Semiparametric varying coefficient model for interval censored data with a cured proportion

List of Figures Figure 4.1 Estimated varying-coefficients with median performance us-ing normal distributions settus-ing I, n = 500.. 42 Figure 4.2 Estimated varying-coefficients with median

VARYING-COEFFICIENT MODEL FOR

INTERVAL CENSORED DATA WITH A

CURED PROPORTION

SHAO FANG

NATIONAL UNIVERSITY OF SINGAPORE

2013

VARYING-COEFFICIENT MODEL FOR

INTERVAL CENSORED DATA WITH A

PROBABILITY NATIONAL UNIVERSITY OF SINGAPORE

2013

2.1 Two-part models with varying-coefficients 12

CONTENTS ii

2.2 Estimation under mixed case interval censoring 14

2.3 Computation 19

2.3.1 Estimations 20

2.3.2 Algorithm 23

Chapter 3 Inference 26 3.1 Asymptotic theory 26

3.2 Estimation of asymptotic variance 28

3.3 Bandwidth and model selection 30

3.3.1 Cross-validation 30

3.3.2 BIC 31

3.3.3 Algorithm for bandwidth selection 33

3.3.4 Algorithm for model selection 34

Chapter 4 Simulations and Data Analysis 36 4.1 Simulations 36

4.2 Data analysis 80

CONTENTS iii

4.2.1 Data and statistical models 80

4.2.2 Normal distribution 84

4.2.3 Logistic distribution 90

4.2.4 Gumbel distribution 92

Chapter 5 Discussion and Further Research Topics 95 5.1 Discussion 95

5.2 Further research topics 96

5.2.1 Modelling survival times of non-cured subjects with the in-verse Gaussian distributions 96

5.2.2 Bayesian two-part models with varying-coefficients using adap-tive regression splines 100

5.2.3 Two-part models with varying-coefficients and random effects 107 5.2.4 Other topics 113

Chapter A Proofs of Theorems 126 A.1 Notations 126

A.2 Conditions 127

CONTENTS iv

A.3 Proofs of theorems 129

List of Figures

Figure 4.1 Estimated varying-coefficients with median performance

us-ing normal distributions (settus-ing I), n = 500 . 42

Figure 4.2 Estimated varying-coefficients with median performance

us-ing normal distributions (settus-ing I), n = 1000 . 43

Figure 4.3 Estimated varying-coefficients with median performance

us-ing logistic distributions (settus-ing I), n = 500 . 45

Figure 4.4 Estimated varying-coefficients with median performance

us-ing logistic distributions (settus-ing I), n = 1000 . 46

List of Figures vi

Figure 4.5 Estimated varying-coefficients with median performance

us-ing Gumbel distributions (settus-ing I), n = 500 . 49

Figure 4.6 Estimated varying-coefficients with median performance

us-ing Gumbel distributions (settus-ing I), n = 1000 . 50

Figure 4.7 Estimated varying-coefficients with median performance

us-ing normal distributions (settus-ing II) 53

Figure 4.8 Estimated varying-coefficients with median performance

us-ing logistic distributions (settus-ing II) 54

Figure 4.9 Estimated varying-coefficients with median performance

us-ing Gumbel distributions (settus-ing II) 55

Figure 4.10 Estimated varying-coefficients with median performance

us-ing normal distributions (settus-ing III) 62

Figure 4.11 Estimated varying-coefficients with median performance

us-ing logistic distributions (settus-ing III) 64

Figure 4.12 Estimated varying-coefficients with median performance

us-ing Gumbel distributions (settus-ing III) 66

Figure 4.13 Estimated varying-coefficients with median performance

us-ing normal distributions (settus-ing IV) 70

Figure 4.14 Estimated varying-coefficients with median performance

us-ing logistic distributions (settus-ing IV) 71

List of Figures vii

Figure 4.15 Estimated varying-coefficients with median performance

us-ing Gumbel distributions (settus-ing IV) 72

Figure 4.16 Typical estimated varying-coefficients in 150 simulations

us-ing normal distributions (settus-ing V) 75

Figure 4.17 Typical estimated varying-coefficients in 150 simulations

us-ing logistic distributions (settus-ing V) 76

Figure 4.18 Typical estimated varying-coefficients in 150 simulations

us-ing Gumbel distributions (settus-ing V) 77

Figure 4.19 Estimated varying-coefficients using normal distributions in

List of Tables ix

Table 4.5 Simulation results using Gumbel distributions under three

cases (setting I), n = 500. 47

Table 4.6 Simulation results using Gumbel distributions under three

cases (setting I), n = 1000 . 48

Table 4.7 Simulation results using normal distributions under three cases

(setting II) 56

Table 4.8 Simulation results using logistic distributions under three cases

(setting II) 57

Table 4.9 Simulation results using Gumbel distributions under three

cases (setting II) 58

Table 4.10 Simulation results using normal distributions under three cases

(setting III) 61

Table 4.11 Simulation results using logistic distributions under three cases

