Estimating Unknown Transition Times Using a Piecewise Nonlinear Mixed-Effects Model in Men with Prostate Cancer

Random changepoint segmented regression with smooth transition

We consider random changepoint segmented regression models to analyse data from a study conducted to verify whether treatment with stem cells may delay the onset of a symptom of amyotrophic lateral sclerosis in genetically modified mice. The proposed models capture the biological aspects of the data, accommodating a smooth transition between the periods with and without symptoms. An additional changepoint is considered to avoid negative predicted responses. Given the nonlinear nature of the model, we propose an algorithm to estimate the fixed parameters and to predict the random effects by fitting linear mixed models iteratively via standard software. We compare the variances obtained in the final step with bootstrapped and robust ones.

Detecting Multiple Random Changepoints in Bayesian Piecewise Growth Mixture Models

Piecewise growth mixture models are a flexible and useful class of methods for analyzing segmented trends in individual growth trajectory over time, where the individuals come from a mixture of two or more latent classes. These models allow each segment of the overall developmental process within each class to have a different functional form; examples include two linear phases of growth, or a quadratic phase followed by a linear phase. The changepoint (knot) is the time of transition from one developmental phase (segment) to another. Inferring the location of the changepoint(s) is often of practical interest, along with inference for other model parameters. A random changepoint allows for individual differences in the transition time within each class. The primary objectives of our study are as follows: (1) to develop a PGMM using a Bayesian inference approach that allows the estimation of multiple random changepoints within each class; (2) to develop a procedure to empirically detect the number of random changepoints within each class; and (3) to empirically investigate the bias and precision of the estimation of the model parameters, including the random changepoints, via a simulation study. We have developed the user-friendly package BayesianPGMM for R to facilitate the adoption of this methodology in practice, which is available at https://github.com/lockEF/BayesianPGMM . We describe an application to mouse-tracking data for a visual recognition task.

A note on estimating single-class piecewise mixed-effects models with unknown change points

Piecewise mixed-effects models are useful for analyzing longitudinal educational and psychological data sets to model segmented change over time. These models offer an attractive alternative to commonly used quadratic and higher-order polynomial models because the coefficients obtained from fitting the model have meaningful substantive interpretation. The current study thus focuses on the estimation of piecewise mixed-effects model with unknown random change points using maximum likelihood (ML) as described in Du Toit and Cudeck (2009). Previous simulation work (Wang & McArdle, 2008) showed that Bayesian estimation produced reliable parameter estimates for the piecewise model in comparison to frequentist procedures (i.e., first-order Taylor expansion and the adaptive Gaussian quadrature) across all simulation conditions. In the current article a small Monte Carlo simulation study was conducted to assess the performance of the ML approach, a frequentist procedure, and the Bayesian approach for fitting linear–linear piecewise mixed-effects model. The obtained findings show that ML estimation approach produces reliable and accurate estimates under the conditions of small residual variance of the observed variables, and that the size of the residual variance had the most impact on the quality of model parameter estimates. Second, neither ML nor Bayesian estimation procedures performed well under all manipulated conditions with respect to the accuracy and precision of the estimated model parameters.

Estimating mono- and bi-phasic regression parameters using a mixture piecewise linear Bayesian hierarchical model

The dynamics of tumor burden, secreted proteins or other biomarkers over time, is often used to evaluate the effectiveness of therapy and to predict outcomes for patients. Many methods have been proposed to investigate longitudinal trends to better characterize patients and to understand disease progression. However, most approaches assume a homogeneous patient population and a uniform response trajectory over time and across patients. Here, we present a mixture piecewise linear Bayesian hierarchical model, which takes into account both population heterogeneity and nonlinear relationships between biomarkers and time. Simulation results show that our method was able to classify subjects according to their patterns of treatment response with greater than 80% accuracy in the three scenarios tested. We then applied our model to a large randomized controlled phase III clinical trial of multiple myeloma patients. Analysis results suggest that the longitudinal tumor burden trajectories in multiple myeloma patients are heterogeneous and nonlinear, even among patients assigned to the same treatment cohort. In addition, between cohorts, there are distinct differences in terms of the regression parameters and the distributions among categories in the mixture. Those results imply that longitudinal data from clinical trials may harbor unobserved subgroups and nonlinear relationships; accounting for both may be important for analyzing longitudinal data.

