IPRC Discussion Paper Series No.15

Transcription

1 IPRC Discussion Paper Series No.5

2 Evaluaion of he 6 revision of he medical paymen sysem in Japan by he Box-Cox ransformaion model and he Hausman es -An analysis of he lengh of he hospial say for caarac operaions- Kazumisu Nawaa Graduae School of Engineering, Universiy of okyo, Bunkyo-ku, okyo , Japan, nawaa@mi..u-okyo.ac.jp Koichi Kawabuchi Graduae School of Medical and Denal Sciences, okyo Medical and Denal Universiy, okyo Medical and Denal Universiy, Bunkyo-ku, okyo, 3-85, Japan Absrac he number of caarac paiens in Japan has increased rapidly wih he aging of he populaion. herefore, conrolling medical expenses by reducing he lengh of he hospial say has become very imporan in he reamen of his disease. In his paper, we evaluae he effecs of he 6 revision of he medical paymen sysem (DPC/PDPS) on he lengh of he hospial say for caarac operaions. For he analysis, he Box-Cox ransformaion model and he Hausman es are used. We analyze he daa colleced from 5 hospials before and afer he revision. he number of paiens in he daa se is 4,9. Keywords: Diagnosis Procedure Combinaion (DPC), DPC/PDPS, inclusive paymen sysem, caarac, lengh of say (LOS), Box-Cox ransformaion model, Hausman es

3 . Inroducion Since Japanese medical expenses have been increasing rapidly wih he aging of he populaion, shorening he average lengh of say (ALOS) by reducing long-erm hospializaions has become an imporan poliical issue in Japan. A new inclusive paymen sysem based on he Diagnosis Procedure Combinaion (DPC) was inroduced in 8 special funcioning hospials (i.e., universiy hospials, he Naional Cancer Cener and he Naional Cardiovascular Cener) in April 3 in Japan []. he DPC Evaluaion Division of he Cenral Social Insurance Medical Council [5] now calls he new inclusive paymen sysem based on he DPC he DPC/PDPS (per diem paymen sysem), and we use his erm and refer o hospials paricipaing in he DPC/PDPS as DPC hospials hroughou his paper. Since April 4, he DPC/PDPS has been gradually exended o general hospials which saisfy he required condiions. he DPC/PDPS has been revised every wo years since hen. According o he DPC Evaluaion Division [6], as of April, 3, a oal of,496 hospials, comprising abou % of he 7,58 general hospials in Japan, had joined he DPC/PDPS. hese hospials had 474,98 beds, which represened more han half of he oal number of beds (899,385 beds) in all general hospials. (he daa for general hospials were obained from he survey of hospials.) Furhermore, addiional 44 hospials were preparing o join he DPC/PDPS (hereafer preparaion hospials). able gives he numbers of hospials and beds by hospial size. he hospial size is measured by he number of beds in each hospial. A clear rend is eviden in hese daa; namely, as he size of he hospials becomes larger, he percenage of he DPC hospials increases. Among small hospials wih fewer han beds, only 5.7% joined he DPC/PDPS, and hese hospials had.3% of heir beds in his caegory. On he oher hand, among large hospials wih 5 or more beds, 65.% were DPC hospials, and hese hospials had more han hree quarers (77.6%) of heir beds in his caegory. he inroducion of he DPC/PDPS was one of he larges and mos imporan revisions of he paymen sysem since he Second World War. o ensure he effecive use of medical resources, i is absoluely necessary o horoughly analyze he DPC/PDPS and he revisions ha have been implemened every wo years. However, sufficien evaluaions of he sysem have no ye been done. Empirical sudies of he lengh of he hospial say (LOS) and of medical paymens using economeric models are necessary o evaluae he sysem correcly. A simple comparison of he ALOS by hospial is no sufficien; differences in ypes of disease mus be considered, and he individual characerisics of paiens and ypes of reamen mus also be considered for

