Hospital Readmission Reduction Strategies Using a Penalty-Incentive Model

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1 Procdings of th 2016 Industrial and Systms Enginring Rsarch Confrnc H. Yang, Z. Kong, and MD Sardr, ds. Hospital Radmission Rduction Stratgis Using a Pnalty-Incntiv Modl Michll M. Alvarado Txas A&M Univrsity Collg Station, TX, U.S.A. Yi Zhang Txas A&M Univrsity Collg Station, TX, U.S.A. Mark Lawly Txas A&M Univrsity Collg Station, TX, U.S.A. Abstract In 2012 th Cntrs for Mdicar and Mdicaid Srvics implmntd a pnalty-only systm to rduc th numbr of hospitals with high radmission rats. W dvlop a pnalty-incntiv modl for hospital radmissions in a basic gam thortic stting btwn an Insurr and Hospital. Th modl assums that th probability of ach patint s radmission is linarly dcrasing with th Hospital s lvl of car, whil th Hospital s tratmnt cost for ach patint is linarly incrasing with th lvl of car. Th Insurr aims to minimiz thir cost whil th Hospital aims to maximiz thir rvnu. Th Insurr dsigns a pnalty-incntiv mchanism that can inspir th Hospital to adopt a propr lvl of car. Th systm is analyzd using cntralizd and dcntralizd control. W idntify th Win-Win rgion for th pnalty-incntiv factor, in which both th Insurr and Hospital ar bttr off undr th proposd mchanism. Additionally, rsults show that th Win-Win rgion dos not xist whn a pnalty-only systm is usd by th Insurr. Kywords hospital radmissions, halth policy, gam thory, win-win rgion 1. Introduction A hospital radmission occurs whn a patint is admittd to a hospital within 30 days aftr bing dischargd from an arlir hospital stay. Th cost of hospital radmission has bn high for many yars [1]. To rduc high radmission rats, th Affordabl Car Act (ACA) stablishd th Hospital Radmission Rduction Program (HRRP). HRRP rquirs th Cntrs for Mdicar and Mdicaid Srvics (CMS) to rduc 30-day hospital radmissions. In 2013, CMS adoptd a pnalty-only mchanism for hospitals that prformd blow th risk-adjustd avrag. Th radmission pnalty was 1% in FY2013, 2% in FY2014, and was raisd to 3% in FY2015. Thr ar currntly 5 conditions in which th radmission pnalty is applid. In this papr, w focus on acut myocardial infarction (AMI). According to [2], thr ar mainly thr typs of stratgis to rduc hospital radmissions: srvic dlivry rform, financing rform, and intgratd srvic and financing rform. Thr ar many rsourcs, supporting or criticizing HRRP which discuss th ffcts and waknsss of this program [3, 4]. [5] dvlop a gam-thortic modl and analyz th ffctivnss of HRRP from an conomic and oprational prspctiv. As shown by [5], th HRRP dos not always provid incntiv for th Hospital to rduc th radmission rat, and comptition btwn hospitals can vn incras th non-incntivizd hospitals. In this papr, w dvlop a pnalty-incntiv mchanism that will rduc th hospital s 30-day radmission rat by inspiring th hospital to adopt an improvd lvl of car. Our solution approach is to driv th Insurr s optimal pnalty-incntiv factor and th Hospital s bst rspons undr an Insurr-lad Stacklbrg stting assuming both agnts ar rational. W idntify th Win-Win rgion for th pnalty-incntiv factor, in which both th Insurr and th Hospital ar bttr off undr th proposd mchanism.

