Latent variable modeling of disability in people aged 65 or more more

Stat Methods Appl (2011) 20:49–63 DOI 10.1007/s10260-010-0148-6 Latent variable modeling of disability in people aged 65 or more Giorgio E. Montanari · M. Giovanna Ranalli · Paolo Eusebi Accepted: 10 September 2010 / Published online: 2 October 2010 © Springer-Verlag 2010 Abstract This paper aims at classifying, on the basis of their disability profile, the population of elderly and quantifying the number of those with a very low level of functioning in a central region of Italy. This is accomplished using a set of variables on the difficulty of accomplishing everyday tasks (Activities of Daily Living, ADL) and functions. This issue is very important for National and Local Health organizations in order to evaluate the need for care, planning services, elaborating policies and allocating resources. Latent class models are applied on data coming from the Italian National Survey on Health Conditions and Appeal to Medicare to extract the latent trait of disability and classify the elder population according to their disability profile. Model selection brings to a classification into four latent classes. Looking at posterior probabilities, classes may be interpreted as follows: elderly without disability, with difficulties in movements, with difficulties in movements and daily tasks, with very low functioning level. Estimates of the amount of population aged 65 or more falling in each class is also provided. Cross-validation shows evidence of the robustness of such classification. Item response theory models are also applied to the items considered to study how functions are lost with increasing levels of disability. In particular, the abilities of climbing stairs and stooping down are those lost first, while those of eating and getting washed are those lost last. The work reported here was supported by a grant awarded to the Department of Economics, Finance and Statistics of the University of Perugia by the Regione Umbria. G. E. Montanari · M. G. Ranalli (B ) Dipartimento di Economia, Finanza e Statistica, Università degli Studi di Perugia, Via Pascoli, 06123, Perugia, Italy e-mail: giovanna.ranalli@stat.unipg.it P. Eusebi Epidemiology Department, Regional Health Authority of Umbria, Via M. Angeloni, 61, 06123, Perugia, Italy 123 50 G. E. Montanari et al. Keywords Categorical variables · Latent variables model · Rasch model · Partial credit model · Activities of daily living 1 Introduction From the latest available data, the proportion of population aged 65 or more in Italy was 20.1% in 2009. This was the largest value among EU25 European countries. Aging of the population is a central issue for policy makers. In particular, assessing a condition of severe disability and estimating the number of people in such a condition is very important for its consequences on Health system organization, policy making and funding. In this work we propose a method for classifying the population of elderly according to profiles of disability in a central region of Italy, made of 11 provinces (LAU1 level), and to estimate the number of people that show each profile in such a population. To this end, we use data routinely collected by the Italian Statistical Institute for more general purposes. The interest on this area comes from the concern shown by the Administrative Department responsible for Health organization and planning of the Region Umbria. It is located in the centre of Italy and shows the second largest proportion of people aged 65 or more in the nation. The area of interest is made of the two provinces of the region Umbria and of all their neighboring provinces. To fulfill the task of classifying the elderly and obtaining statistical significative estimates of the number of people that show each profile in the whole population of elderly, since ah hoc surveys are too expensive, we propose to use data from some existing extensive surveys on the population. To this end, we use data coming from the national survey on Health Conditions and Appeal to Medicare 1999–2000 conducted by the Italian National Institute of Statistics (ISTAT 2002). Also note that the survey is conducted routinely every five years, thereby providing the possibility of a consistent monitoring of the phenomenon over time. The questionnaire employed in such survey is constructed accounting for the International Classification of Impairments, Disabilities and Handicaps (ICIDH) developed by the World Health Organization in 1980. The latter provides a conceptual framework for disability, which is described in three dimensions: Impairment, Disability and Handicap. In the context of health experience an impairment is any loss or abnormality of psychological, physiological or anatomical structure or function. On the other side, a disability is any restriction or lack (resulting from an impairment) of ability to perform an activity in the