4 he same disease. he Box-Cox [3] ransformaion model (hereafer, he BC model) is widely used o examine various problems in survival analysis, such as he LOS. However, since he error erms canno have a normal disribuion excep when he ransformaion parameer is zero, he likelihood funcion under he normaliy assumpion (hereafer, he BC likelihood funcion) is misspecified, and he maximum likelihood esimaor (hereafer, he BC MLE) canno be consisen. Alernaive versions of he BC model have been proposed by various auhors. However, in hese versions he simpliciy of he model is los [4], and so hese alernaives have no been widely used. Alhough he BC MLE is generally inconsisen, he BC MLE can be a consisen esimaor if he small σ condiion described in Bickel and Doksum [] and he error erms are independen and idenically disribued (i.i.d.) random variables. Nawaa [8] proposed a new consisen esimaor of he BC model. However, he esimaor is inconsisen if he error erms are no i.i.d. random variables (hereafer non-i.i.d. case). In his paper, we firs consider a robus esimaor ha is consisen even for he non-i.i.d. case. Using hese esimaors, we consider Hausman [4] ess for he BC MLE; ha is, we can deermine wheher we can use he BC MLE or no for he esimaion of he model. We hen evaluae he effecs of he 6 revision of he DPC/PDPS on he LOS and he medical paymens for caarac operaionsdpc caegory code: ). he number of caarac paiens in Japan has increased rapidly wih he aging of he populaion. According o a survey conduced by he Minisry of Healh, Labour and Welfare [7], nearly 8, caarac operaions are performed annually and nearly.5 billion yen are spen for caarac operaions annually. In he case of caarac operaions, a major change was made concerning he DPC classificaions, he hree periods, and he inclusive paymens deermined by he DPC/PDPS in he 6 revision []. o evaluae he revision, we analyzed he daa se obained from 5 DPC hospials (Hp -5where one-eye caarac operaions were performed boh before and afer he revision and he number of paiens was more han in each period. he number of paiens in he daa se is 4,9.. Esimaors of he BC model. BC model We consider he BC model z = x ' β + u, y, =,,...,, () 3

5 λ y λ, if λ, where z = { log( y ), if λ =, y is he LOS, x and β are he k-h dimensional vecors of he explanaory variables and he coefficiens, respecively, and λ ransformaion parameer. he BC likelihood funcion is given by log L( θ) = log f ( θ), and () log f ( θ ) = logφ{( z x ' β)/ σ} logσ + ( λ ) log y is he where φ is he probabiliy densiy funcion of he sandard normal assumpion, σ is he variance of u and θ ' = ( λ, β ', σ ). he BC MLE is obained as follows: log L =, λ log L β = ( ' ) =, x z x β and (3) σ log L σ = ( z x ' β) 4 σ σ =. log L Le θ ' = ( λ, β', σ ) be he rue parameer value of θ. Since E [ θ ], λ he BC MLE is generally inconsisen. However, if he error erms are i.i.d. random variables and λσ /( + λx ' β) (in pracice, P [ < ] is small enough), he BC MLE is no an only consisen bu also efficien esimaor and small σ asympoics [] of he BC MLE ˆ' ˆ, ˆ ' θ = ( λ β, ˆ σ ) are obained by ˆ ( θ BC θ ) N[, A B( A') ], (4) where A log L = E[ θ ] θ θ ', and log f log = [ f B E ' ] θ. θ θ θ BC y BC BC BC 4

6 . Nawaa s esimaor Nawaa [8] considered he roos of he equaions, log( λx ' β + ) z x ' β λ G ( θ ) = [ [{ + } y }]( ' β ) z z x (5) σ λ λ λx ' β + z x ' β + log( λx ' β + ) + ] g ( θ) =, λ λx ' β + log L =, β log L and =. σ G (θ) is obained by he approximaion of log L / λ. If he firs and hird momens of u are zero, E [ ( θ )] = is obained, and he esimaor G obained by Equaion (5) is consisen. (Hereafer, I refer his esimaor as he N-esimaor.) he asympoic disribuion of he N-esimaor ˆ N N N N θ ' = ( λ, β ', σ ) is given by ( ˆ θ N θ ) N[, C ( θ ) where C = E[ θ ], θ ' D( C') ], (6) D = E ( θ ) ( θ )'], ( θ)' = [ ( θ), ξ ( θ)', ς ( θ)], [ o g ( z x ' β) σ ξ ( θ ) = x ( ' β ), z x and ς ( θ) =. σ σ.3 A robus esimaor he N-esimaor is no consisen for he non-i.i.d. case. In his secion, we consider a robus esimaor which is consisen even for he non-i.i.d. case if E ( u x ) = and E ( u 3 x ) =. Here, we use he firs- and hird-momen resricions and consider he roos of he equaions 3 M ( ϑ) = ( ϑ) =, m ϑ) m( ϑ, x, y ) = ( z ' β ), and (7) m x ( z x ' β) =, ( x where ϑ ' = ( λ, β' ). Noe ha he second equaion in (7) gives he leas-squares esimaor when he value of λ is given. Le ϑ ' = ( λ, β' ). Since E [ M ( ϑ )] =, here exiss a consisen roo among he roos of (7). he 5

