Sample Size Calculation Based on the Semiparametric Analysis of Short-term and Long-term Hazard Ratios. Yi Wang

Transcription

1 Sample Sze Calculaton Based on the Semparametrc Analyss of Short-term and Long-term Hazard Ratos Y Wang Submtted n partal fulfllment of the requrements for the degree of Doctor of Phlosophy under the Executve Commttee of the Graduate School of Arts and Scences COLUMBIA UNIVERSITY 23

2 c 23 Y Wang All Rghts Reserved

3 ABSTRACT Sample Sze Calculaton Based on the Semparametrc Analyss of Short-term and Long-term Hazard Ratos Y Wang We derve sample sze formulae for survval data wth non-proportonal hazard functons under both fxed and contguous alternatves. Sample sze determnaton has been wdely dscussed n lterature for studes wth falure-tme endponts. Many researchers have developed methods wth the assumpton of proportonal hazards under contguous alternatves. Wthout covarate adjustment, the logrank test statstc s often used for the sample sze and power calculaton. Wth covarate adjustment, the approaches are often based on the score test statstc for the Cox proportonal hazards model. Such methods, however, are napproprate when the proportonal hazards assumpton s volated. We develop methods to calculate the sample sze based on the semparametrc analyss of short-term and long-term hazard ratos. The methods are bult on a semparametrc model by Yang and Prentce 25). The model accommodates a wde range of patterns of hazard ratos, and ncludes the Cox proportonal hazards model and the proportonal odds model as ts specal cases. Therefore, the proposed methods can be used for survval data wth proportonal or non-proportonal hazard functons. In partcular, the sample sze formula by Schoenfeld 983) and Hseh and Lavor 2) can be obtaned as a specal case of our methods under contguous alternatves. KEY WORDS: Accrual and follow-up; Contguous alternatves; Cox model; Crossng

4 hazards; Fxed alternatve; Non-proportonal hazard functons; Sample sze; Shortterm and long-term hazard ratos; Survval analyss.

5 Table of Contents Table of Contents Introducton 2 General procedure of sample sze calculaton 7 2. Example General procedure of sample sze and power calculatons The fxed alternatve and the contguous alternatve hypotheses Sample sze calculaton for the Cox proportonal hazards model 6 3. Notatons and assumptons Model specfcaton Sample sze formula under fxed alternatve Sample sze formula under contguous alternatves Sample sze calculaton wth Yang and Prentce s semparametrc model Notatons and Model specfcaton Parameter estmaton Sample sze formula under fxed alternatve Sample sze formula under contguous alternatves Smulaton studes

6 4.5. Smulaton studes to evaluate sample sze formula derved under contguous alternatves Smulaton studes to evaluate sample sze formula derved under fxed alternatve Accrual and follow-up tmes n sample sze calculaton Accrual and followup n sample sze calculaton Example Dscusson Concludng remarks Future work Proofs Proofs of the theorems and corollares n Chapter Proof of theorem Proof of Corollay Proof of Theorem Proof of Corollary Proofs of the theorems n Chapter Lemma and proof Proof of Theorem 4.3. ) Proof of Theorem Bblography 99

7 Lst of Fgures. Kaplan-Meer estmates for the VA lung cancer data Proportonal hazards: Cox model γ = γ 2 = γ) Non-proportonal hazards: proportonal odds model γ 2 = ) Non-proportonal hazards: long-term effect model γ = ) Non-proportonal hazards: crossng hazards γ < and γ 2 >, or, γ > and γ 2 < ) Sample sze for accrual and follow-ups up to 3 months Sample sze for 3 months accrual a = 3) Sample sze for 2 months follow-up f = 2)

8 Lst of Tables 2. Comparson of η and z α/2 + z β ) 2 for commonly assumed α s and β s. 4. Values of η for dfferent α s and β s Emprcal power of calculated sample sze for α =.5, β =.9 and θ =, ) Emprcal power of calculated sample sze for α =.5, β =.9 and θ =, ) Emprcal power of calculated sample sze for α =.5, β =.9 and θ =.4,.4) Emprcal power of calculated sample sze for the Cox model. α =.5, β =.9 and θ = Emprcal power of calculated sample sze for the Cox model. α =.5, β =.9 and θ = Emprcal power of calculated sample sze for the Cox model. α =.5, β =.9 and θ = Emprcal power of calculated sample sze for the Cox model. α =.5, β =.9 and θ = Dfferent accrual dstrbutons as n Mak 26) and Wang et al. 22). 6 v

9 Acknowledgments I would lke to express my deepest grattude to my advsor, Dr. Zhezhen Jn, for hs supervson and support. I would also lke to thank Dr. Bn Cheng, Dr. Robert Taub, Dr. We-Yann Tsa and Dr. Anta Wang for servng on my dssertaton commttee. Last but not least, many thanks to my frends Huahou Chen, We Xong, Wenfe Zhang and Zqang Zhao, who have encouraged and supported me throughout the entre process. v

10 To my famly v

11 CHAPTER. INTRODUCTION Chapter Introducton Sample sze determnaton s mportant n clncal studes, especally at the desgn stage of a tral when researchers want to address some scentfc hypotheses. Sample sze calculaton procedure conssts of two mportant elements: hypotheses of nterest and test statstc. For the sample sze calculaton n a study, the null and alternatve hypotheses need to be determned frst. Then, a proper test statstc must be dentfed along wth the dstrbutons of the test statstc under the null and alternatve hypotheses. Sample sze can then be evaluated wth the pre-specfed type I error, power, effect sze, and desgn effects). When the exact dstrbutons of the test s- tatstc are not avalable, the asymptotc dstrbutons are often used nstead. Type I error, power, sample sze, effect sze, and desgn effects) are related to one another for the pre-specfed hypotheses of nterest and test statstc. Therefore, power can be calculated f sample sze, type I error, effect sze and desgn effects) are known. Sample sze and power calculaton follows bascally the same procedure. Revews on the general procedure for sample sze determnaton and power analyss n clncal trals are gven by many nvestgators e.g., Lachn 98 [25]; Donner 984 [2]; Dupont and Plummer 99 []). The general procedure s revewed and dscussed n detal n Chapter 2. Although sample sze determnaton s the focus n ths dssertaton, power analyss for the same type of studes can be conducted n a smlar manner.

