Bayesian and profile likelihood change point methods for modeling cognitive function over time

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Abstract

Change point models are often used to model longitudinal data. To estimate the change point, Bayesian (Biometrika 62 (1975) 407; Appl. Statist. 41 (1992) 389; Biometrics 51 (1995) 236) or profile likelihood (Statist. Med. 19 (2000) 1555) methods may be used.

We compare and contrast the two methods in analyzing longitudinal cognitive data from the Bronx Aging Study. The Bayesian method has advantages over the profile likelihood method in that it does not require all subjects to have the same change point. Caution must be taken regarding sensitivity to choice of prior distribution, identifiability, and goodness of fit. Analyses show that decline in memory precedes diagnosis of dementia by 7.5–8 years, and individual change points are not needed to model heterogeneity across subjects.

Introduction

Dementia is characterized by accelerated cognitive decline before and after diagnosis as compared to normal aging. Determining the time at which rate of decline begins to accelerate in persons who will develop dementia is important, both in describing the natural history of the disease process, and in identifying the optimal time window for treatments designed to prevent the development of diagnosable disease in those who are in the early stages of accelerated cognitive decline (Smith et al., 1996; Peterson et al., 1999).

The study of dementia is problematic in even the best-designed study. Alzheimer's disease (AD), the most common cause of dementia (Hendrie, 1998), is a progressive degenerative disease that generally presents with increasing cognitive loss over a period of many years. Studies of diagnosed AD cases capture only a portion of the disease's natural history. Diagnosis is made by a medical professional at a clinical examination, when a somewhat arbitrary threshold of cognitive loss is reached. Since that threshold requires decline in social or occupational function, the time of diagnosis is determined in part by social and occupational demands and not just disease severity (McKhann et al., 1984; American Psychiatric Association, 1994). The actual time of diagnosis is also influenced by a patient's schedule of clinic visits. These difficulties result in natural interval censoring of the time of dementia diagnosis. In addition, it is often difficult even for an experienced clinician to distinguish among dementia subtypes (AD, vascular dementia, etc.). As a consequence, misclassification and heterogeneity in dementia subtype contribute to variability. Memory tests provide the most appropriate method for following cognitive status in dementia. Memory loss is the only obligatory cognitive feature in dementia and is often the first feature to develop as dementia unfolds (Grober et al., 1988; Katzman, 1993; American Psychiatric Association, 1994; Peterson et al., 1994).

Prospective studies in which repeated measurements are taken on individuals require that statistical methods appropriate for longitudinal data be used (Diggle et al., 1994). For continuous data such as scores on neuropsychological tests, linear models with random effects have been used for many years (Laird and Ware, 1982) and applied to longitudinal cognitive data (for example, Wilson et al., 1999). Because cognitive decline accelerates when dementia begins, longitudinal studies should consider the possibility of a nonlinear relationship between cognitive function and time. Polynomial functions are well suited for this, but change point models in which an inflection point in the trajectory of cognitive decline is estimated from the data have the important advantage of allowing inferences to be made regarding the time at which the rate of cognitive decline accelerates (Hall et al., 2000).

Change point inference has a long history. Hinkley (1970) described a frequentist approach to change point problems. Smith (1975) developed a Bayesian approach. Both papers were restricted to discrete time analyses where the index of a sequence of random variables corresponding to the change point was estimated. Carlin et al. (1992) extended Smith's approach using Markov chain Monte Carlo (MCMC) methods for change points having continuous support. Lange et al. (1994), and Kiuchi et al. (1995) used MCMC methods for the analysis of longitudinal data in HIV studies. Hall et al. (2000) used a profile likelihood approach in studying longitudinal cognitive function data. The profile likelihood approach requires that all persons with a change point have the same change point, an assumption that may not be appropriate given the clinical heterogeneity of dementia and the problem of interval censoring. In this paper, we compare a profile likelihood approach similar to that of Hall et al. (2000) with a Bayesian approach in the spirit of Kiuchi et al. (1995) in the analysis of cognitive data from the Bronx Aging Study, in order to examine the progression in cognitive decline in persons who develop dementia. In particular, we will examine the common change point assumption.

Section snippets

The Bronx aging study data

The Bronx Aging Study (BAS, Katzman et al., 1989) began in 1980, and enrolled 488 initially community dwelling healthy individuals age 75–85. During 20 years of follow-up, 121 have developed dementia, with the following dementia subtypes based on clinical criteria: 59 probable or possible AD, 30 probable or possible vascular dementia, 16 mixed dementia (usually AD+vascular), 1 Parkinson's disease, and 15 other or unknown (generally multiple causes). Three hundred and sixty-seven persons did not

Profile likelihood

Hall et al. (2000) used both chronological age and time to dementia diagnosis in comparing persons who develop dementia to those who do not, and used a profile likelihood method to develop inferences. A similar approach will be used here to compare age-associated memory decline to disease-associated memory decline among those who develop dementia. Specifically, models of individual regression coefficients and individual change points of the following type will be considered:yikii1xi1ki2x

Model selection

We wish to be able to compare the “reduced model” (MR): τ1=τ2=⋯=τn=τ to the “full model” (MF) in which one or more of the change points differ among subjects. We use two methods that have been proposed for model selection in a Bayesian context: the pseudo-Bayes factor (PSBF) and the posterior Bayes factor (POBF). Let y(−i) denote the vector of responses with the ith subject deleted from the data set and θr denote the vector of unknown parameters for the model Mr(r=1,2). We use L to denote the

Results

We were concerned that results from the Bayesian analyses might be sensitive to the prior distributions; therefore, we ran numerous scenarios as a sensitivity analysis. Table 1 lists the prior choices (shape and scale) for each of the parameter σ2 and the hyperparameters σα2, σβ12, σβ22, σβ32 and σβ42. These three scenarios are chosen as follows: MR1 and MF1 match the profile likelihood results to the mean and variance of the inverse gamma density; MR2 and MF2 reflect more diffuse priors than MR

Discussion

We have demonstrated the use of Bayesian approaches to the estimation of change points in data where follow-up varies enormously. For models in which a common change point is assumed, the results are similar to that obtained via a profile likelihood method. Results were relatively insensitive to the choice of prior distributions for the variance component parameters. Results of model selection criteria show that allowing each subject to have his/her own individual change point does not add to

Acknowledgements

The authors wish to express their appreciation to Dr. Martin Sliwinski, Dr. Herman Buschke, and Ms. Mindy Katz for their assistance.

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    This research supported in part by National Institute on Aging grants AG-03949 and AG-13631.

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