Estimating prediction equations in repeated measures designs

Experimental designs with repeated measures allow response patterns over time (or dose) to be modelled and compared between different homogeneous groups. Issues in data analysis often focus on the pattern of variation of the repeated measures, the appropriateness of a univariate or multivariate analysis, and the shape of the response pattern. An aspect of analysis that is often of equal importance is the development of a regression model for response once the pattern has been characterized. Analysis of variance or multivariate growth curve results often do not include easily interpretable regression equation estimates that can be used for prediction. We present methods and tables that permit simple construction of such predictive equations for repeated measures designs when response is modelled as a polynomial over time with univariate or multivariate analyses.     

A model for incomplete longitudinal multivariate ordinal data

In studies where multiple outcome items are repeatedly measured over time, missing data often occur. A longitudinal item response theory model is proposed for analysis of multivariate ordinal outcomes that are repeatedly measured. Under the MAR assumption, this model accommodates missing data at any level (missing item at any time point and/or missing time point). It allows for multiple random subject effects and the estimation of item discrimination parameters for the multiple outcome items. The covariates in the model can be at any level. Assuming either a probit or logistic response function, maximum marginal likelihood estimation is described utilizing multidimensional Gauss-Hermite quadrature for integration of the random effects. An iterative Fisher-scoring solution, which provides standard errors for all model parameters, is used. A data set from a longitudinal prevention study is used to motivate the application of the proposed model. In this study, multiple ordinal items of health behavior are repeatedly measured over time. Because of a planned missing design, subjects answered only two-third of all items at a given point.     

Validation of pesticide root zone model 3.12: employing uncertainty analysis

Computer models are being increasingly used to provide an efficient cost-effective means of evaluating the fate and behavior of chemicals in the environment. These models can be used in lieu of or in conjunction with field studies. Because of the increasing reliance on models for critical regulatory decision making, the need arose to assess the validity of regulatory models via an analysis of the correlation of model response estimates with measured data. In conjunction with the evaluation of the correlation of model response estimates and measured field data, a rigorous statistically based validation was also warranted. Because of the unique nature of the correlative exercise using modeled and measured data, standard statistical analyses, while informative, failed to encompass factors associated with the uncertainty of measured environmental fate data and potential model inputs. In an effort to evaluate this uncertainty, an initial sensitivity analysis was performed where key model input parameters for runoff and leaching simulations were identified. Once the sensitive input parameters were identified, a Monte Carlo-based preprocessor was developed whereby the sampling distributions of these parameters were used to propagate uncertainty in the input parameters into error in model predictions. Importantly, assumptions about parameter distributions for input into the Monte Carlo tool were made only after a formal detailed site-specific analysis of measured field data. Employing the functionality of the Crystal Ball Pro development environment, the pesticide root zone model (PRZM) 3.12 was run iteratively for 500 trials, and model output was collated and analyzed. The model predictions were considered reasonably accurate for most regulatory requirements, and the model prediction error was considered acceptable.     

Estimating probability of non-response to treatment using mixture distributions.

Repeat measurements of patient characteristics are often used to assess response to treatment. In this paper we discuss a normal mixture model for the observed change in the characteristic of interest in treated patients. The methods described can be used to estimate the overall proportion of non-response to treatment and also the probability that a patient has not responded to treatment given his or her observed change. The model parameters are estimated using maximum likelihood, and the delta method is used to construct a pointwise confidence band for the conditional probability that a patient is a non-responder to treatment. The work was initially motivated by analysis issues in the Fracture Intervention Trial (FIT), a randomized trial of the osteoporosis drug alendronate, and the method is illustrated with data from that study. We also evaluate key aspects of the estimation procedure with two simulation studies. In the first, the data generation model is the assumed normal mixture model, and in the second, the data are generated according to a shifted and scaled central t-distribution model suggested by the FIT data.     

A novel case‐control subsampling approach for rapid model exploration of large clustered binary data

In many settings, an analysis goal is the identification of a factor, or set of factors associated with an event or outcome. Often, these associations are then used for inference and prediction. Unfortunately, in the big data era, the model building and exploration phases of analysis can be time‐consuming, especially if constrained by computing power (ie, a typical corporate workstation). To speed up this model development, we propose a novel subsampling scheme to enable rapid model exploration of clustered binary data using flexible yet complex model set‐ups (GLMMs with additive smoothing splines). By reframing the binary response prospective cohort study into a case‐control–type design, and using our knowledge of sampling fractions, we show one can approximate the model estimates as would be calculated from a full cohort analysis. This idea is extended to derive cluster‐specific sampling fractions and thereby incorporate cluster variation into an analysis. Importantly, we demonstrate that previously computationally prohibitive analyses can be conducted in a timely manner on a typical workstation. The approach is applied to analysing risk factors associated with adverse reactions relating to blood donation.     