Modeling longitudinal data with a random change point and no time-zero: applications to inference and prediction of the labor curve

In some longitudinal studies the initiation time of the process is not clearly defined, yet it is important to make inference or do predictions about the longitudinal process. The application of interest in this article is to provide a framework for modeling individualized labor curves (longitudinal cervical dilation measurements) where the start of labor is not clearly defined. This is a well-known problem in obstetrics where the benchmark reference time is often chosen as the end of the process (individuals are fully dilated at 10 cm) and time is run backwards. This approach results in valid and efficient inference unless subjects are censored before the end of the process, or if we are focused on prediction. Providing dynamic individualized predictions of the longitudinal labor curve prospectively (where backwards time is unknown) is of interest to aid obstetricians to determine if a labor is on a suitable trajectory. We propose a model for longitudinal labor dilation that uses a random-effects model with unknown time-zero and a random change point. We present a maximum likelihood approach for parameter estimation that uses adaptive Gaussian quadrature for the numerical integration. Further, we propose a Monte Carlo approach for dynamic prediction of the future longitudinal dilation trajectory from past dilation measurements. The methodology is illustrated with longitudinal cervical dilation data from the Consortium of Safe Labor Study.

Segmented mixed models with random changepoints: a maximum likelihood approach with application to treatment for depression study

We present a simple and effective iterative procedure to estimate segmented mixed models in a likelihood based framework. Random effects and covariates are allowed for each model parameter, including the changepoint. The method is practical and avoids the computational burdens related to estimation of nonlinear mixed effects models. A conventional linear mixed model with proper covariates that account for the changepoints is the key to our estimating algorithm. We illustrate the method via simulations and using data from a randomized clinical trial focused on change in depressive symptoms over time which characteristically show two separate phases of change.

Deriving benefit of early detection from biomarker-based prognostic models

Many prognostic models for cancer use biomarkers that have utility in early detection. For example, in prostate cancer, models predicting disease-specific survival use serum prostate-specific antigen levels. These models typically show that higher marker levels are associated with poorer prognosis. Consequently, they are often interpreted as indicating that detecting disease at a lower threshold of the biomarker is likely to generate a survival benefit. However, lowering the threshold of the biomarker is tantamount to early detection. For survival benefit to not be simply an artifact of starting the survival clock earlier, we must account for the lead time of early detection. It is not known whether the existing prognostic models imply a survival benefit under early detection once lead time has been accounted for. In this article, we investigate survival benefit implied by prognostic models where the predictor(s) of disease-specific survival are age and/or biomarker level at disease detection. We show that the benefit depends on the rate of biomarker change, the lead time, and the biomarker level at the original date of diagnosis as well as on the parameters of the prognostic model. Even if the prognostic model indicates that lowering the threshold of the biomarker is associated with longer disease-specific survival, this does not necessarily imply that early detection will confer an extension of life expectancy.

Screening for prostate cancer using multivariate mixed-effects models

Using several variables known to be related to prostate cancer, a multivariate classification method is developed to predict the onset of clinical prostate cancer. A multivariate mixed-effects model is used to describe longitudinal changes in prostate specific antigen (PSA), a free testosterone index (FTI), and body mass index (BMI) before any clinical evidence of prostate cancer. The patterns of change in these three variables are allowed to vary depending on whether the subject develops prostate cancer or not and the severity of the prostate cancer at diagnosis. An application of Bayes' theorem provides posterior probabilities that we use to predict whether an individual will develop prostate cancer and, if so, whether it is a high-risk or a low-risk cancer. The classification rule is applied sequentially one multivariate observation at a time until the subject is classified as a cancer case or until the last observation has been used. We perform the analyses using each of the three variables individually, combined together in pairs, and all three variables together in one analysis. We compare the classification results among the various analyses and a simulation study demonstrates how the sensitivity of prediction changes with respect to the number and type of variables used in the prediction process.

