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Multidimensional IRT Models
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Dimensionality of the Latent Structure and Item Selection Via Latent Class Multidimensional IRT Models
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PSYCHOMETRIKA—VOL. 77, NO. 4, 782–802
O
CTOBER 2012
DOI: 10.1007/S11336-012-9278-0
DIMENSIONALITY OF THE LATENT STRUCTURE AND ITEM SELECTION
VIA LATENT CLASS MULTIDIMENSIONAL IRT MODELS
F. BARTOLUCCI,G.E.MONTANARI, AND S. PANDOLFI
DEPARTMENT OF ECONOMICS, FINANCE AND STATISTICS, UNIVERSITY OF PERUGIA
With reference to a questionnaire aimed at assessing the performance of Italian nursing homes on
the basis of the health conditions of their patients, we investigate two relevant issues: dimensionality of
the latent structure and discriminating power of the items composing the questionnaire. The approach
is based on a multidimensional item response theory model, which assumes a two-parameter logistic
parameterization for the response probabilities. This model represents the health status of a patient by
latent variables having a discrete distribution and, therefore, it may be seen as a constrained version of the
latent class model. On the basis of the adopted model, we implement a hierarchical clustering algorithm
aimed at assessing the actual number of dimensions measured by the questionnaire. These dimensions
correspond to disjoint groups of items. Once the number of dimensions is selected, we also study the
discriminating power of every item, so that it is possible to select the subset of these items which is able
to provide an amount of information close to that of the full set. We illustrate the proposed approach on
the basis of the data collected on 1,051 elderly people hosted in a sample of Italian nursing homes.
Key words: EM algorithm, discriminating power, hierarchical clustering, quality of life.
1. Introduction
In the last decades, the progressive aging of the contemporary society has raised the re-
quirement for health assistance, with a rapid increase of the costs of health care, especially for
long term care assistance of elderly people (Anderson & Hussey, 2000). In Italy, in particular,
the population aging has been one of the fastest among developed countries during the last 50
years (Kohler, Billari, & Ortega, 2002). A direct consequence of the increasing costs for health
assistance due to population aging is that governments of industrialized countries have begun to
focus on the rationalization of interventions, in terms of public spending, regulation, and poli-
cies. Public intervention is also crucial to guarantee the accessibility to care facilities of elderly
people with low income.
The above arguments imply that measuring health care quality and comparing nursing home
performance, together with the role of public institutions, represent a reason of concern through-
out the world. In the medical literature, there is also a great debate about the construction of
indicators which measure nursing home performance and the use of such indicators to rank these
types of facilities in a certain geographical area; see Phillips, Hawes, Lieberman, and Koren
(2007). The indicators currently used to evaluate nursing home performance are based on data
coming from surveys which are periodically carried out by public institutions; see, among others,
Hirdes, Zimmerman, Hallman, and Soucie (1998) and Mor, Berg, Angelelli, Gifford, Morris, and
Moore (2003). Often, a ranking of the nursing homes based on these indicators is publicly avail-
able (Harrington, Collier, O’Meara, Kitchener, Simon, & Schnelle, 2003). In the United States,
nursing homes are compared by means of several quality indicators derived either from inspec-
tions or from routinely collected clinical data (Zimmerman, 2003). In particular, the Center for
Medicare and Medicaid Services is engaged in building quality measures for nursing homes using
Requests for reprints should be sent to F. Bartolucci, Department of Economics, Finance and Statistics, University
of Perugia, Perugia, Italy. E-mail: bart@stat.unipg.it
© 2012 The Psychometric Society
782
F. BARTOLUCCI, G.E. MONTANARI, AND S. PANDOLFI 783
descriptive statistics as indicators; these indicators are generally referred to the psycho-physics
conditions of elderly people hosted in these facilities.
One of the main ideas behind the nursing home ranking is that unidimensional criteria to
classify elderly people according to their health status are available (Mor et al., 2003). This
assumption implies that the difference between two subjects in responding to a set of items on the
health conditions only depends on a single latent trait which summarizes these conditions. This
may obviously be a restrictive assumption. For instance, this does not allow us to classify subjects
who show a degenerative health status in relation to a specific pathology, but apparently have an
overall good health status. Violations of this assumption may lead to misleading conclusions
reached on the basis of a unidimensional ranking which relies on a single score assigned to
each facility. Along these lines, dimensionality of health conditions becomes a central issue for
obtaining a consistent and generalizable index for the performance of the nursing homes.
In the present paper, we mainly investigate the dimensionality of the latent structure behind
questionnaires about the health conditions of the patients hosted in nursing homes. Moreover,
we address the issue of item selection on the basis of the discriminating power of the items
composing these questionnaires. The approach is based on a multidimensional item response
theory (IRT) model, proposed by Bartolucci (2007), which assumes a two-parameter logistic
(2PL) parameterization of the probability of “success” in responding to a dichotomous item,
given the ability. Moreover, it assumes that the population is divided into a certain number of
latent classes, with each latent class being associated to a specific value of the latent traits. The
adopted model is then a constrained version of the latent class (LC) model of Goodman (1974).
