Practical application of cure mixture model for longterm censored survivor data from a withdrawal clinical trial of patients with major depressive disorder
 Ichiro Arano†^{1, 2},
 Tomoyuki Sugimoto†^{3},
 Toshimitsu Hamasaki^{3}Email author and
 Yuko Ohno^{2}
DOI: 10.1186/147122881033
© Arano et al; licensee BioMed Central Ltd. 2010
Received: 21 January 2010
Accepted: 23 April 2010
Published: 23 April 2010
Abstract
Background
Survival analysis methods such as the KaplanMeier method, logrank test, and Cox proportional hazards regression (Cox regression) are commonly used to analyze data from randomized withdrawal studies in patients with major depressive disorder. However, unfortunately, such common methods may be inappropriate when a longterm censored relapsefree time appears in data as the methods assume that if complete followup were possible for all individuals, each would eventually experience the event of interest.
Methods
In this paper, to analyse data including such a longterm censored relapsefree time, we discuss a semiparametric cure regression (Cox cure regression), which combines a logistic formulation for the probability of occurrence of an event with a Cox proportional hazards specification for the time of occurrence of the event. In specifying the treatment's effect on diseasefree survival, we consider the fraction of longterm survivors and the risks associated with a relapse of the disease. In addition, we develop a treebased method for the time to event data to identify groups of patients with differing prognoses (cure survival CART). Although analysis methods typically adapt the logrank statistic for recursive partitioning procedures, the method applied here used a likelihood ratio (LR) test statistic from a fitting of cure survival regression assuming exponential and Weibull distributions for the latency time of relapse.
Results
The method is illustrated using data from a sertraline randomized withdrawal study in patients with major depressive disorder.
Conclusions
We concluded that Cox cure regression reveals facts on who may be cured, and how the treatment and other factors effect on the cured incidence and on the relapse time of uncured patients, and that cure survival CART output provides easily understandable and interpretable information, useful both in identifying groups of patients with differing prognoses and in utilizing Cox cure regression models leading to meaningful interpretations.
Background
In a clinical study involving patients with major depressive disorder (MDD), a longterm placebo treatment is not acceptable due to an increased risk of suicide. In such a situation, a randomized withdrawal design is one of the most useful approaches for comparing the longterm efficacy and safety of a drug and a placebo. A typical randomized withdrawal study consists of an initial phase during which all patients are given openlabel active treatment followed by randomization of the responders to continued doubleblind treatment with either active or placebo treatment. The advantage of this study design is the reduced duration of placebo treatment and an earlyescape endpoint, such as a relapse of signs or symptoms of the disease or a lack of efficacy.
Study designs and relapse rates from several randomized withdrawal studies in patients with MDD
Source  Study objectives, design and analysis  Reported relapse rates  