Modeling cholinesterase activity for human dietary risk assessment of carbamate insecticides

This article demonstrates statistical models to quantify the interaction between a carbamate insecticide and acetylcholinesterase. Carbamates are a class of chemicals that inhibit the activity of acetylcholinesterase in humans, an enzyme involved in the regulation of the neurotransmitter acetylcholine. Following exposure to a carbamate insecticide, we specifically address (1) if acetylcholinesterase activity recovers to its level of preexposure activity; (2) the level of inhibition of acetylcholinesterase activity; (3) the recovery time of acetylcholinesterase activity to its preexposure level for a typical individual; and (4) the upper percentiles of the recovery time of acetylcholinesterase activity across individuals. A nonlinear mixed-effects model is fitted to data from a repeated measures experiment conducted with human volunteers randomly assigned to a control and four dose groups. Repeated measurements were taken prior to exposure and at 1, 2, 4, 6, 8, and 21 hours after exposure to the carbamate aldicarb. It was found that full recovery did occur. Inhibition at 1 hour was estimated with maximum inhibition most likely occurring prior to 1-hour postexposure. In addition, recovery was rapid even for sensitive individuals. Given this information, the potential effect from exposure to a carbamate consumed in the diet during a day can be quantitatively assessed.

Hierarchical changepoint models for biochemical markers illustrated by tracking postradiotherapy prostate-specific antigen series in men with prostate cancer

PURPOSE

Biomarkers provide valuable information when detecting disease onset or monitoring disease progression; examples include bone mineral density (for osteoporosis), cholesterol (for coronary artery diseases), or prostate-specific antigens (PSA, for prostate cancer). Characteristics of markers series can then be used as prognostic factors of disease progression, such as the postradiotherapy PSA doubling time in men treated for prostate cancer. The statistical analysis of such data has to incorporate the within and between-series variabilities, the complex patterns of the series over time, the unbalanced format of the data, and the possibly nonconstant precision of the measurements.

METHODS

We base our analysis on a population-based cohort of 470 men treated with radiotherapy for prostate cancer; after treatment, the log(2)PSA concentrations follow a piecewise-linear pattern. We illustrate the flexibility of Bayesian hierarchical changepoint models by estimating the individual and population postradiotherapy log(2)PSA profiles; parameters such as the PSA nadir and the PSA doubling time were estimated, and their associations with baseline patient characteristics were investigated. The residual PSA variability was modeled as a function of the PSA concentration. For comparison purposes, two alternative models were briefly considered.

RESULTS

Precise estimates of all parameters of the PSA trajectory are provided at both the individual and population levels. Estimates suggest greater PSA variability at lower PSA concentrations, as well as an association between shorter PSAdts and greater baseline PSA levels, higher Gleason scores, and older age.

CONCLUSIONS

The use of Bayesian hierarchical changepoint models accommodates multiple complex features of longitudinal data, permits realistic modeling of the variability as a function of the marker concentration, and provides precise estimates of all clinically important parameters. This type of model should be applicable to the study of marker series in other diseases.

Analysis of nonlinear patterns of change with random coefficient models

Nonlinear patterns of change arise frequently in the analysis of repeated measures from longitudinal studies in psychology. The main feature of nonlinear development is that change is more rapid in some periods than in others. There generally also are strong individual differences, so although there is a general similarity of patterns for different persons over time, individuals exhibit substantial heterogeneity in their particular response. To describe data of this kind, researchers have extended the random coefficient model to accommodate nonlinear trajectories of change. It can often produce a statistically satisfying account of subject-specific development. In this review we describe and illustrate the main ideas of the nonlinear random coefficient model with concrete examples.

Semiparametric Bayesian classification with longitudinal markers

We analyse data from a study involving 173 pregnant women. The data are observed values of the β human chorionic gonadotropin hormone measured during the first 80 days of gestational age, including from one up to six longitudinal responses for each woman. The main objective in this study is to predict normal versus abnormal pregnancy outcomes from data that are available at the early stages of pregnancy. We achieve the desired classification with a semiparametric hierarchical model. Specifically, we consider a Dirichlet process mixture prior for the distribution of the random effects in each group. The unknown random-effects distributions are allowed to vary across groups but are made dependent by using a design vector to select different features of a single underlying random probability measure. The resulting model is an extension of the dependent Dirichlet process model, with an additional probability model for group classification. The model is shown to perform better than an alternative model which is based on independent Dirichlet processes for the groups. Relevant posterior distributions are summarized by using Markov chain Monte Carlo methods.

