Posterior maximization and averaging for Bayesian working model choice in the continual reassessment method.

The continual reassessment method (CRM) is a method for estimating the maximum tolerated dose in a dose-finding study. Traditionally, use is made of a single working model or 'skeleton' idealizing an underlying true dose-toxicity relationship. This working model is chosen either by discussion with investigators or published data, before the beginning of the trial or simply on the basis of operating characteristics. To overcome the arbitrariness of the choice of such a single working model, Yin and Yuan (biJ. Am. Statist. Assoc. 2009; 104:954-968) propose a model averaging over a set of working models. Here, instead of averaging, we investigate some alternative Bayesian model criteria that maximize the posterior distribution. We propose three adaptive model-selecting CRMs using the Bayesian model selection criteria, in which we specify in advance a collection of candidate working models for the dose-toxicity relationship, especially initial guesses of toxicity probabilities, and adaptively select the only one working model among the candidates updated by using the original CRM for each working model, based on the posterior model probability, the posterior predictive loss or the deviance information criteria, during the course of the trial. These approaches were compared via a simulation study with the model averaging approach.

Experimental designs for phase I and phase I/II dose-finding studies.

We review the rationale behind the statistical design of dose-finding studies as used in phase I and phase I/II clinical trials. We underline what the objectives of such dose-finding studies should be and why the widely used standard design fails to meet any of these objectives. The standard design is a "memoryless" design and we discuss how this impacts on practical behaviour. Designs introduced over the last two decades can be viewed as designs with memory and we discuss how these designs are superior to memoryless designs. By superior we mean that they require less patients overall, less patients to attain the maximum tolerated dose (MTD), and concentrate a higher percentage of patients at and near to the MTD. We reanalyse some recently published studies in order to provide support to our contention that markedly better results could have been achieved had a design with memory been used instead of a memoryless design.

Dose finding and O6-alkylguanine-DNA alkyltransferase study of cisplatin combined with temozolomide in paediatric solid malignancies.

Cisplatin may have additive activity with temozolomide due to ablation of the DNA repair protein O6-alkylguanine-DNA alkyltransferase (MGMT). This phase I/II study determined recommended combination doses using the Continual Reassessment Method, toxicities and antitumour activity in paediatric patients, and evaluated MGMT in peripheral blood mononuclear cells (PBMCs) in order to correlate with haematological toxicity. In total, 39 patients with refractory or recurrent solid tumours (median age approximately 13 years; 14 pretreated with high-dose chemotherapy, craniospinal irradiation, or having bone marrow involvement) were treated with cisplatin, followed the next day by oral temozolomide for 5 days every 4 weeks at dose levels 80 mg m(-2)/150 mg m(-2) day(-1), 80/200, and 100/200, respectively. A total of 38 patients receiving 113 cycles (median 2, range 1-7) were evaluable for toxicity. Dose-limiting toxicity was haematological in all but one case. Treatment-related toxicities were thrombocytopenia, neutropenia, nausea-vomiting, asthenia. Hearing loss was experienced in five patients with prior irradiation to the brain stem or posterior fossa. Partial responses were observed in two malignant glioma, one brain stem glioma, and two neuroblastoma. Median MGMT activity in PBMCs decreased after 5 days of temozolomide treatment: low MGMT activity correlated with increased severity of thrombocytopenia. Cisplatin-temozolomide combinations are well tolerated without additional toxicity to single-agent treatments; the recommended phase II dosage is 80 mg m(-2) cisplatin and 150 mg m(-2) x 5 temozolomide in heavily treated, and 200 mg m(-2) x 5 temozolomide in less-heavily pretreated children.

Dose-finding designs for HIV studies.

We present a class of simple designs that can be used in early dose-finding studies in HIV. Such designs, in contrast with Phase I designs in cancer, have a lot of the Phase II flavor about them. Information on efficacy is obtained during the trial and is as important as that relating to toxicity. The designs proposed here sequentially incorporate the information obtained on viral reduction. Initial doses are given from some fixed range of dose regimens. The doses are ordered in terms of their toxic potential. At any dose, a patient can have one of three outcomes: inability to take the treatment (toxicity), ability to take the treatment but insufficient reduction in viral load (viral failure), and ability to take the treatment as well as a sufficient reduction of viral load (success). A clear goal for some class of designs would be the identification of the dose leading to the greatest percentage of successes. Under certain assumptions, which we identify and discuss, we can obtain efficient designs for this task. Under weaker, sometimes more realistic assumptions, we can still obtain designs that have good operating characteristics in identifying a level, if such a level exists, having some given or greater success rate. In the absence of such a level, the designs will come to an early closure, indicating the ineffectiveness of the new treatment.

