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In describing high dimensional discrete response data, mathematical and statistical issues arise that require multivariate procedures that are not based on normal distributions, that is, the mathematical representation of high dimensional discrete response data (Event Spaces) requires a representation in lower dimensional parameter spaces consistent with the discrete properties of the Event Space. Mapping discrete responses to latent discrete classes has the limitation of not representing real individual variation within the categories. The use of a fuzzy partition model is proposed which describes individuals in terms of partial membership in multiple latent categories which represents bounded discrete event spaces with significant third and higher order moments. We discuss statistical issues arising in identifying both the deterministic and the stochastic variation of data when applications involve systematic variation due to observed and unobserved variables. We present an empirical Bayes-maximum likelihood estimation scheme for the application of the fuzzy partition models.

As schizophrenia is genetically and clinically heterogeneous, systematic investigations are required to determine whether ICD-10 or DSM-IV categorical diagnoses identify a phenotype suitable and sufficient for genetic research, or whether correlated phenotypes incorporating neurocognitive performance and personality traits provide a phenotypic characterisation that accounts better for the underlying variation. We utilised a grade of membership (GoM) model (a mathematical typology developed for studies of complex biological systems) to integrate multiple cognitive and personality measurements into a limited number of composite graded traits (latent pure types) in a sample of 61 nuclear families comprising 80 subjects with ICD-10/DSM-IV schizophrenia or schizophrenia spectrum disorders and 138 nonpsychotic first-degree relatives. GoM probability scores, computed for all subjects, allowed individuals to be partly assigned to more than one pure type. Two distinct and contrasting neurocognitive phenotypes, one familial, associated with paranoid schizophrenia, and one sporadic, associated with nonparanoid schizophrenia, accounted for 74% of the affected subjects. Combining clinical diagnosis with GoM scores to stratify the entire sample into liability classes, and using variance component analysis (SOLAR), in addition to parametric and nonparametric multipoint linkage analysis, we explored candidate regions on chromosomes 6, 10 and 22. The results indicated suggestive linkage for the familial neurocognitive phenotype (multipoint MLS 2.6 under a low-penetrance model and MLS>3.0 under a high-penetrance model) to a 14 cM area on chromosome 6, including the entire HLA region. Results for chromosomes 10 and 22 were negative. The findings suggest that the familial neurocognitive phenotype may be a pleiotropic expression of genes underlying the susceptibility to paranoid schizophrenia. We conclude that use of composite neurocognitive and personality trait measurements as correlated phenotypes supplementing clinical diagnosis can help stratify the liability to schizophrenia across all members of families prior to linkage, allow the search for susceptibility genes to focus selectively on subsets of families at high genetic risk, and augment considerably the power of genetic analysis.

The World Health Organization, in collaboration with the Indonesian Institute of Health Research and Development, conducted a survey of disability in 14 of 24 Indonesian provinces. This survey was designed to assess the validity of the disablement process as described in WHO's (1980) classification of impairments, disabilities, and handicaps. We analyzed these data with a multivariate procedure which simultaneously identifies subgroups in the surveyed population and the typical attributes of those subgroups. The purpose of the analysis was to (a) determine the association of basic patterns of physical and psychological impairments with disabilities (limitations of the ability to perform certain functions) and handicaps (limitations in the ability to fulfill social roles); (b) assess how those disabilities and handicaps were expressed in urban and rural contexts in a developing country, and (c) determine how the relation of impairments with disabilities and handicaps varied with age.

We introduce a bidirectional associative memory (BAM). The stable points of the memory are naturally interpreted as (non-sharp) concepts – the memory performs association of extents and intents of concepts. We show that this memory is stable and that the set of all stable points forms a complete lattice. We propose a learning algorithm and prove that it enables perfect learning provided the training set forms a consistent conceptual structure. Examples demonstrating the results are presented. Unlike in the case of other associative memories (M. Arbib (Ed.), The Handbook of Brain Theory and Neural Networks, MIT Press, London, 1995) the formal apparatus, architecture, dynamics and convergence proof etc. are based on algebraic structures of fuzzy logic in narrow sense.