7 proof is given in Appendix A. Le ˆ ϑ ' ( ˆ, ˆ R = λr βr' ) be he consisen roo (hereafer, he robus esimaor). Le ψ ( ϑ) = x ( z x ' β ) and ω ( ϑ)' = [ ( ϑ), ψ ( ϑ)' ]. Suppose ha m ω ( ϑ) ϑ converges o a nonsingular marix F in probabiliy and ϑ' E[ ω ( ϑ ) ω ( ϑo )'] converges o a nonsingular marix H. hen he asympoic disribuion of ϑˆ R is given by ( ˆ ϑ R ϑ ) N[, F H( F') ]. he proof is given in Appendix B. Noe ha replacing A, B, C, D, F and H by lim λ A, lim λ B, lim λ C, lim λ D, lim λ F and lim λ H, we can use he same formulas when λ =.. (8) 3. ess of he assumpions 3. A es of he small σ assumpion log L Since G ( θ ) = θ under he small σ and i.i.d. assumpions are λ saisfied, B = D and we ge ( ˆ λ ˆ λ ) N(, δ ), (9) N BC where δ = he firs elemen of ( A C ) B( A C )'. Hence we can perform a more precise es han a es where he asympoic variance is calculaed by a difference of wo variances in he Hausman ype es. Using = ˆ λ ˆ λ )/ ˆ δ as he es saisic, where δˆ is he ( N BC esimaor of δ, we can es he smallσ assumpion; ha is, we can es wheher we can successfully use he BC MLE or no []. Since he rank of he variance-covariance marix of ( ˆ λ ˆ λ ), ( ˆ β ˆ β )'] [ BC N BC N asympoically becomes one, we canno use any elemen of β in he 6

8 Hausman ype es [3]. 3. A es of he i.i.d. assumpion In he previous secion, we consider he BC MLE and he N-esimaors, however, hey are no consisen for a non i.i.d. case even if he small σ assumpion is saisfied. herefore, i is also necessary o es he i.i.d. assumpion using he robus esimaor defined in Secion 3. If boh of he small σ and i.i.d. assumpions are saisfied, ( ˆ λbc λ ) = a' ( θ) + op(), () ( ˆ λn λ ) = c' ( θ) + op(), and ( ˆ λ R λ) = d' ω( ϑ) + op (), where a ', c ' and d ' are he firs rows of he herefore, he second es can be done as follows: A, C and D. i) If he small σ is acceped, we compare he BC MLE and he robus esimaor. he asympoic variance of ˆ λ ˆ λ ) is given by ( BC N a Ba + d' Fd a' E[ ( θ ) ω( )'] d and E ( θ ) ω( )'] is esimaed by ' ϑ [ ϑ [ ( ˆ ) ( ˆ θbc ω ϑbc )'] where ˆ ϑ ˆ, ˆ BC' = ( λbc βbc' ). We use he BC MLE if he i.i.d. assumpion is acceped, and he robus esimaor oherwise. ii) If he small σ assumpion is rejeced, we compare he N-esimaor and he robus esimaor. he asympoic variance of ˆ λ ˆ λ ) is given by ( N R c Bc + d' Hd c' E[ ( θ ) ω( )'] d. We use he N-esimaor he i.i.d. assumpion ' ϑ is acceped and use he robus esimaor oherwise. Noe ha he N-esimaor is no an efficien esimaor, we canno use a difference of wo variances in his case. 4. Daa and he summary of he 6 revision for caarac operaions 7

