Constrained parametric model for simultaneous inference of two cumulative incidence functions

We propose a parametric regression model for the cumulative incidence functions (CIFs) commonly used for competing risks data. The model adopts a modified logistic model as the baseline CIF and a generalized odds‐rate model for covariate effects, and it explicitly takes into account the constraint that a subject with any given prognostic factors should eventually fail from one of the causes such that the asymptotes of the CIFs should add up to one. This constraint intrinsically holds in a nonparametric analysis without covariates, but is easily overlooked in a semiparametric or parametric regression setting. We hence model the CIF from the primary cause assuming the generalized odds‐rate transformation and the modified logistic function as the baseline CIF. Under the additivity constraint, the covariate effects on the competing cause are modeled by a function of the asymptote of the baseline distribution and the covariate effects on the primary cause. The inference procedure is straightforward by using the standard maximum likelihood theory. We demonstrate desirable finite‐sample performance of our model by simulation studies in comparison with existing methods. Its practical utility is illustrated in an analysis of a breast cancer dataset to assess the treatment effect of tamoxifen, adjusting for age and initial pathological tumor size, on breast cancer recurrence that is subject to dependent censoring by second primary cancers and deaths.     

Sensitivity and insensitivity of breast cancer to tamoxifen

Tamoxifen is the endocrine treatment of choice for breast cancer. In several laboratory models in vivo tamoxifen is a tumoristatic agent. When MCF-7 breast cancer cells are inoculated into athymic mice, palpable tumors do not grow unless the animals are treated with estrogen, and tamoxifen inhibits estrogen-stimulated growth. If tamoxifen is stopped, tumors regrow. These results suggest that adjuvant tamoxifen therapy should involve long treatment periods (even lifetime) to prevent tumor recurrence. Unfortunately resistance to therapy and patient relapse inevitably occur, and such disease recurrence involving tamoxifen resistance is difficult to treat successfully. A laboratory model of endocrine therapy failure has been developed. When athymic mice with MCF-7 tumors are treated for 6–8 months with tamoxifen, several tumors grew and continued to grow in tamoxifen-treated mice. These estrogen receptor-positive tumors grow with either tamoxifen or estradiol. Tamoxifen-stimulated tumor growth has been observed in human endometrial tumors implanted into athymic animals. Growth of these tamoxifen-stimulated tumors can be inhibited with the pure antiestrogen ICI 164,384 upon withdrawal of tamoxifen. These data are discussed in terms of treatment strategies for tamoxifen-failed patients.     

On testing dependence between time to failure and cause of failure when causes of failure are missing

The hypothesis of independence between the failure time and the cause of failure is studied by using the conditional probabilities of failure due to a specific cause given that there is no failure up to certain fixed time. In practice, there are situations when the failure times are available for all units but the causes of failures might be missing for some units. We propose tests based on U-statistics to test for independence of the failure time and the cause of failure in the competing risks model when all the causes of failure cannot be observed. The asymptotic distribution is normal in each case. Simulation studies look at power comparisons for the proposed tests for two families of distributions. The one-sided and the two-sided tests based on Kendall type statistic perform exceedingly well in detecting departures from independence.     

Benefit with aromatase inhibitors in the adjuvant setting for postmenopausal women with breast cancer

