A Measure of the Proportion of Treatment Effect Explained by a Surrogate Marker

Randomized clinical trials with rare primary endpoints or long duration times are costly. Because of this, there has been increasing interest in replacing the true endpoint with an earlier measured marker. However, surrogate markers must be appropriately validated. A quantitative measure for the proportion of treatment effect explained by the marker in a specific trial is a useful concept. Freedman, Graubard, and Schatzkin (1992, Statistics in Medicine 11, 167–178) suggested such a measure of surrogacy by the ratio of regression coefficients for the treatment indicator from two separate models with or without adjusting for the surrogate marker. However, it has been shown that this measure is very variable and there is no guarantee that the two models both fit. In this article, we propose alternative measures of the proportion explained that adapts an idea in Tsiatis, DeGruttola, and Wulfsohn (1995, Journal of the American Statistical Association 90 , 27–37). The new measures require fewer assumptions in estimation and allow more flexibility in modeling. The estimates of these different measures are compared using data from an ophthalmology clinical trial and a series of simulation studies. The results suggest that the new measures are less variable.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/65535/1/j.0006-341X.2002.00803.x.pd

Mixture cure model with random effects for the analysis of a multi-center tonsil cancer study

Cure models for clustered survival data have the potential for broad applicability. In this paper, we consider the mixture cure model with random effects and propose several estimation methods based on Gaussian quadrature, rejection sampling, and importance sampling to obtain the maximum likelihood estimates of the model for clustered survival data with a cure fraction. The methods are flexible to accommodate various correlation structures. A simulation study demonstrates that the maximum likelihood estimates of parameters in the model tend to have smaller biases and variances than the estimates obtained from the existing methods. We apply the model to a study of tonsil cancer patients clustered by treatment centers to investigate the effect of covariates on the cure rate and on the failure time distribution of the uncured patients. The maximum likelihood estimates of the parameters demonstrate strong correlation among the failure times of the uncured patients and weak correlation among cure statuses in the same center. Copyright © 2010 John Wiley & Sons, Ltd.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/79409/1/4098_ftp.pd

Optimizing Dynamic Predictions from Joint Models using Super Learning

Joint models for longitudinal and time-to-event data are often employed to calculate dynamic individualized predictions used in numerous applications of precision medicine. Two components of joint models that influence the accuracy of these predictions are the shape of the longitudinal trajectories and the functional form linking the longitudinal outcome history to the hazard of the event. Finding a single well-specified model that produces accurate predictions for all subjects and follow-up times can be challenging, especially when considering multiple longitudinal outcomes. In this work, we use the concept of super learning and avoid selecting a single model. In particular, we specify a weighted combination of the dynamic predictions calculated from a library of joint models with different specifications. The weights are selected to optimize a predictive accuracy metric using V-fold cross-validation. We use as predictive accuracy measures the expected quadratic prediction error and the expected predictive cross-entropy. In a simulation study, we found that the super learning approach produces results very similar to the Oracle model, which was the model with the best performance in the test datasets. All proposed methodology is implemented in the freely available R package JMbayes2

Individualized predictions of disease progression following radiation therapy for prostate cancer.

Background: Following treatment for localized prostate cancer, men are monitored with serial PSA measurements. Refining the predictive value of post-treatment PSA determinations may add to clinical management and we have developed a model that predicts for an individual patient future PSA values and estimates the time to future clinical recurrence. Methods: Data from 934 patients treated for prostate cancer between 1987 and 2000 were used to develop a comprehensive statistical model to fit the clinical recurrence events and pattern of PSA data. A logistic regression model was used for the probability of cure, non-linear hierarchical mixed models were used for serial PSA measurements and a time-dependent proportional hazards model was used for recurrences. Data available up to February 2001 and September 2003 was used to assess the performance of the model. Results: The model suggests that T-stage, baseline PSA, and radiotherapy dosage are all associated with probability of cure. The risk of clinical recurrence in those not cured by radiotherapy is most strongly affected by the slope of the long-transformed PSA values. We show how the model can be used for individual monitoring of a patient’s disease progression. For each patient the model predicts, based upon his baseline and all post-treatment PSA values, the probability of future clinical recurrence in the validation dataset and of 406 PSA measurements obtained 1-2 years after February 2001, 92.8% were within 95% prediction limits from the model. Conclusions: This statistical model presented accurately predicts future PSA values and risk of clinical relapse. This predictive information for each individual patient, which can be updated with each additional PSA value, may prove useful to patents and physicians in determining what post-treatment salvage should be employed

Adaptive prior variance calibration in the Bayesian continual reassessment method

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/98219/1/sim5621.pd

Analysis on binary responses with ordered covariates and missing data

We consider the situation of two ordered categorical variables and a binary outcome variable, where one or both of the categorical variables may have missing values. The goal is to estimate the probability of response of the outcome variable for each cell of the contingency table of categorical variables while incorporating the fact that the categorical variables are ordered. The probability of response is assumed to change monotonically as each of the categorical variables changes level. A probability model is used in which the response is binomial with parameters p ij for each cell ( i , j ) and the number of observations in each cell is multinomial. Estimation approaches that incorporate Gibbs sampling with order restrictions on p ij induced via a prior distribution, two-dimensional isotonic regression and multiple imputation to handle missing values are considered. The methods are compared in a simulation study. Using a fully Bayesian approach with a strong prior distribution to induce ordering can lead to large gains in efficiency, but can also induce bias. Utilizing isotonic regression can lead to modest gains in efficiency, while minimizing bias and guaranteeing that the order constraints are satisfied. A hybrid of isotonic regression and Gibbs sampling appears to work well across a variety of scenarios. The methods are applied to a pancreatic cancer case–control study with two biomarkers. Copyright © 2007 John Wiley & Sons, Ltd.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/56130/1/2815_ftp.pd

Joint partially linear model for longitudinal data with informative drop‐outs

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/136540/1/biom12566.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/136540/2/biom12566_am.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/136540/3/biom12566-sup-0001-SuppData.pd

Modeling intra-tumor protein expression heterogeneity in tissue microarray experiments

Tissue microarrays (TMAs) measure tumor-specific protein expression via high-density immunohistochemical staining assays. They provide a proteomic platform for validating cancer biomarkers emerging from large-scale DNA microarray studies. Repeated observations within each tumor result in substantial biological and experimental variability. This variability is usually ignored when associating the TMA expression data with patient survival outcome. It generates biased estimates of hazard ratio in proportional hazards models. We propose a Latent Expression Index (LEI) as a surrogate protein expression estimate in a two-stage analysis. Several estimators of LEI are compared: an empirical Bayes, a full Bayes, and a varying replicate number estimator. In addition, we jointly model survival and TMA expression data via a shared random effects model. Bayesian estimation is carried out using a Markov chain Monte Carlo method. Simulation studies were conducted to compare the two-stage methods and the joint analysis in estimating the Cox regression coefficient. We show that the two-stage methods reduce bias relative to the naive approach, but still lead to under-estimated hazard ratios. The joint model consistently outperforms the two-stage methods in terms of both bias and coverage property in various simulation scenarios. In case studies using prostate cancer TMA data sets, the two-stage methods yield a good approximation in one data set whereas an insufficient one in the other. A general advice is to use the joint model inference whenever results differ between the two-stage methods and the joint analysis. Copyright © 2008 John Wiley & Sons, Ltd.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/58565/1/3217_ftp.pd