9 4. Daa In his sudy, we use daa from he Secion of Healh Care Economics of okyo Medical and Denal Universiy. he daa were colleced from 86 hospials in Japan from 5 o 7, from April o December of each year. For each paien, he DPC code, daes of hospializaion and discharge from he hospial, dae of birh, sex, placemen afer hospializaion, ICD- code for he principal disease, purpose of hospializaion, presence of concurren disease and he aending reamen if any, and medical paymen amouns (including DPC-based, fee-for-service, and oal paymens) were repored []. In Japan, in addiion o one-eye caarac operaions (in which a single eye is operaed on during a single period of hospializaion), wo-eye caarac operaions (in which boh eyes are operaed on during a single period of hospializaion) are also performed. I is o be expeced ha he wo-eye operaion would require a paien o say in he hospial for a longer period of ime. herefore, we considered paiens who underwen one-eye caarac operaions only (he DPC code for his procedure afer he 6 revision is xx97xx). o evaluae he effec of he 6 revision of he DPC/PDPS, we used a daa se obained from 5 DPC hospials (Hp -5where one-eye caarac operaions were performed boh before (5) and afer he revision (6 and 7, hereafer 6-7) and he number of paiens was more han in each period. For says over days, he per diem paymen was deermined hrough he convenional fee-for-service sysem in any case. herefore, we only analyzed he daa of paiens whose says were less han or equal o days. A oal of 4,9 paiens were analyzed,,5 in 5 and 3,4 in 6-7. In 5, he ALOS was 4.36 days, he median was 4. days, he sandard deviaion was.6 days, he skewness was.94and he kurosis was 3.78 for all,5 paiens. he maximum ALOS by hospial was 6.57 days (Hp ), and he minimum was. days (Hp 5). he maximum was abou 3. imes larger han he minimum, and here were large differences among hospials. In 6-7he ALOS was 4.8 days, he median was 4. days, he sandard deviaion was.8 days, he skewness was.96 and he kurosis was 5.8 for all 3,4 paiens. he maximum ALOS by hospial was 5.87 days (Hp 9), and he minimum was.4 days (Hp 5). he skewness and kurosis values were large in some hospials. he large values imply ha here were paiens who sayed in a hospial for long periods of ime. 4. Summary of he revision for caarac operaions he 6 revision of he DPC/PDPS conained a major change for caarac operaions. Before he revision, differen DPC codes were assigned depending on he 8

10 presence of concurren diseases (wihou concurren diseases 3xx; wih concurren diseases 3xx, and he medical paymens differed accordingly. Afer he revision, caarac operaions were caegorized under jus one DPC code xx97xx independen of he presence of concurren diseases. Furhermore, Periods I and II and he Specific Hospializaion Period were shorened, and he per diem inclusive paymens were revised as well. he per diem inclusive paymen in 5 for paiens wihou concurren diseases was,59 poins up o he hird day of hospializaion,,855 poins for he 4h-6h days, and,577 poins for he 7h-h days. For hose wih concurren diseases, he per diem inclusive paymen was,69 poins up o he hird day,, poins for he 4h-7h days, and,7 poins for he 8h-h days. Afer he revision, he per diem inclusive paymen became,48 poins up o he second day,,787 poins for he 3rd-4h days, and,59 poins for he 5h-8h days for all caarac paiens independen of he presence of concurren diseases. In 5, he inclusive paymens for 7 days of hospializaion for paiens wihou and wih concurren diseases were 4,669 and 5,875 poins, respecively. On he oher hand, he inclusive paymen became,967 poins afer he revision. he inclusive paymens were reduced by,7 poins (.6%) wihou concurren diseases and by,98 poins (8.3%) wih concurren diseases. 5. Resuls of esimaion When we analyzed he LOS, i was necessary o consider he characerisics of he paiens and he ypes of principal disease as he explanaory variables. For he gender of paiens, we used a Female Dummy (: female, : oherwise). he numbers of male and female paiens were,638 and,38, respecively. As a paien becomes older, he LOS ends o increase. herefore, we used Age (he age of he paien) as an explanaory variable. he average and sandard deviaion of he age variable were 73.6 and.5, respecively. o analyze he impac of seasonal climae, we used a Winer Dummy (: winer, : oherwise). he number of paiens reaed in winer was 4. he oher variables represening he characerisics of he paiens were: Concurren (number of concurren diseases), Complicaion (number of complicaions), Urgen Dummy: urgen hospializaion, : oherwise, and Oher Hospial Dummy (: he paien was discharged o anoher hospial, : oherwise). A oal of 359 paiens had concurren diseases. he average number of concurren diseases for hese paiens was.84. A oal of paiens had complicaions, and he average number of complicaions was.7. he numbers of paiens who underwen urgen hospializaion and were discharged o oher hospials were 5 and, respecively. 9