The third-generation aromatase inhibitors, letrozole, anastrozole, and exemestane, have been shown to be effective both as alternatives to tamoxifen in first-line treatment of hormone-sensitive advanced breast cancer in postmenopausal women and following failure of first-line tamoxifen for endocrine therapy. These 3 agents are now being investigated as adjuvant therapy of early breast cancer, as alternative or complementary treatments to the standard, tamoxifen. Three treatment strategies are under investigation: replacement of tamoxifen as adjuvant therapy for 5 years (early adjuvant therapy), sequencing of tamoxifen before or after an aromatase inhibitor during the first 5 years (early sequential adjuvant therapy), or following 5 years of tamoxifen (extended adjuvant therapy). In the first adjuvant trial (Arimidex, Tamoxifen Alone or in Combination [ATAC]), anastrozole was significantly superior to tamoxifen in reducing risk of disease recurrence, and recently, the Breast International Group (BIG) trial BIG 1-98 demonstrated the significant superiority of letrozole over tamoxifen in improving disease-free survival. A large trial (International Collaborative Cancer Group [ICCG] trial 96) investigated sequencing of 2 to 3 years of exemestane after 2 to 3 years of tamoxifen and found that switching to exemestane was significantly superior in disease-free survival compared with continuing on tamoxifen. The Arimidex or Nolvadex (ARNO) and the small ITA (Italian Tamoxifen Arimidex) trials similarly sequenced anastrozole after tamoxifen and also found that sequencing reduced the hazard of recurrence compared with remaining on tamoxifen. Trial MA.17 evaluated extended adjuvant therapy with letrozole vs placebo following 5 years of tamoxifen. Disease-free survival was significantly improved with letrozole vs placebo, irrespective of whether patients had lymph node-positive or node-negative tumors. All 3 aromatase inhibitors were generally well tolerated. Results of these trials indicate that aromatase inhibitors provide important benefits relative to tamoxifen in each of these adjuvant treatment settings, but the optimal approach still needs to be defined. Other trials continue to investigate some of these adjuvant treatment strategies.     

Utilizing Cause-of-Death Information to Estimate Conditional Probabilities of Disease and Death

Conditional probabilities that do not require the assumption of independence among competing risks for identifiability are proposed for the analysis of carcinogenesis bioassay data as a reasonable adjustment for deaths or other removals due to competing risks. These conditional probabilities permit consideration of one type of tumor at a time, but in such a way that inferences are relevant to actual experimental conditions under which other diseases and causes of death are present and operating. The importance of assigning cause of death in bioassays is demonstrated by the fact that it allows the definition and identification of functions useful in the interpretation of carcinogenesis data, without requiring that a disease of interest be independent from competing risks. However, one proposed conditional probability does require sacrifice data for its identifiability.     

Estimation and prediction in a multi-state model for breast cancer

An important aim in clinical studies in oncology is to study how treatment and prognostic factors influence the course of disease of a patient. Typically in these trials, besides overall survival, also other endpoints such as locoregional recurrence or distant metastasis are of interest. Most commonly in these situations, Cox regression models are applied for each of these endpoints separately or to composite endpoints such as disease-free survival. These approaches however fail to give insight into what happens to a patient after a first event. We re-analyzed data of 2795 patients from a breast cancer trial (EORTC 10854) by applying a multi-state model, with local recurrence, distant metastasis, and both local recurrence and distant metastasis as transient states and death as absorbing state. We used an approach where the clock is reset on entry of a new state. The influence of prognostic factors on each of the transition rates is studied, as well as the influence of the time at which intermediate events occur. The estimated transition rates between the states in the model are used to obtain predictions for patients with a given history. Formulas are developed and illustrated for these prediction probabilities for the clock reset approach.     

Analysis of cause of death: Competing risks or progressive illness‐death model?

The analysis of cause of death is increasingly becoming a topic in oncology. It is usually distinguished between disease‐related and disease‐unrelated death. A frequently used approach is to define death as disease‐related when a progression to advanced phases has occurred before, otherwise as disease‐unrelated. The data are often analyzed as competing risks, while a progressive illness‐death model might in fact describe the situation more precisely. In this study, we investigated under which circumstances this misspecification leads to biased estimations of the state occupation probabilities. We simulated data according to the progressive illness‐death model in various settings, analyzed them with a competing risks model and with a progressive illness‐death model and compared them to the true state occupation probabilities. Censoring was either added independently of the status or based on the patients' status. The simulations showed that the censoring mechanism was decisive for the bias while neither the progression hazard nor the Markov property was important. Further, we found a slightly increased standard deviation for the competing risk estimator when censoring was independent of the patients' status. For illustration, both methods were applied to two practical examples of chronic myeloid leukemia (CML): one randomized controlled trial and one registry data set. While in the first case both estimators yielded almost identical results, in the latter case, visible differences were found between both methods.     