With a latent-class model, each individual belongs to a single latent class, which determines the person's set of response probabilities for the observed, or manifest, variables. A more general model, proposed herein, adds a single parameter and involves drawing, separately and independently for each individual, a latent set of mixing weights from a Dirichlet distribution whose dispersion is governed by the added parameter. The person's set of response probabilities then consists of weighted averages of the probabilities for the classes, where the weights are the person's Dirichlet values. The posterior probabilities commonly used under latent-class models generalize under Dirichlet models to posterior expectations, which serve much the same function. We give examples of formulations of the Dirichlet model, along with numerical illustrations using published data. The first two model formulations involve Guttman scaling and panel analysis and effectively have no latent-class models that compete.

The objective of this study was to investigate Grade of Membership analysis (GoM) as a means of profiling the practice styles of individual physicians. GoM uses maximum likelihood techniques to estimate occurrences of management items that define unique practice styles. It also provides statistical estimates (grade of membership coefficients) of how well each individual physician's practice fits the identified styles. Data sources were responses to a mailed survey asking 814 physicians to select from predetermined management options in three scenarios depicting patients with acute low back pain. Four distinct types of practice style were identified: watchful waiting; investigative; guideline based; and physiotherapy focused. Most physicians were characterized by high grades of membership in the two profiles that most closely matched existing guidelines for the management of acute low back pain. GoM successfully identified characteristic practice styles, and the GoM coefficients obtained provided descriptions of individual management approaches. GoM is data driven and less subject to judgment than opinion-based measures of performance. It has several potential applications in identifying physicians for whom some form of intervention would be beneficial.

This study employed grade of membership (GoM) analysis in a clinical setting to determine if the DSM-III-R personality disorder (PD) diagnostic criteria cluster into recognizable disorders resembling the official axis II nosology. The GoM model, based on fuzzy-set theoretic concepts, explicitly examines medical diagnostic systems by quantitatively identifying and characterizing subpatterns of illness within a broad class. A semistructured assessment of 110 outpatients was performed for 12 PDs and their 112 diagnostic criteria. GoM analysis was performed using internal variables of the 112 PD criteria rated as present or absent. Demographic variables, axis I and 11 diagnosis (structured clinical Interview for DSM [SCID]), and treatment response (Global Adjustment Scale [GAS]) information were used as external validators. Four pure types (PT) provided the most satisfactory solution to the data. PT -I is characterized by marked maladaptive personality pathology, which is manipulative, egocentric, impulsive, and alloplastic. PT-II consists primarily of exaggerated socially anxious and detached traits. PT-III is sociably dependent and autoplastic. PT-IV is essentially asymptomatic. GoM provides a more parsimonious handling of the PD criteria than provided by classifying according to DSM categories. The analysis fails to confirm the natural occurrence of any single specific axis II PD or cluster.

Objective: To determine the prevalence and 3-year incidence of dementia in Blacks and Whites age 65 and older in a five-county Piedmont area of North Carolina. Design: Stratified random sample of members of the Duke Established Populations for Epidemiologic Studies of the Elderly (EPESE) (baseline n = 4,136; 55% Black; weighted n ≈ 28,000). Prevalence study members were differentially selected on the basis of score on the Short Portable Mental Status Questionnaire at the second in-person Duke EPESE wave. Incidence study members included all persons with obvious cognitive decline over a 3-year period, and a 10% sample of the remainder. Measurements: Self- and informant report on health history, functional status, and memory. Consortium to Establish a Registry for Alzheimer’s Disease (CERAD) Neuropsychology Battery administered to all subjects, and CERAD Clinical Battery to those with impaired memory. Clinical consensus to determine presence and type of dementia. Results: Prevalence of dementia for persons ⩾68 years old was 0.070 (95% confidence interval = 0.021–0.119) for Blacks and 0.072 (0.022–0.122) for Whites. Rates for Black men (0.078, 0.001–0.155) exceeded those for Black women (0.066, 0.003–0.129), but gender rates for Whites were reversed (men: 0.044, 0.000–0.103), (women: 0.087, 0.015–0.160). Neither race nor gender differences were significant. Prevalence of dementia increased through age 84 and tapered off thereafter. Three-year incidence of dementia was 0.058 (0.026–0.090) for Blacks and 0.062 (0.027–0.097) for Whites. Neither race nor gender differences were significant. Incidence increased through age 84, but moderated thereafter for all but Black men. The proportional representation of different types of dementia varied little by race. Conclusion: Prevalence, 3-year incidence, and types of dementia are comparable in Black and White elderly in the Piedmont area of North Carolina.