2 Alvarado, Zhang, and Lawly 2. Problm Formulation W bgin with a basic gam-thortic modl with an Insurr (.g., Mdicaid/Mdicar) M and a singl hospital H assuming both agnts ar rational. M dtrmins th pnalty-incntiv factor f that influncs th paymnt to H. Givn th pnalty-incntiv factor, H dcids th lvl of car it adopts, y, which is a st of actions that may dcras th radmission rat. Thus, M is th ladr in th gam whil H is th followr. W assum that th probability that a patint radmits to th hospital is a linarly dcrasing function of H s lvl of car and is givn by d y, whr 0 < d 1 and 0 < d. Hr d is th probability of a patint radmitting to th hospital if no additional car is providd, and is th amount th probability of radmission can b rducd by th maximum lvl of car. W will approximat th avrag tratmnt pr patint rcivd by 1 + (d y). Th cost for ach tratmnt incurrd by H is linarly incrasing in y and is givn by a + by, whr a 0 and 0 b a. Hr a is th baslin cost of car for on patint and b is th maximum amount it would cost to provid a highr lvl of car. Th currnt mony that M pays H for ach tratmnt is α. If H sts th lvl of car at y, y [0,1], which implis y fraction of th actions in th action st ar takn, thn th mony that M pays H for ach tratmnt is givn by I(y, f ) = α[(1 φ 1 f )(1 y) + (1 + φ 2 f )y] = α[1 φ 1 f + (φ 1 + φ 2 ) f y], (1) whr f is th pnalty-incntiv factor dcidd by M, and y is th lvl of car st by H. φ 1 and φ 2 ar pnalty and incntiv magnituds. Th rasonabl valus of φ 1 and φ 2 satisfy th following conditions: φ 1 0 and φ 2 0; b ( )α < 1. Th first condition rquirs positiv pnalty/incntiv magnitud. By (1), th mchanism drops to incntiv only typ if φ 1 = 0, drops to pnalty only typ if φ 2 = 0, and drops to symmtric typ if φ 1 = φ 2, i.., th pnalty magnitud and incntiv magnitud ar qual. Th scond condition is ncssary to guarant our mchanism is ffctiv in inspiring H to tak dsird actions. Hospital s Profit Function: H s profit from ach tratmnt is th rvnu from M lss th cost. Thn th profit that H can obtaind from ach patint is givn by Π H (y f ) = {α[1 φ 1 f + (φ 1 + φ 2 ) f y] a by}(1 + d y) (2) Insurr s Cost Function: M pays H for ach tratmnt, and thus th cost that M spnds on ach patint is givn by TC M ( f y) = α[1 φ 1 f + (φ 1 + φ 2 ) f y](1 + d y) (3) Systm-wid Cost Function: Th systm-wid cost is M s cost lss H s profit and is givn by TC(y) = (a + by)(1 + d y) (4) 3. Typs of Control In this sction, w show th optimal solutions for th cntralizd and dcntralizd control, as wll as th do-nothing option. 3.1 Cntralizd Control Undr cntralizd control, th systm-wid cost nds to b minimizd, and th cntralizd problm is givn by min TC(y) (5) s.t. 0 y 1 Proposition 1. Th cntralizd optimal lvl of car y satisfis: If a b + 1 < 1+d thn y = 0; If a b + 1 = 1+d thn y = 0 or 1; If a b + 1 > 1+d thn y = 1. Whn a 1+d b +1 >, th optimal outcom of our mchanism rducs th systm-wid cost, i.., th systm-wid bnfit is improvd. Whn a b d, th st of dsird actions can not fficintly improv th srvic fficintly, and thus,

3 Alvarado, Zhang, and Lawly nds to b rdsignd. In this papr w assum th st of actions is wll-dsignd. Thus, w will only considr th cas that a b + 1 > 1+d. Sinc w assum < d < 1, b a + 1 > 1+d actually implis a > b. 3.2 Dcntralizd Control W assum rational agnts and complt symmtric information. Undr Insurr-lad dcntralizd control, M dtrmins th valu of f, and thn H dtrmins th valu of y. Givn th pnalty/incntiv factor f, H s problm is to solv max Π H (y f ) (6) s.t. 0 y 1 whr Π H (y f ) is givn by (2). W dnot th solution of th abov problm by y ( f ). In ordr to dtrmin th valu of f, M first prdicts y ( f ) and thn solvs th following problm min TC M (y ( f ), f ) = α[1 φ 1 f + (φ 1 + φ 2 ) f y ( f )](1 + d y ( f )) (7) s.t. 0 f Hospital s Lvl of Car For simplification purposs, w us th following notations: (α a) + b(1 + d) A = α[(φ 1 + φ 2 )(1 + d) + φ 1 ] (α a) + b(1 + d 2) B = α[(φ 1 + φ 2 )(1 + d) (φ 1 + 2φ 2 )] (α a) + b(1 + d ) C = α[(φ 1 + φ 2 )(1 + d) φ 2 ] Proposition 2. For any givn f, H s dcntralizd optimal solution for y is givn by 1. If α > a + φ 1b thn 2. If α = a + φ 1b thn 0, 0 f A y [(φ ( f ) = 1 +φ 2 )α f b](1+d) [α(1 φ 1 f ) a] 2[(φ 1 +φ 2 )α f b], A < f < B 1, B f 1. y ( f ) = 0, 0 f < φ 1b ( )α y [0,1], f = φ 1b 1, ( )α φ 1 b ( )α < f 1. (8) (9) (10) 3. If α < a + φ 1b thn { y 0, 0 f C ( f ) = 1, C f Insurr s Pnalty-Incntiv Factor In this sction w solv M s problm in (7). Rcalling Proposition 2, w nd to considr thr cass sparatly. Cas I: α > a + φ 1b is th most complx cas and has bn omittd du to spac limitations. Cas II: α = a + φ 1b has th solution f b = ( )α, y ( f ) = 1, TC M (y ( f ), f ) = (a + b)(1 + d ) and Π H (y ( f ), f ) = 0. Cas III: α < a + φ 1b has th solution f = C, y ( f ) = 1, TC M (y ( f ), f ) = α(1 + φ 2 C)(1 + d ) and Π H (y ( f ), f ) = Π H (1,C) = (α(1 + φ 2 C) a b)(1 + d ). Not that in Cas II and Cas III, our mchanism can achiv cntralizd optimal solution sinc y ( f ) = 1.