(setting III) 63

Table 4.12 Simulation results using Gumbel distributions under three

cases (setting III) 65

Table 4.13 Simulation results using normal, logistic and Gumbel

distri-butions(setting IV) 69

Table 4.14 Simulation results using normal distributions under three cases

(setting V) 78

Table 4.17 Estimation results for Models 1 & 2 using normal distributions 85

Table 4.18 Estimation results for corresponding parametric Models 1 &

2 using normal distributions 86

Table 4.19 Estimation results for Models 1 using logistic distributions 90

Table 4.20 Estimation results for corresponding parametric Model 1

us-ing logistic distributions 92

Table 4.21 Estimation results for Models 1 using Gumbel distributions 93

Table 4.22 Estimation results for corresponding parametric Model 1

us-ing Gumbel distributions 94

ACKNOWLEDGEMENTS

I am very grateful to have had Associate Professor Li Jialiang as my supervisor

He is truly a great mentor, not only for statistics but also for daily life I would like

to thank him for his guidance, encouragement, time, and endless patience Next,

I would like to thank all my friends who make life as a graduate student easier for

me I also wish to express my gratitude to the university and the department forsupporting me throughout the tenure of my NUS Graduate Research Scholarship.Finally, I thank my family for their love and support

SUMMARY

Varying-coefficient models make up an increasing portion of statistical researchand are now applied to censored data analysis in medical studies This research in-corporates such flexible semiparametric regression tools for interval censored datawith a cured proportion A two-part model is adopted to describe the overallsurvival experience for such complicated data To fit the unknown functional com-ponents into the model, the standard local polynomial approach is taken withbandwidth chosen by cross-validation Consistency and asymptotic distribution

of the estimation procedure are established A resampling scheme is proposed forinference A BIC-type model selection method is constructed to recommend an ap-propriate specification of parametric and nonparametric components in the model.Extensive simulations are conducted to assess the performance of our methods Anapplication on some decompression sickness data is used to illustrate our methods

LIST Of NOTATIONS

in hypobaric environments A high grade of venous gas emboli (VGE) can be

a precursor to serious DCS Therefore, it is important to build a model for thetime to onset of grade IV VGE in order to predict the situations in which it ismost likely to occur The HDSD data set has records from volunteer subjectsundergoing denitrogenation test procedures prior to being exposed to a hypobaric

environment Since the onset time of grade IV VGE, if it occurred, was recordedonly as contained within certain time intervals, owing to the nature of the measure-

ment procedure, the attained data were subject to interval censoring Moreover, it

has been suggested that some subjects had finite event times, whereas others hadinfinite event times Individuals with infinite event times are sometimes referred

to as ‘cured’ or ‘immune’ A cure rate model that allows a subgroup of subjects

to be immune to the event of interest is thus warranted (Li and Ma (2010); Ma(2010); Thompson and Chhikara (2003)) Usually a two-part model is adopted forthis type of data The first part models the probability of cure and the secondpart models the survival distribution for susceptible subjects it is very often ofinterest to investigate the covariate effects For the first part of the model, weusually use familiar binary regression models such as the logistic model; for thesecond part, we may choose all kinds of parametric or semiparametric regressionmodels appropriate for survival data

Zhang and Sun (2010b) gives the latest review of statistical analysis for interval

censored data In general, the theoretical justification for interval censored dataanalysis may be difficult because there is a lack of basic tools as simple and elegant

as the partial likelihood theory and the martingale theory for right censored data.However, for likelihood-based estimation under a known class of distributions,

we can still obtain √

n consistency and asymptotic normality For

distribution-free inference procedures for interval censoring data, only √3

n-consistency can be

achieved For more details, one may consult Sun (2006) Because of this benefit,parametric models, such as accelerated failure time models, are usually preferred

in lieu of Cox model for interval censored data, when the distribution is suitable

