Chapter 6 Maps for Rates

In Chapters 4 and 5, the maps pertained to a single variable. In the current chapter, I deal with some special aspects related to the mapping of rates or proportions. In GeoDa, such variables can be constructed on the fly, by specifying a numerator and a denominator. The numerator is typically a type of event, such as the incidence of a disease, and the denominator is the population at risk. Such data have broad applications in public health and criminology. However, several of the principles covered can be equally applied to any ratio, such as an unemployment rate (number of unemployed people in a given region as a share of the labor force), or any other per capita measure (e.g., gross regional product per capita).

The chapter contains three main sections. First, the creation of a raw rate or crude rate map is considered, which is the same as the types of maps considered so far, except that the rate is calculated on the fly. Next, the focus shifts to so-called excess risk maps, which compute a measure that compares the value at each location to an overall average, highlighting extreme observations. Such excess risk is known under different terms in various literatures, such as a standardized mortality rate (SMR) in demography (Preston, Heuveline, and Guillot 2001), or a location quotient (LQ) in regional economics (McCann 2001).

Finally, attention shifts to the important topic of variance instability that pertains to any rate measure, and the associated concept of rate smoothing. In essence, the precision of the rate as an estimate for the underlying risk depends on the size of the denominator. In practice, this means that rates estimated for small populations (e.g., rural areas) may have large standard errors and provide imprecise estimates for the actual risk. This may lead to erroneous suggestions of extreme values, such as the presence of outliers. Rate smoothing techniques use a Bayesian logic to borrow strength and adjust the small area estimates. There is a large literature in statistics dealing with such techniques (e.g., Lawson, Browne, and Rodeiro 2003). Here, the discussion will be limited to the basic principle, illustrated by the most common form of Empirical Bayes smoothing.

I continue to use the Ceará Zika sample data set to illustrate the various features.