Contextual Effects

The notion of groups plays a major role in social science. Does membership of an individual in a particular group have any impact on thoughts or actions of the individual? Does living in a neighborhood consisting of many Democrats mean than an individual has a higher probability of being a Democrat than otherwise would have been the case? Such an impact by the group is known as a contextual effect. The study of these effects is called contextual analysis or MULTILEVEL ANALYSIS. It is also known as the study of HIERARCHICAL models.

If contextual effects exist, it raises the following questions: Why does belonging to a group have an impact, how does this mechanism work, and how should we measure the impact? The first two questions may best be left to psychologists, but the third question has sparked much interest among statisticians and others. In particular, in education, a classroom makes an obvious group for contextual analyses.

When we have data on two variables on individuals from several groups, within each group we can analyze the relationship of a dependent variable Y and an independent variable X. Regression analysis gives a slope and an INTERCEPT for the regression line for each group. It may now be that the slopes and intercepts vary from one group to the next. If this variation in the regression lines is larger than can be explained by randomness alone, then there must be something about the groups themselves that affects the lines.

There can be several reasons why the slopes and intercepts vary. It may be that both depend on the level of the X variable in the group through the group mean. Thus, one possible contextual model can be expressed in the following equations:

where a and b are the intercept and slope in the jth group, and the u and v are residual terms.

It is also possible to substitute for a and b in the equation for the line in the jth group. That gives the following equation:

for the ith individual in the jth group. Thus, we can now run a multiple regression on data from all the groups and get the c coefficients. For this model, c1 measures the effect of X on the level of the individual, c2 measures the effect of X on the level of the group, and c3 measures the effect on Y of the interaction of the individual and the group-level effects. One problem with this analysis is that the residual E depends on the group means of X; another is the COLLINEARITY that exists in the three explanatory variables.

Contextual analysis is not limited to the case in which the group intercepts and slopes depend on the group means of X. They could also depend on a variety of other variables.

Bayesian statistics lends itself particularly well to multilevel analysis because the slopes and intercepts vary from group to group. It may then be possible to specify how the intercepts and slopes within the groups come from specific distributions.

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