Use of hypertonic saline in the treatment of severe refractory posttraumatic intracranial hypertension in pediatric traumatic brain injury.

OBJECTIVES: To evaluate the effect of prolonged infusion of 3% hypertonic saline (514 mEq/L) and sustained hypernatremia on refractory intracranial hypertension in pediatric traumatic brain injury patients. DESIGN: A prospective study. SETTING: A 24-bed Pediatric Intensive Care Unit (Level III) at Children's Hospital. PATIENTS: We present ten children with increased intracranial pressure (ICP) resistant to conventional therapy (head elevation at 30 degrees, normothermia, sedation, paralysis and analgesia, osmolar therapy with mannitol, loop diuretic, external ventricular drainage in five patients), controlled hyperventilation (Pco2, 28-35 mm Hg), and barbiturate coma. We continuously monitored ICP, cerebral perfusion pressure (CPP), mean arterial pressure, central venous pressure, serum sodium concentrations, serum osmolarity, and serum creatinine. INTERVENTIONS: A continuous infusion of 3% saline on a sliding scale was used to achieve a target serum sodium level that would maintain ICP <20 mm Hg once the conventional therapy and barbiturate coma as outlined above failed to control intracranial hypertension. MEASUREMENTS AND MAIN RESULTS: The mean duration of treatment with 3% saline was 7.6 days (range, 4-18 days). The mean highest serum sodium was 170.7 mEq/L (range, 157-187 mEq/L). The mean highest serum osmolarity was 364.8 mosm/L (range, 330-431 mosm/L). The mean highest serum creatinine was 1.31 mg/dL (range, 0.4-5.0 mg/dL). There was a steady increase in serum sodium versus time zero that reached statistical significance at 24, 48, and 72 hrs (p < .01). There was a statistically significant decrease in ICP spike frequency at 6, 12, 24, 48, and 72 hrs (p < .01). There was a statistically significant increase in CPP versus time zero at 6, 12, 24, 48, and 72 hrs (p < .01). There was a statistically significant increase in serum osmolarity versus time zero at 12 hrs (p < .05) and at 24, 48, and 72 hrs (p < .01). Two patients developed acute renal failure and required continuous veno-venous hemodialysis; these were concurrent with an episode of sepsis and multisystem organ dysfunction. Both recovered full renal function with no electrolyte abnormalities at the time of discharge. CONCLUSION: An increase in serum sodium concentration significantly decreases ICP and increases CPP. Hypertonic saline is an effective agent to increase serum sodium concentrations. Sustained hypernatremia and hyperosmolarity are safely tolerated in pediatric patients with traumatic brain injury. Controlled trials are needed before recommendation of widespread use.

Estimating average regression effect under non-proportional hazards.

We present an estimator of average regression effect under a non-proportional hazards model, where the regression effect of the covariates on the log hazard ratio changes with time. In the absence of censoring, the new estimate coincides with the usual partial likelihood estimate, both estimates being consistent for a parameter having an interpretation as an average population regression effect. In the presence of an independent censorship, the new estimate is still consistent for this same population parameter, whereas the partial likelihood estimate will converge to a different quantity that depends on censoring. We give an approximation of the population average effect as integral beta(t)dF(t). The new estimate is easy to compute, requiring only minor modifications to existing softwares. We illustrate the use of the average effect estimate on a breast cancer dataset from Institut Curie. The behavior of the estimator, its comparison with the partial likelihood estimate, as well as the approximation by integral beta(t)dF(t)are studied via simulation.

Another look at two phase I clinical trial designs.

This note is a response to a recent paper by Korn et al. in which two phase I trial designs were compared, the designs in question being the standard design and the CRM design. The authors concluded that: (i) CRM designs will take longer to complete than standard designs; and (ii) CRM designs are less safe than the standard designs. These conclusions followed from a set of simulations for three different dose toxicity situations. The first purpose of this note is to point out that these conclusions lean on false assumptions. The claims are in error. In their comparisons Korn et al. never in fact used CRM, as defined in O'Quigley, Pepe and Fisher, but a modified version. The second purpose of this note is to look at the same cases studied by Korn et al. but this time using a correctly defined CRM model. Using the same comparison tools of Korn et al. we will see that for these situations, and indeed for a very wide class of other situations not presented here, not only are (i) and (ii) not true but that the exact opposite holds.

