# NON-PARAMETRIC APPROACH FOR ESTIMATING MULTI-LEVEL LATENT CLASS ANALYSIS OF ATTITUDES TOWARD MATHEMATICS

Attitudes towards mathematics are multidimensional constructs that can be divided into self-confidence, value of mathematics, enjoyment of mathematics and motivation (Tapia & Marsh, 2005). Each dimension cannot be measured directly but is represented by multi-indicators. As of any research in behavioural and social science area, these indicators are usually in the form of ordinal or ordered categorical data. A researcher has a choice between analyzing the responses to each item separately and to ignore the correlation structure or to treat them as multivariate data. The latter leads to a more efficient analysis (Lawrence et al, 2008). A more common approach in analysing the ordered categorical data is to assign scores to the categories and treat the scores as continuous data (Lawrence et al, 2008). It is common to determine the intrinsic dimension among the larger number of items using factor analysis. Other studies, on the other hand, form an index using subjective rules (TIMSS).

There are considerable debates over the past years around the treatment of ordinal scaled variables as if they were continuous scale (Byrne, 2001). These data are analyzed using either Structural Equation Modeling (SEM) or other statistical techniques such as Analysis of Variance (ANOVA) where the scale of the observed variables is assumed to be continuous and normally distributed.

In order to maintain the nature of the ordinal scale, the most suitable statistical method in identifying subtypes of related cases using a set of categorical variables is the Latent Class Analysis (LCA). These subtypes are referred as latent classes since they are not directly observed, rather they are inferred from multiple observed indicators. LCA assumes the existence of a categorical latent variable that explains the relations between a set of categorical manifest variables.

Given the complexity of data collected in this type of study and the data structure for which observations are not independent of one another, multilevel latent class analysis (MLCA) is needed to account for the clustering effects. MLCA allows latent class intercept to vary across Level 2 units and at the same time examine how Level 2 unit influence the Level 1 latent classes.

Vermunt (2003) proposed both parametric and non-parametric random-coefficient or multilevel latent class analysis (MLCA). The parametric MLCA is a continuous latent variable where the mean(s) from the Level 1 latent class solution varies across communities (Level 2). However, the parametric model can be computationally heavy (Vermunt 2003 and Van Horn et al. 2008) since the random-effects approach makes strong assumptions about the mixing distribution. A nonparametric approach was proposed by Vermunt (2003, 2008) and Asparouhov and MuthÃ©n (2008) where the second latent class model is specified at Level 2. In this approach, the assumption of a multinomial distribution has replaced the assumption of the normal distribution. It also provides more meaningful interpretation in social science studies as it produces a finite number of Level 2 latent classes

The purpose of this paper is to illustrate the use of LCA in identifying distinct groups of students based on their responses to a set of questionnaire items in each of three dimensions of attitudes towards mathematics. The data were obtained from the Trends in Mathematics and Science Study (TIMSS) 2007 for eighth grader students in Malaysia.