A joint model for longitudinal measurements and survival data in the presence of multiple failure types

Summary .   In this article we study a joint model for longitudinal measurements and competing risks survival data. Our joint model provides a flexible approach to handle possible nonignorable missing data in the longitudinal measurements due to dropout. It is also an extension of previous joint models with a single failure type, offering a possible way to model informatively censored events as a competing risk. Our model consists of a linear mixed effects submodel for the longitudinal outcome and a proportional cause-specific hazards frailty submodel ( Prentice et al., 1978 , Biometrics 34, 541–554) for the competing risks survival data, linked together by some latent random effects. We propose to obtain the maximum likelihood estimates of the parameters by an expectation maximization (EM) algorithm and estimate their standard errors using a profile likelihood method. The developed method works well in our simulation studies and is applied to a clinical trial for the scleroderma lung disease.     

Estimation of competing risks with general missing pattern in failure types

In competing risks data, missing failure types (causes) is a very common phenomenon. In this work, we consider a general missing pattern in which, if a failure type is not observed, one observes a set of possible types containing the true type, along with the failure time. We first consider maximum likelihood estimation with missing-at-random assumption via the expectation maximization (EM) algorithm. We then propose a Nelson-Aalen type estimator for situations when certain information on the conditional probability of the true type given a set of possible failure types is available from the experimentalists. This is based on a least-squares type method using the relationships between hazards for different types and hazards for different combinations of missing types. We conduct a simulation study to investigate the performance of this method, which indicates that bias may be small, even for high proportion of missing data, for sufficiently large number of observations. The estimates are somewhat sensitive to misspecification of the conditional probabilities of the true types when the missing proportion is high. We also consider an example from an animal experiment to illustrate our methodology.     

The analysis of failure times in the presence of competing risks.

Distinct problems in the analysis of failure times with competing causes of failure include the estimation of treatment or exposure effects on specific failure types, the study of interrelations among failure types, and the estimation of failure rates for some causes given the removal of certain other failure types. The usual formation of these problems is in terms of conceptual or latent failure times for each failure type. This approach is criticized on the basis of unwarranted assumptions, lack of physical interpretation and identifiability problems. An alternative approach utilizing cause-specific hazard functions for observable quantities, including time-dependent covariates, is proposed. Cause-specific hazard functions are shown to be the basic estimable quantities in the competing risks framework. A method, involving the estimation of parameters that relate time-dependent risk indicators for some causes to cause-specific hazard functions for other causes, is proposed for the study of interrelations among failure types. Further, it is argued that the problem of estimation of failure rates under the removal of certain causes is not well posed until a mechanism for cause removal is specified. Following such a specification, one will sometimes be in a position to make sensible extrapolations from available data to situations involving cause removal. A clinical program in bone marrow transplantation for leukemia provides a setting for discussion and illustration of each of these ideas. Failure due to censoring in a survivorship study leads to further discussion.     

Composite Hazard Functions Reflecting Competing Latent Processes

A method for fitting parametric models to apparently complex hazard rates in survival data is suggested. Hazard complexity may indicate competing causes of failure. A competing risks model is constructed on the assumption that a failure time can be considered as the first passage time of possibly several latent, stochastic processes competing in reaching a barrier. An additional assumption of independence between the hidden processes leads directly to a composite hazard function as the sum of the cause specific hazards. We show how this composite hazard model based on Wiener processes can serve as a flexible tool for modelling complex hazards by varying the number of processes and their starting conditions. An example with real data is presented. Parameter estimation and model assessment are based on Markov Chain Monte Carlo methods. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiple Decrement Analysis with Dependent Risks

In a competing risks problem where a well-defined population is exposed simultaneously to several causes of death, interest has centered on the estimation of the probability of death from a given cause when one or more other causes have been eliminated. A basic component of all available procedures for estimating these probabilities is the assumption that the several causes of death act independently—an unrealistic assumption in the context of human and animal populations. This article considers the estimation of these probabilities assuming the existence ofinterdependencies among the various causes of death. A general formula is derived based on a given set of crude probabilities of death as well as the characteristics of the joint distribution of random variables indicating death from the various causes. This formula identifies alternative assumptions, less restrictive than that of independent risks, which may he used for estimation purposes.     

