A generalized multilevel linear model for likert responses

Abstract

n this study Generalised Linear models to handle Multilevel models with Likert responses are introduced. These include a simple transformation to normality based on the work of Snell (1964), a Discrete Ordinal multilevel (DOML) model (with logistic and complementary log log links, derived from first principles) and an application of Bayesian MCMC multilevel (BML) model to data with likert responses. For completeness the following models were also fitted, Ordinary Least Squares regression (OLS), Fixed effects model (FE), Random effects model (RE), Ordinal regression model (ORD), Generalized Linear Mixed (GLMM) (with nor mal and Poisson distributions)).The research was based on a simulation study which is a standard practice when assessing methodology (Murray, Ryan and Reisinger, 2011). The noisy Data was simulated from fifteen data sets each with n = 1000 observations (based on five β = (β01, β10) combinations and 3 skewness levels). The models discussed above were all fitted, they were required to estimate β01 correctly, i.e., the slope at level one. The results showed the DOML estimate with a complimentary log log link as being a superior model in all scenarios (symmetric, left skewed and right skewed data). Each time the DOML recovered the true value and out performed the other models on the basis of the percentage bias, standardised bias and coverage statistics. The BML model was also fitted, this model performed reasonably well almost at par with the DOML when used with a cloglog link in place of the logistic link which is used in literature (Congdon, 2005; Johnson and Albert, 1999), it however, failed when skewed data sets were used.