多个二项反应变量多水平因子分析模型的原理及应用

目的探讨多个二项反应变量多水平因子分析模型的应用。方法在MLwiN2．02软件中，采用马尔科夫链一蒙特卡罗参数估计方法，通过实例模拟及分析，说明模型的实际应用。结果该模型适用于反应变量为二项分类的、具有层次结构的数据。结论多个二项反应变量多水平因子分析模型有其独特的意义和用途。     

Placebo non‐response measure in sequential parallel comparison design studies

The Sequential Parallel Comparison Design (SPCD) is one of the novel approaches addressing placebo response. The analysis of SPCD data typically classifies subjects as ‘placebo responders’ or ‘placebo non‐responders’. Most current methods employed for analysis of SPCD data utilize only a part of the data collected during the trial. A repeated measures model was proposed for analysis of continuous outcomes that permitted the inclusion of information from all subjects into the treatment effect estimation. We describe here a new approach using a weighted repeated measures model that further improves the utilization of data collected during the trial, allowing the incorporation of information that is relevant to the placebo response, and dealing with the problem of possible misclassification of subjects. Our simulations show that when compared to the unweighted repeated measures model method, our approach performs as well or, under certain conditions, better, in preserving the type I error, achieving adequate power and minimizing the mean squared error. Copyright © 2015 John Wiley & Sons, Ltd.     

Improving measurement in health education and health behavior research using item response modeling: introducing item response modeling

This paper is the first of several papers designed to demonstrate how the application of item response models in the behavioral sciences can be used to enhance the conceptual and technical toolkit of researchers and developers and to understand better the psychometric properties of psychosocial measures. The papers all use baseline data from the Behavior Change Consortium data archive. This paper begins with an introduction to item response models, including both dichotomous and polytomous versions. The concepts of respondent and item location, model interpretation, standard errors and testing model fit are introduced and described. A sample analysis based on data from the self-efficacy scale is used to illustrate the concepts and techniques.     

Identifying response shift statistically at the individual level

Objective The purpose of this study was to explore whether a longitudinal comparison between reported and predicted health could be used as a method of identifying subjects who potentially experienced response shift. Methods A response-shift model was developed using data from a longitudinal study of stroke in which measures of stroke impact were made at study entry and at 1, 3, 6, and 12 months post stroke. Residuals from a random effects model were centered and used to create trajectories. This model was tested against a data set from a study in which the then-test had been administered. Twenty simulated data sets were also generated to examine how much of response shift could be attributed to random error. Results Group-based trajectory analysis identified seven trajectory groups. The majority (67%) of the 387 persons showed no response shift over time, whereas 15% lowered and 13% raised their health over time, disproportionally to that predicted. Conclusion Results of the validation studies were supportive that this methodology identifies response shift, but further research is required to compare results with other methodologies and other predictive models.     

Residual analysis in linear regression models with an interval-censored covariate

Residual analysis is a useful class of techniques for the evaluation of the goodness of a fitted model. Checking the underlying assumptions is important since most linear regression estimators require a correctly specified regression function and independent and identically distributed errors to be consistent. For uncensored data, the examination of the residuals of the fitted model is a standard tool for checking whether or not the underlying model assumptions hold. Such analysis has not been widely developed for censored data. Hillis (Statistics in Medicine 1995; 14:2023-2036) developed a residual plot for model checking when the response variable of a linear model is right-censored, and Gomez et al. (Statistics in Medicine 2003; 22:409-425) proposed residuals in models with interval-censored covariates. In this paper, we propose a new definition of residuals for linear models that incorporate interval-censored covariates. This definition can be also applied when the response variable is interval-censored. These new residuals are shown to perform better in model checking than other types of residuals in this context. We illustrate them with a data set from an AIDS clinical trial study.     