Strategies for breast cancer therapy with antiestrogens

Current clinical research is focused upon the application of adjuvant therapy for the treatment of breast cancer. Combination chemotherapy is the most successful adjuvant therapy for premenopausal patients whereas the antiestrogen tamoxifen (1 or 2 yr) is successful in postmenopausal disease. We have developed a unifying strategy for the treatment of breast cancer. The thesis is based upon the application of continuous adjuvant therapy with tamoxifen in a low estrogen environment. Chemotherapy causes a chemical castration in premenopausal patients. In contrast, tamoxifen causes an increase in steroidogenesis. A combination of both approaches will work against each other until ovarian failure occurs. Patients should be checked for castration to provide a low estrogen environment in which tamoxifen, a competitive antagonist of estrogen action, can effectively work. Laboratory evidence using carcinogen-induced rat mammary tumor models demonstrates the efficacy of long-term therapy. Studies with the human breast cell line MCF-7 grown in athymic mice show that tamoxifen is a tumoristatic agent so that once the therapy is stopped, tumors can be regrown by estrogen administration. Patients should receive continuous tamoxifen therapy to prevent the growth-stimulating effects of adrenal steroids, environmental and phyto-estrogens.     

THE BETA-BINOMIAL MODEL: ACCOUNTING FOR INTER-TRIAL VARIATION IN REPLICATED DIFFERENCE AND PREFERENCE TESTS

Binomial tests are commonly used in sensory difference and preference testing under the assumptions that choices are independent and choice probabilities do not vary from trial to trial. This paper addresses violations of the latter assumption (often referred to as overdispersion) and accounts for variation in inter-trial choice probabilities following the Beta distribution. Such variation could arise as a result of differences in test substrate from trial to trial, differences in sensory acuity among subjects or the existence of latent preference segments. In fact, it is likely that overdispersion occurs ubiquitously in product testing. Using the Binomial model for data in which there is inter-trial variation may lead to seriously misleading conclusions from a sensory difference or preference test. A simulation study in this paper based on product testing experience showed that when using a Binomial model for overdispersed Binomial data, Type I error may be 0.44 for a Binomial test specification corresponding to a level of 0.05. Underestimation of Type I error using the Binomial model may seriously undermine legal claims of product superiority in situations where overdispersion occurs. The Beta-Binomial (BB) model, an extension of the Binomial distribution, was developed to fit overdispersed Binomial data. Procedures for estimating and testing the parameters as well as testing for goodness of fit are discussed. Procedures for determining sample size and for calculating estimate precision and test power based on the BB model are given. Numerical examples and simulation results are also given in the paper. The BB model should improve the validity of sensory difference and preference testing.     

Are aromatase inhibitors superior to antiestrogens?

Aromatase inhibitors (AIs) have been in use to treat metastatic breast cancer for over 25 years. Recently potent and specific AIs have been introduced, which, because of their low toxicity profile, are being used in the adjuvant and neoadjuvant situation and also for the prevention of breast cancer. The two non-steroidal AIs, anastrozole and letrozole, and the steroidal AI, exemestane, have all shown superiority to tamoxifen as first-line treatment for advanced breast cancer. Interestingly, the oestrogen receptor downregulator, fulvestrant, was shown to be equivalent to anastrozole when compared as second-line therapy after the failure of tamoxifen. The first adjuvant AI trial began in 1996 and recruited over 9000 patients (ATAC trial). Anastrozole was compared with tamoxifen and a combination of the two drugs. There were no significant differences between tamoxifen and the combination. However, anastrozole showed about a 20% improvement in disease-free survival in ER+ disease compared with the other treatments. An overall survival analysis will be reported later this year. Two trials have compared 5 years of tamoxifen with 2–3 years of tamoxifen, followed by 2–3 years of AI (one trial (ITA) used anastrozole and another (intergroup) exemestane). Both trials show a disease-free advantage for the switch to AI. In another study (MA17) 5 years of tamoxifen was followed by a randomisation to letrozole or placebo and showed a significant disease-free advantage to the AI. Both letrozole and anastrozole show superiority to tamoxifen when used as a neoadjuvant therapy. Anastrozole significantly reduced contralateral breast cancer compared with tamoxifen, and this has led to two prevention trials: one in women at risk comparing anastrozole with placebo and the other after excision of DCIS comparing anastrozole with tamoxifen (IBIS II). The NCI Canada has also just initiated a trial of exemestane for prevention. Nearly all data available indicate that AIs are superior to tamoxifen. The important question is whether survival is improved when they are used as adjuvant therapy?     