manner or within the range considered normal for a human being, whilst a handicap is a disadvantage for a given individual, resulting from an impairment or a disability, that limits or prevents the fulfillment of a role that is normal (depending on age, sex, and social and cultural factors) for that individual. ICIDH has been long used ever since its introduction for various purposes, such as in health statistics, research, clinical work, and social policy. In recent years, however, such classification has been replaced by WHO with the International Classification of Functioning, Disability and Health (ICF, see http://www.who.int/classifications/icf/en/), which has not been implemented yet in the Italian survey on health conditions although this is on the agenda (Crialesi 2008). In the Italian survey on Health Conditions and Appeal to Medicare, a person is considered disabled if, excluding temporary limitations, he/she expresses the largest 123 Latent variable modeling of disability in people aged 65 or more 51 degree of difficulty in at least one of a set of functions concerning the ability of a person to accomplish everyday tasks such as getting washed and dressed, eating, walking or hearing, even with the aid of tools such as glasses, walking sticks, prostheses. If an elder expresses the maximum degree of difficulty in fulfilling one of these tasks, then he/she is classified to be in a disability condition. From a policy maker perspective, this definition of disability is way too loose, in that an elder may be defined as disabled because he/she cannot hear a TV-show, but he/she can perfectly take care of him/herself without an extra-burden for the National Health Care system. In fact, what is mainly interesting for policy making and planning is to evaluate that level of severe disability that is strongly connected with a status of dependency or self insufficiency. ISTAT (2004), for example, defines an elder as self insufficient when affected by a permanent disability or a chronic disease, that deprives his/her autonomy so much that he/she needs continuous personal assistance to complete basic everyday tasks. Activities of Daily Living (ADLs) have been used extensively to detect disability and, often, together with Instrumental ADLs (IADLs) that include basic activities necessary to reside in the community, such as grocery shopping, telephoning, and housekeeping. The basic approach to measuring disability is by using a summed index, where individual scores on all items are summed to produce a total. One obvious sufficient condition for the summed index (total score) to be valid is that all items equally contribute to some sort of measure of disability. This condition may not hold, however, when ADL measures are considered—someone who cannot climb into a bath is usually not assumed to be as equally disabled as someone who cannot eat—and must, therefore, be carefully checked. Therefore, tools from Item Response Theory are applied to obtain a proper aggregated measure of functional disability. Erosheva (2002) provides an excellent review of application of such models to ADLs. See also Cabrero-Garcìa and Lòpez-Pina (2008) for the use ADLs and IADLs to obtain an aggregated measure of functional disability from a national survey in Spain. Disability-based classification in rehabilitation is conducted using the FIM-Functional Independence Measure Scale (see e.g. Tennant et al. 2004; Poss et al. 2008). Note that IADLs are not surveyed in the Italian survey on Health Conditions and Appeal to Medicare, nor is “continence” which is a primary source and an indicator of dependence in elderly who show an otherwise good mobility. Therefore, we will resort to ADLs and to the other items used to this end in the survey. Section 2 presents the survey employed and the data available in more detail. After, we will first employ Item Response Theory models, and in particular Partial Credit Models (PCMs, Masters 1982), to obtain a measure of disability and to investigate the role of each item to measure the difficulty of accomplishing activities and tasks. Such approach, however, provides a continuous value for the latent variable and, therefore, does not address our issue of classification unless one or more thresholds are somehow imposed. To overcome this issue, in Sect. 3 we employ Latent Class Models (LCMs; Lazarsfeld and Henry 1968; Goodman 1974): they find homogeneous groups (latent classes) from multivariate categorical data. LCMs have been applied in many fields related to social sciences and public health research. D’Uva (2005) emphasizes the role of latent classes for modeling health care utilization. Teresia et al. (1999) estimate the prevalence of cognitive impairment, signs of stroke and Parkinson’s disease. Li et al. (2003) estimate the prevalence of fear of falling in the elderly related to functional 123 52 G. E. Montanari et al. ability and quality of life. LCMs have been used for evaluating diagnostic criteria for autism (Szatmari et al. 1995) and for sorting people into clusters with respect to their symptom profiles related to post-traumatic stress disorders (Breslau et al. 2005). Erosheva (2002) shows an application of LCMs to a combination of binary ADLs and IADLs to classify the elderly. 