12 CHAPTER. INTRODUCTION 2 Sample sze formulae can be dfferent wth dfferent choce of ether the hypotheses of nterest or the test statstc. In sample sze calculaton, two types of alternatve hypotheses are usually consdered for the hypotheses of nterest: fxed alternatve and contguous alternatves. An ntroducton to these alternatves can be found n Chapter 2. For any specfc hypotheses of nterest, the choce of the test statstcs also makes a dfference when sample sze s calculated for a study. There are model-based and non model-based test statstcs. The test statstc should be model-based when some non-bnary effects) s are) of nterest, and/or there s a need for covarate adjustment. When only a categorcal effect often bnary), for example, treatment effect, s of nterest, one can calculate the sample sze wth a non model-based test statstc. The methods for sample sze calculaton n survval analyss have been dscussed extensvely n the lterature. Many authors have focused on developng methods for survval data wth treatment as the effect of nterest e.g., Halpern et al. 968 [2]; Pasternack and Glbert 97 [38]; Pasternack 972 [37]; George and Desu 974 [7]; Palta and McHugh 979 [35]; Palta and McHugh 98 [36]; Wu et al. 98 [5]; Lachn 98 [25]; Rubnsten et al. 98 [4]; Schoenfeld 98 [4]; Makuch and Smon 982 [32]; Schoenfeld and Rchter 982 [43]; Freedman 982 [5]; Schoenfeld 983 [42]; Gal 985 [6]; Palta and Amn 985 [34]; Lachn and Foulkes 986 [26]; Lakatos 988 [28]; Gu and La 999 [9]; Chen et al. 2 [5]; Wang et al. 22 [49]). Collett 23) [9] has dscussed and revewed sample sze calculaton methods n survval analyss. Most of these methods focused on the smple two-sample problem and assumed proportonal hazards. In partcular, early works assumed exponental dstrbuton n survval tmes e.g., Pasternack and Glbert 97 [38]; Pasternack 972 [37]; George and Desu 974 [7]; Palta and McHugh 98 [36]; Lachn 98 [25]; Rubnsten et al. 98 [4]; Makuch and Smon 982 [32]; Schoenfeld and Rchter 982 [43]; Lachn and Foulkes 986 [26]). Among these methods, there are some that are not lmted to two-sample problem or to the assumpton of proportonal hazards.

13 CHAPTER. INTRODUCTION 3 For example, Makuch and Smon 982) [32] provded the sample sze requrement for more than two treatment groups; Lakatos 988) [28] and Lakatos and Lan 992) [29] derved methods that can be used for non-proportonal hazard functons a SAS macro for the approaches are gven by Shh 995 [45]); Wu et al. 98) [5] also employed an approach that allows for tme-dependent dropout and event rates as an extenson of the approach by Halpern et al. 968) [2]; Gu and La 999) [9] derved a formula for clncal studes wth nterm analyses; Chen et al. 2) [5] dscussed sample sze determnaton n jont modelng of longtudnal and survval data; Wang et al. 22) [49] derved a formula based on the cure rate proportonal hazards model, whch ncludes the Cox proportonal hazards model as a specal case. Some researchers developed approaches that can be used for non-bnary covarates). For example, Zhen and Murphy 994) [53] presented ther approach based on the exponental model; Hseh and Lavor 2) [2] derved a method based on the Cox proportonal hazards model. The sample sze formula by Schoenfeld 983) [42] s commonly used n practce. The treatment effect s handled as a bnary covarate n the Cox proportonal hazards model. The formula by Schoenfeld 983) [42] based on the score test statstc for the Cox proportonal hazards model) s the same as that of Schoenfeld 98) [4] based on the logrank test statstc) when there are no tes consdered. Ths s because the logrank test statstc s actually the same as the score test statstc based on the partal lkelhood for the Cox proportonal hazards model when the only covarate s the treatment ndcator under no tes. Many authors derved the same formula under dfferent assumptons, among whch there are Schoenfeld and Rchter 982 [43] and Collett 23) [9]. Hseh and Lavor 2) [2] extended Schoenfeld s result 983) [42] to the case of non-bnary covarate. All of these methods are derved wth the proportonal hazards assumpton under local alternatves. Therefore, the methods are napproprate when the proportonal hazards assumpton s volated. Even f the proportonal hazards assumpton holds, the valdty of the formula may

14 CHAPTER. INTRODUCTION 4 be questonable when the results under the fxed alternatve are more desred. Fgure. presents the Kaplan-Meer estmates for the VA lung cancer tral Kalbflesch and Prentce 22 [23]). Ths s a classc example of possble volaton of the proportonal hazards assumpton. If there s crossng n hazards functons, there must be crossng n survval functons. In practce, people often check the Kaplan-Meer curves to see whether the survval functons cross. When we observe Kaplan-Meer curves nstead of the true survvals, questons are rased as to whether the underlyng survvals are crossed and whether the underlyng hazards are crossed. Approprate methods for crossng hazards can help dagnose the problem. Reasons behnd crossng can be complcated. Conventonal methods such as the logrank test and the score test for the Cox proportonal hazards model would perform poorly n the presence of crossng hazards. Consequently, the sample sze calculaton based on these test statstcs wll be napproprate. Fgure.: Kaplan-Meer estmates for the VA lung cancer data. Survval Functon Treatment Control Tme days) We develop methods to calculate the sample sze for survval data wth non-

15 CHAPTER. INTRODUCTION 5 proportonal hazard functons. Our methods are based on the pseudo score test for a semparametrc model by Yang and Prentce 25) [5]. Ths semparametrc model accommodates a wde range of hazard rato patterns. It models the hazard rato that changes monotoncally over tme. For a two-sample problem, ths model can be used for survval data wth proportonal or non-proportonal even crossed) hazard functons. There are three submodels of specfc nterest: the Cox proportonal hazards model, the proportonal odds model and the long-term effect model. These submodels are dscussed n detal as specal cases of our methods. We obtan the sample sze formulae based on Yang and Prentce s method under dfferent alternatves fxed and contguous alternatves). Sample sze calculaton for the Cox model s revewed n ths dssertaton, wth generalzaton to the result under the fxed alternatve. It can be shown that our sample sze formula for the semparametrc short-term and long-term hazard ratos model reduces to the formula developed by Schoenfeld 983) [42] and Hseh and Lavor 2) [2] under contguous alternatves for the Cox proportonal hazards model. Accrual and follow-up tmes can be ncorporated n sample sze calculaton. Some authors ncluded accrual and follow-up tmes n ther sample sze formulae e.g., Pasternack and Glbert 97 [38]; Pasternack 972 [37]; George and Desu 974 [7]; Lachn 98 [25]; Rubnsten et al. 98 [4]; Schoenfeld and Rchter 982 [43]; Schoenfeld 983 [42]; Donner 984 [2]; Gal 985 [6]; Lachn and Foulkes 986 [26]; Lakatos 986 [27]; Lakatos 988 [28]; Wang et al. 22 [49]). We wll adopt the assumptons n Wang et al. 22) [49] for accrual and follow-up tmes. The sample sze wth accrual and follow-ups are derved n a smlar way. Detals wll be dscussed n Chapter 5. Two mportant technques to prove the results of ths work are countng processes and emprcal processes. These are useful tools to evalulate the asymptotc propertes. Some books dscussed the applcaton of countng processes n survval analyss e.g., Flemng and Harrngton 99 [4]; Andersen et al. 993 []; Kalbflesch and Prentce

