small positive values for zero, but this 

can result in biased parameters 

 There are better methods
62 

Next session- December 4

 Two-part models 

 Regressions with link functions

 Non-parametric statistical tests

 How to determine which method is best?
63 

Reading assignment on cost 

models

Basic overview of methods of analyzing 

costs 

– P Dier, D Yanez, A Ash, M Hornbrook, DY 

Lin. Methods for analyzing health care 

utilization and costs Ann Rev Public Health

(1999) 20:125-144

 HERC@va.gov 
64 

Supplemental reading on Log 

Models

 Smearing estimator for retransformation of log 

models

– Duan N. Smearing estimate: a nonparametric 

retransformation method. Journal of the American 

Statistical Association (1983) 78:605-610.

 Alternatives to smearing estimator

– Manning WG. The logged dependent variable, 

heteroscedasticity, and the retransformation 

problem. Journal of Health Economics (1998) 

17(3):283-295.
65 

Appendix: Derivation of the meaning of 

the parameter in log model 

β is the proportional change in Y for a small change in X

β

β

μβα

=

= =

++=

dx

YdY

YdYdLnYas

dx

YdLn

/

, /

 X YLn