In Zhang and Sun (2010a), clustered interval-censored failure time data with

informative cluster size were analyzed using regression Two methods were posed One was a weighted estimating equation-based procedure Another was awithin-clustered resampling procedure This procedure sampled a single subjectfrom each cluster and transformed the data to the usual univariate failure timedata, which could be analyzed with a generalized linear model Since the obser-vations in the resampled data are independent, we may take the average of allresampled-based estimates

pro-To incorporate the cure rate for interval censoring, Ma (2009, 2010) consideredthe previously mentioned two-part model and studied the theoretical properties ofestimation under the Cox proportional hazards model Li and Ma (2010) used asimilar mixture modelling idea and discussed various location-scale families for thesurvival distribution

Some recent papers have also made contributions towards cure rate modelling

In Xiang et al (2011), the mixture cure model with random effects for clusteredinterval-censored survival data was studied As with research of Li and Ma (2010),random effects were incorporated in both the cure rate part under the logisticregression and PH regression components The authors followed the generalizedlinear mixed model method for estimation The best linear unbiased prediction(BLUP) type log-likelihood was constructed which can be viewed as a penalizedlog-likelihood The key point of the authors’ approach was to define a latentvariable to indicate whether the individual would be cured or not This ran-dom variable was unknown when right-censored and could be treated as missing.Therefore, to find the restricted maximum log-likelihood estimators, the authorsimplemented EM algorithm, which was iterative Their method employed the Coxsemi-parametric PH function form for the survival function component and a self-consistent estimator for the baseline survival function Their approach required nospecification of the parametric survival function form, and there were no potentialproblems due to survival distribution misspecification

Peng and Taylor (2011) used similar models in Xiang et al (2011) They alsoapplied the Gaussian quadrature approximation method for estimation Lu (2010)studied the accelerated failure time model with a cure fraction via kernel-basednonparametric maximum likelihood estimation As with the research of Li and

Ma (2010), logistic regression was used for the cure fraction, and the accelerated

failure time model for the survival function component However, as with thework of Xiang et al (2011), there was no specification of the parametric survivalfunction form The authors also used a latent variable to indicate whether theindividual would be cured or not Therefore, the EM algorithm was employed Tomaximize the corresponding condition expectation in E step, smooth estimation

of the hazard function was needed The author started with a piece-wise constanthazard function, which was asymptotically equivalent to the kernel-smoother Itwas shown that the resulting estimates were consistent and asymptotically normal.The author proposed to consistently estimate the asymptotic covariance matrix byinverting the empirical Fisher information matrix based on the EM-aided numer-ical differentiation method for computing the second derivative of the log profilelikelihood at the maximum Using similar ideas, Zhang et al (2011) proposed anew semiparametric estimation method based on a kernel smoothed approximationwhich is asymptotically equivalent to the profile likelihood function in the acceler-ated hazard model This method leads to smooth estimating equations and is easy

to use Consistency and asymptotic normality of the estimates from this methodare also proved

Zhang and Peng (2012) reviewed two estimation methods for the acceleratedfailure time mixture cure model One was Li and Taylor’s method based on solv-ing the general estimating equation, and the other was the rank-like estimation

method Authors proved that the two estimation methods are asymptoticallyequivalent They also suggested using the rank-like estimation method with aGehan type weight in practice, because of its accuracy and efficiency Peng and Xu(2012) reviewed the recently proposed Box-Cox transformation cure model (BCT).This cure model includes the mixture cure model (MCM) and the bounded cu-mulative hazard cure model (BCH) as special cases The authors gave a similarbiological interpretation for the BCT model to that for the BCH model For modelselection between MCM and BCH, the authors used AIC, the likelihood ratio testand the score test The author’s study showed that AIC was informative and thatboth the likelihood ratio test and the score test had adequate power for modelselection when the sample size was large