Two-sample continual reassessment method.

We discuss an extension of the continual reassessment method (CRM) for use in phase I dose-finding studies. The extension enables the method to be applied to two groups of patients to determine the appropriate dose levels for each group. The method takes the specification of a simple relationship between the dose-toxicity curves for the two groups and runs the CRM on the bivariate model using maximum likelihood. We prove consistency of the method under fairly weak conditions and provide several simulations to give an idea how the method works in practice. We also undertake an evaluation of its performance by considering three possible situations: The first is the two-sample CRM, which directly uses a working model for the relationship between the two groups, carrying out a single trial using this method; the second situation carries out single trials for each of the two groups separately using the original (one-sample) CRM. The third situation is the case where such heterogeneity is ignored and the two groups are pooled into a single group, again using the original (one-sample) CRM. Simulations are carried out under a large class of model misspecifications, both of the dose-toxicity relationships and of the functional form linking the groups, and are backed up by asymptotic results. Our conclusions match intuition: The first scheme gives the most favorable results when the two groups are different but share some features. When the groups are very different, the second scheme performs similarly to the first for finite sample sizes while having some advantages in terms of asymptotic efficiency. The third, as expected, gives the best results in the absence of patient heterogeneity. The two-sample method appears particularly advantageous when there may not be enough subjects in one of the subgroups for it to be feasible to carry out two trials.

Phase I and pharmacokinetic study of intraperitoneal topotecan.

OBJECTIVE: To determine the maximum tolerated dose and pharmacokinetics of topotecan when administered by the intraperitoneal route. METHODS: A dose-escalating Phase I trial was conducted in which fifteen % of the total dose was given as an intraperitoneal bolus in two litres of D5W and the remainder was given as a continuous intraperitoneal infusion over 24 hours. Treatments were given every 21 days. Pharmacokinetic analyses were performed at the recommended phase II dose. RESULTS: Seventeen patients received a total of 43 cycles at 21-day intervals. The maximum tolerated dose was 4 mg/m2 and acute dose-limiting toxicity was neutropenia. Other toxicities included leukopenia, anemia, emesis, fever, and abdominal pain. Although no objective responses were achieved, five of ten patients with ascites had a decrease in fluid accumulation with administration of intraperitoneal topotecan. The recommended phase II dose is 3 mg/m2. Pharmacokinetic analysis performed at a dose of 3 mg/m2 demonstrated that elimination from the peritoneal cavity followed second-order kinetics with k1 = 1.6 hr(-1), k2 = 0.3 hr(-1) and first and second-phase half-lives of 0.49 and 2.7 hours, respectively. Plasma pharmacokinetic behavior was best described by first-order kinetics with k = 0.5 hr(-1) and a half-life of 3.9 hours. The pharmacologic advantage, expressed as the peritoneal to plasma AUC ratio was 31.2. CONCLUSIONS: Intraperitoneal administration of topotecan at 3 mg/m2 results in a substantial increase in drug exposure for the peritoneal cavity without compromising systemic exposure; this may be beneficial for the treatment of patients with ovarian cancer or intraperitoneal carcinomatosis.

Continual reassessment method: a likelihood approach.

The continual reassessment method as described by O'Quigley, Pepe, and Fisher (1990, Biometrics 46, 33-48) leans to a large extent upon a Bayesian methodology. Initial experimentation and sequential updating are carried out in a natural way within the context of a Bayesian framework. In this paper we argue that such a framework is easily changed to a more classic one leaning upon likelihood theory. The essential features of the continual reassessment method remain unchanged. In particular, large sample properties are the same unless the prior is degenerate. For small samples and as far as the final recommended dose level is concerned, simulations indicate that there is not much to choose between a likelihood approach and a Bayesian one. However, for in-trial allocation of dose levels to patients, there are some differences and these are discussed. In contrast to the Bayesian approach, a likelihood one requires some extra effort to get off the ground. This is because the likelihood equation has no solution until we observe a toxicity. Initially then we suggest working with either a standard Up-and-Down scheme or standard continual reassessment method until toxicity is observed and then switching to the new scheme.

A two-stage procedure for survival studies with surrogate endpoints.