2 Results from application of partial credit models As mentioned in the Introduction, data come from the national survey on Health Conditions and Appeal to Medicare 1999–2000 conducted by the Italian National Institute of Statistics (ISTAT 2002). The survey has involved about 60,000 italian households and, in particular, 3,150 people aged 65 or more living in the region of interest made of 11 provinces as described in the Introduction. The population of interest for the survey is that of households; therefore, the elderly who live permanently in a care facility are not included in the analysis. The survey uses a two stage stratified clustered sampling design and provides direct estimates reliable up to the region level (NUTS2). Sampling weights that are adjusted through calibration techniques (Deville and Särndal 1992) to match known population counts at sex-by-age cells are provided with the data (ISTAT 2002). The questionnaire used in the survey is constructed accounting for the ICIDH and disability is evaluated by means of 14 items in the questionnaire that include ADLs. Four types of disability are defined according to the kind of deprived functional autonomy: confinement, difficulties in movement, difficulties in everyday activities and tasks, sensory deprivation. The items (and corresponding categorization) are reported in Table 1. For confinement, a condition of permanent constriction in bed, on a chair or at one’s home due to physical or psychical reasons is intended. People with difficulties in movements show problems in walking, i.e. they can only walk few steps before taking a rest; they cannot climb stairs without stopping; they cannot bend to pick up something from the ground. Difficulties in the ADL are concerned essentially with a lack of independence in accomplishing basic everyday tasks as going to bed, sitting, getting dressed or washed, taking a bath or a shower. Finally, sensory deprivation includes limitations in hearing, e.g. not being able to listen to a TV show even at a high volume, in spite of the use of hearing aid; limitations in seeing, e.g. not being able to recognize a friend at a meter distance; limitations in talking. The items are all ordinal with categories increasing with the difficulty of fulfilling the task and they are grouped according to the aforementioned four types of disability. Let us consider a vector of J ordinal items Y = (Y1 , . . . , Y j , . . . , Y J ) and a sample of n subjects S = {1 . . . , v, . . . , n}. Polytomous Rasch Models are introduced in Rasch (1960) for multiple-choice questions. We refer to the following generalized version of the model proposed by Fischer and Molenaar (1995): P(Y j = h|θv ) = exp(hθv − β j h ) mj l=0 exp(lθv − β jl ) for h = 0, . . . , m j and j = 1, . . . , J, (1) 123 Latent variable modeling of disability in people aged 65 or more Table 1 Items’ description and categorization Type of disability Confinement Item description CONF = type of confinement Categories 0 = No 53 1 = confined to one’s home 2 = confined to a chair 3 = confined to one’s bed Difficulties in movements STAIR = going up and down the stairs DIST = longest walkable distance 0 = More than 200 m. 1 = Less than 200 m. 2 = Only few steps 0 = Yes 1 = With some effort 2 = With a lot of effort 3 = No STOOP = stooping down Difficulties in everyday activities and tasks CHAIR = sitting and standing DRESS = getting dressed and undressed BATH = taking a bath or a shower WASH = washing one’s face and hands EAT = eating cutting one’s food CHEW = chewing BED = getting in and out of bed Same 0 = No effort 1 = With some effort 2 = With the help of others Same Same Same Same Same 0 = Yes 1 = With some effort 2 = With a lot of effort 3 = No Sensory deprivation SEE = seeing and recognizing a friend HEAR = hearing a TV show 0 = Yes 1 = Only at a high volume 2 = No 0 = Yes 1 = Only at short distance 2 = No SPEE = speaking 0 = Yes 1 = With some effort 2 = With a lot of effort 3 = No where h is the response category of the ordinal item Y j , β j h is an unknown parameter denoting the difficulty of scoring h on Y j and θv is the unknown parameter denoting the position of the individual v on the latent continuum. Masters (1982) suggests a version of (1), called Partial Credit Model (PCM), where different categories give the h respondents different credits, i.e. model (1) is constrained as follows: β j h = l=0 α jl 123 54 G. E. Montanari et al. where α jl is the credit awarded for response l. Under the assumption that items are independent of one another given θv , joint maximum likelihood estimation, marginal