We have to clarify that other approaches to study the dimensionality of the latent structure
are typically based on Rasch (1961) type IRT models (see, for instance, Martin-Löf, 1973) which,
however, do not allow for item selection on the basis of their discriminating power. Therefore,
we exploit an approach based on a multidimensional model, which includes, as item parameters,
the discrimination indices. To our knowledge, no other approach allows us to jointly study the
dimensionality and the discriminating power of the items in a simple way.
We study the above issues with reference to a dataset coming from the ULISSE database
(Lattanzio, Mussi, Scafato, Ruggiero, Dell’Aquila, Pedone, et al., 2010), which is collected
within a survey carried out in Italy on the basis of the “InterRAI Minimum DataSet” (MDS)
questionnaire (Hawes, Morris, Phillips, Fries, Murphy, & Mor, 1997). The questionnaire cov-
ers several aspects of the health status of elderly people, such as cognitive conditions, humor
and behavioral disorders, and activities of daily living. These aspects are summarized into eight
different sections of items.
The approach we propose to analyze the data coming from the ULISSE study, or from similar
studies, is based on a series of steps. The first step consists of assessing if the multidimensional
2PL model of Bartolucci (2007) is suitable for the analysis of the data. For this aim, we com-
pare, through an appropriate information index, the LC model, in which the conditional response
probabilities are completely unconstrained, with the multidimensional model formulated by as-
suming a separate latent dimension for each section of the questionnaire used to collect the data.
We then use a hierarchical clustering algorithm (Bartolucci, 2007) in order to divide the items
of the questionnaire into groups that refer to different latent traits or dimensions and then select
the actual number of latent dimensions. Once the number of dimensions has been chosen in this
way, we select a subset of items which is able to provide an amount of information close to that
of the full set. In fact, the number of items included in a questionnaire (such as that used in the
ULISSE study) is usually large, leading to a time consuming and expensive administration, and
possibly inaccurate responses. For this reason, it may be of interest to use only those items which
are actually informative. For this aim, we use the estimated discrimination indices of the items;
moreover, we verify if the found subset of items actually provides an amount of information
close to that of the full set on the basis of the number of subjects for whom the classification in
784 PSYCHOMETRIKA
latent classes changes. As usual, this classification is based on the posterior probabilities of the
latent classes.
The remainder of the paper is organized as follows. In Section 2, we describe the ULISSE
dataset on health conditions of elderly people hosted in nursing homes in Italy. In Section 3,we
briefly review the statistical methodology based on the LC and the multidimensional 2PL model.
In Section 4, we describe in detail the empirical analysis, and in Section 5 we provide a final
discussion about the study.
The proposed approach is developed by using M
ATLAB routines that can be obtained, to-
gether with a simulated dataset of the same structure as the ULISSE dataset, from the website
http://www.stat.unipg.it/bartolucci/multirasch.zip.
2. The ULISSE Dataset
We consider a dataset collected within the ULISSE project (“Un Link Informatico sui Servizi
Sanitari Esistenti per l’Anziano”—“A computerized network on health care services for older
people”), which is an observational multicenter prospective study aimed at describing the elderly
patients who currently receive hospital care, home care, or nursing home care in Italy, as well
as service characteristics and the quality of provided care (Lattanzio et al., 2010). The project
is carried out by the Italian Ministry of Health jointly with the Italian Society of Gerontology
and Geriatrics. Only the data collected in the nursing homes are used for the analysis presented
here. In particular, the ULISSE study is based on a longitudinal survey about the assistance level
provided by 37 randomly chosen nursing homes, distributed across Italy. The survey has been
carried out since 2004 through the repeated administration of a questionnaire (every 6 months)
which is filled out by the nursing assistant of each patient and concerns several aspects of the
everyday life. For our analysis, we consider only the first interview, which covers 1,051 patients,
aged at least 65 years (after discarding records containing missing responses to the items). Only
long stay residents, that is those permanently admitted to the nursing homes, are included in the
study.
Within the ULISSE project, detailed patients’ data are collected by means of the MDS ques-
tionnaire for nursing home care (Morris, Hawes, Murphy, Nonemaker, Phillips, Fries, et al.,
1991; Hawes et al., 1997). This instrument has been validated in Italian and is recommended by
the Italian Geriatrics Society. The MDS questionnaire consists of over 300 items. However, we
singled out 79 among the available items, including only those related to health conditions and
discarding the items related to treatments received by the residents and those for which responses
were evidently inaccurate. The questionnaire is divided into eight sections concerning (for the
complete list of items in each section see Table 1):
• Cognitive Conditions (CC);
• Auditory and View Fields (AVF);
• Humor and Behavioral Disorders (HBD);
• Activities of Daily Living (ADL);
• Incontinence (I);
• Nutritional Field (NF);
• Dental Disorders (DD);
• Skin Conditions (SC).