16 CHAPTER. INTRODUCTION 6 22 [23]). The large sample propertes of emprcal processes were presented n Van der Vaart 998) [47] and Van der Vaart and Wellner 996) [48]. Although dfferent technques may be used to derve the results, countng processes and emprcal processes are powerful tools n survval analyss. We wll use these technques n our dervatons. Ths dssertaton s structured as follows. In Chapter 2, the general procedure of sample sze estmaton s llustrated wth examples. The sample sze formulae are derved for the Cox proportonal hazards model under both fxed and contguous alternatves n Chapter 3. The formula by Schoenfeld 983) [42] and Hseh and Lavor 2) [2] s shown to be a specal case of our formula under contguous alternatves. In Chapter 4, the sample sze formulae are developed based on the semparametrc model by Yang and Prentce 25) [5] for non-proportonal hazard functons. Under certan condtons, the sample sze formula by Schoenfeld 983) [42] and Hseh and Lavor 2) [2] agan turns out to be a specal case of our formula. Smulaton studes are summarzed n the chapter. In Chapter 5, the proposed methods are further generalzed when accrual and follow-up tmes are ncorporated. Chapter 6 gves a summary of the fndngs and dscussons, along wth possble alternatve approaches for sample sze determnaton wth falure-tme endpont. All the proofs are presented n Chapter 7.

17 CHAPTER 2. GENERAL PROCEDURE OF SAMPLE SIZE CALCULATION 7 Chapter 2 General procedure of sample sze calculaton Ths chapter begns wth a smple example to llustrate how the sample sze can be calculated. Smlar dscussons appeared n many works, ncludng Lachn 98) [25], Donner 984) [2] and Dupont and Plummer 99) []. The general procedures for sample sze calculaton and power analyss are summarzed n Secton 2.2. It s mportant to specfy the hypotheses of nterest and dentfy a proper test statstc n sample sze and power calculaton. In ths dssertaton, sample sze formulae are derved under both fxed and contguous alternatves. We gve a bref ntroducton to the two types of alternatve hypotheses fxed and contguous alternatves). 2. Example Let {W,..., W n } be a random sample of sze n, where W s are ndependent and dentcally dstrbuted..d.) random varables wth unknown mean µ and known fnte varance σ 2, for =,..., n. We consder the null hypothess H : µ = µ versus the alternatve hypothess H : µ = µ, where µ, µ R are known and µ µ. In ths example, the Wald test statstc s used to calculate the sample sze. We assume

18 CHAPTER 2. GENERAL PROCEDURE OF SAMPLE SIZE CALCULATION 8 that the type I error s α and the desred power s β n the study. We present the sample sze formula for the followng two cases: random normal sample and random non-normal sample. The major dfference between the two s that the asymptotc dstrbutons, nstead of the exact dstrbutons of the test statstc, are used for the calculaton n the case of non-normal random sample. Case : Random normal sample If W s are..d. random varables from a normal dstrbuton wth unknown mean µ and known fnte varance σ 2, for =,..., n, the Wald test statstc can be used to test the hypotheses of nterest. The test statstc s T w n = n Wn µ ), 2.) σ where W n = n = W /n s the sample mean of {W,..., W n }. The dstrbutons of the test statstc T w n alternatve hypotheses: Under H : µ = µ, 2.) can be derved under the null and T w n N, ). 2.2) Under H : µ = µ, nµ Tn w µ ) σ N, ). 2.3) The results can be used to calculate sample sze wth pre-specfed type I error α, power β, effect sze µ µ, and desgn effect σ 2. By 2.2) and the defnton of type I error, Pr H T w n > c ) = α, where the crtc value c s equal to z α/2 for gven type I error α, wth z α/2 beng the upper α/2 th percentle of the standard normal dstrbuton.

19 CHAPTER 2. GENERAL PROCEDURE OF SAMPLE SIZE CALCULATION 9 By the defnton of power, Pr H T w n > c ) = β, whch mples Pr H T w n > c ) + Pr H T w n < c ) = β. 2.4) For small α and β, one of the tems on the left-hand sde of equaton 2.4) s small and can thus be omtted. Then, sample sze can be calculated based on the prevous results. If µ > µ, t follows from 2.4) that Pr H T w n < c ) = β. Wth some algebra, the followng equaton can be obtaned Pr H T w nµ µ ) n σ nµ µ )) < c = β. σ By 2.3), z β can be approxmated: nµ µ ) c σ = z β. The sample sze s obtaned by substtutng z α/2 for c n the above equaton. n = z α/2 + z β ) 2 µ µ ) 2 /σ 2, 2.5) where z β s the upper β th percentle of the standard normal dstrbuton. If µ < µ, t follows from 2.4) that Pr H T w n > c ) = β. Wth some algebra, the followng equaton can be obtaned Pr H T w nµ µ ) n σ nµ µ )) > c = β. σ

20 CHAPTER 2. GENERAL PROCEDURE OF SAMPLE SIZE CALCULATION By 2.3), z β can be approxmated: nµ µ ) c σ = z β. The sample sze s n = z α/2 z β ) 2 µ µ ) 2 /σ 2, 2.6) where z β s the upper β) th percentle of the standard normal dstrbuton. Remark 2... In fact, 2.5) and 2.6) are equvalent because z β = z β. The sample sze formula s regardless of the relatonshp between µ and µ. n = z α/2 + z β ) 2 µ µ ) 2 /σ 2 2.7) Remark An alternatve way to calculate the sample sze s to consder the dstrbutons of T w n ) 2. The advantage of ths method s to use all the terms nvolved, wthout the omsson we made of the term on the left-hand sde of equaton 2.4). Under H : µ = µ, T w n ) 2 χ 2, 2.8) where χ 2 s the Ch-squared dstrbuton wth degree of freedom. Under H : µ = µ, T w n ) 2 χ 2 η ), 2.9) where χ 2 η ) s the non-central Ch-squared dstrbuton wth degree of freedom and noncentralty parameter η. The noncentralty parameter η can be approxmated by η n = nµ µ ) 2. The sample sze formula can be derved n a smlar manner wthout σ 2 makng the approxmaton n 2.4). Followng the procedure shown n ths secton, the sample sze s gven by n = η µ µ ) 2 /σ 2, 2.)