Competing Risk Analysis of Censored Survival Data

A competing risk model is developed to accommodate both planned Type I censoring and random withdrawals. MLE's, their properties, confidence regions for parameters and mean lifetimes are obtained for a model regarding random censoring as a competing risk and compared to those obtained for the model in which withdrawals are regarded as random censoring. Estimated net and crude probabilities are calculated and compared for the two models. The model is developed for two competing risks, one following a Weibull distribution and the other a Rayleigh distribution, and random withdrawals following a Weibull distribution.     

High-dimensional feature selection in competing risks modeling: A stable approach using a split-and-merge ensemble algorithm

Variable selection is critical in competing risks regression with high-dimensional data. Although penalized variable selection methods and other machine learning-based approaches have been developed, many of these methods often suffer from instability in practice. This paper proposes a novel method named Random Approximate Elastic Net (RAEN). Under the proportional subdistribution hazards model, RAEN provides a stable and generalizable solution to the large-p-small-n variable selection problem for competing risks data. Our general framework allows the proposed algorithm to be applicable to other time-to-event regression models, including competing risks quantile regression and accelerated failure time models. We show that variable selection and parameter estimation improved markedly using the new computationally intensive algorithm through extensive simulations. A user-friendly R package RAEN is developed for public use. We also apply our method to a cancer study to identify influential genes associated with the death or progression from bladder cancer.     

Inference in spline-based models for multiple time-to-event data, with applications to a breast cancer prevention trial

As part of the National Surgical Adjuvant Breast and Bowel Project, a controlled clinical trial known as the Breast Cancer Prevention Trial (BCPT) was conducted to assess the effectiveness of tamoxifen as a preventive agent for breast cancer. In addition to the incidence of breast cancer, data were collected on several other, possibly adverse, outcomes, such as invasive endometrial cancer, ischemic heart disease, transient ischemic attack, deep vein thrombosis and/or pulmonary embolism. In this article, we present results from an illustrative analysis of the BCPT data, based on a new modeling technique, to assess the effectiveness of the drug tamoxifen as a preventive agent for breast cancer. We extended the flexible model of Gray (1994, Spline-based test in survival analysis, Biometrics 50, 640-652) to allow inference on multiple time-to-event outcomes in the style of the marginal modeling setup of Wei, Lin, and Weissfeld (1989, Regression analysis of multivariate incomplete failure time data by modeling marginal distributions, Journal of the American Statistical Association 84, 1065-1073). This proposed model makes inference possible for multiple time-to-event data while allowing for greater flexibility in modeling the effects of prognostic factors with nonlinear exposure-response relationships. Results from simulation studies on the small-sample properties of the asymptotic tests will also be presented.     

Where do selective estrogen receptor modulators (SERMs) and aromatase inhibitors (AIs) now fit into breast cancer treatment algorithms?

The agents used for endocrine therapy in patients with breast cancer have changed markedly over the past decade. Tamoxifen remains the anti-oestrogen of choice, but could be replaced by the oestrogen receptor down-regulator ICI 182780 or by the fixed ring triphenylethylene arzoxifene (previously SERM III) soon. Whilst aminoglutethimide and 4-OH androstenedione were the aromatase inhibitors of choice, they have been replaced by non-steroidal (anastrozole and letrozole) and steroidal (exemestane) inhibitors of high potency and low side effect profile. Previously, often used treatments such as progestogens (megestrol acetate and medroxyprogesterone acetate) and androgens are now rarely used or confined to fourth or fifth line treatments. The LHRH agonist, goserelin, remains the treatment of choice for pre-menopausal patients with advanced breast cancer although recent randomised trials indicate a response, time to progression and survival advantage for the combination of goserelin and tamoxifen compared with goserelin alone.

