A likelihood-based approach for multivariate one-sided tests with missing data

Inequality-restricted hypotheses testing methods containing multivariate one-sided testing methods are useful in practice, especially in multiple comparison problems. In practice, multivariate and longitudinal data often contain missing values since it may be difficult to observe all values for each variable. However, although missing values are common for multivariate data, statistical methods for multivariate one-sided tests with missing values are quite limited. In this article, motivated by a dataset in a recent collaborative project, we develop two likelihood-based methods for multivariate one-sided tests with missing values, where the missing data patterns can be arbitrary and the missing data mechanisms may be non-ignorable. Although non-ignorable missing data are not testable based on observed data, statistical methods addressing this issue can be used for sensitivity analysis and might lead to more reliable results, since ignoring informative missingness may lead to biased analysis. We analyse the real dataset in details under various possible missing data mechanisms and report interesting findings which are previously unavailable. We also derive some asymptotic results and evaluate our new tests using simulations.     

Empirical evaluation of the implementation of the EMA guideline on missing data in confirmatory clinical trials: Specification of mixed models for longitudinal data in study protocols

In confirmatory clinical trials, the prespecification of the primary analysis model is a universally accepted scientific principle to allow strict control of the type I error. Consequently, both the ICH E9 guideline and the European Medicines Agency (EMA) guideline on missing data in confirmatory clinical trials require that the primary analysis model is defined unambiguously. This requirement applies to mixed models for longitudinal data handling missing data implicitly. To evaluate the compliance with the EMA guideline, we evaluated the model specifications in those clinical study protocols from development phases II and III submitted between 2015 and 2018 to the Ethics Committee at Hannover Medical School under the German Medicinal Products Act, which planned to use a mixed model for longitudinal data in the confirmatory testing strategy. Overall, 39 trials from different types of sponsors and a wide range of therapeutic areas were evaluated. While nearly all protocols specify the fixed and random effects of the analysis model (95%), only 77% give the structure of the covariance matrix used for modeling the repeated measurements. Moreover, the testing method (36%), the estimation method (28%), the computation method (3%), and the fallback strategy (18%) are given by less than half the study protocols. Subgroup analyses indicate that these findings are universal and not specific to clinical trial phases or size of company. Altogether, our results show that guideline compliance is to various degrees poor and consequently, strict type I error rate control at the intended level is not guaranteed.     

Optimal nonparametric estimator of the area under ROC curve based on clustered data

Abstract

In diagnostic trials, clustered data are obtained when several subunits of the same patient are observed. Intracluster correlations need to be taken into account when analyzing such clustered data. A nonparametric method has been proposed by Obuchowski (1997 Obuchowski, N. A. 1997. Nonparametric analysis of clustered ROC curve data. Biometrics 53 (2):567–78.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) to estimate the Receiver Operating Characteristic curve area (AUC) for such clustered data. However, Obuchowski’s estimator is not efficient as it gives equal weight to all pairwise rankings within and between cluster. In this paper, we propose a more efficient nonparametric AUC estimator with two sets of optimal weights. Simulation results show that the loss of efficiency of Obuchowski’s estimator for a single AUC or the AUC difference can be substantial when there is a moderate intracluster test correlation and the cluster size is large. The efficiency gain of our weighted AUC estimator for a single AUC or the AUC difference is further illustrated using the data from a study of screening tests for neonatal hearing.     

Detecting diagnostic accuracy of two biomarkers through a bivariate log-normal ROC curve

In biomedical research, two or more biomarkers may be available for diagnosis of a particular disease. Selecting one single biomarker which ideally discriminate a diseased group from a healthy group is confront in a diagnostic process. Frequently, most of the people use the accuracy measure, area under the receiver operating characteristic (ROC) curve to choose the best diagnostic marker among the available markers for diagnosis. Some authors have tried to combine the multiple markers by an optimal linear combination to increase the discriminatory power. In this paper, we propose an alternative method that combines two continuous biomarkers by direct bivariate modeling of the ROC curve under log-normality assumption. The proposed method is applied to simulated data set and prostate cancer diagnostic biomarker data set.     