Rapaport et al. [1]  Escitalopram continuation treatment to prevent relapse; a multicenter, placebo controlled, randomized withdrawal study; 36week randomized treatment; KaplanMeier estimate and logrank test as primary statistical analysis  Escitalopram  26.0%* (109) 
Placebo  40.0%* (116)  
Keller et al. [2]  Longterm efficacy and tolerability of gepirone ER; a multicenter, placebo controlled, randomized withdrawal study; 4044 weeks of randomized treatment; chisquare test; KaplanMeier estimate and logrank test as Primary statistical analysis  Gepirone ER  20.6% (26/126) 
Placebo  28.2% (35/124)  
Kamijima et al.[3]  Efficacy, safety and tolerability of sertraline in the prevention of relapse; a multicenter, placebo controlled, randomized withdrawal study; 16week randomized treatment; KaplanMeier estimate and logrank test as primary statistical analysis  Sertraline  08.5% (10/117) 
Placebo  19.5% (23/118)  
Perahia et al. [4]  Efficacy, safety and tolerability of duloxetine in the prevention of relapse; A multicenter, placebo controlled, randomized withdrawal study; 26week randomized treatment; KaplanMeier estimate, logrank test as primary statistical analysis  Duloxetine  17.4% (23/132) 
Placebo  28.5% (39/137)  
Kocsis et al. [5]  Longterm efficacy and safety of venlafaxine ER in preventing recurrence; a multicenter, placebo controlled, randomized withdrawal study; 12 months randomized treatment; KaplanMeier estimate and logrank test as primary statistical analysis  Venlafaxine ER  23.1%* (129) 
Placebo  42.0%* (129) 
In this paper, two methods of analysis were considered: a semiparametric cure regression (Cox cure regression) and a treebased method, known as survival CART (Classification And Regression Trees) [6]. The two methods are generally used for the analysis of timetoevent censored data and may provide the findings different from the standard Cox regression by assuming that there are cured individuals in the data. The former method combines a logistic formulation for the probability of occurrence of an event with a proportional hazards specification for the time of occurrence of the event [7–13], so that the effects of the treatment and other factors can be interpreted separately into those on the proportion of cured patients and the failure time of uncured patients. In the latter method, termed cure survival CART, the probability of occurrence of an event with the exponential or Weibull distribution for the time of the event occurrence is modelled using binary treestructure. Our approach was to incorporate the LR test statistic from the fitting of the cure survival regressions for recursive partitioning procedures, although the standard survival CART method adapts the method of CART paradigm using the logrank statistic [14]. Cure survival CART was performed to supplement the results of Cox cure regression and to provide a simple and useful interpretation. These findings would provide valuable information for the future treatment of patients with MDD. This paper is structured as follows: first, we describe the models, parameter estimations, and algorithms of the two methods. We then illustrate some aspects of the two methods, using data collected in a sertraline randomized withdrawal study in patients with MDD [3], and end with a discussion.
Methods
Cox cure regression
The model
Suppose that the data (T_{ i }, Δ_{ i }, X_{ i }, Z_{ i }) are available for an individual i = 1,..., n. T_{ i }denotes the time to the occurrence of the event defined by min( , U_{ i }), where and U_{ i }are the random variables of true survival and censoring, respectively. Δ_{ i }denotes the censoring indicator Δ_{ i }= I ( = T_{ i }), where I(·) is the indicator function. X_{ i }and Z_{ i }are the covariate vectors related to the cured incidence and uncured survival, respectively.
where αis the qdimensional parameter corresponding to X_{ i }, which usually consists of the form (1, X_{i1},..., X_{iq1})^{T} with the intercept term X_{i0 }= 1.
Parameter estimation
where , S_{ i }(t) = exp[exp(β^{T} Z_{ i })Λ_{0}(t)] and θ= (α^{T}, β^{T})^{T}
The estimate of (θ, Λ_{0}) is obtained by maximizing l_{ f }over (θ, Λ_{0}). This maximization is performed using the EM (ExpectationMaximization) algorithm, given a suitable stating value (θ^{(0)}, ) of (θ, Λ_{0}). We prepare a suitable θ^{(0)} using the Monte Carlo method [7, 8]. Once we have θ^{(0)}, a suitable corresponding to θ^{(0)} is computed by a NewtonRaphson method discussed in Sugimoto and Hamasaki [13]. Although the optimization technique based on the EM algorithm is computationally fast, it easily fails to converge depending on starting values in Cox cure regression [10]. Therefore, providing an appropriate (θ^{(0)}, ) prudently in advance using these methods, then we go ahead to the EM algorithm.
Then, the Mstep of over (β, Λ_{0}) is replaced by maximizing only over βin treating Λ_{0} as a nonparametric function. Therefore, the Mstep is easily performed by maximizing and over αand β, respectively, via a NewtonRaphson method.
Now, if θ^{(m) }= (α^{(m)}, β^{(m}) are the current estimates obtained by the Mstep for a given w^{(m)}, then current estimate of Λ_{0} is written as . For the next (m +1) th Estep, w^{(m) }is updated to w^{(m+1) }by substituting (θ^{(m}, ) into (θ, Λ_{0}) in (1). Finally, ( ), which provides becomes an estimate of (θ, Λ_{0}) in a series of the EMiteration. From our limited experiences with real data and simulation, it is recommended that a variety of starting value θ^{(0)} be used to ensure that a global maximum is found. For further discussions, please see Sugimoto and Goto [8], Peng and Dear [9], Sy and Taylor [10] and Sugimoto et al. [12]
Cure survival CART
In general, treebased methods provide classifications of patients with differing prognoses that may help clarify the association of patient characteristics and survival. In this section, a treebased method for censored data including longterm survivors is developed, based on cure survival regression. With respect to the cost of calculations in estimating the parameters, rather than the straightforward use of the semiparametric Cox cure regression discussed in the previous section, the fitting of parametric cure regression is more reasonable, where the exponential distribution is assumed to be an underlying distribution for the latency time of relapse. However, since an assumption of exponential distribution is restrictive, the Weibull distribution is also considered. First, the model will be described and then the algorithm will be discussed.
The model
The major area of interest was the examining of cure rates and the identification of MDD patients with different prognoses. Parameter transformations such as log{c_{ h }/(1  c_{ h })} = α_{ h }, and were adapted and the EM algorithm, described in the previous section, is used to estimate parameters c_{ h }, μ_{ h }and ρ_{ h }. For the Weibull model, it is recommended that a suitable starting point of ρ_{ h }is sought in advance.
The algorithm
The treebased method requires the splitting, pruning, and selection of a pruned subtree to be specified [6]. Since highdimensional data may give a complicated treestructure, the following specifications were considered to simplify the tree structure, and the results were subsequently interpreted.
Splitting
A split satisfying over s ∈ is selected, where the set is composed of all possible binary splitting rules considered at node h. The two new daughter nodes, l (h) and r (h), determined by are generated, and the tree as a whole grows. Such a generation of nodes is performed successively at any node until a prespecified stopping condition (e.g., the maximal number of terminal nodes or the minimal number of observations in terminal nodes) is satisfied.
Pruning
The "Pruning procedure" [6] determines the order how to remove superfluous subtrees of a largest tree to build a good fitting simple treebased model. This process is repeated until only a small fraction of the data remains. The final tree expresses a logical rule representing an extreme outcome group.
where indicates that is a subtree of ; and is the smallest optimally pruned subtree such that for every optimally pruned subtree .
The pruning algorithm is necessary for finding optimal subtrees because the number of possible subtrees grows very rapidly as a function of tree size.
Selection of a pruned subtree
The selection of a pruned subtree can be based on a resampling technique (crossvalidation) to correct for overoptimism due to split point optimization [6]. The most popular method for obtaining treebased estimates of prediction error (or expected deviance) is Vfold crossvalidation. The data are divided up into V sets and subsamples of about equal size, and trees are grown from the subsamples . For any γ, an optimally pruned subtree (γ) and MLEs and in are obtained.
Let γ_{k* }be the value of that minimizes R^{ cv }(γ) of , k = 0,1,2,⋯. Then a tree corresponding to γ_{k* }is selected.
We adopt V = 10 because a smaller value of V is preferred to V = n in the application of CART [18]. While 10fold crossvalidation is a standard method for selecting tree size, it is subject to considerable variability. Therefore, in the application to this data, we performed 500 replications of 10fold crossvalidation and then determined γ_{k* }to minimize the average of 500 pairs of . We will be able to use the "1SE (Standard Error)" rule [6] to choose a simpler tree, where such an SE is directly estimated by the variation of R^{ cv }(γ) with 500 replications in this application.
Results
Study design and major results of a sertraline randomized withdrawal study
A multicenter, placebocontrolled, randomized withdrawal study was used to evaluate the efficacy and safety of sertraline in Japanese patients with MDD [3]. Following a 1week observational period for washout, only those patients who responded after 8 weeks of openlabel sertraline treatment were randomly assigned to receive one of two subsequent 16week doubleblind treatments with sertraline or a placebo. Patients who did not respond after 8 weeks of the openlabel treatment were discontinued from the study. The primary variable was a relapse of the disease during the doubleblind phase, which was defined as either (i) having a score of Hamilton rating scale for depression (HAMD) (17 items) of 18 point or greater and a clinical global impression (compared to baseline of the openlabel phase) of "nochange" or "worse", during two consecutive visits, or (ii) being unable to continue treatment due to insufficient efficacy.
Result of standard logistic and Cox regressions to a sertraline randomized withdrawal study in patients with MDD
95% CI  

