All the above mixture model is estimated by maximum-likelihood. The EM algorithm (Muth´en and Shedden, 1999) is implemented to obtain maximum-likelihood estimates.
The mixture model allows Y to be missing at random (Little and Rubin, 1987). It should be noted that mixture models in general are prone to have multiple local maxima of the likelihood and the use of several different sets of starting values in the iterative procedure is strongly recommended.
With maximum-likelihood estimation, we compute information criteria which are use-ful for comparing non-nested models. The Akaike information criterion (AIC) is defined as
AIC = −2 log L + 2T, (17)
where T is the number of free model parameters (Akaike, 1987) and log L =PNi=1log P r(Yi|xi), Yi = (Yi1, · · · , YiM)T, being the log likelihood function. The Bayesian information criterion (Schwartz, 1978) is defined as
BIC = −2 log L + T ln N. (18)
where N is the number of observations. The model with the smallest AIC or BIC value is taken to be the best one.
On the other hand, we performed latent class analysis (LCA) with number of latent classes varying from two to eleven for selecting the best number of classes by AIC and BIC criteria. We selected the best model to consider how the AIC and BIC value to change, when the number of latent classes of LCA varied from two to eleven, and consider the stability of the model, which is the number of fixed parameters and the latent prevalence of each class, with number of latent classes varying from two to eleven.
The degree to which the latent classes are clearly distinguishable by the data and the model can be assessed by using the estimated posterior probabilities for each individual in each class. By classifying each individual into his/her most likely class, a J x J table can be constructed with rows corresponding to individuals who have the highest probability for that class and the entries are average probabilities in each class. For individuals in each row, the column entries give the average and conditional probabilities. This will be
referred to as a classification table (Nagin, 1999). High diagonal is given by the entropy measure (Ramaswamy et al., 1993),
EJ = 1 −
PN i=1
PJ
j=1(−ˆpijln ˆpij)
N ln J , (19)
where ˆpij denotes the estimated posterior probability for individual i in class j. Entropy values range from zero to one, where entropy values close to one indicate clear classifi-cations in that the entropy decreases for probability values that are not close to zero or one.
3 Latent Structure of PANSS
3.1 Background
According to the criteria of the Diagnostic and Statistical Manual of Mental of Disorders (4th ed., DSM-IV; American Psychiatric Association, 1994), schizophrenia is a psychotic disorder characterized by several sets of symptoms. Many studies have examined the structure of symptoms in schizophrenia. Since Crow proposed the two-factor concept of schizophrenia in 1980 (Crow et al., 1980), researchers began to produce evidence for a syndromic dichotomy (negative-positive) (Bilder et al.,1985; Cornblatte et al., 1985; An-dreasen and Grove, 1986; Kay and Sevy, 1990; Mortimer et al., 1990; Dollfus et al., 1991;
Peralta et al., 1992; Bell et al., 1994a; White et al., 1994). Positive symptoms, such as hal-lucinations and delusions, represent a behavioral excess generally considered psychotic. In contrast, negative symptoms, like blunted affect and passive social withdrawal, represent a deficiency in normal behavior. Till now, many of these investigations have developed the symptom structures from Crow’s original two-dimension distinction, and researchers have found that more than two components are required to describe the symptoms in Schizophrenia (Liddle, 1987; Arndt et al., 1991; Andreason et al.,1995; Lindenmayer et al., 1995; Lenzenweger and Dworkin, 1996; Johnstone and Frith, 1996). For instance, Liddle (1987) has proposed the disorganization symptoms. A recent study suggested that a four-factor model fit as well as two- and three- factor models (Dollfus and Everitt, 1998).
However, the study was limited by the heterogeneity of patients in acute and stabilized phases and its lack of validation by follow-up data.
Some instruments were developed for measuring and quantifying different symptom dimensions, such as the Assessment of Negative Symptoms (SANS; Andreasen, 1983), the Scale for the Assessment of Positive Symptoms (SAPS; Andreasen, 1984) and the Positive and Negative Syndrome Scale (PANSS; Kay et al., 1987). The SANS and SAPS were designed to measure Positive and Negative syndromes. These instruments may be
limited in their potential to identify schizophrenia subtypes because of the prior selection of symptoms. The PANSS is a more extensive assessment of the symptom phenomenology of schizophrenia. It was developed by Kay et al. used the Brief Psychiatric Rating Scale (BPRS; Overall and Gorham, 1962) and the Psychopathology Rating Schedule (PRS;
Singh and Kay, 1975). The PANSS provides well-defined operational criteria for symptom assessment yielding good to excellent rater reliability. It demonstrates better inter-rater reliability and greater predictive power than the BPRS (Bell et al., 1992) and has been an effective research tool in a wide range of studies (Kay and Sevy, 1990).
A number of studies performed exploratory factor analyses (EFA; Lin et al., 1996, 1998), confirmatory factor analysis (CFA; Dollfus and Everitt, 1998), or cluster analysis (Dollfus et al., 1996) for unraveling the structure of the PANSS items. White et al. (1997) fitted 20 previously proposed models to data from a sample of 1,233 schizophrenics for attempt to reconcile the different research finds. They concluded that none of these models fitted the data adequately, then they derived a new ”pentagonal” model retaining only 25 items of the PANSS, which were labeled: Positive, Negative, Dysphoric mood, Activation, and Autistic preoccupation, and it’s presently proposed in the manual for the PANSS (Kay et al., 2000).
However, the study by White et al. did not finish the argument surrounding the factor structure of the PANSS. Critics argued that the structure of the PANSS items may not be best represented by five components (Emsley et al., 2003), and the proposed pentagonal model had inadequate goodness of fit in other samples (Lykouras et al., 2000; Fitzgerald et al., 2003). Differences in patient characteristics and symptom ensembles assessed might partly account for the discrepancies. In addition, the inclusion of patients at different stages of the disease may constitute another source of bias. This study was conducted in schizophrenic patients at various progressive stages of the disease. We conducted a study in two distinct populations of schizophrenic patients, one in the acute, and the other in the chronic stage.
The aim of the study reported in this article is to examine the structure of the PANSS items by using the regression extension of latent class analysis (RLCA, Huang and Bandeen-Roche, 2004), which is useful for classifying subjects based on their responses to a set of categorical items. First, the number of classes for two distinct phases of the disease will be selected based on AIC and BIC criteria. Second, according to the number of classes obtained in first step, the regression extension of latent class analysis (RLCA, Huang and Bandeen-Roche, 2004) will be performed to classify schizophrenic patients at two distinct phases (acute and chronic) of the disease. In addition, we will perform RLCA with demographic variables, environmental factors or neuropsychological variables to explore the relation between the latent class and demographic variables, environmental factors or neuropsychological variables. On the other hand, the structure of the PANSS in this study is compared with the structure of the PANSS in the previous studies.