Erratum: Estimating lifetime or episode-of-illness costs under censoring

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  • HEALTH ECONOMICSHealth Econ. 20: 125126 (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): 10101028.

    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 Wij5M(aj)M(aj1) be the actual total costs accumulated by individual i for interval j dened

    by time points (aj1, aj). Wij5 0 for individuals who die before interval j.Let

    mij EWij EXfEWijjXig

    EXfEIVi4aj Wij1Iaj1oViaj WijjXig Assump 1 : Wij 0 ifViaj1EXfEIVi4aj;Ci >aj Yij1Iaj1oViaj WijjXi;Ci4aj1g; Assump 2 : Ci

    aVi;Wi1; ::WikjXi

    EXPrVi4ajjXi;Ci4aj EYijjVi4aj;Xi;Ci4aj

    1Praj1oViajjXi;Ci4aj1 EWijjaj1oViaj;Xi;Ci4aj1

    ( ); under Assump 2

    Let

    Uij Vi aj1 if aj1oViaj.Also let

    Zij Ci aj1 ifaj1oCiaj.Therefore,

    EWijjaj1oViaj; Xi;Ci4aj1 EU:k jak1oViakfEWijjaj1oViaj;Xi;Ci4aj1;Uijg EU:k jak1oViak and Zij>UijfEYijjaj1oViaj;Xi;Ci4aj1;Uijg;

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

    Wij;Uijjaj1oViaj;Xi;F 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 nalpartial month in their model, which holds true if censoring happens at the beginning of the nal 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, mij5

    EXPrVi4ajjXi;Ci4aj EYijjVi4aj;Xi;Ci4aj1Praj1oViajjXi;Ci4aj1

    EU:k jak1oViak and Zij>UijfEYijjaj1oViaj;Xi;Ci4aj1;Uijg

    ( )

    EXfSjXi m1jXi1Sj1Xi hjXi m2jXig Under Assump 2

    Estimators:Expressions for PrVi4ajjXi;Ti4aj Sj1Xi 1 hjXi SjXi and Praj1oViajjXi;Ci4

    aj1 Sj1Xi hjXi are estimated using data (Xi,Vi, and Ci) and a survival accelerated failure timemodel to obtain S^jXi; S^j1Xi; h^jXi.

    Expression for EYijjVi4aj;Xi;Ci4aj is estimated with a cost model in Xi for all who survive andare not censored in interval j to obtain m^2jXi.

    And nally, expression for EU:k jak1oViak and Zij>UijfEYijjaj1oViaj;Xi;Ci4aj1;Uijg is estimatedusing 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

    m^2jXi.By the law of large numbers, S^jXi, S^j1Xi; h^jXi; m1jXi, and m^2jXi converge in probability to

    Sj(Xi), Sj1Xi; hjXi, m1j(Xi) and m2j(Xi) respectively. It then follows from Slutskys theorem thatm^jX E^XfS^jXi m^1jXi1S^j1Xi h^jXi m^2jXig 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: 113128.

    Liu L, Wolfe RA, Kalbeisch JD. 2007. A random effects model for censored medical costs. Statistics inMedicine 26: 139155.

    ERRATUM126

    Copyright r 2010 John Wiley & Sons, Ltd. Health Econ. 20: 125126 (2011)

    DOI: 10.1002/hec

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