A two-stage procedure for survival studies with surrogate endpoints is proposed. The objective of the procedure is to reduce the duration of a survival study relative to classical procedure. A surrogate endpoint is an event which is related to survival time and may occur earlier during follow up. In the first stage, all patients are followed to the primary endpoint in order to evaluate the strength of the relationship between the surrogate endpoint and survival. In the second stage, follow up is terminated on patients who reach the surrogate endpoint. Indirect inferences on the survival endpoint is now possible by virtue of the first stage analysis. We present methods for data collected in the two-stage procedure, for estimating the survivorship function, S(t), and for comparing two treatment groups using a non-parametric permutation test. The methods are applied to the results of a study of resected lung cancer.

Integral evaluation for continual reassessment method.

In this paper we show how the exact analytical solutions to the numerical integrals required in the implementation of the continual reassessment method can be evaluated. The formulas have been given but, with increasing sample size, rapidly become unwieldy. We develop an algorithm which uses the numerical binary representation of the subset size and, in the process, enables us to easily identify those members of the subset needed in the integral evaluation. The complex combinatorial problem of selecting the set of all subsets with a particular characteristic is then greatly simplified. We describe the nature and type of all the variables and parameters used in a PASCAL procedure.

Predictive capability of proportional hazards regression.

A measure of the predictive capability of a proportional hazards regression is derived. The measure is based on the residuals appropriate to proportional hazards regression. A population version is presented and can be seen not to depend on the censoring mechanism under the provision that any such censoring be independent or conditionally independent of the failure mechanism given the covariate. For the special case of a Weibull regression model, for which the covariate distribution follows binary, uniform, normal, or exponential laws, we derive analytic results. These alone give credence to the measure which can be seen to reflect strength of regression effect, as quantified by the parameter estimate, although on a scale between 0 and 1, independently of the intercept or shape parameter of the particular Weibull law and only weakly dependent on the covariate distribution. Extensions to partial and multiple measures of predictive ability are straightforward. An example is provided.

Estimating the probability of toxicity at the recommended dose following a phase I clinical trial in cancer.

The problem of point and interval estimation following a Phase I trial, carried out according to the scheme outlined by O'Quigley, Pepe, and Fisher (1990, Biometrics 46, 33-48), is investigated. A reparametrization of the model suggested in this earlier work can be seen to be advantageous in some circumstances. Maximum likelihood estimators, Bayesian estimators, and one-step estimators are considered. The continual reassessment method imposes restrictions on the sample space such that it is not possible for confidence intervals to achieve exact coverage properties, however large a sample is taken. Nonetheless, our simulations, based on a small finite sample of 20, not atypical in studies of this type, indicate that the calculated intervals are useful in most practical cases and achieve coverage very close to nominal levels in a very wide range of situations. The relative merits of the different estimators and their associated confidence intervals, viewed from a frequentist perspective, are discussed.

Methods for dose finding studies in cancer clinical trials: a review and results of a Monte Carlo study.

We discuss some of the statistical approaches to the design and analysis of phase I clinical trials in cancer. An attempt is made to identify the issues, particular to this type of trial, that should be addressed by an appropriate methodology. A brief review of schemes currently in use is provided together with our views of the extent to which any particular scheme addresses the main issues. Some simulations are provided together with graphical illustration of the operating characteristics of the particular methods. It appears that the continual reassessment method is preferable to other contending schemes.

The problem of a covariate-time qualitative interaction in a survival study.

We introduce a test for the equality of two survival distributions against the specific alternative of crossing hazards. Although this kind of alternative is somewhat rare, designing a test specifically aimed at detecting such departures from the null hypothesis in this direction leads to powerful procedures, upon which we can call in those few cases where such departures are suspected. Furthermore, the proposed test and an approximate version of the test are seen to suffer only moderate losses in power, when compared with their optimal counterparts, should the alternative be one of proportional hazards. Our interest in the problem is motivated by clinical studies on the role of acute graft versus host disease as a risk factor in leukemic children and we discuss the analysis of this study in detail. The model we use in this work is a special case of the one introduced by Anderson and Senthilselvan (1982. Applied Statistics 31, 44-51). We propose overcoming an inferential problem stemming from their model by using the methods of Davies (1977, Biometrika 64, 247-254; 1987, Biometrika 74, 33-43) backed up by resampling techniques. We also look at an approach relying directly on resampling techniques. The distributional aspects of this approach under the null hypothesis are interesting but, practically, its behaviour is such that its use cannot be generally recommended. Outlines of the necessary asymptotic theory are presented and for this we use the tools of martingale theory.

