Semi-parametric Inference for Cure Rate Models 1

Transcription

1 Semi-parametric Inference for Cure Rate Models 1 Fotios S. Milienos jointly with N. Balakrishnan, M.V. Koutras and S. Pal University of Toronto, This research is supported by a Marie Curie International Outgoing Fellowship within the 7th European Community Framework Programme.

2 Outline 1 Cure rate models 2 Piecewise linear approximation 3 Simulation study 4 Illustrative example 5 Conclusions

3 Introduction The primary aim of this work is the study of survival times or more generally, the times till the occurrence of an event. This event may be: the occurrence or recurrence of a disease; the return to prison or rearrest of a released prisoner; the death of a patient; etc

4 Introduction The primary aim of this work is the study of survival times or more generally, the times till the occurrence of an event. This event may be: the occurrence or recurrence of a disease; the return to prison or rearrest of a released prisoner; the death of a patient; etc Using the models found in traditional survival analysis, every item will experience the event of interest, at some point.

5 Introduction The primary aim of this work is the study of survival times or more generally, the times till the occurrence of an event. This event may be: the occurrence or recurrence of a disease; the return to prison or rearrest of a released prisoner; the death of a patient; etc Using the models found in traditional survival analysis, every item will experience the event of interest, at some point. population Traditional survival analysis the event of interest... after some time...

6 Introduction Cure rate models allow for a proportion of items which will never experience the event of interest.

7 Introduction Cure rate models allow for a proportion of items which will never experience the event of interest. Cure rate models the event of interest (susceptibles) population... after some time... cured items (non-susceptibles) These items are called cured or non-susceptibles or long-term survivors or immune or immortals etc.

8 Introduction The cure rate models have been extensively studied during the last decades. We can find more than 200 publication the last years and manymany applications... The reason is the great improvement of treatments for a range of diseases, but also to the fact that in many social phenomena a number of individuals are not susceptible to the event of interest. Applications may be found in biomedical studies, criminology, finance, industrial reliability etc (see the monographs by Maller and Zhou, 1996; Ibrahim, Chen and Sinha, 2005).

9 Binary cure rate model Boag (1949) and Berkson and Gage (1952) are the first works (to the best of my knowledge) which took into account the existence of two subgroups in the population: the susceptibles and the non -susceptibles (cured).

10 Binary cure rate model Boag (1949) and Berkson and Gage (1952) are the first works (to the best of my knowledge) which took into account the existence of two subgroups in the population: the susceptibles and the non -susceptibles (cured). Based on this assumption, the population survival function of the timeto-event T, is given by the following mixture model S P (t)=p(t > t)=p 0 + (1 p 0 )S(t) where p 0 is the probability of a patient to be cured, and S(t) = P(T > t susceptibles) is the survival function of susceptibles. Note that lim t S P (t)=p 0.

11 Binary cure rate model What problems do we have? What is the distribution of susceptibles S(t) = P(T > t susceptibles)? How the covariates can be incorporated into the model (in p 0 or/and S(t) = P(T > t susceptibles))? How to estimate the parameters of the model?

12 Binary cure rate model S P (t)=p(t > t)=p 0 + (1 p 0 )S(t) What problems do we have? What is the distribution of susceptibles S(t) = P(T > t susceptibles)? How the covariates can be incorporated into the model (in p 0 or/and S(t) = P(T > t susceptibles))? How to estimate the parameters of the model?

13 Binary cure rate model S P (t)=p(t > t)=p 0 + (1 p 0 )S(t) What problems do we have? What is the distribution of susceptibles S(t) = P(T > t susceptibles)? How the covariates can be incorporated into the model (in p 0 or/and S(t) = P(T > t susceptibles))? How to estimate the parameters of the model?

14 Binary cure rate model S P (t)=p(t > t)=p 0 + (1 p 0 )S(t) What problems do we have? What is the distribution of susceptibles S(t) = P(T > t susceptibles)? How the covariates can be incorporated into the model (in p 0 or/and S(t) = P(T > t susceptibles))? How to estimate the parameters of the model?