21 CHAPTER 2. GENERAL PROCEDURE OF SAMPLE SIZE CALCULATION where η s derved such that χ 2, α = χ 2, β η ). χ 2, α s the upper α th percentle of the Ch-squared dstrbuton wth degree of freedom, and χ 2, β η ) s the upper β) th percentle of the non-central Ch-squared dstrbuton wth degree of freedom and noncentralty parameter η. Note that 2.7) and 2.) dffer only n the numerator. If the approxmaton n 2.4) s approprate, z α/2 + z β ) 2 should be close to η. We compare these values for small α s and β s n Table 2.. The two values are almost dentcal for dfferent small values of α and β. Thus, we conclude that the sample sze formula 2.7) wth normal approxmaton) can be used nstead of 2.). Table 2.: Comparson of η and z α/2 + z β ) 2 for commonly assumed α s and β s. α β η z α/2 + z β ) Remark When the varance σ 2 s unknown, the sample varance ˆσ 2 can be used nstead n the sample sze calculaton. Consequently, σ s substtuted wth ˆσ n the test statstc Tn w 2.). The sample sze formula has the form of 2.7) wth σ beng replaced by ˆσ, and z α/2 and z β beng replaced by t α/2 n ) and t β n ), the upper α/2th and βth percentles of t-dstrbuton wth n degrees of freedom. Case 2: Random non-normal sample Suppose that W s are..d. random varables from a non-normal dstrbuton wth unknown mean µ and known fnte varance σ 2, for =,..., n. The sample sze

22 CHAPTER 2. GENERAL PROCEDURE OF SAMPLE SIZE CALCULATION 2 calculaton procedure remans largely the same as n Case. The dfference s that the asymptotc dstrbutons should be used nstead of the exact dstrbutons under the null and alternatve hypotheses. The test statstc n 2.) can stll be used n ths case. The asymptotc dstrbutons should dffer from 2.2) and 2.3) only n the asymptotc context;.e., the results are vald when n. Because the formula obtaned s based on the large sample theorem, the method may not be good for small sample sze. We present the asymptotc dstrbutons of the test statstc Tn w 2.) under the null and alternatve hypotheses: By the classc central lmt theorem, under H : µ = µ, T w n d N, ) as n. 2.) Under H : µ = µ, nµ Tn w µ ) σ For a gven type I error α, by defnton we have: d N, ) as n. 2.2) Pr H T w n > c ) = α, where the crtc value c s equal to z α/2. By the defnton of power, we derve the followng: Pr H T w n > c ) = β, whch mples Pr H T w n > c ) + Pr H T w n < c ) = β. 2.3) We adopt the same normal approxmaton method as n the random normal case. Wthout loss of generalty, we assume µ > µ. 2.3) can be smplfed as Pr H T w n < c ) = β.

23 CHAPTER 2. GENERAL PROCEDURE OF SAMPLE SIZE CALCULATION 3 It follows that Pr H T w nµ µ ) n σ nµ µ )) < c = β. σ By 2.2), nµ µ ) c σ = z β. The sample sze s obtaned by replacng c wth z α/2 n the above equaton. n = z α/2 + z β ) 2 µ µ ) 2 /σ ) Remark The sample sze formula 2.4) s derved based on the asymptotc dstrbutons of the test statstc 2.). As a result, t may not be vald for small samples. Although 2.7) and 2.4) have the same expressons, the method needs to be used wth cauton when n s small. 2.2 General procedure of sample sze and power calculatons The example n the prevous secton llustrates the general procedure n sample sze calculaton. There are several mportant steps n sample sze calculaton. The key s to choose the approprate hypotheses of nterest and test statstc. Sample sze s related to a number of factors such as type I error, power, effect sze, and desgn effects). To make a connecton among these factors, we need to nvestgate the dstrbutons or asymptotc dstrbutons of the test statstc under the null and alternatve hypotheses. Specfcally, the sample sze calculaton procedure can be summarzed as follows. Sample sze calculaton procedure: Step. Specfy the hypotheses of nterest null and alternatve hypotheses).

24 CHAPTER 2. GENERAL PROCEDURE OF SAMPLE SIZE CALCULATION 4 Step 2. Choose a test statstc. Step 3. Derve the asymptotc) dstrbutons of the test statstc under the null and alternatve hypotheses. Step 4. Lnk the sample sze to type I error, power, effect sze, and desgn effects) by Step 3. Type I error, power, sample sze, effect sze, and desgn effects) are related. If only one of the elements s unknown, then t can be determned by the other quanttes. Power can be calculated n a smlar way as sample sze. In general, power can be calculated by the followng steps. Power calculaton procedure: Step. Specfy the hypotheses of nterest null and alternatve hypotheses). Step 2. Choose a test statstc for the hypotheses of nterest. Step 3. Derve the asymptotc) dstrbutons of the test statstc under the null and alternatve hypotheses. Step 4. The power can be lnked to type I error, sample sze, effect sze, and desgn effects) by Step 3. Agan, the two key components here are the hypotheses of nterest and the test statstc. Dfferent formulae may be derved f we choose dfferent alternatve hypothess or test statstc. In ths dssertaton, we derve methods under dfferent alternatve hypotheses. The followng secton ntroduces two commonly used alternatve hypotheses n sample sze calculaton and power analyss.

25 CHAPTER 2. GENERAL PROCEDURE OF SAMPLE SIZE CALCULATION The fxed alternatve and the contguous alternatve hypotheses We adopt the notatons n the example of Secton 2.. Although the example shows the dervaton of sample sze formula under the fxed alternatve hypothess, other alternatve hypotheses can also be consdered, especally when the random sample s non-normal and the asymptotc dstrbuton s dffcult to obtan under the fxed alternatve. For the null hypothess of nterest H : µ = µ, the followng alternatve hypotheses are often consdered n sample sze and power calculaton:. Fxed alternatve: H : µ = µ. 2. Contguous alternatves : H n : µ = µ n = µ + h n, where h R. When the class of contguous alternatves s consdered n sample sze determnaton, the asymptotc dstrbuton of the test statstc needs to be derved under the contguous alternatves. Then, the sample sze can be determned by lettng µ = µ n. Ths actually assumes a small effect sze.e., µ and µ are close). By settng µ = µ n, the sample sze n s a functon of a real-valued h and the effect sze µ µ. Ths s useful when the sample sze formula s derved based on the contguous alternatves. Although sample sze dervaton based on the fxed alternatve s often more desrable, many researchers use the methods under the contguous alternatves when the alternatve dstrbuton of the test statstc s dffcult to obtan. If the assumpton of the closeness of µ and µ s made, one should be cautous when usng the correspondng formula. It s dffcult to actually quantfy the adequacy of the closeness assumpton n practce. Some would use the formulae obtaned under the contguous alternatves wthout checkng the approprateness of the assumpton. In Chapter 3 and 4, we show n detal the dervatons of the sample sze formulae under these two types of alternatves for the Cox model and for the semparametrc model by Yang and Prentce 25) [5].