Well-developed methods for right-censored data may be extended to analyze theinterval-censored data In Chen and Sun (2010), the authors employed an additivehazards model to describe the survival time for studying interval-censored failuretime data A multiple imputation approach was used for inference This algorithmimputes censoring times by sampling from the current estimate of the conditionaldistribution of the error which changes interval-censored data to right-censoreddata The authors then used the ready estimation method for the right-censoreddata for inference This approach can be generalized to time-dependent covariates.However, the imputation procedure may be improper

In this thesis, nonparametric smoothing is included for regression analysis tomake the model more flexible Specifically, varying-coefficients are incorporatedwhich is a research subject that has been focused on in much recent nonparametricliterature However, not many attempts have yet been made to apply state-of-the-art nonparametric fitting techniques to interval censoring Fan and Gijbels(1996) is a good introduction to nonparametric regression models When high-dimensional data are presented, many powerful methods are employed to avoidthe so-called “curse of dimensionality” Among them, a useful extension of classi-cal linear models is varying-coefficient models One advantage of these models isthat the coefficient functions can be easily estimated using a simple local regres-sion one-step procedure However, when different coefficient functions have differ-ent degrees of smoothness, the one-step estimation method is not optimal Fanand Zhang (1999) proposed a two-step estimation procedure for varying-coefficientmodels to deal with this situation Fan and Huang (2005) studied profile likelihoodinferences on semi–parametric varying–coefficient partially linear models, includ-ing the profile likelihood ratio test, the Wald test and asymptotic normality Xia

et al (2004) gave an efficient two-step procedure for semivarying-coefficient modelsbased on minimizing the semi-local least squares estimation procedure The idea

is to estimate parametric coefficients first and then estimate the standard purevarying-coefficient model Wang and Xia (2009) proposed shrinkage estimation

of the varying-coefficient model Their method is called the kernel least absolute

varying-Breiman and Friedman (1985) introduced the backfitting algorithm, a simpleiterative procedure, to fit a generalized additive model The backfitting algorithmmethod smoothes regression residuals for additive models to update functionalestimates one by one Usually, the backfitting algorithm is equivalent to the Gauss–Seidel method algorithm for solving a certain linear system of equations (Hastie andTibshirani (1990), Chapter 5) The theory of the backfitting algorithm method ismuch harder than that of the profile estimation method One may refer to Opsomer(2000) for asymptotic properties of backfitting estimators.

Also relevant to censored data are the following close–knit publications Fan

et al (1997) considered the proportional hazards regression model with a metric risk effect Maximum local likelihood estimations for parametric baseline

hazard functions and maximum local partial likelihood estimations for metric baseline hazard functions were studied Cai et al (2008) and Lu and Zhang(2010) implemented varying-coefficient models under the Cox regression model forright censored data Li and Lee (2011) and Li and Zhang (2011) studied paramet-ric regression with varying-coefficients and thresholding effects Besides varying-coefficient models, other less complicated threshold regression (TR) models arealso studied in Lee et al (2010) and Yu et al (2009), where nonparametric cubicB-splines are adopted

nonpara-The Cox model, or the proportional hazards (PH) regression model, is a class

of widely used survival models In fact, the PH model can be treated as a reducedmathematical form of a TR model Lee and Whitmore (2009) provide more details.Therefore, TR may be considered as an alternative regression approach when there

is evidence or suspicion that the PH property does not hold Even when the PHfeature does hold reasonably well, using a TR modelling framework to look into thehazard structure may lead to a better understanding of the patterns of risk beingexhibited, because the underlying stochastic processes may provide more insightinto the mechanism It should be noted that the PH model is a semiparametricmodel Therefore, some properties of parametric models are different from those ofthe PH model — the consistency and asymptotic normality, for instance In Zhang

et al (2010), the authors proposed a spline-based sieve semiparametric maximum

likelihood approach to fit the Cox model with interval-censored data This proach approximated the baseline cumulative hazard function using a monotoneB-spline function The method is efficient Asymptotic normality of the estimateswas proved with the method to estimate the standard error of the estimates InWang et al (2010), the authors proposed an estimating equation based approach

ap-to regression analysis for interval-censored failure time data using the additivehazards model There was no need to estimate the baseline hazard function forthis method The method is efficient with established asymptotic properties InPerdon´a and Louzada-Neto (2011), the authors proposed a general hazard modelwhich generalized a large number of families of cure rate models The estimationprocedure was based on the maximum-likelihood-estimation method