Evaluating markers for the early detection of cancer: overview of study designs and methods

BACKGROUND

The field of cancer biomarker development has been evolving rapidly. New developments both in the biologic and statistical realms are providing increasing opportunities for evaluation of markers for both early detection and diagnosis of cancer.

PURPOSE

To review the major conceptual and methodological issues in cancer biomarker evaluation, with an emphasis on recent developments in statistical methods together with practical recommendations.

METHODS

We organized this review by type of study: preliminary performance, retrospective performance, prospective performance and cancer screening evaluation.

RESULTS

For each type of study, we discuss methodologic issues, provide examples and discuss strengths and limitations.

CONCLUSION

Preliminary performance studies are useful for quickly winnowing down the number of candidate markers; however their results may not apply to the ultimate target population, asymptomatic subjects. If stored specimens from cohort studies with clinical cancer endpoints are available, retrospective studies provide a quick and valid way to evaluate performance of the markers or changes in the markers prior to the onset of clinical symptoms. Prospective studies have a restricted role because they require large sample sizes, and, if the endpoint is cancer on biopsy, there may be bias due to overdiagnosis. Cancer screening studies require very large sample sizes and long follow-up, but are necessary for evaluating the marker as a trigger of early intervention.

Critical analysis of prostate-specific antigen doubling time calculation methodology

BACKGROUND

Prostate-specific antigen (PSA) doubling time (PSADT) has emerged as an important surrogate marker of disease progression and survival in men with prostate carcinoma. The literature is replete with different methods for calculating PSADT. The objective of the current study was to identify the method that best described PSA growth over time and predicted disease-specific survival in men with androgen-independent prostate carcinoma.

METHODS

PSADT was calculated for 122 patients with androgen-independent prostate carcinoma using 2 commonly used methods: best-line fit (BLF) and first and last observations (FLO). Then, PSADT was calculated by using both a random coefficient linear (RCL) model and a random coefficient quadratic (RCQ) model. Statistical analysis was used to compare the ability of the methods to fit the patients' PSA profiles and to predict disease-specific survival.

RESULTS

The RCQ model provided the best fit of the patients' PSA profiles, as determined according to the significance of the added parameters for the RCQ equation (P < or = 0.002). The PSADT estimates from the FLO method, the RCL model, and the RCQ model were highly significant predictors (P < 0.001) of disease-specific survival, whereas estimates from the BLF method were not found to be significant predictors (P = 0.66). PSADT estimates from the RCQ and RCL models provided an improved correlation of disease-specific survival (both R(2) = 0.55) compared to the FLO (R(2) = 0.11) and BFL (R(2) = 0.003) methods.

CONCLUSIONS

Random coefficient methods provided a more reliable fit of PSA profiles than other models and were superior to other available models for predicting disease-specific survival in patients with androgen-independent prostate carcinoma. The authors concluded that consideration should be given to applying the RCL or RCQ models in future assessments of PSADT as a predictive parameter.

Screening by Prostate-Specific Antigen and Digital Rectal Examination in Relation to Prostate Cancer Mortality

BACKGROUND

The potential role of prostate cancer screening in reducing mortality is uncertain. To examine whether screening with the prostate-specific antigen (PSA) test or digital rectal examination is associated with reduced prostate cancer mortality, we conducted a population-based case-control study in 4 health maintenance organizations.

METHODS

Cases were 769 health plan members who died because of prostate adenocarcinoma during the years 1997-2001. We randomly selected 929 controls from the health plan membership and matched them to cases on health plan, age, race, and membership history. Medical records were used to document all screening tests in the 10 years before and including the date on which prostate cancer was first suspected.

RESULTS

Among white participants, 62% of cases and 69% of controls had a least 1 screening PSA test or digital rectal examination (odds ratio = 0.73; 95% confidence interval = 0.55-0.97). The corresponding proportions for blacks were 59% and 61% (1.0; 0.59-1.4). Most screening tests were digital rectal examinations; therefore, in the subgroup with no history of PSA screening, the association between digital rectal screening and prostate cancer mortality was similar to the overall association (0.65 [0.48-0.88] among whites; 0.86 [0.53-1.4] among blacks). Very few men received screening PSA without screening digital rectal examination (6% of cases and 7% of controls among whites).