Nonparametric tests of association between survival time and continuously measured covariates: the logit-rank and associated procedures.

O'Brien's logit-rank procedure (1978, Biometrics 34, 243-250) is shown to arise as a score test based on the partial likelihood for a proportional hazards model provided the covariate structure is suitably defined. Within this framework the asymptotic properties claimed by O'Brien can be readily deduced and can be seen to be valid under a more general model of censoring than that considered in his paper. More important, perhaps, it is now possible to make a more natural and interpretable generalization to the multiple regression problem than that suggested by O'Brien as a means of accounting for the effects of nuisance covariates. This can be achieved either by modelling or stratification. The proportional hazards framework is also helpful in that it enables us to recognize the logit-rank procedure as being one member of a class of contending procedures. One consequence of this is that the relative efficiencies of any two procedures can be readily evaluated using the results of Lagakos (1988, Biometrika 75, 156-160). Our own evaluations suggest that, for non-time-dependent covariates, a simplification of the logit-rank procedure, leading to considerable reduction in computational complexity, is to be preferred to the procedure originally outlined by O'Brien.

Homogeneity of relative risk against two-step alternatives.

In this paper the computational aspects of testing the null hypothesis of homogeneity of relative risk against two-step alternatives are examined. This representation is the same as that introduced by Anderson and Senthilselvan (Appl. Stat. 31 (1982) 44-51), i.e. a two-step model. Such alternatives may be used to represent decay in effect or, perhaps, inversion of the regression effect or crossing hazards. For such models inferential aspects are slightly more involved than for instance with proportional hazards models having fixed effects, even when time dependent as in O'Quigley and Pessione (Biometrics 45 (1989) 135-144). The necessary techniques for carrying out tests based on the two-stage model have recently been developed (O'Quigley and Pessione (Biometrics (1990) (in press] and in this paper we outline the necessary steps to be taken in the construction of algorithms to implement the proposed procedures. Programs enabling analyses based on the assumption of homogeneity of risk are very widely available. These include software packages such as BMDP, SAS, SPSS and GLIM. In the output of these packages, as well as that from most other standard routines, is contained all the necessary information to carry out the tests proposed by O'Quigley and Pessione. Here we detail the explicit formulae needed for carrying out the calculations in practice. The special cases of crossing hazards are considered in detail.

Gynecological abnormalities following allogeneic bone marrow transplantation.

Forty-four post-pubertal women were studied 261-4628 days after allogeneic transplantation to determine the nature and degree of gynecological abnormalities following bone marrow transplantation. Evaluations included pelvic examinations, exfoliative cytology, serum gonadotropin levels, direct preparations for micro-organisms, and microbial cultures. Pelvic abnormalities were detected in 35 of 44 (80%) women and resembled atrophic changes known to occur after ovarian failure. Findings included reduced vaginal elasticity and rugal folds, pale tissues, small vaginal, uterine and cervical size, atrophic vulvovaginitis, introital stenosis, and loss of pubic hair. Atrophic abnormalities were noted in 33 of 36 recipients of total body irradiation (TBI) compared to two of eight women not prepared with TBI (p = 0.02). Vasomotor symptoms were reported in 67% of TBI recipients compared to 38% of those not given TBI. Elevated serum gonadotropin levels suggested that TBI had caused the ovarian failure. Recognition of these gynecological abnormalities can lead to earlier hormone replacement, alleviating unnecessary discomfort and improving the well-being of the marrow transplant recipient.

Continual reassessment method: a practical design for phase 1 clinical trials in cancer.

This paper looks at a new approach to the design and analysis of Phase 1 clinical trials in cancer. The basic idea and motivation behind the approach stem from an attempt to reconcile the needs of dose-finding experimentation with the ethical demands of established medical practice. It is argued that for these trials the particular shape of the dose toxicity curve is of little interest. Attention focuses rather on identifying a dose with a given targeted toxicity level and on concentrating experimentation at that which all current available evidence indicates to be the best estimate of this level. Such an approach not only makes an explicit attempt to meet ethical requirements but also enables the use of models whose only requirements are that locally (i.e., around the dose corresponding to the targeted toxicity level) they reasonably well approximate the true probability of toxic response. Although a large number of models could be contemplated, we look at a particularly simple one. Extensive simulations show the model to have real promise.