Semiparametric Mixed Effects Models for Unsupervised Classification of Italian Schools

Abstract

Summary The main purpose of the paper is to improve research on school effectiveness by applying a new strategy for uncovering subpopulations of schools that differ in terms of distribution of student outcomes. We propose a semiparametric mixed effects model with an expectation–maximization algorithm to estimate its parameters and we apply it to the Italian Institute for the Educational Evaluation of Instruction and Training data of 2013–2014 as a tool for the identification of latent subpopulations of schools. The semiparametric assumption provides the random effects of the mixed effects model to be distributed according to a discrete distribution with an (a priori) unknown number of support points. This modelling induces an automatic clustering of schools (the higher level of hierarchy), where schools within the same cluster share the same random effects. The latent subpopulations of schools identified may then be exploited through the use of multinomial models that include school level features. The novelties introduced by this paper are twofold: first, the semiparametric expectation–maximization algorithm is an innovative method that could be used in many classification problems; second, its application to education data represents a new approach to study school effectiveness.

Evaluating class and school effects on the joint student achievements in different subjects: a bivariate semiparametric model with random coefficients

“…Following the idea presented in Masci et al (2019), we relax the parametric assumptions about the coefficients of the random effects and we assume the bivariate coefficients 1 i = (δ 1,i δ 2,i ) to follow a bivariate discrete distribution S * , assuming M × K mass points (C 11 , . .…”

“…S * can then be interpreted as the mixing distribution that generates the density of the stochastic model in (3). The ML estimatorŜ * of S * can be obtained following the theory of mixture likelihoods in Lindsay (1983a, b), as explained in Masci et al (2019). In particular, in Lindsay (1983a, b), the authors prove the existence, discreteness and uniqueness of the semiparametric maximum likelihood estimator of a mixing distribution, in the case of exponential family densities.…”

“…In this paper, we extend the statistical model presented in Masci et al (2019) and we propose a study that is innovative from a methodological and an interpretative point of view. We extend the Expectation-Maximization algorithm for semi-parametric models with random coefficients (SPEM algorithm) presented in Masci et al (2019) to the bivariate case, i.e. to the case of a bivariate response variable (which, in our case, is the test score in reading and mathematics).…”

“…Our paper aims at estimating the class effect in the context of within-school heterogeneity. We follow the approach presented in Masci et al (2019), where the authors apply a multilevel linear model to estimate the school effect, but, instead of following a classical parametric approach, they follow a semi-parametric approach: they develop a semi-parametric mixed-effects (two-level, where students are nested within schools) model able to identify a latent structure among the highest level of the hierarchy (schools). They cluster schools standing on the evolution of their student achievements across years.…”

“…The proposed methodology is new to the literature. The semi-parametric mixedeffects linear model in Masci et al (2019) on which we base our multivariate model enters in the research line about the identification of subpopulations of the Growth Mixture Models (GMM) (Muthén 2004;Muthén and Shedden 1999;Nagin 1999) and of Latent Class Mixture Models (LCMM) (McCulloch et al 2002;Vermunt and Magidson 2002), but with the novelty that it does not need to fix a priori the number of latent subpopulations to be identified. Moreover, being the existing methods specified in the Structural Equation Modeling (SEM) framework, they are still relatively limited when covariates are group-specific.…”