Standard logistic regression  Estimates  SE  Lower  Upper  pvalue 
Intercept  1.686  1.687  1.621  4.992  0.3178 
Treatment  0.968  0.429  0.152  1.848  0.0239 
Baseline HAMD score at OP  0.007  0.062  0.111  0.134  0.9096 
Baleline HAMD score at DP  0.100  0.068  0.240  0.029  0.1443 
Gender  1.842  0.575  3.123  0.817  0.0014 
Age  0.026  0.020  0.013,  0.068  0.2020 
The number of episodes  0.174  0.139  0.016  0.531  0.2111 
Duration from the 1st episode  0.002  0.003  0.008  0.005  0.5575 
Duration of this episode  0.002  0.022  0.041  0.046  0.9296 
Interval from the previous episode  0.004  0.007  0.008  0.019  0.5553 
Complication  0.572  0.413  0.241  1.388  0.1654 
Maximum loglikelihood  81.603  
AIC  185.206  
95% CI  
Standard Cox regression  Estimates  SE  Lower  Upper  pvalue 
Treatment  0.845  0.388  1.604  0.086  0.0293 
Baseline HAMD score at OP  0.005  0.053  0.099  0.109  0.9222 
Baseline HAMD score at DP  0.063  0.059  0.052  0.180  0.2847 
Gender  1.639  0.543  0.575  2.702  0.0025 
Age  0.023  0.018  0.058  0.012  0.1977 
The number of episodes  0.152  0.130  0.406  0.102  0.2399 
Duration from the 1st episode  0.002  0.003  0.004  0.007  0.5729 
Duration of this episode  0.005  0.019  0.033  0.042  0.8038 
Interval from the previous episode  0.003  0.006  0.014  0.009  0.6274 
Complication  0.518  0.358  1.221  0.184  0.1482 
Maximum loglikelihood*  180.917  
AIC  381.834 
Application of Cox cure regression and cure survival CART to sertraline data
Cox cure regression, including influential covariates for both cured incidence and uncured survivals, provides flexibility in model building. However, this approach may open discussions on the possibility of an overparameterization of models and the identifiability between the parameters of cured incidence and uncured survivals [10], although the parameters of the standard cure model are identifiable in some sense [19]. In our situation, there was more scientific and practical interest in estimating the cured incidence as the objective of the study was to show how well the drug prevents an eventual episode of recurring illness (cured incidence), compared with placebo and, further, how other covariates influence cured incidence. Thus, in selecting the covariates, we considered that, giving priority to cured incidence over uncured survivals, a minimal number or no covariate in uncured survivals may be appropriate, and then we suggested the following guidelines on variable selection in our situation: (1) treatment is always factored into cured incidence, (2) either a minimal number or no covariates are included into uncured survivals, (3) a covariate already included into cured incidence is not included into uncured survivals. Following these guidelines, the "best" subset of all possible combinations of covariates can be selected by a minimum value of Akaike's information criterion (AIC) [20], given by is (θ, Λ_{0}) that maximizes l_{ f }(θ, Λ_{0}), discussed in the previous Methods section.
Result of best subset for Cox cure regression
95% CI  

