Introduction

Observed densities and true densities: The small number problem

The true density is the probability that an individual in the area has the disease. The observed density is an estimate of the true density. Especially when measured for small areas, chance variations can be large and the true density may lie anywhere within a large range of observed density values. The problem is explained very well by MacDorman et al. (1994, p.244-245) from which the following quote comes:

"An area's observed infant mortality rate should be considered an estimate of the true underlying mortality rate. (The number of infant deaths in an area varies by chance, depending on the number of births and the probability of infant death (the true infant mortality rate). As is the case with any estimate, the infant mortality rate is subject to chance variation. If the area has very few births, the observed infant mortality rate may be very different from the true rate. Thus, if rates for two areas are compared in a given year and one (or both) of the area's rates is based on a small number of births, it would not be unusual for the findings to be reversed the following year.

"Therefore, a method is needed to assess the adequacy of the observed infant mortality rate as an estimate of its true value. The most common method is the use of confidence intervals. Calculation of confidence intervals is explained in detail elsewhere (see Kleinman and Kiely, 1991; Gardner and Altman, 1989; and Armitage and Berry, 1987). Basically, a 95% confidence interval is defined so that the probability is 95% that the true rate is included in the interval. If the interval is very wide, the true rate is not estimated with much precision. The interval generally becomes narrower as the number of births on which the rate is based increases. Two common methods of increasing the numbers of births are to combine years and to combine smaller areas into larger ones.

"Although aggregation over years and areas permits us to compute stable rates, loss of information occurs. Combining heterogeneous areas to obtain a stable rate may be more misleading than helpful. Combining years involves the assumption that in each of the years, the ranking of the areas is the same--that is, annual changes in the rates are the same for all areas.

"The stability issue is especially important when comparing areas or determining whether real changes have occurred over time within an area. In these situations, confidence limits should be used to assess the magnitude of the differences. Two areas (or two time periods for one area) can be compared by using the absolute difference in their rates or by using the ratio of their rates. The ratio of rates (or relative risk as it is sometimes called) is usually preferred because it allows for comparison of areas or time over a wide range of rates."

Source: M.F. MacDorman, D.L. Rowley, S.Lyasu, J.L. Kiely, P.G. Gardner and M.S. Davis. 1994. "Infant Mortality" in L.S. Wilcox and J.S. Marks, eds. From Data to Action: CDC's Public Health Surveillance for Women, Infants, and Children. Atlanta: Public Health Service, U.S. Department of Health & Human Services, Centers for Disease Control, pp.231-249.

Kernels and Windows

Illustrated here are spatial filters of 0.4m and 0.6m.

Window Widths

Choice of window width

Window width is the area over which density is estimated. There is no 'correct' window width; as Scott (1992, p. 161) notes: "It should be emphasized that from an exploratory point of view, all choices of the bandwidth h lead to useful density estimates. Large bandwidths provide a picture of the global structure in the unknown density, including general features such as skewness, outliers, clusters, and location. Small bandwidths, on the other hand, reveal local structure which may or may not be present in the true density. Furthermore, the optimality of h is dependent not only on the choice of metric Lp but also on the feature in the density to be emphasized."

Can permit the interval width to vary to accomplish "equivalent smoothing." This could be done by finding the interval such that equal sample size is in the filter; (Scott, 1992, p. 146).

"Adaptive kernels"......but on the whole, good adaptive estimation is a difficult task.Ó (Scott, 1992, p. 137).

"However, the power of the interactive approach to bandwidth selection should not be underestimated." (Silverman, 1978, p. 161). See also, the test graph method.

Scott, D. W. 1992. Multivariate Density Estimation: Theory, Practice, and Visualization. New York, J. Wiley.

Choice of Kernel

Choice of kernel: "the quality of a density estimate is now widely recognized to be primarily determined by the choice of smoothing parameter, and only in a minor way by the choice of kernel. ..." (Scott, 1992, p. 133).

Choice of Grid Size

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Boundaries