Multivariate data with missing observations

In this paper, multivariate data with missing observations, where missing values could be by chance or by design, are considered for various models including the growth curve model. The likelihood equations are derived and the consistency of the estimates established. The likelihood ratio tests are explicity derived.     

Optimal composite scores for longitudinal clinical trials under the linear mixed effects model

Clinical trials of chronic, progressive conditions use rate of change on continuous measures as the primary outcome measure, with slowing of progression on the measure as evidence of clinical efficacy. For clinical trials with a single prespecified primary endpoint, it is important to choose an endpoint with the best signal‐to‐noise properties to optimize statistical power to detect a treatment effect. Composite endpoints composed of a linear weighted average of candidate outcome measures have also been proposed. Composites constructed as simple sums or averages of component tests, as well as composites constructed using weights derived from more sophisticated approaches, can be suboptimal, in some cases performing worse than individual outcome measures. We extend recent research on the construction of efficient linearly weighted composites by establishing the often overlooked connection between trial design and composite performance under linear mixed effects model assumptions and derive a formula for calculating composites that are optimal for longitudinal clinical trials of known, arbitrary design. Using data from a completed trial, we provide example calculations showing that the optimally weighted linear combination of scales can improve the efficiency of trials by almost 20% compared with the most efficient of the individual component scales. Additional simulations and analytical results demonstrate the potential losses in efficiency that can result from alternative published approaches to composite construction and explore the impact of weight estimation on composite performance. Copyright © 2016. The Authors. Pharmaceutical Statistics Published by John Wiley & Sons Ltd.     

Criteria for reporting noncompartmental estimates of half‐life and area under the curve extrapolated to infinity

Various criteria have been proposed for determining the reliability of noncompartmental pharmacokinetic estimates of the terminal disposition phase half‐life (t_1/2) and the extrapolated area under the curve (AUC_extrap). This simulation study assessed the performance of two frequently used reportability rules: the terminal disposition phase regression adjusted‐r² classification rule and the regression data point time span classification rule. Using simulated data, these rules were assessed in relation to the magnitude of the variability in the terminal disposition phase slope, the length of the terminal disposition phase captured in the concentration‐time profile (data span), the number of data points present in the terminal disposition phase, and the type and level of variability in concentration measurement. The accuracy of estimating t_1/2 was satisfactory for data spans of 1.5 and longer, given low measurement variability; and for spans of 2.5 and longer, given high measurement variability. Satisfactory accuracy in estimating AUC_extrap was only achieved with low measurement variability and spans of 2.5 and longer. Neither of the classification rules improved the identification of accurate t_1/2 and AUC_extrap estimates. Based on the findings of this study, a strategy is proposed for determining the reportability of estimates of t_1/2 and area under the curve extrapolated to infinity.     

Bayesian estimation of varying-coefficient models with missing data,with application to the Singapore Longitudinal Aging Study

Motivated by the Singapore Longitudinal Aging Study (SLAS), we propose a Bayesian approach for the estimation of semiparametric varying-coefficient models for longitudinal continuous and cross-sectional binary responses. These models have proved to be more flexible than simple parametric regression models. Our development is a new contribution towards their Bayesian solution, which eases computational complexity. We also consider adapting all kinds of familiar statistical strategies to address the missing data issue in the SLAS. Our simulation results indicate that a Bayesian imputation (BI) approach performs better than complete-case (CC) and available-case (AC) approaches, especially under small sample designs, and may provide more useful results in practice. In the real data analysis for the SLAS, the results for longitudinal outcomes from BI are similar to AC analysis, differing from those with CC analysis.     

