Generalization of Random Intercept Multilevel Models

The concept of random intercept models has been generalized for k-levels in multilevel models. The random variation in intercepts at individual level is marginally split into components by incorporating higher levels of hierarchy in the single level model. This showed a control in the random variation in intercepts by incorporating the higher levels of hierarchy in the model.


Introduction
A random intercept model is one in which only the intercept of the model vary across the different levels of the model and extracts the information that how the values of response variable vary across the diverse nature of higher levels units in a hierarchical population.In such models, the random variation due to all the hierarchies is taking into account as we split overall random variation into variations due to all the levels.So, sometimes these models are also named as "Variance components Models" (Longford, 1986), "Hierarchical linear Models" (Raudenbaush and Bryk, 1986Bryk, , 2002)), and "Random Coefficient Models" (de Leeuw and Kreft, 1986, Moulton, 1986, Longford, 1992, Goldstein, 1995, Snijders and Bosker, 1999).

Random Intercept Model for K-Levels
The concept of random intercept models in a multilevel model developed by Goldstein (1986), Khan andKamal (2012a,b, 2013) has been extended for k-levels.Suppose where, 0  is an intercept term representing the average value of all the intercepts.The term 2 i  is a Level-2 residual term representing the variation of intercept terms across the second level units with .Model (2.2) is a random intercept multilevel model as it includes the variation of both the levels.Since partitions the individual variation into two components (one is for Level-1 units and the other is due to Level-2 units) so, it is also called random coefficient two-level model. Similarly, value of a response variable y measured at Level-1 where the subscript 3 i refers to Level-3 3 m units.Again, a three-level random intercept model may be defined as, where, 0  is an intercept term representing the average value of all the intercepts across Level-2 and Level-3 units.The term 3 i  is a Level-3 residual term representing the variation of intercept terms across the third level units with .Model (2.3) is a random intercept three-level multilevel model as it includes the variation due to all three levels of hierarchy in the population.
We generalize the random intercept multilevel model for k-levels by method of induction.Suppose be the observation of a response variable measured at where the subscript refers to units, refers to units, refers to units and so on the subscript represents units measured across .If we assume the variation of response variable across only in the intercept term then a k-levels model may be define as, The model (2.7) can be written in matrices form as, (2.10) where, is a vector of units of response variable, is a vector of ones, is a vector of units, The terms are the variances across units respectively.We assume that the covariances between the residuals are zero.Also the matrices of residuals have the following assumptions: (2.12) ,..., ,..., ,..., , , ,..., for h=1,2,...,k ( ) ..., (5.11.9) , for f h, where f,g=1,2,...,k If is known then the Generalized Least Square (GLS) estimates of can be obtained by using the relation, (2.13)If is known and is unknown then the estimates of can be obtained again by using GLS estimator, ( ˆŵith cov( ) (5.11.10) (5.11.14) (5.11.15) and so on, (2.23) where is an identity matrix and is a matrix of ones.Furthermore, is a direct sum operator.Thus consistent estimates of can be obtained by using in .
We estimate the random intercept model for single level, 2-levels, 3-levels, 4-levels and 5-levels by using a higher education data collected from five universities in Pakistan.The educational structure in universities in Pakistan is hierarchical where students are nested within teachers, teachers nested within directors/chairpersons, directors/chairpersons nested within deans, deans nested within universities and the universities.The response variable is student grade point average (GPA) score collected from 40000 students registered in five universities from Pakistan.Random intercept models defined across Level-1,Level-2,Level-3,Level-4& Level-5 are as follows: (2.28) where, where, 0  is the average intercept and the terms , , , & .The iterative generalized least square estimates of the parameters under (2.24), (2.25), (2.26), (2.27) and (2.28)   of hierarchy in the model.Again, 96.7% of the variation in Level-2 units is due to incorporating third level in the model.This suggests the contribution of department level units in the random variation of grades is more when compared to variation due to teachers.Again when we included the fourth level in the model .This information tells the fact that students grades vary faculty (Dean) wise and their impact is comparatively more highlighted as compare to variation due to department.Finally, a major contribution in random variation at Level-4 is due to Level-5 units and students grades vary from university to university.

Conclusions
The random variation in intercepts at individual level is marginally divided into higher levels when consider higher levels of hierarchy in the single level model.So, one can control the random variation in intercepts by incorporating the higher levels in the model.The decreasing trend of -2*loglikelihood confirms the better fit of the more nested model.

 1 i
at individuals' level, where the subscript refers to units.A simple model relating the response variable with the intercept term may be defined asis an intercept term representing the average value of the response variable 1 i y and is fixed.The term is a residual term representing the random departure of values from the fixed intercept 0 random variations of Level-1,..., Level-5 units respectively under model (2.24), (2.25), (2.26), (2.27) and (2.28) and represent the random departure from the