11 Principal Disease Dummies based on he ICD- codes were used o analyze he effecs of principal diseases. he base of he dummy variables was H5. (senile incipien caarac). he number of paiens wih H5. was,75he number wih H5. (senile nuclear caarac) was 77he number wih H5. (senile caarac, morgagnian ype) was 9he number wih H5.8 (oher senile caarac) was 48he number wih H6. (infanile and juvenile caarac) was 3he number wih H6.8 (oher specified caarac) was 5and he number wih H6.9 (unspecified caarac) was,664. Foureen Hospial Dummies (: Hp k, : oherwise) were used o represen he influence of he hospial. he base of he hospial dummy variables was Hp4, where he number of paiens was larges. o analyze he impac of he 6 revision of he DPC/PDPS, which is he main purpose of his sudy, a 6-7 Dummy: 6-7; oherwisewas used. Afer he revision, he exisence of concurren diseases no longer affeced he inclusive paymen. o analyze his effec, we added he produc of he 6-7 Dummy and Concurren o he explanaory variables. he value of he empirical hazard funcion (=number of paiens leaving on he -h day/ number of paiens saying ha morning) showed wo peaks as shown in Figure, one on he fifh day and he oher on he eighh day (one week afer he hospializaion). herefore, we added he Day 8 Dummy (: LOS is more or equal o 8 days; : oherwise). he ransformaion parameer ends o be underesimaed when he LOS consiss of a mixure of wo differen disribuions, and his variable was excluded. Since he expeced signs of he esimaors were posiive for Concurren and Complicaion and negaive for 6-7 Dummy and (6-7 Dummy Concurren), he one-ailed es was employed for hese variables. he wo-ailed es is used for oher variables. Some hospials were preparaion hospials in some pars of he sample period. We also added he Preparaion Dummy (: preparaion hospial; oherwise). hus x ' β of Equaion () becomes ij x ij' β = β + Female Dummy+ β Age + β 3 Winer Dummy+ β 4 Concurren () + β 5 Complicaion + β 6 Urgen Dummy + β 7 Oher Hospial Dummy + β Dummy + β 9 (6-7 Dummy Concurren) + β j-h Principal Disease Dummy + j β Hp k k Dummy + β Day 8 Dummy + β m Preparaion Dummy ables, 3 and 4 presen he resuls of he esimaion by he BC MLE, N-esimaor and robus esimaors. he esimaes of he ransformaion

12 parameers were λˆ BC =.54, λˆ N =.4654, and λˆ R =.6634 which were significanly smaller han.; his resul implied ha some paiens remained in he hospial for a long period of ime. We firs esed he small σ assumpion. We obained d ˆ / n =.44. Hence, he value of = ( λ λ )/ dˆ was herefore, he small σ N BC assumpion was rejeced a he % significance level in eiher case. I hen esed he i.i.d. assumpion. he value of V ˆ λ ˆ λ ) was.9 and ( N BC = ˆ λ ˆ λ )/ V( λ λ ) = 6.88, so he i.i.d. assumpion was also rejeced a he ( R N R N % significance level, indicaing ha he BC MLE could no be used in his sudy. he remainder of his paper is hus an analysis of he resuls of he robus esimaor. he esimae of λˆ R was significanly smaller han.; ha implies some paiens remained in he hospial for a long period of ime. he esimaes of he Female Dummy and Age were posiive and significan a he 5% and% level, respecively. ha implies ha he LOS becomes longer if a paien is female and he age becomes higher. he esimaes of Concurren and Complicaion were posiive bu no significan a he 5% level, so we did no admi he effecs of hese variables in his sudy. he esimaes of Winer, Urgen, and Oher Hospial Dummies were no significan a he 5% level. he esimaes of he H6. and H6.8 Dummies were posiive and significan a he % level. On he oher hand, he esimaes for he oher ypes of diseases were no significan a he 5% level. For he esimaes of he Hospial Dummies, he maximum was.34(hp), he minimum was -. (Hp5), and he difference beween he maximum and minimum values was.35 and was significanly large compared o he oher ypes of variables. his means ha here remained large differences among hospials even if he influence of facors such as paien characerisics and ypes of principal diseases was eliminaed. he esimae of he Day 8 Dummy was.43, and is -value was his means ha many paiens lef he hospial afer one-week hospializaion. hese facs imply ha i may be possible for some hospials o reduce he LOS hrough he inroducion of clinical pahs and he proper managemen of hospializaion schedules [6]. he -value of Preparaion Dummy was -.78 and he difference beween he DPC and preparaion hospials was no admied. he esimae of he 6-7 Dummy was negaive bu no significan a