Bayesian ideas can be applied to analyze interval-censored data HDSD wasanalyzed using Bayesian models in Thompson and Chhikara (2003) Pennel et al.(2009) used Markov chain Monte Carlo (MCMC) to fit TR models with randomeffects and nonproportional hazards This approach included a data augmenta-tion method which avoided complicated posterior distributions and made it moretractable in the Bayesian framework Cancho et al (2011) employed a Bayesiananalysis using MCMC methods for right-censored survival data suitable for pop-ulations with a cure rate The authors modeled the cure rate under the negativebinomial distribution as a special case of the promotion time cure model Lopes and

Bolfatine (2012) studied the promotion cure rate model with random effects BothBayesian and classical estimation methods were implemented The Bayesian ap-proach was implemented using MCMC The classical approach used the restrictedmaximum likelihood estimators

This thesis is organized as follows Chapter 2 covers the modelling ology where we present the two-part model It demonstrates the two-part modelwith varying-coefficients, applications to interval censored data and the estima-tion method Chapter 3 is the inference chapter where asymptotic theorems areprovided A particular resampling method is applied to estimate the asymptoticvariance A modified BIC version of the model selection method is also introduced.Simulations and real data analysis are provided in Chapter 4 and discussions andfurther research topics follow in Chapter 5 The proofs of asymptotic theorems areincluded in the Appendix

Modelling Methodology

A few notations are required to introduce our methodology Assume n pendent subjects are observed in a sample For the ith observation (i = 1, · · · , n), denote p i to be the probability of being susceptible and so 1− p i is the probability

inde-of being cured Let S0 be the proper survivor function for susceptible subjects,

with S0(0) = 1 and S0(∞) = 0 Denote T to be the event time of interest and t to

be its realized value Under the two-part model assumption, the survival function

2.1 Two-part models with varying-coefficients 13

for the failure time of the ith subject is

Denote (M i T , N i T)T to be the vector of covariates associated with p i Under alogistic regression model for binary outcome, i.e being cured or not cured, wehave

1 + exp{−(M T

i γ(U i ) + N T

i θ) } ,

where γ( ·) = (γ1(·), · · · , γ r(·)) T is an r-dimensional vector of unknown functions,

and θ = (θ1, · · · , θ s)T is an s-dimensional vector of unknown parameters Note

that coefficients of M are varying according to an index variable U and those of

Here α( ·) = (α1(·), · · · , α p(·)) T is a p-dimensional vector of unknown functions,

and β = (β1, · · · , β q)T is a q-dimensional vector of unknown parameters Note

that coefficients of X are varying according to U and those of Z are constant F

2.2 Estimation under mixed case interval censoring 14

is a cumulative distribution function for a standardized location-scale distribution

where σ > 0 is the scale parameter Examples of F include the standard

nor-mal distribution, logistic distribution and Gumbel distribution The details are as

follows Supposing the random variable X is a member of the location-scale tributions with the location parameter µ and the scale parameter σ, there exists a standardized random variable Z that is Z = X σ −µ, which has the same distribution

dis-under all values of the parameters µ and σ The density of the distribution is

f X (x) = σ1f Z

(x −µ

σ

) The cumulative distribution function of the distributions is

F X (x) = F Z(x −µ

σ

)

In practice, the two sets of covariates (M , N ) and (X, Z) can have zero, partial

or full overlap In this thesis we explicitly treat these covariates as two differentsets This kind of arrangement is especially plausible when two independent sources

of predictors affect the cure probability and the failure time separately

censor-ing

In a lifetime study the event time t i may not be observable for the ith subject.