A Bayesian hierarchical nonlinear model for assessing the association between genetic variation and drug cytotoxicity

Non‐tumor cell‐based model systems have recently gained interest in pharmacogenetic research as a hypothesis generating tool. The hypotheses generated from these model systems can be followed up in functional studies, or tested in individuals taking the same investigational agents. The current cellular phenotypes (e.g. cytotoxicity) of interest in these studies are based on the effects of an individual dosage of a drug on the cell lines, or a summary of results at many dosages of a drug (e.g. dose that inhibits 50 per cent of cell growth, GI 50 ). A more complete analysis of the impact of genetic variation on all aspects of the dose–response curve may lend additional insight into the pharmacogenomics of a particular drug. This paper illustrates the use of a Bayesian hierarchical nonlinear model for the analysis of pharmacogenomic data with cytotoxicity endpoints. The model is illustrated with cytotoxicity and expression data collected on cell lines from a pharmacogenomic study of the drug gemcitabine. By completing an analysis based on the entire dose–response curve, we were able to detect additional genes that affect not only the GI 50, but also the slope of the curve, which reflects the therapeutic index of the drug. Simulation studies also demonstrate that in comparison with the analyses based on the commonly used summary measure GI 50, investigation of the impact of genetic variation on all aspects of the cytotoxicity dose–response curve is more informative, and more powerful with respect to detecting the effect of gene expression on cytotoxicity. Copyright © 2009 John Wiley & Sons, Ltd.     

A mixed effects model for the analysis of ordinal longitudinal pain data subject to informative drop-out

We extend the model of Pulkstenis et al. that models binary longitudinal data, subject to informative drop-out through remedication, to the ordinal response case. We present a selection model shared-parameter approach that specifies mixed models for both ordinal response and discrete survival time to remedication. In this fashion, the random parameter present in both models completely characterizes the relationship between response and time to remedication inducing their conditional independence. With a log-log link function for both response and study 'survival', as well as specification of a log-gamma distribution for the random effect, we obtain a closed-form expression for the marginal log-likelihood of response and time to remedication that does not require approximation or numerical integration techniques. A data analysis is performed and simulation results presented which support the consistency of parameter and standard error estimates.     

Analysis of crossover designs with multivariate response

Crossover designs involve observing the same response variate under different experimental conditions for each subject. Univariate methods are commonly used for analysis of data arising in these designs, but multivariate procedures offer a more general approach. The general multivariate linear model provides a natural framework for the simplest data structure as well as more complex settings with two or more response variates and measurements repeated over time. Multivariate models for crossover designs provide a unified approach that clarifies specification of hypotheses, assumptions required, and testing procedures in a wide class of applications that include longitudinal data as a special case. We focus on the 2 × 2 crossover design, but also describe models for analysing more complex crossover designs.     

AN APPLICATION OF BAYESIAN POPULATION PHARMACOKINETIC/PHARMACODYNAMIC MODELS TO DOSE RECOMMENDATION

Population pharmacokinetic data consists of dose histories, individual covariates and measured drug concentrations with associated sampling times. Population pharmacodynamic data consist of dose histories, covariates and some response measure. Population analyses, whether they be pharmacokinetic or pharmacodynamic attempt to explain the variability observed in the recorded measurements and are increasingly being seen as an important aid in drug development. In this paper a general Bayesian population pharmacokinetic/pharmacodynamic model is described and an analysis of data for the drug recombinant hirudin is presented. The model we use allows for both outliers and censoring in the concentration data and outlying individual pharmacokinetic parameters. We attempt to address directly important questions such as recommended dose size using predictive distributions for response.     

Analyzing longitudinal data with patients in different disease states during follow‐up and death as final state

This paper considers the analysis of longitudinal data complicated by the fact that during follow‐up patients can be in different disease states, such as remission, relapse or death. If both the response of interest (for example, quality of life (QOL)) and the amount of missing data depend on this disease state, ignoring the disease state will yield biased means. Death as the final state is an additional complication because no measurements after death are taken and often the outcome of interest is undefined after death. We discuss a new approach to model these types of data. In our approach the probability to be in each of the different disease states over time is estimated using multi‐state models. In each different disease state, the conditional mean given the disease state is modeled directly. Generalized estimation equations are used to estimate the parameters of the conditional means, with inverse probability weights to account for unobserved responses. This approach shows the effect of the disease state on the longitudinal response. Furthermore, it yields estimates of the overall mean response over time, either conditionally on being alive or after imputing predefined values for the response after death. Graphical methods to visualize the joint distribution of disease state and response are discussed. As an example, the analysis of a Dutch randomized clinical trial for breast cancer is considered. In this study, the long‐term impact on the QOL for two different chemotherapy schedules was studied with three disease states: alive without relapse, alive after relapse and death. Copyright © 2009 John Wiley & Sons, Ltd.     