15 Binary cure rate model S P (t)=p(t > t)=p 0 + (1 p 0 )S(t) What problems do we have? What is the distribution of susceptibles S(t) = P(T > t susceptibles)? How the covariates can be incorporated into the model (in p 0 or/and S(t) = P(T > t susceptibles))? How to estimate the parameters of the model? Parametric, semi-parametric and non-parametric approaches can be found in the relevant literature.

16 Binary cure rate model Assumptions for the distribution of susceptibles: Exponential distribution (e.g. Berkson and Gage, 1952; Mould, 1973; Ghitany and Maller, 1992; Ghitany, 1993; Farewell, 1977a; Balakrishnan and Pal, 2013c) S(t)=P(T > t susceptibles)=exp( λt),λ>0; Weibull distribution (e.g. Farewell, 1977b, 1982, 1986; Bentzen et al.,1989; Struthers and Farewell, 1989; Balakrishnan and Pal, 2013b,c) S(t)=exp( λt) α,λ,α>0;

17 Binary cure rate model Assumptions for the distribution of susceptibles: Cox s proportional hazard model (e.g. Kuk and Chen, 1992; Sy and Taylor, 2000, 2001; Peng and Dear, 2000; Fang, Li and Sun, 2005; Zhao et al., 2014) h(t)=h 0 (t)exp(x β),λ>0; Kaplan-Meyer estimator (e.g. Taylor, 1995; Laska and Meisner, 1992; Maller and Zhou, 1992); Generalized F distribution (e.g. Peng, Dear and Denham, 1998); Log-normal distribution (Balakrishnan and Pal, 2013a) and others.

18 Binary cure rate model The estimation depends on the available data (censoring, truncation etc): MLE and EM algorithm; A marginal likelihood estimation approach (especially in Cox s PH model); An iterative least squares method (Berkson and Gage, 1952).

19 Binary cure rate model The estimation depends on the available data (censoring, truncation etc): MLE and EM algorithm; A marginal likelihood estimation approach (especially in Cox s PH model); An iterative least squares method (Berkson and Gage, 1952). Logistic regression model for p 0 (logistic-binary mixture model), i.e. p 0 = p 0 (x;β)= 1 1+exp(x β).

20 Competing cause scenario A number of competing causes left alive after a treatment, say M; let W 1,W 2,...,W M be the random variable for the time-to-event due to ith (i=1,2,...,m) competing cause. The random variables W 1,W 2,...,W M are assumed i.i.d. with cdf F(t)= 1 S(t) and are also independent of M.

21 Competing cause scenario A number of competing causes left alive after a treatment, say M; let W 1,W 2,...,W M be the random variable for the time-to-event due to ith (i=1,2,...,m) competing cause. The random variables W 1,W 2,...,W M are assumed i.i.d. with cdf F(t)= 1 S(t) and are also independent of M. Therefore, the population time-to-event T is given by with P(W 0 = )=1. T = min{w 0,W 1,...,W M },

22 Competing cause scenario Then its survival function has the following form S P (t)=p(t > t)=p(w 0 > t,w 1 > t,...,w M > t) = P(M = 0)+P(W 1 > t,w 2 > t,...,w M > t,m 1) = P(M = 0)+ S(t) i P(M = i) i=1

23 Competing cause scenario Then its survival function has the following form S P (t)=p(t > t)=p(w 0 > t,w 1 > t,...,w M > t) = P(M = 0)+P(W 1 > t,w 2 > t,...,w M > t,m 1) = P(M = 0)+ S(t) i P(M = i) i=1 For example, let M follow a Poisson distribution, with mean θ (0, ); then, S P (t)=exp( θ)+ S(t) i exp( θ) θi i! = exp( θf(t)) i=1 e.g. Tsodikov, Ibrahim and Yakovlev (2003) and Ibrahim et al. (2005).

24 Competing cause scenario S P (t)=p(m = 0)+ S(t) i P(M = i) i=1 Again, the next questions must be answered: what is the distribution of M (the number of competing causes); what is the distribution of W i (time-to-event due to ith competing cause); how the covariates can be incorporated into the model; how to deal with the estimation of the parameters.

