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Modelling effectiveness of internally heterogeneous organizations in the presence of survey non-response: an application to the ULTRA study.

Levels of turbulence and uncertainty in family medicine practices participating in the ULTRA study are measured via a survey from heterogeneous viewpoints within each practice, including clinicians, nursing staff and office staff. In order to examine the effect of the average practice 'chaos' score on screening provided to patients, the analysis needed to account for large amounts of missing survey responses. Patterns of missing data included portions of missing responses within a job group (e.g. one out of three clinicians) to missing responses from whole subgroups within a practice (e.g. all three clinicians) to a whole practice of missing survey responses (e.g. all clinicians, nursing staff and office staff). The expectation-maximization (EM) algorithm, using a hierarchical model for covariate information, is used to handle the missing data. It was found that the varied patterns of missingness of survey responses among the heterogeneous subgroups within the practices had large effects on estimated effects of practice chaos on patient care.

Cholesterol↗

Data exploration in meta-analysis with smooth latent distributions.

Meta-analysis with discrete outcomes is interpreted as the estimation (in one or two dimensions) of a non-parametric smooth latent distribution of event probabilities (or rates). A simple but efficient EM algorithm is presented. A fine grid is used and fast smoothing is done by penalized least squares. Data exploration is the primary goal, but the estimated distribution can also be used to compute useful statistics of treatment effects.

Algorithms↗

A classification of Scottish infants using latent class analysis.

This paper illustrates the use of latent class analysis to classify 50,000 infants into a small number of classes or case types, as a preliminary to a study of the allocation of neonatal hospital resources throughout Scotland. Information, extracted from a detailed neonatal discharge record, was summarized by 11 clinical and diagnostic catagorical variables. Statistical models incorporating 1 to 6 latent classes were then estimated using the EM algorithm. The 4 class model was chosen because it provided a good description of the data and the resulting classes had a medical interpretation. The factors influencing the choice of model are discussed and goodness of fit tests are presented. The stability of the classes was also investigated using random halves of the data and an earlier comparable data set.

Classification↗

The use of an extended baseline period in the evaluation of treatment in a longitudinal Duchenne muscular dystrophy trial.

A trial of Duchenne muscular dystrophy involved tracking boys of all ages through a one-year baseline period, followed by a one-year trial of leucine versus placebo treatment. In this paper we develop a model for a total-muscle-strength score that uses the data of the extended baseline period in the evaluation of the leucine treatment. The model is based on a polynomial growth curve in age whose coefficients can vary according to treatment or phase. Maximum likelihood estimates of the parameters of the model are obtained from use of the EM algorithm. We propose tests for the adequacy of the model as well as for treatment effects. A quadratic model appears the most parsimonious fit to the data and there is no evidence of any leucine effect on scores. We examine the asymptotic power of the test for treatment effect and compare it with that of a simpler analysis.

Adolescent↗

A comparative study of three methods for analysing longitudinal pulmonary function data.

We compare three methods of longitudinal analysis of pulmonary function data. Our data set is taken from a study of exposure to toluene diisocyanate (TDI) vapours in a new manufacturing plant. The first two methods are a two-stage weighted regression method and maximum likelihood estimation via the EM algorithm, and these give very similar results. The third method, regression with an autoregressive error structure, was not successfully implemented, and in our view needs better documentation.

Adult↗

The analysis of titration studies in phase III clinical trials.

Clinical trials commonly employ the titration design for certain drugs such as antihypertensives. In a Phase III trial the design has purposes distinct from those of a Phase I or II trial, as well as from those of a trial with a parallel design. In this paper we compare the titration design with the usual parallel design in their respective purposes for Phase III trials, explore the relevant questions addressed, and examine typical data from such trials. We also discuss work which focuses primarily on the Phase I or II titration trials. We formulate the problem in the framework of one-way contingency table augmented with incomplete data and obtain the maximum likelihood estimates of the parameters and their estimated variances/covariances via the EM algorithm. An example of a Phase III study of an antihypertensive agent illustrates the proposed procedure.

Analysis of Variance↗

Adjusting for age-related competing mortality in long-term cancer clinical trials.

Mortality related to causes other than the treated disease may have a significant impact on overall survival in long-term clinical trials. We present a model that adjusts for age-related competing mortality when cause of death is missing or only partially available. Through use of a piecewise exponential survival model, we extend relative survival methods to continuous follow-up data, allowing the competing mortality to differ from that of the general population by a scale parameter. An EM algorithm provides a simple way to compute the maximum likelihood estimators (MLEs) and to test hypotheses using widely available software. We compare the bias and relative efficiency of this model to a piecewise exponential Cox model for overall survival. Theoretical results are confirmed by simulations and illustrated with data from a clinical trial in colorectal cancer. This example also shows how age-related and disease-related mortality can be confounded in an analysis of overall survival. We conclude with a discussion of the advantages and disadvantages of the model.