Note that, in order to apply the approach proposed in this paper, item responses were di-
chotomized so that one category (labeled by 1) indicates the presence of a specific health prob-
lem, whereas the other category (labeled by 0) indicates its absence.
F. BARTOLUCCI, G.E. MONTANARI, AND S. PANDOLFI 785
TABLE 1.
Description of the full set of items.
Item Section Item description
01 CC does not recall what recently happened (5 minutes)
02 CC does not recall some elements of the remote past
03 CC does not recall the actual season
04 CC does not recall where his/her room is
05 CC does not recall the names and faces of the staff
06 CC does not recall where he/she is
07 CC does not decide about his/her daily activities
08 CC gets easily sidetracked
09 CC shows episodes of altered perception or awareness of surrounding
10 CC shows episodes of disorganized speech
11 CC has periods of restlessness movements
12 CC shows lethargic spans
13 CC his/her cognitive conditions change during the day
14 AVF shows hearing deficiency
15 AVF does not make himself/herself understood
16 AVF has not a clear language
17 AVF is not able to understand others
18 AVF is not able to see in conditions of adequate lighting
19 HBD makes negative statements
20 HBD makes repetitive questions
21 HBD makes repetitive verbalizations
22 HBD shows persistent anger with himself or others
23 HBD shows self deprecation disesteem
24 HBD expresses fears that are not real
25 HBD believes himself/herself to be dying
26 HBD complains about his/her health
27 HBD has repeated events of anxiety
28 HBD shows unpleasant mood in morning
29 HBD suffers from insomnia/has problems with sleep
30 HBD has expressions of sad-faced
31 HBD has tears easily
32 HBD shows repetitive movements
33 HBD abstains from activities of interest
34 HBD shows reduced local interactions
35 HBD wanders aimlessly
36 HBD uses offensive language
37 HBD is physically aggressive
38 HBD has a socially inappropriate behavior
39 HBD refuses assistance
40 ADL needs support in moving to/from lying position
41 ADL needs support in moving to/from bed, chair, wheelchair
42 ADL needs support in walking between different points within the room
43 ADL needs support in walking in the corridor
44 ADL needs support in walking into the nursing home ward
45 ADL needs support in walking outside the nursing home ward
46 ADL needs support for dressing
47 ADL needs support for eating
48 ADL needs support for using the toilet room
49 ADL needs support for personal hygiene
786 PSYCHOMETRIKA
T
ABLE 1.
(Continued)
Item Section Item description
50 ADL needs support for taking full-body bath/shower
51 ADL shows balance problems
52 ADL shows loss of mobility in the neck
53 ADL shows loss of mobility in the arm including shoulder or elbow
54 ADL shows limitations in the movements of the hand including wrist or finger
55 ADL shows loss of mobility in the leg and hip
56 ADL shows loss of mobility in the foot and ankle
57 ADL shows limitations in other movements
58 I has fecal incontinence
59 I has urinary incontinence
60 I has stipsi
61 I has diarrhea
62 I has fecaloma
63 NF has chewing problem
64 NF has swallowing problem
65 NF suffers from mouth pain
66 NF complains about the taste of many foods
67 NF complains of being hungry
68 NF leaves the food on his/her plate
69 DD has debris (soft, easily movable substances) present in mouth prior to going to bed at night
70 DD some/all natural teeth lost/does not have or does not use dentures (or partial plates)
71 DD has broken, loose, or carious teeth
72 DD has inflamed gums (gingiva); swollen or bleeding gums; oral abscesses; ulcers or rashes
73 DD has dentures or removable bridge that is not daily cleaned by resident or staff
74 SC shows pressure ulcer
75 SC shows stasis ulcers
76 SC had an ulcer that was resolved or cured
77 SC has skin problems/injuries/abrasions/burns
78 SC has one or more foot problems
79 SC has foot infections
From an initial descriptive analysis of the dataset, we observe that most of the sample is
composed by women (71 %). Moreover, the age distribution shows a high proportion of patients
who are 85 years old or over.
3. The Statistical Methodology
In the following, we briefly review the LC model (Goodman, 1974) and the multidimen-
sional IRT model proposed by Bartolucci (2007). Then we illustrate the proposed strategy which
is aimed at assessing the actual number of dimensions measured by the items of the questionnaire
and to reduce the number of these items.
3.1. Statistical Models
Let n denote the size of the sample of subjects we observe and let J denote the number of
items to which they should respond. For a given subject, we observe a vector y =(y
1
,...,y
J
) of
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