13 he 5% level. However, he esimae of he produc of he 6-7 Dummy and Concurren was negaive and a he 5% level. his means ha he 6 revision seems o have had he expeced effec on he LOS for he presence of concurren diseases. 6. Conclusion In his paper, we analyzed he effec of he 6 revision of he DPC/PDPS on he LOS and medical paymens for single-eye caarac operaions (DPC caegory code ) in Japan using he BC model. he Hausman es for wheher we could use he BC MLE or no was used. We used he daa of 4,7 paiens colleced from 5 DPC hospials where caarac operaions were repored boh before and afer he revision and where more han paiens underwen he operaions in each period. We found ha boh small σ and i.i.d. assumpions were rejeced and concluded ha i is no proper o use he BC MLE for his daa se. We found ha gender and age affeced he LOS. As principal diseases, we found ha H6. and H6.8 were significan. he ALOSs were significanly differen among hospials, despie he fac ha he influence of paien characerisics was eliminaed. he esimae of he Day 8 Dummy was significan, and is value was much larger han hose of he oher variables. he esimae of he 6-7 Dummy and (6-7 Dummy) Concurren) was negaive and significan. he 6 revision seems o have had he expeced effec on he LOS for he presence of concurren diseases I migh have a significan impac on he medical paymen for he caarac operaions. he reducion in medical paymens resuled in a reducion of hospial income. For some hospials, he reducions were large, and hese hospials migh face financial difficulies as a resul of he revision. Paiens could face serious difficulies if hese hospials were o go bankrup. herefore, o improve he DPC/PDPS, we mus consider facors such as regional condiions [5], and we also need o perform he same analysis for oher diseases. hese are subjecs for fuure sudies. Appendix A: Proof of he consisency he proof of he consisency of he esimaor is given using a modificaion of Nawaa [9]. he following assumpions are made: Assumpion. {( x, u )} are independen bu no necessarily idenically disribued. he disribuion of u may depend on x. Assumpion. { u } follow disribuions in which he suppors are bounded from below; ha is, f ( u) = if u a for some a > where f (u) is he

14 probabiliy (densiy) funcion. For any, he following momen condiions are saisfied: (i) E ( u x ) =, (ii) ( 3 6 E u x ) =, and (iii) δ < E( u x ) < δ for some δ < δ <. < Assumpion 3. { x } are independen, and δ 3 < E( x ) < δ 4 for some δ < δ <. he disribuions of x } and he parameer space of β are < 3 4 { resriced so ha inf x ( λ x' β + ) > a λ and inf x, ϑ ( λ x' β + ) > c for some c > in he neighborhood of ϑ ' = ( λ, β' ). Assumpion 4. ω ( ϑ) ϑ' ϑ converges o a nonsingular marix F in probabiliy and E[ ω ( ϑ ) ω ( ϑo )'] converges o a nonsingular marix H. Assumpion 5. (i) and x z x x x ' converges o a nonsingular marix in probabiliy converges o a non-sochasic vecor in probabiliy, and (ii), x z, zx x ' and 3 3 z and heir firs derivaives converge o (vecors of) coninuous funcions of λ in probabiliy in he neighborhood of λ. When λ is given, β is uniquely esimaed by he leas-squares mehod. Le ˆ β ( λ) be he esimaor. Le 3 h ( λ ) = M{ λ, ˆ( β λ)} = { z x '( xsxs') ( xszs)}. () Under Assumpion 5, s ˆ P β ( λ) && β( λ) plim( x ') ( x x z ). (3) from heorem 3..7 of Amemiya []. herefore, h( λ) plim M{ λ, β( λ)} = plim { z x ' β( λ)} exiss and a coninuous funcion of λ in he neighborhood of λ. From heorema 3..5 of Amemiya [], s 3 (4)