26 CHAPTER 3. SAMPLE SIZE CALCULATION FOR THE COX PROPORTIONAL HAZARDS MODEL 6 Chapter 3 Sample sze calculaton for the Cox proportonal hazards model 3. Notatons and assumptons In ths chapter, we generalze the sample sze calculaton methods for the Cox proportonal hazards model under the fxed and the contguous alternatve hypotheses. The test statstc used for the Cox model s the score statstc based on the partal lkelhood. The formula for the fxed alternatve s complcated. We propose a method that smplfes the calculaton under the fxed alternatve. Ths method s evaluated through smulaton studes n Chapter 4. It s shown that our formula for the contguous alternatves reduces to that derved by Schoenfeld 983) [42] and Hseh and Lavor 2) [2] under certan condtons. We defne notatons for rght-censored falure tme data n the followng. Let T be the survval tme, C be the censorng varable, X = mnt, C) be the observed tme and = IT C) be the event ndcator, where I ) s the ndcator functon takng value f the condton s satsfed, otherwse. For rght-censored survval data, the observed tme s ether the survval tme or the censorng tme, whchever occurs frst. For smplcty, only one covarate Z s consdered. Suppose that the observed data are

27 CHAPTER 3. SAMPLE SIZE CALCULATION FOR THE COX PROPORTIONAL HAZARDS MODEL 7 X, Z, ), for =,..., n. We denote f Z ) as the probablty densty functon pdf) or probablty mass functon pmf), F Z ) as the cumulatve dstrbuton functon cdf) and S Z ) as the survval functon, λ Z ) as the hazard functon and Λ Z ) as the cumulatve hazard functon of survval tme T. In a two-sample problem, wth Z beng bnary whch takes value for the control group and for the treatment group, S ) denotes the survval functon of the treatment group and S ) denotes that of the control group. Accordngly, λ ) and λ ) are the hazard functons of the treatment group and control group, respectvely. Let b a denote a,b]. Let N t) = IX t) be the countng process of the number of observed events on, t], and Y t) = IX t) be the at rsk process at t. If the th ndvdual s stll at rsk and has not yet faled at tme t, then Y t) =, for =,..., n and t, τ], where τ > s a fnte tme pont at whch the study ends. Let t denote the left-contnuous pont of t. Denote χ 2 k as the Ch-squared dstrbuton of k degrees of freedom, and χ2 k,ω as the upper ωth percentle of the Ch-squared dstrbuton χ 2 k. Also let χ2 k η) be the non-central Ch-squared dstrbuton of k degrees of freedom wth noncentralty parameter η, and χ 2 k,ω η) be the upper ωth percentle of the non-central Chsquared dstrbuton χ 2 k η). We consder the followng regularty condtons: A) Condtonng on Z, T s ndependent of C. A2) θ Θ, where Θ s compact and Θ R p. A3) Z has bounded support. A4) S T Z t) and S C Z t) are contnuously dfferentable n t, τ]. A5) f T Z t) and f C Z t) are unformly bounded n t, τ]. A6) PrC τ) = PrC = τ) >.

28 CHAPTER 3. SAMPLE SIZE CALCULATION FOR THE COX PROPORTIONAL HAZARDS MODEL 8 A7) PrT > τ) >. A8) PrT C Z) > almost surely under F Z. Non-nformatve censorng s assumed condton A)). Note that θ = θ,,..., θ p ), the parameter of nterest, s a p vector, and Θ s ts parameter space. In ths chapter, p = ; that s, there s only one parameter of nterest θ n the Cox proportonal hazards model, based on whch we conduct sample sze calculaton. Denote S C Z ) and f C Z ) as the survval functon and the pdf, respectvely, of C condtonng on Z of survval tme T. Denote S T Z ) and f T Z ) as the survval functon and the pdf, respectvely, of T condtonng on Z. The condtons A2) A5) are techncal assumptons, whch are needed n the dervaton of the asymptotc propertes. The condton A6) mples that any patents alve by the end of the study at tme τ are consdered to be censored. The condton A7) ndcates that the probablty of an ndvdual survvng after τ s postve. The condton A8) mples that there s a postve probablty of observng an event for any possble value of Z. All the condtons dscussed n ths chapter are adopted throughout ths dssertaton unless otherwse specfed. The followng addtonal condtons are consdered n some of the results n ths dssertaton. C) C s ndependent of Z. C2) Z s bnary and PrZ = ) = ρ, where ρ, ). Condtons C) C2) are used wherever specfed. The condton C2) apples to the case when Z s the treatment group ndcator e.g., Z = f n the treatment arm and Z = f n the control arm), and ρ s the proporton of ndvduals n the treatment arm.

29 CHAPTER 3. SAMPLE SIZE CALCULATION FOR THE COX PROPORTIONAL HAZARDS MODEL Model specfcaton For smplcty, we consder the case wth one covarate Z n the Cox proportonal hazards model λ T Z t Z, θ) = exp{zθ}λ t). For all θ Θ R, the partal lkelhood for the Cox model s ) n exp{z θ} L p θ) = = l RX ) exp{z lθ} n exp{z θ} = n j= Y, jx ) exp{z j θ}) where RX ) s the rsk set at tme X, for =,..., n. = The correspondng log partal lkelhood s n [ τ n )] log L p θ) = Z θ log Y j t) exp{z j θ} dn t). = The score functon U n wth respect to θ and the correspondng observed nformaton matrx V c n based on the partal lkelhood are derved as follows: = j= U n θ) = log L pθ) θ n [ τ n j= Z Z ] jy j t) exp{z j θ} n j= Y dn t), 3.) jt) exp{z j θ} = = V c n θ) = 2 log L p θ) θ 2 n n j= Z2 j Y j t) exp{z j θ} n n j= Y jt) exp{z j θ} j= Z ) 2 jy j t) exp{z j θ} n j= Y dn t). jt) exp{z j θ} = 3.2) 3.3 Sample sze formula under fxed alternatve In ths secton, we derve the sample sze formula based on the score test statstc under the fxed alternatve. We consder the followng hypotheses of nterest: the null