To identify heterogeneity in Alzheimer's disease (AD) we analyzed clinical and neuropsychological data obtained from the first 718 AD patients submitted to the Consortium to Establish a Registry for Alzheimer's Disease (CERAD). These patients, who met stringent exclusion criteria, were entered by 23 tertiary US medical centers and had the advantages of uniform evaluation, substantial size and reduction in selection biases reflecting investigator interests. A Grade of Membership (GoM) analysis was used. GoM, based in fuzzy set theory, identifies the underlying pure types in the data. GoM recognizes that individuals need not fall crisply into a particular type, but that their characteristics may fit different pure types to different extents. Clinical and neuropsychological data obtained at entry were analyzed. Six clinical pure types were identified: AD with parkinsonism; AD with depressive symptomatology; AD with mild language problems (parkinsonism and depressive symptoms absent); AD presenting with impai...

Categorical data of high (but finite) dimensionality generate sparsely populated J-way contingency tables because of finite sample sizes. A model representing such data by a "smooth" low dimensional parametric structure using a "natural" metric would be useful. We discuss a model using a metric determined by convex sets to represent moments of a discrete distribution to order J. The model is shown, from theorems on convex polytopes, to depend only on the linear space spanned by the convex set—it is otherwise measure invariant. We provide an empirical example to illustrate the maximum likelihood estimation of parameters of a particular statistical application (Grade of Membership analysis) of such a model.

The enormous amount of information representation by the brain suggests that fuzzy logic principles may be useful, especially for complex functions. A fuzzy logic model is presented which is based on the known properties of neuroanatomy and physiology.

In order to determine whether the Mini-Mental State Examination (MMSE) assesses multiple areas of cognitive functioning, a grade of membership analysis, carried out initially on data from a representative sample of community residents (see Part I), was repeated on data from 718 carefully characterized Alzheimer’s disease (AD) cases. The findings from both normal and demented samples differed little. In AD cases three pure types were identified which varied in level of cognitive impairment rather than by aspect of cognitive functioning. The presence of the originally proposed five aspects of cognitive functioning was not confirmed, although subsets (selected orientation items; three recall items; the two naming items) were identified. Nevertheless, since problems with learning and recall are a function of AD, and may underlie ability to respond to the items of the MMSE, summary MMSE score may be a sound indicator of severity of cognitive impairment, even if specific areas of cognitive functioning cannot be identified.

The Mini-Mental State Examination (MMSE), a brief, widely used measure of cognitive status, was designed to assess functioning in five areas (orientation, registration, attention and calculation, recall and language). Using a grade of membership analysis, which does not rely on a multivariate normal distribution or assume an underlying continuum for dichotomously scored variables, we examined data from a sample of 1317 community residents 18 years of age and older who made two or more errors on the MMSE. Our findings suggest that for these primarily cognitively intact persons, the items of the MMSE varied in level of difficulty; multiple distinct areas of cognitive functioning were not identified. The results of confirmatory testing using clinic populations with cognitive deficit are given in the accompanying article.