Under a general setting of mixed case interval censoring, we can only observe

2.2 Estimation under mixed case interval censoring 15

(L i , R i], where 0 ≤ L i < t i ≤ R i ≤ ∞ Let δ i be the censoring indicator: δ i equals

1 if ti is interval censored (for which Ri < ∞) and 0 if t iis right censored (for which

R i =∞) The observed data consist of {(L i , R i , δ i , C i ) : i = 1, , n }, where C i is

the union of all the covariates in the model, i.e C i =∪

func-Therefore, the log-likelihood for a single observation is

l (α,γ) i = δilog{S(L i) − S(R i) } + (1 − δ i) log S(Li)

= δ i log p i + δ ilog{S0(L i)− S0(R i)} + (1 − δ i) log [1− p i {1 − S0(L i)}]

= δ ilog{S0(L i)− S0(R i)} + (1 − δ i) log{p i [p −1 i − {1 − S0(L i)}]} + δ i log p i

= δ ilog{S0(L i)− S0(R i)} + (1 − δ i ) log [p −1 i − {1 − S0(L i)}] − log p −1

2.2 Estimation under mixed case interval censoring 16

When unknown functions are present, a local-likelihood usually has to be

con-structed We consider the local polynomial approximation technique (Fan and

Gijbels (1996), Carroll et al (1997), Fan et al (1997), Cai et al (2008)) For

ob-served index variable U i close to the point u, by the first-order Taylor’s expansion,

we have

α(U i) ≈ α(u) + (U i − u) ˙α(u)

= {u i(u) ⊗ I p } T

a(u), γ(U i) ≈ γ(u) + (U i − u) ˙γ(u)

= {u i (u) ⊗ I r } T b(u),

where u i (u) = (1, (U i − u)) T , a(u) = (α(u) T , ˙ α(u) T)T , b(u) = (γ(u) T , ˙ γ(u) T)T

Here ˙f ( ·) denotes the first derivative of the function f(·) and the symbol ⊗ denotes

the Kronecker product

The combined local log-likelihood function of the observed data can be written

2.2 Estimation under mixed case interval censoring 17

The kernel reflects that the above model is only applicable to the neighborhood

around u and weighs down smoothly the contribution of remote data points In this thesis, the Epanechnikov kernel K(t) = 0.75(1 − t2)+ is used for estimation

Estimation based on maximizing the above combined local log-likelihood isdifficult and crude because too many parameters are involved Therefore, to makethe estimation efficient, we may consider a profile estimation approach Suppose

the constant parameters σ, β and θ are known and we maximize the above local

log-likelihood function to obtain ˆa(u) and ˆ b(u) and use the first p entries of ˆ a(u)

and the first r entries of ˆ b(u) as the local maximum likelihood estimates ˆ α(u) and

2.2 Estimation under mixed case interval censoring 18

Initially, for each distinct grid point ui, we maximize (2.2) with respect to

all parameters to obtain initial estimates ˆα( ·) = ˆ α0(·) and ˆγ(·) = ˆγ0(·) using

linear interpolation to obtain the full function estimates While ˆβ, ˆ θ, ˆ σ are also

estimated for each distinct grid point, they are not used to update initial parameter

estimates Initial parameter estimates are updated by maximizing the global

log-likelihood (2.3) shown below with initial function estimates given

The estimates for the constant parameters σ, β and θ can be updated by

maximizing the log-likelihood

2.3 Computation 19

There may be a technical concern for experts on regression smoothing thatthe cure rate part and the survival part should not receive the same amount ofsmoothing and different bandwidth parameters may be needed for the functionalestimates in order to achieve the estimation efficiency This is actually what weimplement in this thesis and more details are provided in the next section Thismakes the computation algorithm similar to the back-fitting algorithm for additivemodels (Breiman and Friedman (1985)) One may refer to Opsomer (2000) forasymptotic properties of backfitting estimators

The previous section introduced a general idea for obtaining the parameter andfunction estimates based on maximizing log-likelihood More details for the actualimplementation are provided here

2.3 Computation 20

2.3.1.1 Initial estimation for both varying-coefficients and parameters

By maximizing the local likelihood (2.2) with respect to all the parameters,

we obtain a set of estimates This is an one-step estimation used for the initialvarying-coefficient estimates in the algorithm below

Initially, for each distinct grid point u i, we maximize (2.2) with respect toall parameters to obtain initial estimates ˆα( ·) = ˆ α0(·) and ˆγ(·) = ˆγ0(·) using

linear interpolation to obtain the full function estimates While ˆβ, ˆ θ, ˆ σ are also

estimated for each distinct grid point, they are not used to update initial parameterestimates Initial parameter estimates are updated by maximizing the global log-likelihood (2.3) shown below with initial function estimates given