CONCLUSIONS

Digital rectal screening was associated with a reduced risk of death due to prostate cancer in our population. Because of several data limitations, this study could not accurately estimate the effect of PSA screening separate from digital rectal examination.

Predicting time to prostate cancer recurrence based on joint models for non-linear longitudinal biomarkers and event time outcomes

Biological markers that are both sensitive and specific for tumour regrowth or metastasis are increasingly becoming available and routinely monitored during the regular follow-up of patients treated for cancer. Obtained by a simple blood test, these markers provide an inexpensive non-invasive means for the early detection of recurrence (or progression). Currently, the longitudinal behaviour of the marker is viewed as an indicator of early disease progression, and is applied by a physician in making clinical decisions. One marker that has been studied for use in both population screening for early disease and for detection of recurrence in prostate cancer patients is PSA. The elevation of PSA levels is known to precede clinically detectable recurrence by 2 to 5 years, and current clinical practice often relies partially on multiple recent rises in PSA to trigger a change in treatment. However, the longitudinal trajectory for individual markers is often non-linear; in many cases there is a decline immediately following radiation therapy or surgery, a plateau during remission, followed by an exponential rise following the recurrence of the cancer. The aim of this article is to determine the multiple aspects of the longitudinal PSA biomarker trajectory that can be most sensitive for predicting time to clinical recurrence. Joint Bayesian models for the longitudinal measures and event times are utilized based on non-linear hierarchical models, implied by unknown change-points, for the longitudinal trajectories, and a Cox proportional hazard model for progression times, with functionals of the longitudinal parameters as covariates in the Cox model. Using Markov chain Monte Carlo sampling schemes, the joint model is fit to longitudinal PSA measures from 676 patients treated at Massachusetts General Hospital between the years 1988 and 1995 with follow-up to 1999. Based on these data, predictive schemes for detecting cancer recurrence in new patients based on their longitudinal trajectory are derived.

Modeling the impact of the decline in distant stage disease on prostate carcinoma mortality rates

BACKGROUND

The incidence of distant stage prostate carcinoma was relatively flat until 1991 and then started declining rapidly. This decline probably was caused by the shift to earlier stage disease associated with the rapid dissemination of prostate specific antigen (PSA) screening. Prostate carcinoma mortality rates started falling at approximately the same time. In this article, the authors model the potential impact of this stage shift on prostate carcinoma mortality rates given various assumptions concerning the survival of patients with screen-detected local-regional disease.

METHODS

The authors used the CAN*TROL 2 computer model to shift each deficit in the number of patients with distant stage disease to local-regional stage disease and modeled the implications on mortality using a set of base, optimistic, and pessimistic survival assumptions. A base survival assumes that a patient with screen-detected local-regional disease of a certain histologic grade has the same prognosis as a patient with clinically detected local-regional disease of same grade (i.e., an assumption of no length bias for patients with screen-detected disease), whereas the optimistic and pessimistic scenarios assume that survival is better or worse, respectively, than the base survival (i.e., complete cure for patients with favorable grade for the optimistic scenario and no improvements in survival for patients with unfavorable grade for the pessimistic scenario).

RESULTS

Model results were compared with observed mortality trends. Rising age-adjusted mortality rates peaked in 1991 for white males and in 1993 for black males and then fell 21% and 13% for white males and black males, respectively, from 1990 through 1999. Under the modeled stage-shift intervention, mortality rates would fall 18%, 8%, and 19% for both white males and black males under the base, pessimistic, and optimistic assumptions, respectively.

CONCLUSIONS

It is impossible to know what the mortality trends would have been in the absence of the introduction of PSA screening. However, under the base assumption, it appears that the decline in distant stage disease can have a fairly sizable and rapid impact on population mortality. The optimistic scenario is not much improved over the base scenario, which is indicative of the facts that the survival of patients diagnosed with clinical local-regional prostate carcinoma is quite good and that further survival improvements can have only a marginal impact. Under the pessimistic scenario, it appears that something else must be responsible for much of the decline in mortality. Screening trial results from the United States and Europe may verify and isolate the size of any mortality benefit associated with PSA screening. Trial results eventually can be put back into these population models to help quantify the impact of screening, treatment, and other factors on population trends.