The analysis of very small samples of repeated measurements I: An adjusted sandwich estimator

The statistical analysis of repeated measures or longitudinal data always requires the accommodation of the covariance structure of the repeated measurements at some stage in the analysis. The general linear mixed model is often used for such analyses, and allows for the specification of both a mean model and a covariance structure. Often the covariance structure itself is not of direct interest, but only a means to producing valid inferences about the response. Existing methods of analysis are often inadequate where the sample size is small. More precisely, statistical measures of goodness of fit are not necessarily the right measure of the appropriateness of a covariance structure and inferences based on conventional Wald‐type procedures do not approximate sufficiently well their nominal properties when data are unbalanced or incomplete. This is shown to be the case when adopting the Kenward–Roger adjustment where the sample size is very small. A generalization of an approach to Wald tests using a bias‐adjusted empirical sandwich estimator for the covariance matrix of the fixed effects parameters from generalized estimating equations is developed for Gaussian repeated measurements. This is shown to attain the correct test size but has very low power. Copyright © 2010 John Wiley & Sons, Ltd.     

Bayesian inference for multivariate meta‐analysis Box–Cox transformation models for individual patient data with applications to evaluation of cholesterol‐lowering drugs

In this paper, we propose a class of Box–Cox transformation regression models with multidimensional random effects for analyzing multivariate responses for individual patient data in meta‐analysis. Our modeling formulation uses a multivariate normal response meta‐analysis model with multivariate random effects, in which each response is allowed to have its own Box–Cox transformation. Prior distributions are specified for the Box–Cox transformation parameters as well as the regression coefficients in this complex model, and the deviance information criterion is used to select the best transformation model. Because the model is quite complex, we develop a novel Monte Carlo Markov chain sampling scheme to sample from the joint posterior of the parameters. This model is motivated by a very rich dataset comprising 26 clinical trials involving cholesterol‐lowering drugs where the goal is to jointly model the three‐dimensional response consisting of low density lipoprotein cholesterol (LDL‐C), high density lipoprotein cholesterol (HDL‐C), and triglycerides (TG) (LDL‐C, HDL‐C, TG). Because the joint distribution of (LDL‐C, HDL‐C, TG) is not multivariate normal and in fact quite skewed, a Box–Cox transformation is needed to achieve normality. In the clinical literature, these three variables are usually analyzed univariately; however, a multivariate approach would be more appropriate because these variables are correlated with each other. We carry out a detailed analysis of these data by using the proposed methodology. Copyright © 2013 John Wiley & Sons, Ltd.     

Rapid model exploration for complex hierarchical data: application to pharmacokinetics of insulin aspart

We consider situations, which are common in medical statistics, where we have a number of sets of response data, from different individuals, say, potentially under different conditions. A parametric model is defined for each set of data, giving rise to a set of random effects. Our goal here is to efficiently explore a range of possible ‘population’ models for the random effects, to select the most appropriate model. The range of possible models is potentially vast, because the random effects may depend on observed covariates, and there may be multiple credible ways of partitioning their variability. Here, we consider pharmacokinetic (PK) data on insulin aspart, a fast acting insulin analogue used in the treatment of diabetes. PK models are typically nonlinear (in their parameters), often complex and sometimes only available as a set of differential equations, with no closed‐form solution. Fitting such a model for just a single individual can be a challenging task. Fitting a joint model for all individuals can be even harder, even without the complication of an overarching model selection objective. We describe a two‐stage approach that decouples the population model for the random effects from the PK model applied to the response data but nevertheless fits the full, joint, hierarchical model, accounting fully for uncertainty. This allows us to repeatedly reuse results from a single analysis of the response data to explore various population models for the random effects. This greatly expedites not only model exploration but also cross‐validation for the purposes of model criticism. © 2015 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd.     

Generalized finite mixture of multivariate regressions with applications to therapeutic biomarker identification

Finite mixtures of regressions have been used to analyze data that come from a heterogeneous population. When more than one response is observed, accommodating a multivariate response can be useful. In this article, we go a step further and introduce a multivariate extension that includes a latent overlapping cluster indicator variable that allows for potential overdispersion. A generalized mixture of multivariate regressions in connection with the proposed model and a new EM algorithm for fitting are provided. In addition, we allow for high-dimensional predictors via shrinkage estimation. This model proves particularly useful in the analysis of complex data like the search for cancer therapeutic biomarkers. We demonstrate this using the genomics of drug sensitivity in cancer resource.