erratum: estimating lifetime or episode-of-illness costs under censoring

HEALTH ECONOMICSHealth Econ. 20: 125–126 (2011)Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/hec.1684

ERRATUM

ANIRBAN BASUa,b,� and WILLARD G. MANNINGc

aSection of Hospital Medicine, the Department of Medicine, and Center for Health and the Social Sciences,University of Chicago, Chicago, IL, USA

bThe National Bureau of Economic Research, Cambridge, MA, USAcHarris Graduate School of Public Policy Studies, and the Department of Health Studies of the Biological Sciences Division,

University of Chicago, Chicago, IL, USA

This erratum refers to: Basu, A. and Manning, W. G. Estimating lifetime or episode-of-illness costsunder censoring. Health Economics 2010; 19(9): 1010–1028.

Changes are denoted in bold italics. We thank Dr. Lei Liu for helping us with the changes.Formal sketch of proof for our proposed estimator:Let Wij 5M(aj)�M(aj�1) be the actual total costs accumulated by individual i for interval j defined

by time points (aj�1, aj). Wij 5 0 for individuals who die before interval j.Let

mij ¼EðWijÞ ¼EXfEðWijjXiÞg

¼EXfEðIðVi4ajÞ �Wij1Iðaj�1oVi�ajÞ �WijjXiÞg ðAssump 1 : Wij ¼ 0 ifVi�aj�1Þ

¼EXfEðIðVi4aj;Ci >ajÞ �Yij1Iðaj�1oVi�ajÞ �WijÞjXi;Ci4aj�1g; ðAssump 2 : CiaðVi;Wi1; ::WikÞjXiÞ

¼EX

PrðVi4ajjXi;Ci4ajÞ �EðYijjVi4aj;Xi;Ci4ajÞ

1Prðaj�1oVi�ajjXi;Ci4aj�1Þ �EðWijjaj�1oVi�aj;Xi;Ci4aj�1Þ

( ); ðunder Assump 2Þ

Let

Uij ¼ Vi � aj�1 if aj�1oVi�aj.

Also let

Zij ¼ Ci � aj�1 ifaj�1oCi�aj.

Therefore,

EðWijjaj�1oVi�aj; Xi;Ci4aj�1Þ

¼ EU:k jak�1oVi�akfEðWijjaj�1oVi�aj;Xi;Ci4aj�1;UijÞg

¼ EU:k jðak�1oVi�akÞ and Zij>UijfEðYijjaj�1oVi�aj;Xi;Ci4aj�1;UijÞg;

where the last equality is strictly not true but can be used as an approximation assumingFðUijjUijoZijÞ ’ FðUijÞ(Assump 3) given Zij

‘ðWij;UijÞjaj�1oVi�aj;Xi;FðÞ is a distribution function.

Such an approximation is often inconsequential when the support of Uij(i.e. the length of the intervals) issmall. Similar ad-hoc methods were used by Liu et al. (2007), who ignored the medical cost for the finalpartial month in their model, which holds true if censoring happens at the beginning of the final month. Linet al. (1997) adopted similar schemes in estimating the lifetime cost from cost history data.

*Correspondence to: Section of Hospital Medicine, the Department of Medicine, and Center for Health and the Social Sciences,University of Chicago, Chicago, USA. E-mail: abasu@medicine.bsd.uchicago.edu

Copyright r 2010 John Wiley & Sons, Ltd.

Thus, mij 5

EX

PrðVi4ajjXi;Ci4ajÞ � EðYijjVi4aj;Xi;Ci4ajÞ1Prðaj�1oVi�ajjXi;Ci4aj�1Þ

� EU:k jðak�1oVi�akÞ and Zij>UijfEðYijjaj�1oVi�aj;Xi;Ci4aj�1;UijÞg

( )

¼ EXfSjðXiÞ � m1jðXiÞ1Sj�1ðXiÞ � hjðXiÞ � m2jðXiÞg ðUnder Assump 2Þ

Estimators:Expressions for PrðVi4ajjXi;Ti4ajÞ ¼ Sj�1ðXiÞ � ½1� hjðXiÞ� ¼ SjðXiÞ and Prðaj�1oVi�ajjXi;Ci4

aj�1Þ ¼ Sj�1ðXiÞ � hjðXiÞ are estimated using data (Xi,Vi, and Ci) and a survival accelerated failure timemodel to obtain SjðXiÞ; Sj�1ðXiÞ; hjðXiÞ.

Expression for EðYijjVi4aj;Xi;Ci4ajÞ is estimated with a cost model in Xi for all who survive andare not censored in interval j to obtain m2jðXiÞ.

And finally, expression for EU:k jðak�1oVi�akÞ and Zij>UijfEðYijjaj�1oVi�aj;Xi;Ci4aj�1;UijÞg is estimated

using another cost model in Xi and Uij for all who are observed to die in interval j and then averagedover the empirical distribution of Uij over all subjects from all intervals (Assump 4: Uij

‘Xij), to obtain

m2jðXiÞ.By the law of large numbers, SjðXiÞ, Sj�1ðXiÞ; hjðXiÞ; m1jðXiÞ, and m2jðXiÞ converge in probability to

Sj(Xi), Sj�1ðXiÞ; hjðXiÞ, m1j(Xi) and m2j(Xi) respectively. It then follows from Slutsky’s theorem thatmjðXÞ ¼ EXfSjðXiÞ � m1jðXiÞ1Sj�1ðXiÞ � hjðXiÞ � m2jðXiÞg converges in probability to mj(X).

REFERENCES

Lin DY, Feuer EJ, Etzioni R, Wax Y. 1997. Estimating medical costs from incomplete follow-up data.Biometrics 53: 113–128.

Liu L, Wolfe RA, Kalbfleisch JD. 2007. A random effects model for censored medical costs. Statistics inMedicine 26: 139–155.

ERRATUM126

Copyright r 2010 John Wiley & Sons, Ltd. Health Econ. 20: 125–126 (2011)

DOI: 10.1002/hec