We examined the use of proxies in samples of persons aged 65 years and over from the 1982 and 1984 National Long-Term Care Surveys (NLTCS). The NLTCS are designed to describe the Medicare-enrolled elderly population, their health and functioning, hospital, home health, and institutional use. The NLTCS, being longitudinal, allows trends in functional and health status to be examined as well as the changing character of community-based and institutional services used by chronically disabled persons aged 65 years and older. In analyses of proxy responses there was little evidence of differences in accuracy between self- and proxy reports in persons with different health and functional characteristics. The amount and type of proxy reporting did depend on the health and functional characteristics of the sample person. The cognitively impaired, and the frail elderly, had high levels of proxy use as well as small differences in the accuracy of reporting service use and program enrollment. The results are consistent with methodological studies of proxy reporting in health surveys of other populations [1,2].

Grade of Membership (GoM) analysis, a multivariate classification technique based on fuzzy-set mathematics, was applied to the demographic, history, and mental-state data on 53 dementia praecox cases and 134 manic-depressive insanity cases admitted to Kraepelin's University Psychiatric Clinic in Munich in 1908. The original data recorded by Kraepelin and his collaborators on special Zählkarten (counting cards) were rated and coded in terms of the Present State Examination (PSE) Syndrome Check List. The statistical analysis resulted in a high degree of replication of Kraepelin's clinical entities. However, the dichotomy of dementia praecox and manic-depressive insanity was not fully supported. The catatonic syndrome tended to occupy an intermediate position between the two major psychoses. The possibility is discussed that catatonia in Kraepelin's time shared certain clinical features with the later diagnostic groupings of schizoaffective disorder, cycloid psychoses, and other "atypical" forms of psychotic illnesses.

Fuzzy sets and fuzzy state modeling require modifications of fundamental principles of statistical estimation and inference. These modifications trade increased computational effort for greater generality of data representation. For example, multivariate discrete response data of high (but finite) dimensionality present the problem of analyzing large numbers of cells with low event counts due to finite sample size. It would be useful to have a model based on an invariant metric to represent such data parsimoniously with a latent “smoothed” or low dimensional parametric structure. Determining the parameterization of such a model is difficult since multivariate normality (i.e., that all significant information is represented in the second order moments matrix), an assumption often used in fitting the most common types of latent variable models, is not appropriate. We present a fuzzy set model to analyze high dimensional categorical data where a metric for grades of membership in fuzzy sets is determined by latent convex sets, within which moments up to order J of a discrete distribution can be represented. The model, based on a fuzzy set parameterization, can be shown, using theorems on convex polytopes [1], to be dependent on only the enclosing linear space of the convex set. It is otherwise measure invariant. We discuss the geometry of the model's parameter space, the relation of the convex structure of model parameters to the dual nature of the case and variable spaces, how that duality relates to describing fuzzy set spaces, and modified principles of estimation.

Grade of membership analysis (GoM) may have particular relevance for genetic epidemiology. The method can flexibly relate genetic markers, clinical features, and environmental exposures to possible subtypes of disease termed pure types even when population allele frequencies and penetrance functions are not known. Hence, GoM may complement existing strategies that sometimes fail in the presence of heterogeneity or when case definition is not well established. To illustrate the method, individuals in the Seattle data set were evaluated with respect to affection status, age at onset, pedigree, sex, and genetic markers on chromosomes 19 and 21. Seven pure types were found which we have designated as: Early Onset, Late Onset, Probable, and Unaffected 1 to Unaffected 4

A number of procedures have been proposed to attack different inference problems for data drawn from a survey with a complex sample design (i.e., a design that entails unequal weighting). Most procedures either are based on finite-population assumptions or require the specification of an explicit model using a superpopulation rationale. Herein we propose some relatively simple approximate procedures that are based on a superpopulation model. They provide valid variance estimators, test statistics, and confidence intervals that allow for sample design effects as expressed by design weights and other weights. The procedures do not rely on conditioning on model elements such as covariates to adjust for design effects. Instead, we obtain estimators by rescaling sample weights to sum to the equivalent sample size (equal to sample size divided by design effect). Using weighted estimators for superpopulation models, we obtain approximations to confidence bounds on the mean for simple sampling situations...