Cured incidence  Estimates  SE  Lower  Upper  pvalue 
Intercept  1.571  2.870  4.053  7.196  0.5840 
Treatment  1.177  0.429  0.335  2.018  0.0061 
Baseline HAMD score at DP  0.122  0.071  0.262  0.018  0.0869 
95% CI  
Uncured survivals  Estimates  SE  Lower  Upper  pvalue 
Gender  1.953  0.535  0.905  3.001  0.0003 
Complication  0.967  0.418  1.798  0.159  0.0193 
Maximum loglikelihood  179.578  
AIC  369.155 
Result of best subset for standard logistic and Cox regressions
95% CI  

Standard logistic regression  Estimates  SE  Lower  Upper  pvalue 
Intercept  2.865  0.848  1.203  4.527  0.0007 
Treatment  1.021  0.421  0.196  1.846  0.0153 
Baseline HAMD score at DP  0.107  0.066  0.236  0.222  0.1036 
Gender  1.737  0.563  2.841  0.633  0.0021 
The number of episodes  0.133  0.114  0.091  0.357  0.2443 
Complication  0.585  0.407  0.213  1.384  0.1507 
Maximum loglikelihood  81.868  
AIC  177.736  
95% CI  
Standard Cox regression  Estimates  SE  Lower  Upper  pvalue 
Treatment  0.913  0.379  1.655  0.171  0.0159 
Gender  1.511  0.534  0.465  2.557  0.0046 
The number of episodes  0.131  0.109  0.345  0.083  0.2296 
Maximum loglikelihood*  183.520  
AIC  373.040 
From the results obtained by the two cure survival CARTs, refined Cox cure regression was reperformed; the model included treatment, the baseline HAMD score at DP and the interaction between the treatment and the baseline HAMD score at DP into cured incidence, and gender and complication into uncured survivals. The baseline HAMD score at DP was categorized into two groups: baseline HAMD score at DP >6 ( = 1) and baseline HAMD score at DP ≤ 6 (= 0).
Result of refined Cox cure regression
95% CI  