The newer treatments have led to questions concerning the optimum sequence of agents to use in advanced breast cancer and as neo-adjuvant and adjuvant therapy in relation to surgery. Two trials of anastrozole compared with tamoxifen and one trial of letrozole compared with tamoxifen indicate that the new triazole aromatase inhibitors have a significant advantage over the anti-oestrogen with respect to time to progression and survival. Similarly, triazole aromatase inhibitors give faster and more complete responses compared with tamoxifen when used in post-menopausal women before surgery.

Major research questions remain with respect to the aromatase inhibitors used as adjuvant therapy. Anastrozole is being tested alone or in combination with tamoxifen compared with tamoxifen in the ‘so-called’ ATAC trial. Over 9000 patients have been randomised to this important study: the results will be available late-2001. A similar study comparing letrozole and tamoxifen started recently under the auspices of the Breast International Group. Importantly, this trial is also comparing the sequence of tamoxifen followed by letrozole (or vice versa). A similar trial of exemestane given after 2–3 years of tamoxifen compared with 5 years of tamoxifen is recruiting well as is a study comparing letrozole (or placebo) for 5 years after 5 years of adjuvant tamoxifen. These studies may show that aromatase inhibitors are superior to tamoxifen or that a sequence is preferable.

ICI 182780 causes complete oestrogen receptor down-regulation leading to a the lack of agonist activity of the drug. Two trials of ICI 182780 compared with anastrozole for advanced disease will report later this year and a comparison with tamoxifen next year. Arzoxifene (SERM III) is being tested against tamoxifen. These studies are likely to result in new anti-oestrogens being introduced into the clinic.

Most of our endocrine treatments deprived the tumour cell of oestradiol. In vitro experiments with MCF-7 cells indicate that tumour cells can adapt and then grow in response to low oestrogen concentrations in the tissue—culture medium. Importantly, the cells were shown to apoptose in response to high oestrogen concentrations. A recent clinical trial has demonstrated a high response rate to stilboestrol given after a median of four previous oestrogen depriving endocrine therapies. These data and the newer treatments available indicate a need to re-think our general approach to endocrine therapy and endocrine prevention.     

Comparison of mastectomy with tamoxifen for treating elderly patients with operable breast cancer.

STUDY OBJECTIVE--Comparison of tamoxifen and mastectomy in treatment of breast cancer in elderly patients. DESIGN--Randomised trial of treatment of operable breast cancer by wedge mastectomy or tamoxifen, with median follow up 24 and 25 months respectively (range 1-63). SETTING--University hospital; most patients from primary catchment area. PATIENTS--135 consecutive patients with breast cancer aged over 70 with operable tumours (less than 5 cm maximum diameter); 68 were allocated to tamoxifen group and 67 to mastectomy group. Histological diagnosis by biopsy. Two incorrect randomisations in each group. Patient characteristics similar in the two groups and all under care of one surgical team. INTERVENTIONS--Mastectomy group received wedge mastectomy plus excision of symptomatic axillary lymph nodes. Tamoxifen group received continuous treatment with tamoxifen 20 mg twice daily. Patients in tamoxifen group received wedge mastectomy if there was sign of local progression. Those in mastectomy group received further excision or radiotherapy for locoregional recurrence and when local treatments had been exhausted or metastatic disease diagnosed they received tamoxifen. END POINT--Treatment efficacy was assessed by local control of disease and by survival. MAIN RESULTS--Mortality from metastatic cancer in tamoxifen group was 7 (10.6%) and in mastectomy group 10 (15.3%) (NS). There was no difference in survival between the two groups. In mastectomy group 70% remained alive and free of local recurrence at 24 months; in tamoxifen group only 47% remained alive and free of local progression. In mastectomy group locoregional recurrence occurred in 16 patients and metastatic disease in 13; in tamoxifen group locoregional progression occurred in 29 patients and metastatic disease in seven. CONCLUSIONS--As a high proportion of patients treated with tamoxifen eventually required surgery treatment of elderly patients with breast cancer should include mastectomy. Optimum treatment may include both mastectomy and tamoxifen.