Often environmental hazards are assessed by examining the spatial variation of disease-specific mortality or morbidity rates. These rates, when estimated for small local populations, can have a high degree of random variation or uncertainty associated with them. If those rate estimates are used to prioritize environmental clean-up actions or to allocate resources, then those decisions may be influenced by this high degree of uncertainty. Unfortunately, the effect of this uncertainty is not to add "random noise" into the decision-making process, but to systematically bias action toward the smallest populations where uncertainty is greatest and where extreme high and low rate deviations are most likely to be manifest by chance. We present a statistical procedure for adjusting rate estimates for differences in variability due to differentials in local area population sizes. Such adjustments produce rate estimates for areas that have better properties than the unadjusted rates for use in making statistically based decisions about the entire set of areas. Examples are provided for county variation in bladder, stomach, and lung cancer mortality rates for U.S. white males for the period 1970 to 1979.

Univariate [Blackorby and Donaldson (1978), Bourguignon (1979), Shorrocks (1980, 1984), Foster (1983)] and multidimensional [Kolm (1973, 1977), Atkinson and Bourguignon (1982), Ram (1982), Massoumi (1986, 1989)] measures of social welfare and economic inequality have been proposed. We consider statistical and measurement issues in empirically evaluating individual and population welfare using a procedure producing ‘fuzzy’ state descriptions which exhibit properties proposed for social welfare functions. We discuss the model's properties and its use in estimating the parameters of a multidimensional welfare process. Analyses of the U.S. elderly disabled population are presented - a group with high service needs important for federal health policy.

Cancer is often reported as contributing to the risk of noncancer causes of death. The age variation of these reports was studied using U.S. data on all causes of death listed on death certificates for 1968 to 1986. The occurrence of cancer as a nonunderlying cause of death increased with age and was higher for treatable and slowly growing tumor types. These patterns persisted even if the cancer manifested changes in occurrence. Nonunderlying occurrences were highest in the 85 to 94 age group and were correlated with cancer survival. This suggests increased importance of cancer as a cause of death and a comorbid condition among oldest-old persons. The high rate of occurrence as an associated cause of death suggests that if life expectancy increases due to declines in circulatory and other chronic disease mortality, cancer could become the preeminent cause of death in the United States.

Many diseases have both genetic and environmental determinants. Some require both, and the disease phenotype then appears only when a vulnerable genotype is expressed after interaction with environmental factors. The detection of such environmental factors has received little prior consideration in diseases with genetic causes. In particular, case-control studies of such diseases may compare exposures among cases, who have the susceptible genotype, and controls who mostly lack it. The authors explored the likely results of such studies, using the example of Alzheimer's disease as an illness where environmental factors may interact with a necessary susceptible genotype to accelerate disease expression. They found that case-control studies of environmental factors in complex genetic diseases will usually produce an odds ratio that differs little from the relative risk among susceptible individuals. In rare situations, however, the discrepancy may be gross. The statistical power of such studies also agrees well with familiar published estimates, suggesting that little power is lost even though the controls are mostly not susceptible. Power may be increased, however, in studies of common illnesses with genetic determinants when the case-control method is applied among discordant monozygotic twins.

Analyzing multivariate clinical data to identify subclasses of patients being treated for a specific disease may improve patient management and increase understanding of the behavior of disease under clinical conditions. In some cases, patients have been classified on prognostic characteristics using standard risk assessment procedures (e.g.. Cox’ regression). This requires long term follow-up, differentiates patients only on attributes relevant to survival, and assumes that patients are sampled from a common population. Other approaches involve the use of clustering algorithms to classify patients into categories based on multiple clinical attributes. We illustrate the use of a multivariate statistical procedure to directly characterize patients on multiple clinical characteristics. The procedure is designed to analyze discrete response data with parameters representing individual differences within groups. Its use is illustrated for patients with Stage I melanoma in determining how age is related to treatment response in different patient groups.