25 Competing cause scenario Assumptions for the distribution of the number of competing causes M: Poisson distribution (e.g. Yakovlev et al. 1993; Cantor and Shuster, 1992; Yakovlev, Cantor and Shuster, 1994; Chen, Ibrahim and Sinha, 1999; Hashimoto, Ortega et al., 2013); Negative binomial distribution (e.g. Castro, Cancho and Rodrigues, 2009; Ortega et al., 2014); COM-Poisson distribution (e.g. Rodrigues et al., 2009; Balakrishnan and Pal, 2012, 2013a,b,c); Compound weighted Poisson distribution (e.g. Rodrigues et al., 2011); Geometric distribution (e.g. Louzada et al., 2014);

26 Unified represantion Taking into account the relation S P (t)= S(t) i P(M = i) i=0 we can say that, both of the two classes of cure rate models (the binary and competing cause model) can be expressed as S P (t)=e[s(t) M ]=G M (S(t)), where the expectations is taken w.r.t. M and G M is the probability generating function of M (see e.g. Tsodikov et al., 2003).

27 Our semi-parametric approach In this work, we are going to study both the logistic-binary mixture and the logistic-poisson mixture models. The common hazard function h(t) of the time-to-event due to ith competing cause is approximated by a piecewise linear function. h( t ) φ N φn 1 φ 2 φ 1 φ 0 τ 0 τ1 τ 2 τ N 1 τ N t

28 Our semi-parametric approach The piecewise linear approximation (PLA), for the common hazard function h(t) of W i, is based on the following decisions: the number of lines, N, to be used; the selection of cut points τ 0 < τ 1 <...<τ N 1 < τ N for forming the line segments. Of course, we further assume that the PLA is continuous function at cut points.

29 Hence, Our semi-parametric approach h L (t)= N (c j + s j t)i [τj 1,τ j ](t), j=1 where I [τj 1,τ j ](t)=1 if and only if t [τ j 1,τ j ] and c j = φ j τ j φ j φ j 1 τ j τ j 1, s j = φ j φ j 1 τ j τ j 1, for j= 1,2,...,N, with φ j = c j + s j τ j and φ 0 = c 1 + s 1 τ 0. h( t ) φ N φn 1 φ 2 φ 1 φ 0 τ 0 τ1 τ 2 τ N 1 τ N t

30 Data and Estimation The MLE and EM algorithm is carried out for estimating the parameters of the model. We consider the scenario where the time-to-event is subject to noninformative random right censoring.

31 Data and Estimation : non-cured/ susceptibles : cured start time end of study

32 Data and Estimation Denoting with C i and T i the censoring time and lifetime of the ith individual, respectively, we then observe and δ i = I(T i C i ), i.e. δ i = Y i = min{t i,c i } { 1, if Yi is a time-to-event 0, if Y i is a censoring time,,i=1,2,...,n.

33 Data and Estimation ( y1, δ 1) = (exact lifetime,1) ( y2, δ 2) = (censoring time,0) ( y3, δ 3) = (censoring time,0) ( y4, δ 4) = (exact lifetime,1) ( y5, δ 5) = (censoring time,0) ( y6, δ 6) = (censoring time,0) ( y7, δ 7) = (exact lifetime,1) ( y8, δ 8) = (censoring time,0) ( y9, δ 9) = (censoring time,0) start time end of study

34 Data and Estimation: likelihood function From n pairs of times and censoring indicators (y 1,δ 1 ),...,(y n,δ n ), the likelihood function can be written as L=L(θ; x, y,δ) n f P (y i, x i ;θ) δ i S P (y i, x i ;θ) 1 δ i. i=1

35 Data and Estimation: likelihood function From n pairs of times and censoring indicators (y 1,δ 1 ),...,(y n,δ n ), the likelihood function can be written as L=L(θ; x, y,δ) Thus, the likelihood becomes n f P (y i, x i ;θ) δ i S P (y i, x i ;θ) 1 δ i. i=1 L(θ; x, y,δ) n (1 p 0 (x i,β)) δ i f U (y i, x i ;θ) δ i [p 0 (x i,β)+(1 p 0 (x i,β))s U (y i, x i ;θ)] 1 δ i, i=1 where S U and f U are the probability density and survival function of susceptibles, respectively.