15 Le h '( λ) = plim h ( λ) = h( λ). (5) h '( λ) = dh dλ. hen / M [ { λ, ˆ( β λ)} M { λ, ˆ( β λ)} ˆ( β λ) + ] λ ˆ( β λ) λ = 3 { z (6) % λ λ x ' ˆ( β λ)} # { y log( y ) z } x '( xsxs ') {log( ys ) ys zs} xs ] if λ, and $ λ s s λ 3 h '( λ) = lim h '( λ) = { z x ' ˆ( β λ)} [{log( y )} x '( xsxs') {log( ys)} xs ] λ s s if λ =. herefore, dh ( λ) / dλ converges o h '( λ) = dh / dλ, which is a coninuous funcion λ, in he neighborhood of λ under Assumpion 5. When λ = λ, he model becomes an ordinary regression model and ˆ( β λ ) is consisen. Hense 3 h( λ ) = plim G ( θ) = lim u. (7) Since E ( 3 ) =, we ge u h ( λ ) =, (8) by heorem 3.3. of Amemiya []. Because h (λ) and h '( λ) are coninuous funcions of λ a λ =, we can rea he λ = case he same as he λ case. From Assumpion 5, h '( λ) is coninuous in he neighborhood of λ and h ( λ ) ' does no become zero excep in very special cases. Consequenly, we can assume ha h '( λ), and ha here exiss δ > such ha sign h'( λ)} = sign{ h'( λ )} 4 { and h '( λ) γ h'( λ ) > if λ [ λ δ, λ + δ ]. By he mean value heorem, for any ε (, δ ), * ** h ( λ + ε) = h( λ + ε) h( λ ) = '( λ ) ε and h( λ ε) = h( λ ε) h( λ ) = '( λ ) ε (9) h * h * * where λ and λ are values in λ ε, λ + ]. herefore, [ ε sign h( λ ε)} sign{ h( λ + )}, h ( λ ε) > γε, and h ( λ + ε) > γε. () { ε P P Since h ( λ ε ) $ $ h( λ ε ) and h ( λ + ε ) ## h( λ + ε ),

16 P [ sign{ h ( λ ε )} sign{ h ( λ + ε)}, h ( λ ε ) >, and h ( λ + ε ) > ]. () Here, h (λ) is a coninuous funcion of λ in he neighborhood of λ. From he inermediae value heorem, h ( λ) = for some λ λ ε, λ + ] if [ ε sign h ( λ ε)} sign{ h ( λ + )}, h ( λ ε) > and h ( λ + ε ) >. herefore, { ε P[ here exiss λˆ such ha h ( ˆ) λ = and ˆ λ [ λ ε, λ + ] ]. () ε Since () holds for any ε (, δ ), h ( λ) = has a consisen roo of λ. Since ˆ β( ˆ λ) is obained by he leas-squares mehod, i is a consisen esimaor when ˆ λ "" P λ. Hence, here exiss a consisen roo among he roos of (7). Appendix B: Proof of he asympoic disribuion Since we ge where ' M ( ϑ) $ ω( ϑ) = ω ( ϑ) = %, ( ) " % ψ ϑ & "# 5 (3) ˆ ˆ ω ϑ ϑ ) = [ * ] ( ϑ ), (4) ϑ ϑ' ( R * ϑ is some value beween ϑˆ and ϑ. Here, 3 & u # ω ( ϑ ) = $!. (5) % xu " herefore, E[ ω ( ϑ )] =. Since he variables { ω ( ϑ )} are independen and he Lindberg condiion is saisfied under Assumpions, 3 and 4 we obain ω ( ϑ ) N(, H), (6) from heorem in Amemiya []. Since & 3 # $ ( z x ' β ) { z log( y ) z} 3 ( z x ' β ) x ' λ! ω / ϑ = $!, (7) $ x! { z log( y ) z} x x ' $ % λ!"

