1. Introduction to Multilevel Analysis Presented by Vijay Pillai

2. 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 the two different units are classes and students. one

3. Just another way to show the hierarchical structure 2

4. 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 models Multilevel linear models Random effect models 3

5. Glossary of terms Multilevel 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 factor represents a random sample of all possible levels. 4

6. ON ML MODELS Basically 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

7. In regression we also make assumptions about the residuals. For example, residuals are normally distributed, with mean0 and variance no multi collinearity, etc Of course, this model works well, when we have a homogeneous population- such as a single community. But what if we have observations from multiple communities ? 6

8. 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

9. 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 on So, 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 -8

10. 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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