By maximizing the local likelihood (2.4) with respect to the unknown functions

for the location-scale part if the parameters θ, β and σ and the functions to be

estimated for the cure rate part are known, we obtain a set of estimates for unknownfunctions in the location-scale part and use linear interpolation to obtain the full

2.3 Computation 22

function estimates

2.3.1.3 Local estimation for varying-coefficients of the cure rate part

As with (2.2), one can define the combined local log-likelihood functions of theobserved data for the cure rate part

By maximizing the local likelihood (2.5) with respect to the unknown

func-tions for the cure rate part if the parameters θ, β and σ and the funcfunc-tions to be

estimated for the location-scale part are known, we obtain a set of estimates for

2.3 Computation 23

unknown functions in the cure rate part and use linear interpolation to obtain thefull function estimates

2.3.1.4 Global estimation for parameters

The estimations of constant parameters σ, β, θ are updated via a profile

likeli-hood approach Maximizing the above log-likelilikeli-hood (2.3), L u,n (β, θ, σ; ˆ α, ˆ γ) leads

to updated parameter estimates, denoted by ˆβ, ˆ θ and ˆ σ.

The following is an iterative algorithm to implement our estimation procedure

Step 0 (Initialization) Choose initial estimates ˆα( ·) = ˆ α0(·) and ˆγ(·) = ˆγ0(·)

obtained by maximizing (2.2) with respect to all parameters for each distinctgrid point and using linear interpolation to obtain the full function estimates.While ˆβ, ˆ θ, ˆ σ are also estimated for each distinct grid point, they are

not used to update initial parameter estimates Next obtain ˆβ0, ˆθ0, ˆσ0 byglobally maximizing (2.3) with initial function estimates given Set ˆβ = ˆ β0,ˆ

θ = ˆ θ0, ˆσ = ˆ σ0

Step 1 (Local estimation for the location-scale part) Fix ˆβ, ˆ θ, ˆ σ, ˆ γ( ·) at their

2.3 Computation 24

current values For u = u i, obtain the local estimates by local estimationusing gradient information to obtain ˆa(u i) for i = 1, , N with the corre- sponding bandwidth h1 Use linear interpolation to obtain the full functionestimates ˆα( ·).

Step 2 (Local estimation for the cure rate part) Fix ˆβ, ˆ θ, ˆ σ, ˆ α( ·) at their current

values For u = u i, obtain the local estimates by local estimation usinggradient information to obtain ˆb(u i ) for i = 1, , n with the corresponding bandwidth h2 Use linear interpolation to obtain the full function estimatesˆ

γ( ·).

Step 3 (Global estimation) Update ˆβ, ˆ θ and ˆ σ by the global MLE with ˆ α(U i) and

ˆ

γ(U i) fixed at the estimates obtained from the previous steps.

Step 4 Repeat Steps 1, 2 and 3 until convergence.

Step 5 Fix ˆβ, ˆ θ, ˆ σ and ˆ γ( ·) at their current values The final estimate of α(u i)

is ˆα(u i ), the first p entries of ˆ a(u) obtained by local estimation for the

location-scale part with the bandwidth h opt1 for i = 1, · · · , N Use linear

interpolation to obtain the full function estimates

Step 6 Fix ˆβ, ˆ θ, ˆ σ and ˆ α( ·) at their current values The final estimate of γ(u i) is

ˆ

γ(u i ), the first r entries of ˆ b(u i) obtained by local estimation for the cure

rate part with the bandwidth h opt2 for i = 1, · · · , N Use linear interpolation

to obtain the full function estimates

2.3 Computation 25

In practice, one may choose N ≥ 20 equispaced points in the domain of U and the

plots of functional estimates should display sufficiently satisfactory visual results

To relax the constraint on the parameter, one may estimate log σ instead of σ.

The bandwidths hopt1 and hopt2 are usually chosen to be slightly larger than

h1 and h2 respectively to make the final functional estimates smoother For the

bandwidth h used in the initial one-step estimation, one may choose h1 = h2 sinceonly crude estimators are needed at Step 0

The Epanechnikov kernel K(t) = 0.75(1 − t2)+ is used for estimation in thisthesis The convergence criterion used in this thesis is that the average of thesquared differences between two consecutive estimates is less than or equal to

0.001 In our experiences it usually takes 20 to 30 iterations with our algorithm

before convergence is declared