36 Data and Estimation: likelihood function Let us now assume that I i = { 1, if the ith individual is susceptible 0, if the ith individual is cured,,i=1,2,...,n. Note that the values of I i for censored items are not observable.

37 Data and Estimation: likelihood function Let us now assume that I i = { 1, if the ith individual is susceptible 0, if the ith individual is cured,,i=1,2,...,n. Note that the values of I i for censored items are not observable. Then, the complete likelihood function can be written as L c = L c (θ; x, y,δ) (1 p 0 (x i,β)) f U (y i, x i ;θ) p 0 (x i,β) 1 I i i 1 i 1 i 0 [(1 p 0 (x i,β))s U (y i, x i ;θ)] I i i 0 (note that i 1 I i = 1).

38 Data and Estimation: EM algorithm For the E-step of the EM algorithm, we have where w (z) i = E[I i θ (z), O]= exp(x i β(z) )S U (y i, x;θ (z) ) 1+exp(x i β(z) )S U (y i, x;θ (z) ), Binary case : S U (t, x;θ)=s L (y i ;φ (z) ), Poisson case : S U (t, x;θ)= exp( θ(x,β)f L(t;φ)) exp( θ(x,β)). 1 exp( θ(x,β))

39 Numerical results A set of data of size n=300 is generated, in which the random variables W i follow a Weibull distribution. We assume that: 1. we have only one covariate with 4 possible values/groups (x {1,2,3,4}); 2. the censoring times C i follow Exponential distributions;

40 Numerical results A set of data of size n=300 is generated, in which the random variables W i follow a Weibull distribution. We assume that: 1. we have only one covariate with 4 possible values/groups (x {1,2,3,4}); 2. the censoring times C i follow Exponential distributions; We fix the cured proportions for the first and fourth group to 0.30 and 0.15, respectively; therefore, based on the equations 0.30= 1 1+exp(β 0 + β 1 1),0.15= 1 1+exp(β 0 + β 1 4) we have that β 0 = and β 1 =

41 Numerical results The cut points τ 0,τ 1,...,τ N 1 of the PLA are taken to be the sample percentiles of the uncensored data (i.e. 0,1/N,2/N,...,(N 1)/Nth percentiles); the last point τ N is the maximum of Y i. The initial values for β 0 and β 1 can be given by replacing the cured proportions with the observed censoring proportions; the initial values for φ 0,φ 1,...,φ N may be computed using a set of estimates through φ j = ĥ(τ j),j = 0,1,...,N; in our case, the Kaplan-Meier estimator was used. Moreover, we assume for each group that the censoring proportions exceed the cured proportions by c= 0.10 or c= 0.20.

42 Numerical results Table: Logistic-binary mixture model: Weibull distribution and low censored proportion (c =.10; n = 300) Cured proportions for Groups 1-4:(.3,.242,.192,.15) Censored proportions for Groups 1-4:(.3,.242,.192,.15)+c,c=.10 Sample size for Groups 1-4: (60,105,65,70),n=300 Parameters:β 0 =.5515,β 1 =.2958,α=1.5,γ=1.5(parameters of the distribution) ˆφ i,(ˆα, ˆγ) N (ˆβ 0, ˆβ 1 ) ˆp 0 l i=0 i=1 i=2 i=3 i=4 Mean 1 (.542,.310) (.301,.240,.189,.148) (.542,.311) (.301,.239,.188,.148) (.543,.311) (.301,.239,.188,.148) (.543,.311) (.301,.239,.188,.148) (.546,.296) (.304,.244,.194,.154) (1.504,1.510) RMSE 1 (.362,.148) (.050,.030,.031,.038) 2 (.362,.148) (.050,.030,.031,.038) 3 (.362,.148) (.050,.030,.031,.038) 4 (.362,.148) (.050,.030,.031,.038) (.377,.144) (.053,.031,.028,.035).007 Std 1 (.363,.148) (.050,.030,.031,.038) (.364,.148) (.050,.030,.031,.038) (.363,.148) (.050,.030,.031,.038) (.363,.148) (.050,.030,.031,.038) (.378,.145) (.053,.031,.028,.034) (.081,.070)