Markers for early detection of cancer: statistical guidelines for nested case-control studies

BACKGROUND

Recently many long-term prospective studies have involved serial collection and storage of blood or tissue specimens. This has spurred nested case-control studies that involve testing some specimens for various markers that might predict cancer. Until now there has been little guidance in statistical design and analysis of these studies.

METHODS

To develop statistical guidelines, we considered the purpose, the types of biases, and the opportunities for extracting additional information.

RESULTS

The following guidelines: (1) For the clearest interpretation, statistics should be based on false and true positive rates - not odds ratios or relative risks (2) To avoid overdiagnosis bias, cases should be diagnosed as a result of symptoms rather than on screening. (3) To minimize selection bias, the spectrum of control conditions should be the same in study and target screening populations. (4) To extract additional information, criteria for a positive test should be based on combinations of individual markers and changes in marker levels over time. (5) To avoid overfitting, the criteria for a positive marker combination developed in a training sample should be evaluated in a random test sample from the same study and, if possible, a validation sample from another study. (6) To identify biomarkers with true and false positive rates similar to mammography, the training, test, and validation samples should each include at least 110 randomly selected subjects without cancer and 70 subjects with cancer.

CONCLUSION

These guidelines ensure good practice in the design and analysis of nested case-control studies of early detection biomarkers.

Statistical models for longitudinal biomarkers of disease onset

We consider the analysis of serial biomarkers to screen and monitor individuals in a given population for onset of a specific disease of interest. The biomarker readings are subject to error. We survey some of the existing literature and concentrate on two recently proposed models. The first is a fully Bayesian hierarchical structure for a mixed effects segmented regression model. Posterior estimates of the changepoint (onset time) distribution are obtained by Gibbs sampling. The second is a hidden changepoint model in which the onset time distribution is estimated by maximum likelihood using the EM algorithm. Both methods lead to a dynamic index that represents a strength of evidence that onset has occurred by the current time in an individual subject. The methods are applied to some large data sets concerning prostate specific antigen (PSA) as a serial marker for prostate cancer. Rules based on the indices are compared to standard diagnostic criteria through the use of ROC curves adapted for longitudinal data.

Longitudinal data analysis (repeated measures) in clinical trials

Longitudinal data is often collected in clinical trials to examine the effect of treatment on the disease process over time. This paper reviews and summarizes much of the methodological research on longitudinal data analysis from the perspective of clinical trials. We discuss methodology for analysing Gaussian and discrete longitudinal data and show how these methods can be applied to clinical trials data. We illustrate these methods with five examples of clinical trials with longitudinal outcomes. We also discuss issues of particular concern in clinical trials including sequential monitoring and adjustments for missing data. A review of current software for analysing longitudinal data is also provided. Published in 1999 by John Wiley & Sons, Ltd. This article is a US Government work and is the public domain in the United States.

Cancer surveillance series: interpreting trends in prostate cancer--part III: Quantifying the link between population prostate-specific antigen testing and recent declines in prostate cancer mortality

BACKGROUND

The objective of this study was to investigate the circumstances under which dissemination of prostate-specific antigen (PSA) testing, beginning in 1988, could plausibly explain the declines in prostate cancer mortality observed from 1992 through 1994.

METHODS

We developed a computer simulation model by use of information on population-based PSA testing patterns, cancer detection rates, average lead time (the time by which diagnosis is advanced by screening), and projected decreased risk of death associated with early diagnosis of prostate cancer through PSA testing. The model provides estimates of the number of deaths prevented by PSA testing for the 7-year period from 1988 through 1994 and projects what prostate cancer mortality for these years would have been in the absence of PSA testing.

RESULTS

Results were generated by assuming a level of screening efficacy similar to that hypothesized for the Prostate, Lung, Colorectal, and Ovarian (PLCO) Cancer Screening Trial. Under this assumption, the projected mortality in the absence of PSA testing continued the increasing trend observed before 1991 only when it was assumed that the mean lead time was 3 years or less. Projected mortality trends in the absence of PSA screening were not consistent with pre-1991 increasing trends for lead times of 5 years and 7 years.

CONCLUSIONS

When screening is assumed to be at least as efficacious as hypothesized in the PLCO trial, it is unlikely that the entire decline in prostate cancer mortality can be explained by PSA testing based on current beliefs concerning lead time. Only very short lead times would produce a decline in mortality of the magnitude that has been observed.