Statistical models are essential in research to investigate relationship between variables, make predictions, classify items, and identify latent structures in a data set. Statistical models can be used to make predictions on population growth, life expectancy, fertility rates and economic growth. These models can also be used to identify significant risk factors of developing cancer, hypertension, diabetes and cardiovascular disease; or risk factors leading to smoking, alcohol and drug addiction. Moreover, statistical models can be used in biology and agriculture to classify species and crop varieties; and in psychology to assess the relationship between latent variables, such as prosocial behaviour, behavioural problem and resilience.
The seminal contribution of Nelder and Wedderburn (1972) to generalised linear models (GLM) proved a useful generalisation of classical normal regression models. All (GLM) are characterised by three components.
Firstly the response outcomes are assumed independent and follow a distribution that is a member of the exponential family; secondly, the linear predictor which describes the pattern of the systematic effect assumes that the explanatory variables enter as a linear combination of their parameters; thirdly, the predicted values are related to the linear predictor through a known link function. However, the assumption of independence is often unrealistic when the data is longitudinal or clustered. Clustered data arise when observations are made on subjects within the same selected group. Typical examples include family members nested in households, students nested in classrooms, patients nested in hospitals, and employees nested in departments.
Multilevel modelling is undoubtedly one of the most popular techniques that overcome the limitation of the independence assumption. In fact these models accommodate nested data embedded in hierarchical structures, where random effects can be incorporated at the different levels of the hierarchy. Similar to generalised linear models, multilevel models accommodate any distribution that is a member of the exponential family.
For instance, the normal or gamma distribution is used for continuous responses; the binomial distribution is used for dichotomous responses; the poisson distribution is used for count data; while the multinomial distribution is used for multichotomous responses. Unlike linear regression models, multilevel models accommodate both fixed and random effects. Moreover, these models allow for the inclusion of both individual-level covariates and cluster-level covariates, while adjusting for random effects associated with each cluster. In the last two decades, the increased interest in multilevel models has led to a wider range of applications, particularly in education, psychology and biomedical research.
Liberato Camilleri is a member of the Department of Statistics & Operations Research within the Faculty of Science of the University of Malta.
Sound Bites
• From the international scene:
In recent developments, Goldstein and Longford developed software that allowed the use of several types of discrete outcomes for multilevel models. Pinheiro and Bates contributed accurate approximations to maximum likelihood estimation using Gauss-Hermite quadrature, which is implemented in the MIXOR and SAS PROC NLMIXED software. Raudenbush and Yosef developed the HLM program, which uses a high order Laplace transform, while Rabe-Hesketh et al. developed the GLLAMM software, which uses adaptive Gauss-Hermite quadrature. The most recent development in estimation procedures focus on Bayesian methods.
• From the local scene:
The author of this page together with other researchers applied these statistical models in several research areas, particularly education and psychology:
– Social, emotional and behaviour difficulties in Maltese schools: A multilevel model. Journal of Research in Educational Sciences, Volume 2, Issue 1, Number 3, p.3-15.
– Examining the structural validity of the strengths and difficulties questionnaire (SDQ) in a multilevel framework. Xjenza Journal, Volume 6, Issue 1, p.16-24.
– Using multilevel random coefficient models to assess students’ spelling abilities. Proceedings of the 2010 ESM Conference p.449-454.
For more sound bites, listen to Radio Mocha: Mondays at 7pm on Radju Malta and Thursdays at 4pm on Radju Malta 2 (https://www.fb.com/RadioMochaMalta).
DID YOU KNOW?
The following results were elicited from studies carried out on Maltese students, using multilevel models.
• A local study that measured the between-school variation in reading comprehension showed a drop from 50 per cent to 26 per cent between 2009 and 2018. This significant reduction in the between-school variation is attributed to the fact that in 2011 the Junior Lyceum and common entrance examinations, which stratified schools by students’ academic performance, were abolished.
• A local study that measured reading enjoyment showed that female students enjoy reading more than males in all school types. Moreover, the reading enjoyment index increased by 0.08 between 2009 and 2018. This positive trend in reading enjoyment may be related to a change in how information is extracted. In this age of digital media, Maltese students may be reading fewer books but more online material, including chats with their friends and online articles.
• A local study showed that Maltese male students use ICT for leisure more than females. However, female students are more likely to use digital devices to participate in social media and to chat online, while males are more likely to use these devices for collaborative online gaming.
• A local study carried on children aged six to 16 years showed that females outperform males in spelling and reading comprehension and the gap augments with an increase in age.
For more trivia, see: www.um.edu.mt/think