43 Data and Estimation

44 Numerical results Table: Logistic-binary mixture model: Weibull distribution and high censored proportion (c=.20; n=300) Cured proportions for Groups 1-4:(.3,.242,.192,.15) Censored proportions for Groups 1-4:(.3,.242,.192,.15)+c,c=.20 Sample size for Groups 1-4: (60,105,65,70),n=300 Parameters:β 0 =.5515,β 1 =.2958,α=1.5,γ=1.5(parameters of the distribution) ˆφ i,(ˆα, ˆγ) N (ˆβ 0, ˆβ 1 ) ˆp 0 l i=0 i=1 i=2 i=3 i=4 Mean 1 (.460,.343) (.314,.244,.187,.144) (.460,.348) (.313,.242,.185,.141) (.460,.349) (.313,.242,.185,.141) (.462,.349) (.312,.242,.185,.141) (.457,.336) (.316,.247,.190,.148) (1.526,1.503) RMSE 1 (.491,.183) (.073,.045,.039,.044) 2 (.492,.185) (.072,.044,.039,.044) 3 (.490,.185) (.072,.044,.039,.044) 4 (.489,.185) (.072,.044,.039,.044) (.503,.193) (.071,.040,.036,.043).010 Std 1 (.484,.177) (.072,.045,.039,.044) (.485,.178) (.071,.044,.039,.043) (.483,.178) (.071,.044,.039,.043) (.483,.178) (.071,.044,.039,.043) (.496,.190) (.069,.040,.036,.043) (.099,.092)

45 Data and Estimation

46 Numerical results Table: Logistic-Poisson mixture model: Weibull distribution and low censored proportion (c=.10; n=300) Cured proportions for Groups 1-4:(.3,.242,.192,.15) Censored proportions for Groups 1-4:(.3,.242,.192,.15)+c,c=.10 Sample size for Groups 1-4: (60,105,65,70),n=300 Parameters:β 0 =.5515,β 1 =.2958,α=1.5,γ=1.5(parameters of the distribution) ˆφ i,(ˆα, ˆγ) N (ˆβ 0, ˆβ 1 ) ˆp 0 l i=0 i=1 i=2 i=3 i=4 Mean 1 (.593,.307) (.292,.231,.182,.143) (.584,.303) (.294,.235,.185,.146) (.585,.303) (.294,.235,.185,.146) (.585,.303) (.294,.235,.185,.146) (.585,.303) (.294,.235,.185,.146) (1.477,1.543) RMSE 1 (.371,.143) (.051,.032,.029,.035) 2 (.365,.141) (.051,.031,.028,.035) 3 (.365,.140) (.051,.031,.028,.035) 4 (.365,.140) (.051,.031,.028,.035) (.365,.140) (.051,.031,.028,.035) (.005,.01) Std 1 (.369,.143) (.051,.030,.028,.035) (.365,.141) (.051,.030,.028,.034) (.364,.141) (.051,.030,.028,.034) (.364,.140) (.051,.030,.028,.034) (.365,.140) (.051,.030,.028,.035) (.067,.093)

47 Data and Estimation

48 Numerical results One more quantity of interest is the probability P(cured T > t)= p 0 (x β) p 0 (x β)+(1 p 0 (x β))s U (t;θ). Hence, in the next Table we mention the mean value (and sample standard deviation, in parentheses) of the above probability at the point t 0.95 for which P(cured T > t 0.95 )=0.95 (at each of the four possible values of our covariate; i.e. for x=1,2,3,4).