17 ω( ϑ) ϑ' ϑ P *!! F, from heorem 4..4 in Amemiya []. From heorem 4..3 in Amemiya [], he asympoic disribuion of ϑˆ R is given by Equaion (8). Acknowledgemens: his sudy was suppored by a Gran-in-Aid for Scienific Research Analyses of Japanese Medical Informaion and Policy Using Large-scale Individual-level Survey Daa (No. 436 and No ) of he Japan Sociey of Science. References []. Amemiya, Advanced Economerics, Harvard Universiy Press, Cambridge, MA, 985. [] P. J. Bickel, K. A. Doksum, An analysis of ransformaions revisied, J. Am. Sa. Assoc. 76 (98) [3] G.E.P. Box, D.R. Cox, An analysis of ransformaions, J. Roy. Sais. Soc. Ser. B 6 (964) -5. [4] DPC Evaluaion Division, Cenral Social Insurance Medical Council, Heisei 4 nendo kaie ni mukea DPC seido (DPC/PDPS) no aiou ni suie (Concerning he seps for he revision of he DPC Sysem (DPC/PDPS)), (in Japanese). [5] DPC Evaluaion Division, Cenral Social Insurance Medical Council. DPC aishou hyouin Junbi hyouin no gennjouni suie, (Curren siuaions of DPC hospials and preparing hospials), 3 (in Japanese). [6] J. Hausman, Specificaion es in economerics, Economerica 46 (978) 5-7. [7] Minisry of Healh, Labour and Welfare, Paien Survey 6, 8. [8] K. Nawaa, A new esimaor of he Box-Cox ransformaion model using momen condiions, Econ. Bull. 33 (3) [9] K. Nawaa, Robus esimaion based on he firs- and hird-momen resricions of he power ransformaion model, IPRC Working Paper No. 8, Universiy of okyo, 3. hp://ipr-cr..u-okyo.ac.jp/jp/libraries/dp/dp8.pdf. [] K. Nawaa, A new es for he Box-Cox ransformaion model, Econ. Bull. 34 (4) [] K. Nawaa, K., M. Ii, H. oyama,. akahashi, Evaluaion of he Inclusive 6

18 Paymen Sysem Based on he Diagnosis Procedure Combinaion wih respec o Caarac Operaions in Japan, Healh (9) [] K. Nawaa, K. Kawabuchi, Bekijo henkan moderu ni yoru shinryou 6 nendo houshuu kaiei ni omonau DPC minaoshi no zaiin nissuu heno eikyou no bunseki (An analysis of he 6 DPC revision associaed wih he reform of he medical paymens on he lengh of hospial say for caarac surgeries), Iryo keizai kenkyu (Jpn. J. Healh Econ. Policy) (3) 8-3 (in Japanese). [3] K. Nawaa, M. McAleer, he maximum number of parameers for he Hausman es when he esimaors are from differen ses of equaions, Econ. Le. 3 (4) [4] M. H. Showaler, A Mone Carlo invesigaion of he Box-Cox model and a nonlinear leas squares alernaive, Rev. Econ. Sa. 76 (994) [5] P. Sivey, he effec of waiing ime and disance on hospial choice for English Caarac Paiens, Healh Econ., [6] J. M. H. Vissers, J. D. Van Der Bij, R. J. Kusers, oward decision suppor for he waiing liss: an operaions managemen view, Healh Care Manag. Sci. 4 ()

19 able. Numbers of hospials and beds by hospial size * ** * ** Source: DPC Evaluaion Division (3). *: As of April 3. **: survey daa. 8

20 able. Resuls of esimaion (BC MLE) λ 9

21 able 3. Resuls of esimaion (N-Esimaor) ** * ** ** ** ** ** ** ** ** + ** ** ** ** ** ** ** ** ** λ **.

22 able 4. Resuls of esimaion (Robus esimaor) *: significan a he 5% level, +: significan a he 5% level (one-ailed es), **: significan a he % level

23