30 CHAPTER 3. SAMPLE SIZE CALCULATION FOR THE COX PROPORTIONAL HAZARDS MODEL 2 hypothess H : θ = θ and the alternatve hypothess H : θ = θ, where θ, θ Θ and θ θ. The score test statstc based on the partal lkelhood s T c n = U nθ ) V c n θ ). 3.3) To determne the asymptotc propertes of the test statstc T c n, the asymptotc dstrbutons of U n θ ), the score functon evaluated at θ should be derved frst. Theorem 3.3. gves the asymptotc dstrbutons of U n θ ) under the null and alternatve hypotheses. We also defne the followng: v c θ θ ) = lm n E θ v c θ θ ) = lm n E θ e c θ θ ) = lm n E θ v c θ θ ) = lm n Var θ [ ] n V n c θ ), [ ] n V n c θ ), [ ] n U nθ ), [ n U n θ ) ]. The expressons of v c θ θ ), e c θ θ ) and v c θ θ ) can be found n Secton 7... That of v c θ θ ) can be found n Secton Theorem Under condtons A) A8), the followng results hold: ) ) Under H : θ = θ, n U n θ ) d N, vθ c θ ) as n. ) Under H : θ = θ, n U n θ ) ) ne c θ θ ) d N, vθ c θ ) as n. Now we need to derve the asymptotc propertes of the test statstc to calculate the sample sze. Ths s not dffcult snce we have already derved the asymptotc propertes of U n θ ) n Theorem We wll only need to know the lmt of n V c n θ ) under the alternatve hypothess. Ths s shown n the proof of Corollary n Secton Corollary gves the correspondng results for the test statstc T c n. Corollary Under condtons A) A8), the results follow: ) Under H : θ = θ, Tn c N, ) as n. ) Under H : θ = θ, Tn c n ec θ θ ) v c θ θ ) d d N, vc θ θ ) v c θ θ ) ) as n.

31 CHAPTER 3. SAMPLE SIZE CALCULATION FOR THE COX PROPORTIONAL HAZARDS MODEL 2 Now we have obtaned the asymptotc propertes of the test statstc Tn c under both the null and alternatve hypotheses. The sample sze can be lnked to type I error, power, effect sze, and desgn effect based on the above results. Result gves the sample sze formula for the Cox model under the fxed alternatve hypothess. Result The hypotheses of nterest to test are H : θ = θ versus H : θ = θ, wth type I error α and power β. Under condtons A) A8), by Corollary ), the followng relatonshp can be establshed: Pr H T c n > c ) = α, where c = z α/2 s the crtc value. The dervaton of the sample sze s smlar as shown n the example of Chapter 2. By the defnton of power, we derve the followng: Pr H T c n > c ) = β, whch mples Pr H T c n > c ) + Pr H T c n < c ) = β. Wthout loss of generalty, we assume that the mean of the test statstc Tn c s negatve under the alternatve hypothess H. The followng result follows wth asymptotc normal approxmaton: Pr H T c n < c ) = β. Then, { v c Pr θ θ ) H v c θ θ ) = β. Tn c ) n ec θ θ ) v c θ θ ) < v c θ θ ) v c θ θ ) c n ec θ θ ) v c θ θ ) )}

32 CHAPTER 3. SAMPLE SIZE CALCULATION FOR THE COX PROPORTIONAL HAZARDS MODEL 22 Based on Corollary ), z β can be approxmated: v c θ θ ) c ) n ec θ θ ) = z v c θ θ ) v c β. θ θ ) The sample sze s obtaned by replacng c wth z α/2 n the above equaton. n = ) v c 2 z α/2 + θ θ ) z v c β θ θ ) e c θ θ )) 2 /v c θ θ ) 3.4) The formula 3.4) can be used for the sample sze calculaton for the Cox proportonal hazards model under the fxed alternatve. It s a functon of type I error, power, and desgn effects e c θ θ ), v c θ θ ) and v c θ θ ). The formula derved here s general as t s based on the score statstc of the Cox model under the fxed alternatve. However, the form of v c θ θ ) s usually complcated and may be dffcult to derve. In the next secton, we consder a method that smplfes the expresson of v c θ θ ) under contguous alternatves. 3.4 Sample sze formula under contguous alternatves The sample sze formula n Secton 3.3 s derved for the null hypothess H : θ = θ versus the fxed alternatve hypothess H : θ = θ. The lmtng varance of the test statstc T c n has a rather complex form. In ths secton, we dscuss the formula derved based on a sequence of alternatves, the contguous alternatves. The asymptotc dstrbuton of the test statstc wll be evaluated under the contguous alternatves H n : θ = θ n = θ + h n, where h R s some fxed real value. The power under the fxed alternatve s approxmated by that under the contguous alternatves. Let θ = θ n = θ + h n, then the sample sze n s related to h, θ and θ, snce h = nθ θ ). By dong so, t s actually assumed that θ s close to θ ;.e., the effect sze s small. When a sequence of contguous alternatves s consdered,

33 CHAPTER 3. SAMPLE SIZE CALCULATION FOR THE COX PROPORTIONAL HAZARDS MODEL 23 the lmtng dstrbuton of T c n has a smpler form. We show that the sample sze formula by Schoenfeld 983) [42] and Hseh and Lavor 2) [2] can be obtaned as a specal case of our method under certan condtons when θ =. Theorem 3.4. gves the asymptotc dstrbuton of U n θ ) under the contguous alternatves H n : θ = θ n = θ + h n. Theorem Under condtons A) A8), the followng result holds. Under H n : θ = θ n = θ + h n, ) U n θ ) d N hv c n θ θ ), vθ c θ ), as n. The proof of Theorem 3.4. can be found n Secton The followng Corollary gves the asymptotc dstrbuton of the test statstc T c n under the contguous alternatves. The proof s straghtforward by applyng Slutsky s theorem. Corollary Under condtons A) A8), the followng result holds. Under H n : θ = θ n = θ + h n, T c n ) d N h vθ c θ ), as n. Corollary s proved n Secton Now we have obtaned the asymptotc dstrbutons of the test statstc under both the null and alternatve hypotheses. The sample sze can be derved by known type I error, power, effect sze, and desgn effect based on the above results. Result gves the sample sze formula for the Cox model based on the contguous alternatves. Result The hypotheses of nterest to test are H : θ = θ versus H : θ = θ, wth type I error α and power β. Under condtons A) A8), by Corollary ), the followng relatonshp can be establshed: Pr H T c n > c ) = α,