Models of gender differences in human mortality and aging depend on assumptions about temporal rates of physiological change. Simple models like the Gompertz fail to describe the mortality of either males or females at late ages. This suggests a need for biologically more detailed models to represent the age dependency of human mortality as well as gender differences in that age dependence. By modeling the sex-specific interaction of time-varying covariates with multiple dimensions of mortality selection, one can more accurately describe the age dependence of mortality and more complex physiological aging patterns. The multivariate model of aging changes is used to describe gender differences using data from (a) a longitudinal study of physiological changes and mortality and (b) a nationally representative longitudinal survey of changes in function and mortality.

BACKGROUND Elderly residents of nursing homes often have multiple comorbidities and functional limitations. The status of 4,525 residents of complex and standard care units in 177 nursing homes where the nursing home was determined to provide adequate care, and from 14 specialized Alzheimer's units, was evaluated on 111 measures of medical condition, functional status, psychological well-being and cognitive performance in a demonstration study assessing quality of care in six states. Detailed measurements were also made of the types and amounts of services used (in minutes per day) by the residents. METHODS Given the number of health measures, and the possibility of assessment error, a multivariate analytic procedure called Grade of Membership (GoM) was used. This procedure identified profiles of health and functioning measures to identify the characteristics of clinically distinct groups of nursing home residents. RESULTS The analysis identified 11 profiles of health and functioning characteristics which described the 111 resident measurements. The 11 profiles predicted differentials in nursing home length of stay, and service use by various classes of caregivers. The GoM profiles described the data better than several other classification procedures applied to the same data. CONCLUSIONS In nursing homes, elderly and oldest-old residents often have multiple comorbidities and disabilities. A multivariate procedure was able to identify the fundamental dimensions describing residents' variation on a number of health measures. These profiles predicted differences in service use so they had predictive validity. Thus, multivariate procedures may help identify clinically distinct groups in studies where complex measures are made.

A model of human health history and aging, based on a multivariate stochastic process with both continuous diffusion and discrete jump components, is presented. Discrete changes generate non-Gaussian diffusion with time varying continuous state distributions. An approach to calculating transition rates in dynamically heterogeneous populations, which generalizes the conditional averaging of hazard rates done in "fixed frailty" population models, is presented to describe health processes with multiple jumps. Conditional semi-invariants are used to approximate the conditional p.d.f. of the unobserved health history components. This is useful in analyzing the age dependence of mortality and health changes at advanced age (e.g., 95+) where homeostatic controls weaken, and physiological dynamics and survival manifest nonlinear behavior.

Models of mortality and aging depend on assumptions about physiological change even if they are not made explicit. Standard models, like the Gompertz, often fail to describe mortality at extreme ages, suggesting a need for biologically more detailed and flexible models. One solution is to model the interaction of time-varying covariates with mortality to better describe the age dependence of mortality, test hypotheses about the relation of physiological change and mortality, and use longitudinal data to generalize assumptions about physiological change. This model is applied to (a) a 34-year follow-up of risk factors and mortality and (b) a 9.5-year follow-up of function and mortality from longitudinal surveys of the U.S. elderly population.

SynopsisThe Grade of Membership (GoM) model is a classification procedure which allows a person to be a member of more than one diagnostic class. It simultaneously quantifies the degrees of membership in classes while generating the discrete symptom profiles or ‘pure types’ describing classes. The model was applied to the symptomatology, history, and follow-up of 1065 cases in the WHO International Pilot Study of Schizophrenia. The model produced an eight diagnostic class or ‘pure type’ solution, of which five were related to the diagnostic concepts of schizophrenia and paranoid disorder, two types were affective disorders, and one asymptomatic type. A subtype of paranoid schizophreniform disorder found primarily in developing countries was identified. There was a strong association between pure types and the original clinical and computer generated (CATEGO) diagnoses. A GoM based psychiatric classification might more clearly identify core disease processes than conventional classification models by filtering the confounding effects of individual heterogeneity from pure type definitions.

