Prospective crime mapping in operational context Final report

e ij = n .j x n i.nWhere, e ij is the expected value for each cell, n ij is the observed, and n i and nj are the row andcolumn totals respectivelyThe results of the analysis can be interpreted by inspecting adjusted residuals, computed foreach cell using the following formula (see, Agresti, 1996, pp31-32):r ij =n ij – e ij[e ij (1-row proportion of n i+ )(1-column proportion of n +j )] 1/2Thus, the adjusted residuals are a measure of the difference between the observed andexpected values for each cell. The adjusted residuals have a mean value of zero and astandard deviation of one, hence adjusted residual scores exceeding values of two (or evenmore stringently three) are considered statistically significant. The null hypothesis is thatevents are not clustered in space and time. Negative values indicated that there are fewerevents occurring within a particular space-time interval.The majority of research concerned with crime (Johnson and Bowers, 2004; Townsley et al.,2003) has used the Knox approximation to examine space-time clustering. However, analternative approach, for which the independence of observations is not a requirement, mayalso be computed (Johnson et al., 2006). This approach uses Monte-Carlo simulation togenerate an expected distribution, rather than using the marginal totals (Besag and Diggle,1977). To do this, the data are permuted, in effect mixing up the dates and locations acrossthe events. Thus, the dates are randomly shuffled using a pseudo-random number generator,whilst the spatial locations remain fixed. The hypothesis is that if there is statisticallysignificant space-time clustering in the data then there should be more events observed tooccur close in space and time than for 95 per cent of the random permutations generated.The process of generating permutations is repeated (iterated) a number of times, typicallyaround 999, a new contingency table generated each time and compared with thecontingency table for the observed distribution. The probability that the observed value foreach cell occurred on the basis of chance may be calculated using the following formula (seeNorth, 2002):n − rank + 1p =n + 1Where n is the number of simulations, and rank is the position of the observed value in a rankordered array for that cellOne possible reservation about the p-values generated using the Monte-Carlo approach (andthe Knox residuals) is that they are not as readily interpretable as one might like.Interpretation of the results can be aided by deriving a simple measure of effect size. 2 In thiscase, a Knox ratio, which contrasts the observed value for each cell with the average‘simulated’ value for that cell (or the expected value derived using the marginal totals of thetable), was computed. Thus, a Knox ratio of 2.0 so derived would indicate that twice as manyburglaries occurred within a particular distance and time of each other than would on averagebe expected on the basis of a chance distribution. A value of one would indicate that theresult conformed to what would be expected on the basis of chance, and values of less thanone that fewer burglaries occurred within a particular distance and time of each other thanone would expect. Which measure of central tendency is best used to compute the odds ratiodepends upon the distribution of the simulated data. If the data are skewed, or there are2 The Knox residuals can be interpreted in the same way to some extent but are not as simple to understand. Insimple terms, the larger the Knox residual the larger the effect.5