49 Numerical results Table: The sample mean of P(cured T > t 0.95 ) (n=300,c=.10) and for each possible value of covariate x. Weibull (α = 1,γ = 1.5)/Binary Weibull (α = 1.5,γ = 1.5)/Binary PLA PLA x N = 1 N = 2 N = 3 N = 4 Par* N = 1 N = 2 N = 3 N = 4 Par* 1.946(.02).945(.02).945(.02).946(.02).947(.02).971(.02).968(.01).967(.02).968(.02).947(.02) 2.947(.02).945(.02).945(.02).946(.02).947(.02).971(.03).967(.02).966(.02).967(.02).946(.02) 3.946(.02).944(.02).944(.02).945(.02).946(.02).970(.03).964(.02).964(.02).964(.02).945(.03) 4.944(.03).942(.03).942(.03).943(.03).944(.02).968(.04).961(.03).961(.03).961(.03).943(.03) Weibull (α = 1,γ = 1.5)/Poisson Weibull (α = 1.5,γ = 1.5)/Poisson 1.935(.04).933(.04).931(.04).931(.04).941(.03).958(.02).931(.04).932(.04).932(.04).931(.03) 2.933(.04).930(.04).928(.04).928(.04).940(.03).960(.02).928(.04).930(.04).931(.04).929(.03) 3.930(.04).926(.05).925(.05).925(.05).939(.03).962(.03).925(.05).927(.05).928(.05).926(.04) 4.926(.05).921(.05).920(.05).920(.06).936(.04).963(.03).921(.06).923(.06).924(.06).921(.05) *The respective estimations using the parametric approach.

50 Melanoma data The study took place from 1991 to 1995, and follow-up was conducted until 1998 (n=427); the data, taken from Ibrahim et al. (2005), present the survival times (in years) until the patient s death or the censoring time. In our application, the nodule category is the only covariate with 4 possible values x = 1,2,3,4 (lymph nodes involved in the disease) and the group sizes are n 1 = 111,n 2 = 137,n 3 = 87 and n 4 = 82, with respective censoring proportions 67.6%,61.3%,52.9% and 32.9%. The results from the PLA will be compared with those of the parametric approach.

51 Melanoma data Table: The estimation of the logistic-binary mixture model. N = 1 N = 2 N = 3 N = 4 N = 5 N = 6 Par 1 Par 2 p (.046).615(.046).617(.046).615(.046).615(.046).615(.046).632(.048).667(.040) p (.042).503(.042).504(.042).502(.042).503(.042).502(.042).506(.041).557(.030) p (.053).422(.053).423(.053).421(.052).421(.052).421(.052).379(.047).440(.033) p (.049).274(.049).275(.049).273(.049).273(.049).273(.049).267(.056).330(.046) φ φ φ φ φ φ φ l AIC BIC *Par 1 : the parametric Log-normal/logistic-binary mixture model of Balakrishnan and Pal (2013a) Par 2 : the parametric Weibull/logistic-binary mixture model of Balakrishnan and Pal (2013b)

52 Melanoma data P( M = 0 T > t) x=1 x=2 x=3 0.4 x=4 t Figure: The probability P(M = 0 T > t) = P(cured T > t), for each nodule category.

53 Similar works Larson and Dinse (1985) used a piecewise constant approximation for the baseline hazard function of the Cox s PH model (under a different competing cause scenario). In Lo et al. (1993), the baseline hazard was determined now by a piecewise continuous linear function (for the Bernoulli model and a different estimation approach). Chen and Ibrahim (2001) studied the Poisson mixture model where the common cdf of competing causes was approximated by a piecewise constant function (using different estimation approach). A piecewise constant approximation under Bayesian framework is met in Ibrahim, Chen and Sinha (2001) and Kim et al. (2007).

54 Conclusions The accuracy of PLA for estimating the regression coefficients is very close to that gained by a parametric approach; a similar remark made by Taylor (1995), through a KM-based non-parametric approach. The last line of our PLA seems to be significantly affected by the underlying censoring mechanism. In Lo et al. (1993) was found that the choice of cut points had a minimal effect on the results which is in accordance with our findings.

55 Conclusions The suggested non-parametric approach is quite flexible and the larger the N the more non-parametric the model is. We suggest to choose small to moderate values of N and trying to check the robustness of the estimates of the regression coefficients. No more lines can offer better approximations if the additional estimates of φ s lie very close to existing lines. Finally, we noted that more than 4 or 5 lines do not offer any improvement using either the maximum likelihood or MSE criterion.

56 Future work Future work includes: assuming that the number of competing causes follows a wider class of discrete distributions; under a fully non-parametric framework about the number of competing causes; a Cox proportional hazard model approach using different estimation methods; the asymptotic properties of these models.

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