Multiple regression estimates average relationships between response (eg educational attainment) and predictor variables e.g. socio-economic status, gender, previous (baseline) ability.

The above graph illustrates a typical linear regression relationship, in this case between outcome attainment and prior-ability among a sample of students. The red line shows that on average an increase in prior ability is associated with an increase in outcome attainment.

A fundamental assumption of this regression model is that the residuals (the distance of the data points from the red regression line) are independent. However, data often have a multilevel structure which violates this assumption.

In this example students are grouped within schools. If we believe that the process of student selection by schools or the education given by schools may influence outcome attainment, then two students within a particular school will tend to be more similar than two students from different schools.

The pupils at two schools are highlighted in the above graph to illustrate this point. If we ignore the nesting of pupils within schools - that is, we analyse the data as though all pupils were independent - then we will tend to underestimate the standard errors of the regression coefficients. This problem, called "misestimated precision", means that we will tend to find too many relationships to be statistically significant.

Generally we are interested not only in the average relationship (the red line) but in how this relationship varies from school to school.

Multilevel modelling provides a powerful framework for exploring how average relationships vary across hierarchical structures.

Next Section: Hierarchical Structures