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Multilevel Modeling

8. A Graphical Introduction

Figure 5

Figure of six graphs with varying intercepts and slopes as discussed in text.

The different patterns in Figure 5 are achieved by allowing the average (fixed) ‘intercept’ and the average (fixed) ‘slope’ to vary (be random) across neighborhoods. Multilevel models specify the different intercepts and slopes for each context as coming from a distribution at a higher level.

The different forms of relationships represented in Figures 5(c)-(f) are a result of how the intercepts and slopes are associated. Graphical models represented in Figures 5(c)-(f) are also called random-slopes models, since the patterns are achieved by allowing the fixed slope to vary across neighborhoods. Figure 5(b), meanwhile, is the simplest form of multilevel modeling and is referred to as a random-intercepts model, as only intercepts are allowed to vary across neighborhoods.

For instance in Figure 5(c), the relationship between poor health and age is strongest in neighborhoods (a steeper slope) where poor health rates are quite high for average age groups (a high intercept). Stated differently, there is a positive association between the intercepts and the slopes. In Figure 5(d) high intercepts are shown to be associated with shallower slopes, that is, a negative association between the slopes and the intercepts. The complex criss-crossing in Figure 5(e) results from a lack of pattern between the intercepts and the slopes such that the health achievement rates of a neighborhood at average age tells us nothing about the direction and magnitude of the poor health/age relationship. The distinctive feature of Figure 5(f) results from the slopes varying around zero. In other words, while typically there is no poor health/age relationship, in some neighborhoods the slope is positive; in others it is negative. In this case, a single-level model would reveal no relationship whatsoever between poor health and age, and as such the ‘average’ relationship would not occur anywhere.

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