Adolescent↗

A method of non-parametric back-projection and its application to AIDS data.

The method of back-projection has been used to estimate the unobserved past incidence of infection with the human immunodeficiency virus (HIV) and to obtain projections of future AIDS incidence. Here a new approach to back-projection, which avoids parametric assumptions about the form of the HIV infection intensity, is described. This approach gives the data greater opportunity to determine the shape of the estimated intensity function. The method is based on a modification of an EM algorithm for maximum likelihood estimation that incorporates smoothing of the estimated parameters. It is easy to implement on a computer because the computations are based on explicit formulae. The method is illustrated with applications to AIDS data from Australia, U.S.A. and Japanese haemophiliacs.

Acquired Immunodeficiency Syndrome↗

Computational aspects of analysing random effects/longitudinal models.

Random effects and longitudinal models are becoming increasingly popular in the analysis of many types of data, including medical and biopharmaceutical, because of their richness and flexibility. They can be, however, difficult to fit using traditional statistical tools. Fortunately, there now exists a burgeoning collection of newer computational methods that can be applied to draw inferences with such models. This review attempts to provide an introduction to some of these techniques by describing them as extensions of the EM algorithm, currently a standard tool for the analysis of longitudinal and random effects models. For clarity of exposition, the extensions are classified into three types: large-sample iterative; large-sample simulation, and small-sample simulation.

Longitudinal Studies↗

Methods for the analysis of informatively censored longitudinal data.

This paper describes the problem of informative censoring in longitudinal studies where the primary outcome is rate of change in a continuous variable. Standard approaches based on the linear random effects model are valid only when the data are missing in a non-ignorable fashion. Informative censoring, which is a special type of non-ignorably missing data, occurs when the probability of early termination is related to an individual subject's true rate of change. When present, informative censoring causes bias in standard likelihood-based analyses, as well as in weighted averages of individual least-squares slopes. This paper reviews several methods proposed by others for analysis of informatively censored longitudinal data, and outlines a new approach based on a log-normal survival model. Maximum likelihood estimates may be obtained via the EM algorithm. Advantages of this approach are that it allows general unbalanced data caused by staggered entry and unequally-timed visits, it utilizes all available data, including data from patients with only a single measurement, and it provides a unified method for estimating all model parameters. Issues related to study design when informative censoring may occur are also discussed.

Linear Models↗

A latent class model for repeated measurements experiments.

Standard models for the analysis of repeated measurements assume a common response profile for all experimental units within a treatment group. However, in many applications this under-represents the nature of the response. There may be several distinct modes of response within a group (for example, responders versus non-responders to a given treatment), or there may be a set of distinct response profiles which are common to all the treatment groups. In these situations the effect of treatment can be characterized both by the shape of the fitted profiles and by estimating the proportion of cases who exhibit each particular response profile. This paper describes how such experiments may be analysed through the introduction of a latent variable into the standard model. Maximum likelihood estimation is straight-forward using the EM algorithm. Model choice requires some care, but good-fitting models can be identified via inspection of residuals and the use of empirical semi-variogram plots. Once the number of distinct profiles has been determined, treatment effects can be investigated using likelihood-ratio statistics. The approach is illustrated with a re-analysis of a dataset first described by Grizzle and Allen.

Animals↗

Studying the relationship between change and initial value in longitudinal studies.

Blomqvist's problem of studying the relationship between change and initial value in a linear growth curve setting is reformulated from a random effects model perspective. First, a maximum likelihood estimate of the between-individual covariance matrix for a simple linear regression model with stochastic parameters is obtained via an EM algorithm as discussed by Laird and Ware. Second, the regression coefficient of the individual-specific slopes on the individual-specific intercepts is estimated as a ratio of elements of the between-individual covariance matrix as discussed by Zucker et al. Then a Fieller's type confidence interval for this ratio is proposed. Discussion is facilitated by recognizing the Laird-Ware model as a special case of a more general model discussed by Hocking.

Algorithms↗

Using time of first positive HIV test and other auxiliary data in back-projection of AIDS incidence.

Estimation of HIV incidence by the method of back-projection typically uses data on the time of diagnosis of AIDS cases, together with known information about the incubation distribution of AIDS. This paper discusses back-projection using auxiliary data on AIDS cases, particularly the time of first positive HIV test. We discuss the possibility that certain types of auxiliary data, including time of first positive test, can be useful in back-projection because they provide extra information about the incubation period of AIDS cases. Under a back-projection model, theoretical efficiency calculations are given comparing back-projection with and without the time of first positive HIV test of AIDS cases. These calculations suggest that such data have the potential to significantly improve HIV incidence estimates, particularly in the recent past. Smoothed non-parametric estimates of both HIV incidence and time-dependent testing rates are described. These can be obtained using the EM algorithm, in conjunction with a smoothing step or a penalized likelihood. The benefit of these methods in practice needs to be assessed as such data become available.

