Bithell's linear risk score method: GeoMed Software Requirements

Ho: The case counts (xi) in the areas are independently Poisson distributed with means equal to tex2html_wrap_inline122 0iei, where the eiare predetermined, area-specific expectations. For an unconditional test of specified spatial relationship, the relative risk tex2html_wrap_inline122 0i is assumed to be constant and equal to 1. Running the unconditional version of the test is equivalent to assuming that the rates used to calculate the ei are appropriate for the study region. For a conditional test of specified spatial relationship, tex2html_wrap_inline122 0i is assumed to be constant but not necessarily equal to 1. The rates used to calculate the ei are not assumed to be appropriate for the study region when running the conditional version of the test.

Ha: Each case is scored with a risk score given by the logarithm of the relative risk for the case’s subregion of residence. Relative risk functions (RRF) can be used to completely specify a pattern of risk and the alternative spatial model used to specify case risk scores. The multiplicative risk model provides linear risk scores based on inverse distance from a point source.

Test Statistic: let tex2html_wrap_inline122 1i denote the risk score for region i under the alternative and let tex2html_wrap_inline122 0i be the corresponding score under the null hypothesis. The most powerful test statistic is:

 

Here t0 is a constant chosen to reflect the correct type I error ( tex2html_wrap_inline102 ) for the test. But in practice, tex2html_wrap_inline122 0i will not be known and a test based on the sum over all cases can be used:

 

where i(j) indicates the region of interest of the jth case, j=1,2,...,n. Each case is assigned a risk score given by the logarithm of the relative risk for the subregion of residence, and these scores are summed over all subjects. Because of the linear structure of the statistic T, Bithell calls a test of this kind a linear risk score (LRS) test. Conditional tests concentrate entirely on the detection of spatial pattern (and can be applied even when the calculated expectations may not be accurate). However, a positive test outcome can result from a deficit of cases remote from the focus in the conditional case. The conditional test (and the Monte Carlo randomization process) is based on the multinomial distribution. In the unconditional version (in which the calculated expectations are appropriate) an overall increase in incidence in the (entire study) area contributes to positive results as well as the tendency for this excess to be concentrated near the focus. The unconditional test (and the Monte Carlo randomization process) is based on the Poisson distribution.