maximum likelihood estimation and conditional maximum likelihood estimation are all feasible (Fischer and Molenaar 1995). We perform item analysis with Winsteps (Linacre and Wright 1998) for assessing scale validity and reliability, e.g. the latent construct is unidimensional, individuals are well differentiated in the measured domain, no evidence of Local Dependence and Differential Item Functioning is detected, internal consistency holds. First, we investigate point-measure correlations (Olsson et al. 1982), which estimate the correlation between the latent variable and the single item response, and find no evidence of negative or zero values. Then, the goodness of fit of the Rasch model is tested with item fit indexes (Bond and Fox 2007). On the basis of the estimated person and item parameters, the expected response of a subject to an item is computed. The similarity between the observed and expected responses to any item is reported by the software, through two fit mean-square statistics: the outlier-sensitive fit statistic (item outfit) and the information-weighted fit statistic (item infit). The estimate produced by the item outfit is relatively more affected by unexpected responses different from a person’s measure, i.e. it is more sensitive to unexpected observations by persons on items that are relatively very easy or very hard for them. The item infit has each observation weighted by the information and, on the other side, is relatively more affected by unexpected responses closer to a person’s measure, i.e. it is more sensitive to unexpected patterns of observations by persons on items that are roughly targeted on them. The expected value for both statistics is one. For infit and outfit values greater/smaller than one indicate more/less variation between the observed and the predicted response patterns, a range of 0.5–1.5 is generally acceptable (Bond and Fox 2007). Out of the 14 items in Table 1 only nine are selected that show infit and outfit values between 0.50 and 1.50. In particular, CONF, CHEW, HEAR, SEE and SPEE are removed from the analysis. This is in line with the literature that raises concerns when merging “motor” and “cognitive” items (Linacre et al. 1994). The selected nine items show pointmeasure correlation coefficient values between 0.69 (EAT) and 0.92 (STAIR), in so doing providing a good correlation with the personal parameter. Items thresholds are well ordered. Person separation index is an indicator of the scale power for separating persons in different strata and is directly related to the scale reliability (Wright and Masters 1982). Person separation index is greater than 2.0 (2.31) and proves a reliability of the scale greater than 80%. Unidimensionality is another suitable property of the scale, and is tested by a Principal Components decomposition of the standardized residuals for the items (Wright 1996). Unidimensionality is supported by observing that 93% of variance in observations is explained by measures and the unexplained variance in first contrasts is less than 2 (1.9) in eigenvalue units. When items are used to form a scale they need to have internal consistency. Cronbach alpha tests whether items all measure the same thing, so they should be correlated with one another. Cronbach alpha takes value 0.95 supporting the reliability of the selected items. Differential Item Functioning (Holland and Wainer 1993) indicates that one group of respondents is scoring better than another group of respondents on an item given the same amount of latent trait. Analysis with Bonferroni multiple- 123 Latent variable modeling of disability in people aged 65 or more DIST STAIR 0 0 1 1 1 1 1 1 2 1 1 -2 0 2 2 2 4 2 1 2 2 3 2 2 2 3 55 STOOP 0 BED CHAIR 0 0 DRESS 0 BATH WASH EAT 0 0 0 -4 Rasch personal parameter Fig. 1 The level of the Rasch personal parameter at which it becomes more probable to choose category h + 1 instead of h is plotted for each item. Numbers are categories for each selected item as reported in Table 1 significance-test correction does not show evidence of Differential Item Functioning with respect to sex and age classes. PCM provides a measure of threshold difficulties specific for each item. Categories characteristics curves are common tools that depict, for each item, the probabilities of choosing a category rather than an adjacent one, given the person parameter. Figure 1 reports the values of the personal parameter at which it becomes more probable to choose the subsequent category for all items. For example, a personal parameter of about −1.4 is enough to make it more probable to be unable to walk more than 200 m (category 1 of item DIST). From this plot, then, we are able to detect those abilities that are lost first and those that are lost only when the level of the personal score (disability in our case) is already very high. As a consequence, we may say that the first difficulties shown are connected