A skew-normal random effects model for longitudinal ordinal categorical responses with missing data

Missing values are common in longitudinal data studies. The missing data mechanism is termed non-ignorable (NI) if the probability of missingness depends on the non-response (missing) observations. This paper presents a model for the ordinal categorical longitudinal data with NI non-monotone missing values. We assumed two separate models for the response and missing procedure. The response is modeled as ordinal logistic, whereas the logistic binary model is considered for the missing process. We employ these models in the context of so-called shared-parameter models, where the outcome and missing data models are connected by a common set of random effects. It is commonly assumed that the random effect follows the normal distribution in longitudinal data with or without missing data. This can be extremely restrictive in practice, and it may result in misleading statistical inferences. In this paper, we instead adopt a more flexible alternative distribution which is called the skew-normal distribution. The methodology is illustrated through an application to Schizophrenia Collaborative Study data [19 D. Hedeker, Generalized linear mixed models, in Encyclopedia of Statistics in Behavioral Science, B. Everitt and D. Howell, eds., John Wiley, London, 2005, pp. 729–738. [Google Scholar]] and a simulation.     

Analysis of a design under the competing effects model with missing values

The analysis of doubly-balanced incomplete block designs (also known as 3-designs) with missing values using a competing effects model is discussed and illustrated with a numerical example.     

A novel quantification of information for longitudinal data analyzed by mixed‐effects modeling

Nonlinear mixed‐effects (NLME) modeling is one of the most powerful tools for analyzing longitudinal data especially under the sparse sampling design. The determinant of the Fisher information matrix is a commonly used global metric of the information that can be provided by the data under a given model. However, in clinical studies, it is also important to measure how much information the data provide for a certain parameter of interest under the assumed model, for example, the clearance in population pharmacokinetic models. This paper proposes a new, easy‐to‐interpret information metric, the “relative information” (RI), which is designed for specific parameters of a model and takes a value between 0% and 100%. We establish the relationship between interindividual variability for a specific parameter and the variance of the associated parameter estimator, demonstrating that, under a “perfect” experiment (eg, infinite samples or/and minimum experimental error), the RI and the variance of the model parameter estimator converge, respectively, to 100% and the ratio of the interindividual variability for that parameter and the number of subjects. Extensive simulation experiments and analyses of three real datasets show that our proposed RI metric can accurately characterize the information for parameters of interest for NLME models. The new information metric can be readily used to facilitate study designs and model diagnosis.     

Estimation of the additive hazards model with interval-censored data and missing covariates

The additive hazards model is one of the most commonly used regression models in the analysis of failure time data and many methods have been developed for its inference in various situations. However, no established estimation procedure exists when there are covariates with missing values and the observed responses are interval-censored; both types of complications arise in various settings including demographic, epidemiological, financial, medical and sociological studies. To address this deficiency, we propose several inverse probability weight-based and reweighting-based estimation procedures for the situation where covariate values are missing at random. The resulting estimators of regression model parameters are shown to be consistent and asymptotically normal. The numerical results that we report from a simulation study suggest that the proposed methods work well in practical situations. An application to a childhood cancer survival study is provided. The Canadian Journal of Statistics 48: 499–517; 2020 © 2020 Statistical Society of Canada     

Modeling a cross-sectional response variable with longitudinal predictors: an example of pulse pressure and pulse wave velocity

We wish to model pulse wave velocity (PWV) as a function of longitudinal measurements of pulse pressure (PP) at the same and prior visits at which the PWV is measured. A number of approaches are compared. First, we use the PP at the same visit as the PWV in a linear regression model. In addition, we use the average of all available PPs as the explanatory variable in a linear regression model. Next, a two-stage process is applied. The longitudinal PP is modeled using a linear mixed-effects model. This modeled PP is used in the regression model to describe PWV. An approach for using the longitudinal PP data is to obtain a measure of the cumulative burden, the area under the PP curve. This area under the curve is used as an explanatory variable to model PWV. Finally, a joint Bayesian model is constructed similar to the two-stage model.     