4 Alvarado, Zhang, and Lawly 3.3 Do-Nothing Modl In th situation of do-nothing, w hav f = y = 0. Dnot H s profit by Π 0 H and M s cost by TC0 M in this situation. Thn w hav Π 0 H = (α a)(1 + d) = (α a)m; TCM 0 = α(1 + d) = αm. 4. Th Win-Win Rgion [ [ Th Win-Win rgion for f to achiv cntralizd optimization is givn by 1φ2 (α a) α(1+d ) + α ], b 1 ( φ2 1+d ) ]. Not that whn b a + 1 > 1+d w hav (α a) α(1+d ) + α b < 1+d. Thus, th Win-Win rgion is not mpty if φ 2 > 0. In pnalty-only cas, w hav φ 2 = 0, th Win-Win [ rgion is mpty. ] Givn that th Win-Win rgion in not mpty, M will always st f at th lft boundary, i.., f = φ 1 (α a) 2 α(1+d ) + α b, and thn H chooss y ( f ) = 1. Thn, H will not los, and M will gt maximum bnfit incras, whil th cntralizd optimization achivs. 5. Numrical Analysis Th rsults of th numrical analysis of th pnalty-incntiv modl ar givn in this sction. Th minimum, maximum, and avrag of for radimission probabilitis, paymnt, and cost of 1738 hospitals for AMI wr takn from a 3-yar priod as rportd by [6] and ar summarizd in Tabl 1. This sampl siz includs all hospitals that had complt data ntris. Sinc data omissions appar random, th authors assum this numbr is rprsntativ of th population. Tabl 1: Statistics from for AMI Probability (d,) Paymnt (α) Cost (a,b) Minimum $16656 $11366 Avrag $21865 $19512 Maximum $32014 $30654 Th first numrical analysis (NA-1) uss data for th currnt pnalty-only modl mployd by CMS. Th paramtrs chosn for th scond numrical analysis (NA-2) ar consistnt with an incntiv-only modl. Th numrical analysis rsults wr computd using MATLAB R2015b. 5.1 NA-1 Th first numrical analysis (NA-1) sts th modl paramtrs to lvls consistnt with thos found in th data and rflcts th pnalty-only policy whr th incntiv factor φ 2 = 0 and th pnalty factor φ 1 = 3 rprsnts th FY2015 pnalty lvl. With 96.6% of th hospitals having radmission probabilitis of 30% or lss, this was takn as th upprbound radmission probability of d whil was chosn such that th diffrnc, d, was qual to th minimum probability of radmission. Th paymnt valu α was st to th avrag paymnt valu from Tabl 1. Th baslin cost a was st to th avrag cost and b was chosn such that th sum a + b was quivalnt to th maximum cost from Tabl 1. A summary for all modl paramtrs is givn in Tabl 2. Tabl 2: Modl Paramtrs for Numrical Analysis (NA) - 1 Probability Paymnt Cost Pnalty-Incntiv d =.3000 α=21865 a=19512 φ 1 =3 =.2494 b=11142 φ 2 =0 d =.0606 a + b = Th rsults of a snsitivity analysis for paramtrs φ 1 and φ 2 ar shown in Figurs 1 and 2 rspctivly. Th cost of th cntralizd and do-nothing modls ar givn in th first row and th dcisions of ths rspctiv modls ar givn in th scond row. Notic that th paramtrs in NA-1 mt th first condition in Proposition 1 and thus th paramtrs in NA-1 ar not wll-dsignd. As a rsult, th dcntralizd solution rsults ar invalid. Th figurs indicat thr is no Win-Win rgion for th currnt policy and modl paramtrs bcaus th cntralizd solution is always y = 0 for all valus of φ 1 and φ 2.