34 CHAPTER 3. SAMPLE SIZE CALCULATION FOR THE COX PROPORTIONAL HAZARDS MODEL 24 where c = z α/2 s the crtc value. Wthout loss of generalty, we assume that the mean of the test statstc Tn c s negatve under H : θ = θ. By Corollary and the defnton of power, we derve the followng: Pr H T c n > c ) = β, whch s equvalent to Pr H T c n > c ) + Pr H T c n < c ) = β. Wth asymptotc normal approxmaton, Pr H T c n < c ) = β. Based on Corollary 3.4.2, we assume that Tn c follows an asymptotc normal dstrbuton wth mean approxmated by h vθ c θ ), where h = nθ θ ), and asymptotc varance equal to. Thus, we approxmate the dstrbuton of T c n under H by that under H n. It follows that Pr H T c n h vθ c θ ) < c h vθ c θ ) ) = β, and c h vθ c θ ) = z β. Based on the fact that h = nθ θ ), the sample sze formula can be obtaned by substtutng c wth z α/2 n the above equaton. n = zα/2 + z β ) 2 v c θ θ ) θ θ ) ) The formula 3.5) can be used for sample sze calculaton for the hypotheses of nterest H : θ = θ versus H : θ = θ, wth type I error α and power β. It

35 CHAPTER 3. SAMPLE SIZE CALCULATION FOR THE COX PROPORTIONAL HAZARDS MODEL 25 s calculated based on the contguous alternatves H n : θ = θ n = θ + h n. The formula s a functon of α, β, θ, θ, and the desgn effect v c θ θ ). We notce that the lmtng varances of the score statstc under the null and alternatve hypotheses are both v c θ θ ). Therefore, the sample sze formula under the fxed alternatve can be smplfed. We wll dscuss the method n the followng remark. Remark It s shown that the asymptotc varance of n U n θ ) s v c θ θ ) under the contguous alternatves see Theorem 3.4. and ts proof n Secton 7..3). Ths result ndcates that the lmtng varance of n U n θ ) under the alternatve hypothess s close to v c θ θ ) when the alternatve s close to the null hypothess. Let us assume a smple lnear relatonshp between the two lmtng varances v c θ θ ) and v c θ θ ); that s, v c θ θ ) = φv c θ θ ), where φ >. When the effect sze s small, that s, when θ s close to θ, v c θ θ ) v c θ θ ). Thus, t s expected that φ s close to for small effect sze. Dfferent values of φ can be explored to obtan the best approprate sample sze. If we replace v c θ θ ) wth φv c θ θ ) n 3.4, the sample sze formula becomes: n = ) φv c 2 z α/2 + θ θ ) z v c β θ θ ) e c θ θ )) 2 /v c θ θ ) 3.6) Dfferent values of φ can be nvestgated f we use 3.6) nstead of 3.4). The value of φ can be determned through a seres of smulaton studes. In ths way, the calculaton s greatly smplfed wthout dervng the explct form of the lmtng varance of the score statstc under the fxed alternatve. In practce, t s often of nterest to derve the sample sze for the null hypothess of no effect θ = ) versus some small effect under the alternatve hypothess. The followng result gves the sample sze formula for ths knd of specal case when θ =. Result The hypotheses of nterest to test are H : θ = versus H : θ = θ, wth type I error α and power β. Under condtons A) A8) and C), the sample sze formula based on the contgu-

36 CHAPTER 3. SAMPLE SIZE CALCULATION FOR THE COX PROPORTIONAL HAZARDS MODEL 26 ous alternatves H n : θ = θ n = h n s n = zα/2 + z β ) ) VarZ) Pr θ= { : = }) θ 2 We denote Pr θ= { : = }) as the event probablty under the null hypothess of θ =. Result s derved as a drecton applcaton of Result Note that wth the addtonal condton C), vθ c can be smplfed when θ =. 3.7) s thus the same as the sample sze formula by Hseh and Lavor 2) [2]. The followng equaton can be obtaned from 3.7): npr θ= { : = }) = zα/2 + z β ) 2 VarZ) θ 2. Let D θ= denote the expected number of events under the null hypothess H : θ =. Therefore, D θ= = npr θ= { : = }) = z α/2+z β) 2 VarZ) θ 2. In partcular, when the covarate s bnary condton C2)), VarZ) = ρ ρ). Then the sample sze formula 3.7) becomes n = zα/2 + z β ) ) ρ ρ) Pr θ= { : = }) θ 2 Then D θ=, the expected number of events under the null hypothess H : θ =, can be calculated by z α/2+z β) 2 ρ ρ) θ 2. These results are the same as what Schoenfeld 983) [42] has derved. Ths representaton of D θ= has appeared n many works, ncludng Schoenfeld 98) [4], Schoenfeld and Rchter 982) [43] and Collett 23) [9].

37 CHAPTER 4. SAMPLE SIZE CALCULATION WITH YANG AND PRENTICE S SEMIPARAMETRIC MODEL 27 Chapter 4 Sample sze calculaton wth Yang and Prentce s semparametrc model The methods dscussed n Chapter 3 rely on the proportonal hazards assumpton. When there s presence of non-proportonalty of the hazard functons, an approprate model should be consdered for sample sze calculaton. We derve the sample sze formulae based on the semparametrc model by Yang and Prentce 25) [5]. Ths model can be used for non-proportonal hazard functons, wth the Cox proportonal hazards model and the proportonal odds model beng two specal cases. Thus, the sample sze calculaton based on ths model can be used for survval data when the proportonal hazards assumpton does not hold. The frst secton of ths chapter ntroduces the semparametrc model for short-term and long-term hazard ratos Yang and Prentce 25 [5]). The sample sze formulae based on ths model under fxed and contguous alternatves wll follow. The specal cases for testng the null hypothess of no treatment effect are dscussed for the three submodels: Cox proportonal hazards model, proportonal odds model, and long-term effect model. The last secton provdes smulaton results for the proposed methods under varous scenaros.