with the ability to climb stairs and stoop down. On the other side, the ability of taking a bath is the first that is lost completely, followed by those connected with movements. Finally, the ability to eat, get washed and to stand up from a chair are those lost last. This is in agreement with literature that shows that the most difficult items are the first to be lost (Tesio et al. 2002; Stineman et al. 2003). The output of the PCM is a score on a continuous scale. The aim of the paper is to obtain a classification of the elderly according to different levels of disability. Therefore, some cut-off thresholds should be determined. To this end we employ two methodologies: ROC curves and CART classification through regression trees. The first method seeks for a cut-off point for the raw score that best partitions some external binary variables (see e.g. Zweig and Campbell 1993). To this end, we use presence of morbidities as diabetes, osteoporosis, Parkinson’s and Alzheimer’s diseases, presence of home sanitary care, Physical Component Summary (PCS12) and Mental Component Summary (MCS12). These two latter indexes are obtained from 12 items that provide an assessment of the generic health status of individuals: larger (smaller) values are associated with better (worse) physical or mental conditions. In particular, a value of 50 denotes a good health condition, while a value of 20 denotes 123 56 G. E. Montanari et al. a poor condition. We define dichotomized versions for PCS12 and MCS12 after we attained the maximum value of the Area Under the Curve. The cut-off points vary: 2 for diabetes and osteoporosis, 5 for PCS12, 4 for MCS12, 6 for Parkinson and Alzheimer’s diseases. In alternative, we employed CART classification through regression trees. The method seeks for thresholds for the raw score that best partition (in terms of least-squares) some external continuous or discrete variables. To this end, we use age, PCS12, presence of morbidities as diabetes, osteoporosis, Parkinson’s and Alzheimer’s diseases, presence of home sanitary care. The raw score ranges from 0 to 20. From the analysis, no unique value for one or more thresholds could be found. Almost all variables isolate the “healthy” group with 0 score, corresponding to relatively younger elder, with a higher PCS12, without morbidities and with no need of extra assistance. However, the other threshold, which is more significant in our perspective, varies a lot, taking values from 6 (for PCS12) to 16 (for presence of home sanitary care). For all these reasons we resort in the next section to latent class models. 3 Results from the application of LCMs In Latent Class Models (LCMs; Lazarsfeld and Henry 1968) the categories of the latent variable C are latent classes which are denoted by ck for k = 1, . . . , K . The ordinal responses to items are generated by class-specific probabilities. The basic assumption of a LCM is that items are independent of one another given the latent class memberships (Goodman 1974). Therefore, LCMs do not require a unidimensional latent variable and model patterns as opposed to the raw score. A LCM can be defined as K P(Y = h) = k=1 P(C = ck )P(Y = h|C = ck ), (2) where P(C = ck ) is the probability of belonging to class k and h is a response pattern. By local independence, the probability of a response pattern can be assumed to be the product of the probabilities of the single responses. Linear constraints can be imposed on the probability that item j takes value h, P(Y j = h|C = ck ), that give an equivalent reparameterization of (2) where P(Y j = h|C = ck ) = exp(α j hk ) , mj ł=0 exp(α jlk ) for h = 0, . . . , m j and j = 1, . . . , J. (3) If we impose the Equidistance condition, we obtain the following linear constraints: α j hk = μ jk + (m j − 2h)τ jk , (4) which impose constant differences between α parameters for adjacent response categories. The parameters of a LCM are usually estimated by means of the EM algorithm (Dempster et al. 1977). The final step in a traditional LCM analysis is to use the estimated parameters of the model to classify cases into the appropriate latent classes. The 123 Latent variable modeling of disability in people aged 65 or more 57 33000 34000 35000 36000 37000 38000 LCM LCM equidistance 2 3 4 5 6 7 8 number of latent classes Fig. 2 BIC values for competing LCMs based on a different number of latent classes classification problem of assigning respondents to latent classes may be approached from a Bayesian point of view. In fact, posterior probabilities P(C = ck |Y = h) = P(Y = h|C = ck )P(C = ck ) P(Y = h) (5) can be estimated once we have estimates of class sizes, item parameters and parameters of the conditional score distributions. The estimate of ck is given by the class with the largest posterior probability given an observed response pattern. Successive LCMs are fitted to the data with an increasing number of classes until the simplest model