Introduction to Multilevel Analysis

Presented by

Vijay Pillai

A GENERAL INTRODUCTION

In Hierarchical data one unit is nested with in the other unit.These units are also called levels

Level -1 represents the smallest unit of measurement Eg.: students

Level -2 represents a larger unit of measurement Eg.: Class

The level -1 units are said to be nested within level -2 units

Probably, the most common educational example is when thetwo different units are classes and students.

one

Just another way to show the hierarchical structure

2

class

Student student student

In the last figure there were two levels.

There is no reason why their can’t be 3 or 4 (Multi.)ML models are also called

Mixed modelsMultilevel linear models Random effect models

3

Glossary of termsMultilevel data –Data that have some intergroup membership

Fixed effect: A condition in which the levels of a factor include all levels of interest to the researcher

Random effect: A condition in which the levels of a factorrepresents a random sample of all possible levels.

4

0 1i i iy X r

0

1

ir

ON ML MODELSBasically ML models are regression models.Well, we all know the basic OLS regression model.

where

is the intercept ,

is the slope and

is the residual.5

In regression we also make assumptions about the residuals.

For example, residuals are normally distributed, with mean0 and variance

2 no multi collinearity, etcOf course, this model works well, when we have a homogeneous population- such as a single community.But what if we have observations from multiplecommunities ?6

Each community then has its own regression line (with a intercept and a slope),

Now , the population we have may longer be homogenous.

We need a notation to indicate which community we are talking about

We will use a new subscript j to indicate which community we are talking about

We will have a total of j communities in our sample.7

0 1ij j j ij i jy X r

0 j

1 j

So now the our regression line for the ith person in the jth community is

Where

is the intercept for the jth community, is the slope for the jth community, so onSo , if we randomly select communities and compute the regression line for each community

-we can consider the intercept as a random variable-we can consider the slope as a random variable

- Both the intercept and slope can then be predicted by other properties of the communities

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0 . ..j ijand

0

1

0 00 01

1 10 11

j

j

j j

j j

W u

W u

ML models fit a regression model for each of the

- called the Level – 2 regression model.Level -2 regression models are expressed as follows.

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