Cured incidence  Estimates  SE  Lower  Upper  pvalue 
Intercept  0.753  1.374  1.940  3.447  0.5835 
Treatment  30.954  10.001  11.353  50.555  0.0020 
Baseline HAMD score at DP  0.233  0.646  1.449  1.033  0.7183 
Treatment × Baseline HAMD score at DP  30.180  13.497  56.634  3.725  0.0254 
95% CI  
Uncured survivals  Estimates  SE  Lower  Upper  pvalue 
Gender  1.926  0.535  0.878  2.974  0.0003 
Complication  0.895  0.469  1.814  0.024  0.0563 
Maximum loglikelihood  177.499  
AIC  366.998 
Discussion
The results obtained by Cox cure regression and cure exponential CART agree with findings reported by several authors [21–23]. For example, Nierenberg et al. [23] reported that a greater number of residual symptoms and higher HAMD scores are associated with a higher probability of relapse. Although there are several inconsistencies regarding the gender difference in the course of the relapse, Kuehner [21] reported that female patients have a higher risk of earlier occurrence of relapse. In addition, comorbidity of MMD with other illnesses has been widely reported [22]. The variable of complication used in the analysis was a binary variable of "yes" or "no", but original data included more detailed information on the number and type of complications for each patient. Therefore, further investigation on the effect of the number and type of complications on uncured survivals (or cured incidence) is necessary. Furthermore, the cutoff point for the baseline HAMD score at DP at 6 points suggested by cure survival CART was nearly equal to the HAMD definition of full remission (a score of 7 points or less). It is generally considered that the final goal of treatment for MDD is to achieve and maintain remission and the prevention of relapse.
Cox cure regression (or survival cure CART) generally requires a longterm followup [10, 24]. The sertraline study discussed in this paper was intended to contribute to a new drug application in Japan, so the length of the followup period (16 weeks) in the study was minimized to merely detect the drug's effect, in order to reduce the duration of unnecessary exposure of patients to the drug or placebo. However, in the application of Cox cure regression to real data, before the formal analysis, an assessment of whether or not the length of followup is sufficient would be useful for interpreting the result. To confirm this for the sertraline data, the q_{ n }test discussed by Maller and Zhou [24] was performed for the sertraline and placebo groups, constructed by the estimated cure rates and censoring distribution for this data. For the sertraline group, the observed value of 0.08547 of q_{ n }was between 94% and 96% critical points of the test, which supported that the length of followup for the sertraline group was acceptably minimal and the data had levelled off. On the other hand, for the placebo group, the observed value of 0.0085 of q_{ n }was much smaller than the value of 0.068 for the 95% point, which did not support that the length of followup for the placebo group was sufficient and that the data had levelled off. According to the results of two q_{ n }tests, we could conclude that the length of followup period in the sertraline study was sufficient at least to detect the drug's effect compared with placebo.
In variable selection for the fitting of Cox cure regression to the sertraline data, we gave priority to cured incidence over uncured survivals. However, if there was more scientific interest in when the illness may recur rather than in the eventual cure, giving priority to uncured survivals over cured incidence could be appropriate. There are several aspects of variable selection, depending on the applications of interest.
In the paper, we discussed the two methods of cure Cox regression and cure survival CART. As described in Method section, the former method is a semiparametric regression, but the latter method use a parametric cure regression. Although the cure survival CART output provided information in refining Cox cure regression leading to meaningful interpretations for the sertraline data, note that there is the potential inconsistency between the two regressions when they consider both for data as the estimates from the semiparametric Cox cure regression and those from the parametric Cox cure regression are usually sensitive to the baseline distribution assumption. Our future challenge is to develop the semiparametric cure survival CART with fewer amounts of computations.
Conclusions
In this study, a semiparametric cure regression was used to investigate the latency time of recurrence observed in a sertraline randomized withdrawal study in patients with MDD. In specifying the treatment's effect on diseasefree survival, account was taken of the fraction of longterm survivors and the risks associated with the relapse of the disease. In addition, a treebased method, i.e., the cure survival CART, was used to analyze the time to event data in order to identify groups of patients with differing prognoses. The following are the main findings: (1) Cox cure regression reveals facts on who may be cured, and how the treatment and other factors effect on the cured incidence and on the relapse time of uncured patients. (2) Cure survival CART output provides easily understandable and interpretable information, useful both in identifying groups of patients with differing prognoses and in utilizing Cox cure regression leading to meaningful interpretations.
The methods discussed in this paper could be applied to the development of stratification schemes for future clinical studies and the identification of patients suitable for studies involving therapy targeted at a specific prognostic group. This is would be beneficial as it is often desirable to understand the correlation between a patient's characteristics and relapse times to aid in the design of clinical studies.
Abbreviations
 CART:

Classification and regression trees
 CI:

Confidence interval
 Doubleblind phase:

DP
 EM:

ExpectationMaximization
 HAMD:

Hamilton rating scale for depression
 Likelihood ratio:

LR
 MDD:

Major depressive disorder
 MLE:

Maximum likelihood estimate
 Openlabel phase:

OP
 SE:

Standard error.
Declarations
Acknowledgements
The authors are grateful to two referees and editors for their constructive comments and valuable suggestions. The authors also thank Pfizer Japan for giving permission to use the data from the sertraline randomized withdrawal trial for this research. This research was financially supported by a program grant from The Ministry of Education, Culture, Sports, Science and Technology, Japan (GrantinAid for Scientific Research (C), NO.20500255) and The 2008 Inamori Research Grants Program.
Authors’ Affiliations
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