5 Alvarado, Zhang, and Lawly Figur 1: Rsults for NA-1, Snstivity φ 1 Figur 2: Rsults for NA-1, Snstivity φ NA-2 Th scond numrical analysis (NA-2) considrs an incntiv-only policy. To satisfy conditions for a Win-Win rgion, th paramtrs wr changd to thos shown in Tabl 3 to crat a st of wll-dsignd actions and satisfy Cas III for th Insurr. Spcifically, α and b wr rducd. Additionally, φ 1 was st to 0 for an incntiv-only modl and φ 2 was st to 5 to rprsnt a 5% incntiv lvl. A summary for all modl paramtrs in NA-2 is givn in Tabl 2. Tabl 3: Data usd for Numrical Analysis (NA) - 2 Probability Paymnt Cost Pnalty-Incntiv d =.3000 α=19511 a=19512 φ 1 =0 =.2494 b=3902 φ 2 =5 d =.0606 a + b = A Win-Win rgion in ths figurs if vidnt whn th hospital H s dcision y = 1 in both th cntralizd and dcntralizd controls. In Figur 3, th Win-Win rgion only occurs whn was α < a, which implis that th baslin cost to th Hospital is lss than th Insurr paymnt. Thus, th hospital should incur vn mor costs and provid a highr lvl of car to rciv a highr rimbursmnt. Finally, Figur4 shows that th Win-Win rgion xists whn th incntiv lvl is high, or φ 2 > Conclusions W hav dvlopd a pnalty-incntiv modl with conditions for a Win-Win rgion. Th modl and rsults show that th currnt policy is inffctiv in inspiring an improvd lvl of car. Th numrical rsults also rval a fw prliminary obsrvations. First, th paymnts α should b lss than th baslin cost a. Thr is not a Win-Win rgion for a pnalty-only modl. Thr can only b a Win-Win rgion for a modl that includs incntivs φ 2 > 0. Thrfor, CMS should considr changing th policy from a pnalty-only to an incntiv-only modl. In futur rsarch, w plan to analyz policy changs basd on conditions othr than AMI. W will also analyz policy changs basd on hospital

6 Alvarado, Zhang, and Lawly Figur 3: Rsults for NA-2, Snstivity α Figur 4: Rsults for NA-2, Snstivity φ 2 charactristics (.g.,% of Mdicaid patints, hospital siz, tc.) to dtrmin if crtain hospitals do not bnfit from th proposd changs to th radmission policy. Rfrncs [1] Kripalani, S., Thobald, C.N., Anctil, B., and Vasilvskis, E.E. Rducing hospital radmission rats: currnt stratgis and futur dirctions. Annual rviw of mdicin, 65: , [2] Ston, J., Hoffman, G.J., t al. Mdicar hospital radmissions: issus, policy options and PPACA. In Rport for Congrss, [3] Grhardt, G., Yman, A., Hickman, P., Olschlagr, A., Rollins, E., and Brnnan, N. Data shows rduction in mdicar hospital radmission rats during Mdicar Mdicaid Rs Rv, 3(2):E1-E11, [4] Joynt, K.E., and Jha, A.K. Charactristics of hospitals rciving pnaltis undr th hospital radmissions rduction program. Jama, 309(4): , [5] Zhang, D.J., Park, E., Gurvich, I., Van Mighm, J.A., Young, R.S., and Williams, M.V. Hospital radmissions rduction program: An conomic and oprational analysis. Availabl at SSRN , [6] CMS. Cntrs for Mdicar and Mdicaid Srvics ; Accssd 01-Octobr-2015.

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