38 CHAPTER 4. SAMPLE SIZE CALCULATION WITH YANG AND PRENTICE S SEMIPARAMETRIC MODEL Notatons and Model specfcaton Yang and Prentce 25) [5] developed a semparametrc model that can accommodate dfferent short-term and long-term hazard ratos. In the smple two-sample case, the model has the followng form: λ t) λ t) = γ γ 2 γ + γ 2 γ )S t). 4.) The hazard rato n 4.) s a functon of two postve parameters γ and γ 2 and the baselne survval functon S. It s monotone n t for fxed γ and γ 2, snce S t) s nonncreasng n t. Notce that γ = lm t λ t) λ t) and γ λ 2 = lm t) t λ. Ths s t) because S t) goes to when t goes to, and S t) goes to when t goes to. Therefore, γ can be nterpreted as the short-term hazard rato and γ 2 as the longterm hazard rato. The model reduces to the Cox proportonal hazards model when γ = γ 2 = γ. The hazard rato for the Cox model s constant over tme, and the correspondng hazard functons are proportonal. When γ 2 =, the model becomes the proportonal odds model. The proportonal odds model has non-constant hazard rato, and the correspondng hazard functons are not proportonal. The treatment effect would fade away over tme for the proportonal odds model. Another specal case of nterest s what we call the long-term effect model, n whch γ =. Ths means that there s no treatment effect at the begnnng, but the effect shows up later on. The long-term effect model has non-constant hazard rato, and the correspondng hazard functons are non-proportonal. The model 4.) can be used for a wde range of hazards patterns, ncludng crossng hazards. When there s crossng n the hazard functons, γ < and γ 2 >, or, γ > and γ 2 <. Examples wth baselne hazard functon λ t) = / + t) correspondng baselne survval S t) = / + t) and odds of baselne survval S t) S t) = t) are shown n the followng Fgures. The hazard rato, correspondng hazard functons and survval functons are plotted under dfferent scenaros. Fgure 4. gves two examples of constant hazard rato. These are examples of the Cox

39 CHAPTER 4. SAMPLE SIZE CALCULATION WITH YANG AND PRENTICE S SEMIPARAMETRIC MODEL 29 proportonal hazards model. When the common hazard rato s less than, the treatment s more favorable. When the common hazard rato s great than, the treatment s not effectve. Fgure 4.2 shows the examples of the proportonal odds model, n whch γ 2, the long-term effect, s. Fgure 4.3 shows two examples of the long-term effect model, where γ, the short-term effect, s. The correspondng survval functons of the treatment and the control are close to each other n each scenaro of the long-term effect model, but start to devate from each other after a whle. Fgure 4.4 gves examples of the crossng hazard functons. These are the cases when the treatment effect s dfferent at dfferent stages of a study. The treatment may be benefcal at the begnnng, but turns out to do harm later on, or, the treatment may not be favorable at the begnnng, but appears to be more effectve as the study contnues. Crossng n hazard functons occurs when γ < and γ 2 >, or, γ > and γ 2 <. Notce that n each of the crossng hazards example, both of the hazard functons and the survval functons cross, but the crossng ponts are dfferent. We wll use the same notatons and regularty condtons as n Secton 3.. The regularty condtons are A) A8). Let θ = log γ be the parameter of nterest. We suppose that θ Θ, where Θ s a compact subset of R p. In the model by Yang and Prentce 25) [5], p = 2. The parameter of nterest s θ = θ, θ 2 ), whch s a real-valued vector.

40 CHAPTER 4. SAMPLE SIZE CALCULATION WITH YANG AND PRENTICE S SEMIPARAMETRIC MODEL 3 Fgure 4.: Proportonal hazards: Cox model γ = γ 2 = γ). γ =.5, γ 2 =.5 Hazard Rato Hazard Functon Treatment Control Survval Functon Treatment Control Tme Tme Tme γ =2, γ 2 =2 Hazard Rato Hazard Functon Treatment Control Survval Functon Treatment Control Tme Tme Tme

41 CHAPTER 4. SAMPLE SIZE CALCULATION WITH YANG AND PRENTICE S SEMIPARAMETRIC MODEL 3 Fgure 4.2: Non-proportonal hazards: proportonal odds model γ 2 = ). γ =.8, γ 2 = Hazard Rato Hazard Functon Treatment Control Survval Functon Treatment Control Tme Tme Tme γ =2, γ 2 = Hazard Rato Hazard Functon Treatment Control Survval Functon Treatment Control Tme Tme Tme

42 CHAPTER 4. SAMPLE SIZE CALCULATION WITH YANG AND PRENTICE S SEMIPARAMETRIC MODEL 32 Fgure 4.3: Non-proportonal hazards: long-term effect model γ = ). γ =, γ 2 =.5 Hazard Rato Hazard Functon Treatment Control Survval Functon Treatment Control Tme Tme Tme γ =, γ 2 =.8 Hazard Rato Hazard Functon Treatment Control Survval Functon Treatment Control Tme Tme Tme

43 CHAPTER 4. SAMPLE SIZE CALCULATION WITH YANG AND PRENTICE S SEMIPARAMETRIC MODEL 33 Fgure 4.4: Non-proportonal hazards: crossng hazards γ < and γ 2 >, or, γ > and γ 2 < ). γ =.6, γ 2 =2.5 Hazard Rato Hazard Functon Treatment Control Survval Functon Treatment Control Tme Tme Tme γ =2.5, γ 2 =.5 Hazard Rato Hazard Functon Treatment Control Survval Functon Treatment Control Tme Tme Tme

44 CHAPTER 4. SAMPLE SIZE CALCULATION WITH YANG AND PRENTICE S SEMIPARAMETRIC MODEL Parameter estmaton The model has the followng form after re-parameterzatons and the ncorporaton of a covarate Z: λ T Z t Z, θ) = e Zθ + e Zθ 2R t) dr t), 4.2) dt where θ j = logγ j ), for =, 2. Note that θ = θ, θ 2 ) Θ R 2 s a real-valued vector of two dmensons. Now θ = logγ ) can be nterpreted as the logarthm of the short-term hazard rato and θ 2 = logγ 2 ) as the logarthm of the long-term hazard rato. The true parameter θ s a vector that conssts of two elements: θ and θ 2,.e., θ = θ, θ 2 ). The compact parameter space Θ s a subset of R 2. We also defne R as the odds of the baselne survval functon,.e., R t) = S t) S, for t, τ]. t) The hazard rato s then a functon of θ, θ 2 and R. The model reduces to the Cox proportonal hazards model when θ = θ 2, to the proportonal odds model when θ 2 =, and to the long-term effect model when θ =. If θ and θ 2 have dfferent sgns.e., one s postve and the other s negatve), then there s crossng n hazard functons. If R t) n the model 4.2) s known, the lkelhood and log lkelhood of θ = θ, θ 2 ) can be wrtten as follows under non-nformatve censorng: L n θ, R ) = logl n θ, R ) = n λ T Z X Z, θ) S T Z X Z, θ), = n logλ T Z t Z, θ) dn t) n = = λ T Z t Z, θ)y t) dt. The score functon wth respect to θ s loglnθ,r ). If R θ s unknown, t can be consstently estmated by ˆR n, where ˆR n t, θ) has the followng close-from expresson Yang and Prentce 25 [5]) ˆR n t, θ) = t ˆP n t, θ) ˆP n s, θ) dˆλ n, s, θ), 4.3)