# Hierarchical Linear Modeling

Hierarchical Linear Modeling:

Hierarchical linear modeling is an approach to analysis of hierarchical (nested) data – i.e. data represented by categories, sub-categories, …, individual units (e.g. school -> classroom -> student).

At the first stage, we choose a linear model (level 1 model) and fit it to individual units in each group separately using conventional regression analysis . At the second stage, we consider estimates of the level 1 model parameters as dependent variables which linearly depend on the level 2 independent variables. The level 2 independent variables characterize groups, not individuals. We find level 2 regression parameters by a method of linear regression analysis.

There may be more than 2 levels in this process, provided there are more than two levels in the hierarchy of groups or categories, e.g. district -> school -> classroom -> student.

Technically, hierarchical linear models are such models that for any term all effects of lower order are also included in the model.

For example, for two-way KxN contingency table s, the independence model

 Mij = m+ liX + ljY;   i=1,…,K,   j=1,…,N

and the saturated model

 Mij = m+ liX + ljY + lijXY;    i=1,…,K,   j=1,…,N

are the only hierarchical linear models that involve both variables.

The following model is an example of a non-hierarchical model:

 Mij = m+ ljY + lijXY;   i=1,…,K,   j=1,…,N.

This model is non-hierarchical since it does not contain the term liX but contains the higher order effect lijXY .

If the linear relationship is postulated between the logarithm of the dependent variable(s) and parameters of the model, the models are named “hierarchical loglinear models”.

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