is found that provides an adequate fit. LCMs have been fitted using the 14 items reported in Table 1, given that unidimensionality is not a requirement here. We fitted LCMs with WinMira software (Davier 1994) and tested different LCMs with two up to eight classes. Figure 2 reports BIC values for the seven LCMs fitted. LCMs are also fitted imposing equidistance as in (4). It can be seen that BIC stabilizes after using 4 classes for both types of models. In addition, the simplification imposed using equidistance provides a smaller BIC, so that our final model is a LCM with four classes and parameters constrained via equidistance. People belonging to each class have been labeled in the following way: elderly without disability, with difficulties in movements, with difficulties in movements and daily tasks, with very low functioning level. Such interpretation of the latent classes comes from the estimated conditional probabilities of answering a certain category of each item, given that the elder belongs to a certain latent class. Table 2 shows such conditional probabilities for each latent class and item category. Note that all probabilities are very close to one for the smallest category of each item in the first class. No limitation in the activities of daily living is reported by people belonging to this class, so that we label them as being without disability. In the second class, on the 123 58 G. E. Montanari et al. Table 2 Conditional probabilities of item responses for each of the four latent classes Item Category 0 1 2 3 Item Category 0 1 2 3 cl1—Without disabilities CONF DIST STAIR STOOP BED CHAIR DRESS BATH WASH EAT CHEW HEAR SIGHT SPEE CONF DIST STAIR STOOP BED CHAIR DRESS BATH WASH EAT CHEW HEAR SIGHT SPEE 0.996 0.992 0.952 0.960 1.000 1.000 0.999 0.990 0.999 0.999 0.961 0.912 0.976 0.988 0.984 0.611 0.198 0.260 0.870 0.928 0.872 0.666 0.997 0.984 0.741 0.771 0.922 0.967 0.004 0.008 0.045 0.040 0.000 0.000 0.001 0.008 0.001 0.001 0.034 0.077 0.019 0.009 0.016 0.375 0.701 0.622 0.127 0.072 0.128 0.264 0.003 0.016 0.226 0.200 0.072 0.026 0.000 0.000 0.003 0.000 0.000 0.000 0.000 0.002 0.000 0.000 0.003 0.011 0.006 0.002 0.000 0.014 0.101 0.116 0.003 0.000 0.000 0.070 0.000 0.000 0.030 0.029 0.006 0.004 0.003 0.002 0.001 0.002 0.001 0.000 0.001 0.000 0.000 0.000 cl3—Difficulties in movements and daily tasks CONF DIST STAIR STOOP BED CHAIR DRESS BATH WASH EAT CHEW HEAR SIGHT SPEE CONF DIST STAIR STOOP BED CHAIR DRESS BATH WASH EAT CHEW HEAR SIGHT SPEE 0.784 0.199 0.043 0.047 0.166 0.359 0.242 0.127 0.744 0.682 0.494 0.644 0.733 0.830 0.145 0.010 0.000 0.003 0.000 0.009 0.026 0.021 0.114 0.198 0.290 0.526 0.547 0.460 0.198 0.523 0.293 0.310 0.761 0.627 0.642 0.446 0.221 0.268 0.369 0.280 0.227 0.123 0.244 0.041 0.000 0.006 0.061 0.238 0.081 0.013 0.338 0.342 0.340 0.257 0.290 0.287 0.017 0.278 0.478 0.475 0.073 0.014 0.117 0.427 0.034 0.050 0.119 0.076 0.040 0.032 0.310 0.949 0.083 0.043 0.939 0.753 0.893 0.966 0.548 0.460 0.252 0.218 0.163 0.165 0.088 0.118 0.916 0.947 0.015 0.301 0.017 0.187 0.168 0.001 cl2—With difficulties in movements cl4—With very low functioning level other end, we find larger probabilities for the items corresponding to difficulties in movements. Limitations in walking, in going up and down the stairs and in stooping down are reported by people in this class. Probabilities for the third class suggest that people belonging to this class report some or greater difficulty in walking without taking a rest, going up and down the stairs, stooping down, taking a bath or a shower, getting dressed and undressed. These people can be considered as showing difficulties in movements and daily tasks. Finally, probabilities in the fourth class are much larger for the highest categories of all items, but CHEW and those related to sensory deprivation (HEAR, SIGHT and SPEE). People in this class are likely confined and report 123 Latent variable modeling of disability in people aged 65 or more Table 3 Amount of elderly belonging to the four latent classes in the study area as of 01.01.2000 Latent class Without disability With difficulties in movements With difficulties in movements and daily tasks With very low functioning level Total (elderly) 7 5 3 1 59 Number of units 457, 157 148, 580 82, 621 49, 033 737, 391 % of elderly 60.8 21.2 11.4 6.6 100.0 % of population 13.1 4.6 2.5 1.4 21.6 ES S BA TH W AS H EA EW IS BE O EA SE O AI AI ST R C ST C -3 -5 cl1 -7 cl2 cl3 cl4 Fig. 3 Location parameters μ jk for each item and category (final LCM with 4 classes) the largest degree of difficulty for most activities, so that we label them as being with very low functioning level. Through posterior probabilities (5), it is now possible to attach to every respondent the probability of belonging to each of the four latent classes, given its response pattern. This allows to classify each respondent with the class for which the posterior probability is largest