Two-step and likelihood methods for HIV viral dynamic models with covariate measurement errors and missing data

HIV viral dynamic models have received much attention in the literature. Long-term viral dynamics may be modelled by semiparametric nonlinear mixed-effect models, which incorporate large variation between subjects and autocorrelation within subjects and are flexible in modelling complex viral load trajectories. Time-dependent covariates may be introduced in the dynamic models to partially explain the between-individual variations. In the presence of measurement errors and missing data in time-dependent covariates, we show that the commonly used two-step method may give approximately unbiased estimates but may under-estimate standard errors. We propose a two-stage bootstrap method to adjust the standard errors in the two-step method and a likelihood method.     

Asymptotic bias in the linear mixed effects model under non-ignorable missing data mechanisms

Summary.  In longitudinal studies, missingness of data is often an unavoidable problem. Estimators from the linear mixed effects model assume that missing data are missing at random. However, estimators are biased when this assumption is not met. In the paper, theoretical results for the asymptotic bias are established under non-ignorable drop-out, drop-in and other missing data patterns. The asymptotic bias is large when the drop-out subjects have only one or no observation, especially for slope-related parameters of the linear mixed effects model. In the drop-in case, intercept-related parameter estimators show substantial asymptotic bias when subjects enter late in the study. Eight other missing data patterns are considered and these produce asymptotic biases of a variety of magnitudes.     

A semiparametric stochastic mixed effects model for bivariate cyclic longitudinal data

We propose a flexible semiparametric stochastic mixed effects model for bivariate cyclic longitudinal data. The model can handle either single cycle or, more generally, multiple consecutive cycle data. The approach models the mean of responses by parametric fixed effects and a smooth nonparametric function for the underlying time effects, and the relationship across the bivariate responses by a bivariate Gaussian random field and a joint distribution of random effects. The proposed model not only can model complicated individual profiles, but also allows for more flexible within-subject and between-response correlations. The fixed effects regression coefficients and the nonparametric time functions are estimated using maximum penalized likelihood, where the resulting estimator for the nonparametric time function is a cubic smoothing spline. The smoothing parameters and variance components are estimated simultaneously using restricted maximum likelihood. Simulation results show that the parameter estimates are close to the true values. The fit of the proposed model on a real bivariate longitudinal dataset of pre-menopausal women also performs well, both for a single cycle analysis and for a multiple consecutive cycle analysis. The Canadian Journal of Statistics 48: 471–498; 2020 © 2020 Statistical Society of Canada     

Bounding sample size projections for the area under a ROC curve

Studies of diagnostic tests are often designed with the goal of estimating the area under the receiver operating characteristic curve (AUC) because the AUC is a natural summary of a test's overall diagnostic ability. However, sample size projections dealing with AUCs are very sensitive to assumptions about the variance of the empirical AUC estimator, which depends on two correlation parameters. While these correlation parameters can be estimated from the available data, in practice it is hard to find reliable estimates before the study is conducted. Here we derive achievable bounds on the projected sample size that are free of these two correlation parameters. The lower bound is the smallest sample size that would yield the desired level of precision for some model, while the upper bound is the smallest sample size that would yield the desired level of precision for all models. These bounds are important reference points when designing a single or multi-arm study; they are the absolute minimum and maximum sample size that would ever be required. When the study design includes multiple readers or interpreters of the test, we derive bounds pertaining to the average reader AUC and the ‘pooled’ or overall AUC for the population of readers. These upper bounds for multireader studies are not too conservative when several readers are involved.     

On the optimalizes of the mbgdd's under mixed effects model

In the present paper, under the assumption of a mixed effects model with random block effects, the type 1 optimality of the most balanced group divisible designs of type 1 has been established within the general class of all proper and connected block designs with k<v.     

Pairwise- and marginal-likelihood estimation for the mixed Rasch model with binary data

A marginal–pairwise-likelihood estimation approach is examined in the mixed Rasch model with the binary response and logit link. This method belonging to the broad class of composite likelihood provides estimators with desirable asymptotic properties such as consistency and asymptotic normality. We study the performance of the proposed methodology when the random effect distribution is misspecified. A simulation study was conducted to compare this approach with the maximum marginal likelihood. The different results are also illustrated with an analysis of the real data set from a quality-of-life study.