In Europe and most developed and developing countries of the world, increasing levels of resources are required to fund health care. This is due to a variety of fundamental causes, including the aging of populations, which leads to increasing per capita use of health care services, the fact that higher per capita income also leads to increasing use of health care services, and technical advances which allow physicians to treat previously unbeatable diseases and conditions. This has increased interest in defining equitable methods for resource allocation and payment for health care services. This chapter focuses on inpatient hospital services, although taking ambulatory care or long-term care (LTC) into account may be equally important over the long-term.

The Grade of Membership (GOM) model is a general multivariate procedure for analyzing high dimensional discrete response data. It does this by estimating, using maximum likelihood principles, two types of parameters. One describes the probability that a person who is exactly like one of the K analytically defined types has a particular response on a given variable. The second describes each individual's degree of membership in each of the K types. This "partial" membership score reflects the logic of the fuzzy partitions (rather than of discrete groups) that are employed in the analyses. By modifying the probability structure of the basic model we show how the procedure can be applied to a number of different types of data and analytic problems. The utility of the different GOM models for different types of aging research is discussed.

The results of a comparison of two typologies of sibling relationships in old age are reported. Both analyses rely on the same data collected in individual interviews with adults over the age of 65. The first typology was constructed using constant comparative analysis; the second relied on the grade of membership (GOM) technique. This is the first time GOM has been used to create a taxonomy of human relationships based on psychosocial variables. The two typologies are compared in terms of number of types generated, the characteristics of each type, factors influencing typological construction, and the utility of empirical results. Implications for additional use of the GOM technique to study late-life sibling relations are discussed.

Isolated cold stress testing (ICST) has been used to assess cold stress performance or digital thermoregulation, but statistical analysis of the results has been limited to visual comparisons of data. In this prospective study, 11 patients who underwent complete digital replantation were followed serially with ICST at 6 weeks, 3 months, 6 months, 1 year, and 2 years postoperatively, and the results were analyzed quantitatively. For that analysis, we devised a mathematical method that provided a cooling and a warming coefficient to fit the data. Differences in these cooling and warming coefficients were then regressed against time after replantation, patient age, number of digits replanted, hand dominance, and clinical evidence of sensory recovery. There was a significant correlation between response to isolated cold stress testing (thermoregulation) and degree of sensory recovery (p less than or equal to 0.02). This method for quantitative analysis of isolated cold stress testing data allows objective evaluation of digital blood flow patterns based on temperature, thereby providing a reliable and objective assessment of the recovery of thermoregulation in the replanted human digit.

The applicability of the theory of partially observed finite-state Markov processes to the study of disease, morbidity, and disability is explored. A method is developed for the continuous updating of parameter estimates over time in longitudinal studies analogous to Kalman filtering in continuous valued continuous time stochastic processes. It builds on a model of filtering of incompletely observed finite-state Markov processes subject to mortality due to Yashin et al. The method of estimation is based on maximum likelihood theory and the incompleteness in the observation of the process is dealt with by applying missing information principles in maximum likelihood estimation.

One likely consequence of the aging of the U.S. population is the growth of the number of persons in nursing homes. As the numbers of persons in nursing homes increase so will the amount of resources required to keep them in those homes. This will make it increasingly important to understand the dynamics of nursing home utilization so that we can more effectively plan the allocation of resources. Unfortunately, we lack direct information on the dynamics of nursing home utilization both because of the expense of implementing longitudinal studies to gather such information and because the available data on current residents are inappropriate to study the dynamics of utilization because of several types of bias. Demographic methods are presented that can be applied to survey data to remove sample biases from the data and permit study of the dynamics of the utilization of facilities for the total U.S. nursing home population and for various of its components.