Acquired Immunodeficiency Syndrome↗

The analysis of repeated-measures data on schizophrenic reaction times using mixture models.

Reaction times for schizophrenic individuals in a simple visual tracking experiment can be substantially more variable than for non-schizophrenic individuals. Current psychological theory suggests that at least some of this extra variability arises from an attentional lapse that delays some, but not all, of each schizophrenic's reaction times. Based on this theory, we pursue models in which measurements from non-schizophrenics arise from a normal linear model with a separate mean for each individual, whereas measurements from schizophrenics arise from a mixture of (i) a component analogous to the distribution of response times for non-schizophrenics and (ii) a mean-shifted component. We fit four mixture models within this framework, where the distinctions between models arise from assumptions about the variance of the shifted observations and the exchangeability of schizophrenic individuals. Some of these models can be fit by maximum likelihood using the EM algorithm, and all can be fit using the ECM algorithm, where the covariance matrices associated with the parameters are calculated by the SEM and SECM algorithms, respectively. Bayesian model monitoring using posterior predictive checks is invoked to discard models that fail to reproduce certain observed features of the data and to stimulate the development of better models.

Algorithms↗

Re-evaluating design specifications of longitudinal clinical trials without unblinding when the key response is rate of change.

The design of clinical trials often requires knowledge of quantities such as between- and within subject variances about which only imprecise information exists. To provide assurance that the study has the desired power to detect a minimum clinically meaningful difference between treatment groups. Gould, Gould and Shih, and Shih have recommended obtaining relevant information from the trial at an interim stage without unblinding. Wittes and Brittain provided a similar recommendation, but viewed the portion up to the interim stage as an (internal) pilot study and required unblinding. This paper considers the problem of re-evaluating the design specifications in longitudinal clinical trials when the key response is the rate of change (slope). The proposed method aims to re-evaluate the sample size and study duration in a way that maintains the trial's blinding, using an EM algorithm. Simulation results show that the effect on type I error rate in negligible, but the potential gain in power can be substantial. The procedure is simple to use in practice, as it does not unblind patients' treatment identifications, and, since it does not unveil the relative efficacy of treatments, it fulfils the requirement of a valid 'administrative' (interim) analysis.

Algorithms↗

Assessing human fertility using several markers of ovulation.

In modelling human fertility one ideally accounts for timing of intercourse relative to ovulation. Measurement error in identifying the day of ovulation can bias estimates of fecundability parameters and attenuate estimates of covariate effects. In the absence of a single perfect marker of ovulation, several error prone markers are sometimes obtained. In this paper we propose a semi-parametric mixture model that uses multiple independent markers of ovulation to account for measurement error. The model assigns each method of assessing ovulation a distinct non-parametric error distribution, and corrects bias in estimates of day-specific fecundability. We use a Monte Carlo EM algorithm for joint estimation of (i) the error distribution for the markers, (ii) the error-corrected fertility parameters, and (iii) the couple-specific random effects. We apply the methods to data from a North Carolina fertility study to assess the magnitude of error in measures of ovulation based on urinary luteinizing hormone and metabolites of ovarian hormones, and estimate the corrected day-specific probabilities of clinical pregnancy. Published in 2001 by John Wiley & Sons, Ltd.

Algorithms↗

Analysis of change in the presence of informative censoring: application to a longitudinal clinical trial of progressive renal disease.

The rate of change in a continuous variable, measured serially over time, is often used as an outcome in longitudinal studies or clinical trials. When patients terminate the study before the scheduled end of the study, there is a potential for bias in estimation of rate of change using standard methods which ignore the missing data mechanism. These methods include the use of unweighted generalized estimating equations methods and likelihood-based methods assuming an ignorable missing data mechanism. We present a model for analysis of informatively censored data, based on an extension of the two-stage linear random effects model, where each subject's random intercept and slope are allowed to be associated with an underlying time to event. The joint distribution of the continuous responses and the time-to-event variable are then estimated via maximum likelihood using the EM algorithm, and using the bootstrap to calculate standard errors. We illustrate this methodology and compare it to simpler approaches and usual maximum likelihood using data from a multi-centre study of the effects of diet and blood pressure control on progression of renal disease, the Modification of Diet in Renal Disease (MDRD) Study. Sensitivity analyses and simulations are used to evaluate the performance of this methodology in the context of the MDRD data, under various scenarios where the drop-out mechanism is ignorable as well as non-ignorable.

Algorithms↗