among the four. Furthermore, since each unit in the sample has a sampling weight attached to it coming from the sampling design, we can estimate the number of elder people belonging to each class though a weighted summation of such posterior probabilities over all sample units. Table 3 reports the distribution of the elderly in the target population according to the four latent classes. For example, we estimate that about 6.6% of the population aged 65 or more shows a very low functioning level (they represent 1.4% of the whole population). From the final LCM with four classes we also have the estimates of the parameters in (4) for each item. Figure 3 shows the value of the location parameters μ jk for each item and each class. It shows that there is the same item difficulty ordering across classes. In addition, curves are well separated denoting that latent classes are well defined. Also note that the small variability over the four classes shown by the last four items (CHEW, HEAR, SEE, SPEE) is another hint of their poor discrimination power. The dispersion parameters τ jk give an indication of the dispersion around the mean of the threshold parameters. We do not report the detailed values here. However, D C H SP D O H H EE N -1 T D R R R P E F T 123 60 1.2 cl1 cl2 cl3 G. E. Montanari et al. cl4 1 0.8 0.6 0.4 0.2 0 AI R ST O O P BE D C H AI R D R ES S BA TH W AS H T C H EW C O N F D IS T H EA EA SE Fig. 4 Scaled expected item response for each item and class we note that they take negative values for items in the first class—making smaller categories more probable than larger ones—while positive values are associated with the three other classes—making larger categories more probable than smaller ones. Now, consider the expected item response for each class; it is calculated as the mean category weighted with the conditional probabilities. It represents a sort of the most likely category response for that item in that class. Figure 4 gives the expected item response for each class once re-scaled in the interval 0–1 to make it comparable across items with a different number of categories. The ordering of the four curves shows how it becomes more probable to respond with a higher category with moving from one class to the next with increasing disability. In addition, curves are well separated denoting that latent class are well defined. Those items for which the difference between consecutive curves is larger have more discriminating power; these are items surveying disability in movements and in ADL. The item related to confinement discriminates especially going from the third to the fourth class, while items related to sensory deprivation seem to have a poor discrimination power. Table 4 aims at validating to some extent the classification made by showing the average age and the prevalence of some attributes in the four latent classes. It can be noted that average age increases with increasing levels of disability. More importantly, the prevalence of people affected by Parkinson’s or Alzheimer’s disease increases considerably with increasing disability, so much that 37.6% of those belonging to the fourth class suffer from one of such neuro-degenerative diseases. The incidence of blind or deaf people is particularly large in the last class—31.9 and 15.5%, respectively—compared to the average, as that of people showing mental or motion deficiencies—37.1 and 82.3%, respectively. Finally, the last two rows show the prevalence of elder people who are under assistance, being it financially provided by the family itself (elderly assistance) or by the public sanitary system (home sanitary care). The last class shows values way above the average for both types of assistance: 33.8% of elderly in this class has elderly assistance, while 26.3% has home sanitary care. Finally PCS12 decreases steadily from the first to the fourth class, going from a value of 47.1 to 26.2. 123 ST SP EE R E Latent variable modeling of disability in people aged 65 or more Table 4 Mean age and prevalence (%) of some attributes in the four latent classes Variables Without disability With difficulties With difficulties With very low in movements in movements functioning level and daily tasks 75.9 1.4 1.1 7.8 1.6 2.3 2.5 0.1 37.9 80.2 18.8 8.5 9.6 10.3 20.3 8.9 11.9 28.4 80.6 37.6 31.9 15.5 37.1 82.3 33.8 26.3 26.2 61 All elderly Age (mean) Parkinson, Alzheimer Blindness Deafness Mental deficiency Motion invalidity Elderly assistance Home sanitary care PCS(12 mean) 72.5 1.6 1.2 0.9 0.2 0.6 1.1 0.5 47.1 74.5 5.7 3.9 1.9 3.8 8.0 4.2 3.2 42.1 Another tool to investigate the robustness of the classes is that of cross-validation. In particular, the sample of elderly has been randomly divided into two groups of dimension 1,575 each. One group has been taken as the so-called training set, while the other as the validation set. A LCM has been fitted using observations from the training set and parameters from this model have been used to classify observations from the validation set. Note that 425 observations from the validation set could not be classified since they showed different response patterns from those observed in the training set. Such classification has been compared with the one obtained fitting the model on the validation set. This exercise resulted with only 9 observations out of the remaining 1,150 from the validation set having a different classification, providing a classification error rate of only 0.8%. This is a quite reassuring result on the robustness of the classification obtained. We also fitted the model on analogous data from other regions of Italy and the same classes were found, with very similar sizes. 4 Conclusions In this paper latent variable models have been employed to extract profiles of disability using data from the national survey on Health Conditions and Appeal to Medicare 1999–2000, conducted routinely by the Italian National Institute of Statistics. The focus is on a central region of Italy that shows a large proportion of people aged 65 or more. Fourteen items that include the Activities of Daily Living are employed and described in Table 1. Partial Credit Models are first used to obtain a measure of disability using a selection of nine items that support the assumption of a single latent trait. From such a model, we are able to detect those abilities that are lost first and those that are lost only when the level of the personal score (disability in our case) is already very high: the first difficulties shown are connected with the ability to climb stairs and stoop down. On the other side, the ability of taking a bath is the first that is lost completely, followed by those connected with movements. Finally, the ability to eat, get washed and to stand up from a chair are those lost last. 123 62 G. E. Montanari et al. The output of a Partial Credit Model is a score on a continuous scale, while the aim of the paper was to obtain a classification of the population, so that latent class models were then employed. Model selection through BIC brought to a good classification into four latent classes. Table 2 reports conditional probabilities describing the profile of each class and Fig. 4 depicts the scaled expected item response for each class. Latent class models do not assume any hierarchy across classes, yet the four classes found prove to reflect some ordering in terms of disability and dependence. Therefore, classes reflect independent profiles of functional deficits, helping to depict the profile of a person’s needs. In the meanwhile, they also represent gross windows of increasing overall burden of care. The national survey uses a complex sampling design with a sample size that provides good estimates at a regional level. We therefore used sampling weights attached to each unit in the sample to estimate the amount of population belonging to each of the four latent classes. Table 3 reports the population distribution in the four classes. In particular, 6.6% of the population aged 65 or more may be classified as with very low functioning level (1.4% of the whole population). The robustness of the classification obtained has been checked using cross-validation: classification error is about 0.8%. In addition, the interpretation of the classes found has been validated by looking at the distribution of other variables related to health status and care needs in the four classes. Results are reported in Table 4. The work developed here provides a tool that allows to classify a population according to different profiles of disability. This is done using data from a national survey, that makes it also possible to compare different territories—e.g. North vs South, different Regions—on a common basis. Such an exercise has been impossible before, since disability was only measured on a personal, individual basis and using different scales by different Regions. The work done has also some limitations, from which we envision new directions for future research. In particular, there is no clear evidence on the care required by people in each class: ad-hoc surveys are needed that link response patterns from the items employed in the National Survey to treatment needs to properly allocate resources according to these data. In addition, the Italian health system has many functions managed at a local (regional) level. Each Region is organized in Health Districts. Such areas represent unplanned domains, i.e. subpopulations not accounted for in the sampling designs, that may have small sample sizes and for which, therefore, reliable direct estimates cannot be computed. Small area estimation models should therefore be built that link the latent trait of disability with socio-demographic information available for all units in the sub-populations from censuses or administrative archives. 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