Research Notes and Comments I 347

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1 Research Notes and Comments I 347 mum-likelihood estimation of the constant and does not need to be applied a posteriori. Overall, the replacement of the lognormal model by the Poisson model provides a more homogeneous family of trip distribution models. LITERATURE CITED Batty, M., and S. Mackie (1972). The Calibration of Gravity, Entropy, and Related Models of Spatial Interaction. Enoironment and Planning, 4, Flowerdew, R. (1982). Fitting the Lognormal Gravity Model to Heteroscedastic Data. Geographical Analysis, 14, Flowerdew, R., and M. Aitkin (1982). A Method of Fitting the Gravity Model Based on the Poisson Distribution. Journal of Regional Science, 22, Fotheringham, A. S. (1983). A New Set of Spatial Interaction Models: The Theory of Competing Destinations. Enoironment and Planning A, 15, Haworth,/. M., ~d P. J. Vincent (1979). The Stochastic Disturbance Specification and Its Implications or Log Linear Regression. Enoironment and Planning A, 11, Heien, P. M. (1968). A Note on Log-Linear Regression. Journal of the American Statistical Association, 63, Nelder, J. A,, and R. W. M. Wedderburn (1972). Generalized Linear Models. Journal of the Royal Statistical Society A, 135, Powell, M. J. D. (197). A FORTRAN Subroutine for Solving Systems of Non-Linear A1 ebraic Equations. In Numerical Methods for Non-Linear Algebraic Equations, edited by P. Ra%inowitz. London: Gordon and Breach. Stron e, W. B., and R. R. Schultz (1978). Heteroscedasticity and the Gravity Model. GeographicafAnalysis, 1, A Higher Moment for Spatial Statistics by C. Michael Costanzo, Lawrence J. Hubert, and Reginald G. GoUedge* As discussed by Cliff and Ord (1981), and Hubert, Golledge, and Costanzo (1981), many of the indexes used to evaluate spatial autocorrelation can be defined as special cases of the general cross-product statistic A = C WijCij ij Assuming there are n observations on a single variable X, say xi,..., x, realized in locations Z1,..., Z, respectively, then Wij is a measure of spatial separation between Zi and 4, and C, is some measure of similarity (or proximity) between the values xi and xj. For example, C, could be defined as (xi - xj)z to give the basis of Geary s (1954) statistic or as the absolute difference between xi and xj to give the statistic suggested by Royaltey, Astrachan, and Sokal(l975). Further connections to the literature are outlined in Hubert, Golledge, and Costanzo (1981), along with a suggestion of alternative definitions of Cij (e.g., proximity could be a function of several variables and defined as a correlation or distance measure over *Partial support for this research was provided by the National Institute of Justice through grant 82-IJ-CX-19. C. Michael Costanzo is a doctoral student in geography, Lawrence J. Hubert is professor of education, and Reginald G. Golledge is professor of geography, University of CaZi$ornia, Santa Barbara. GeographicalAnaZysis, Vol. 15, No. 4 (Oct. 1983) 1983 Ohio State University Press

2 348 I Geographical Analysis the latter). For an additional discussion of A in a geographical context, the reader is referred to Glick (1979). Under the usual randomization model, the typical procedure for assessing the significance of A has been to use a normal approximation, possibly complemented with a Monte Carlo simulation. Judging by some recent work, the applicability of the normal approximation is, in general, questionable. For instance, Mielke and his associates (e.g., Mielke, Berry, and Johnson 1976) and Ascher and Bailar (1982) have found that the reference distributions for A under randomization are seriously skewed for several very reasonable situations. Furthermore, and contrary to an optimistic hope for asymptotic normality, the skewness parameter in these cases does not converge to as n + m. As one dramatic illustration, the statistic A can be used to assess group differences by letting the matrix {Wj.} contain -1 values that represent the given group structure in block-diagonal drm. Unfortunately, if the group sizes are equal, the skewness parameter is bounded away from zero asymptotically. Even in those instances where normality may be a reasonable asymptotic approximation, seriously skewed distributions often are encountered when n ii moderate in size. Obviously, the true significance level also may be at variance with that generated from the normal. This latter fact is noted in some detail by Cliff and Ord (1981), who then proceed to outline a Monte Carlo approach for assessing the relative size of A when an assumption of normality may lead to difficulty. The Pearson Type III Approximation One of the standard ways of approximating a distribution, when the first two moments are not sufficient to give an adequate representation with the normal, is through curve-fitting with moments of the third or higher order. As noted by Cliff and Ord (1981, p. 53), however, -this has not been pursued to date because of the practical difficulties of evaluating the higher order moments numerically, even when the theoretical forms are known." Some recent work by Mielke (1979) and Mielke, Berry, and Brier (1981) appears to have overcome at least some of these difficulties. These authors suggest a fit to the reference distribution for A under randomization through a Pearson Type I11 (gamma) function that, in turn, is based on an exact skewness parameter y obtained from a computationally eecient formula given in Mielke (1979, and errata). While efficient, the formula is much too long to be reproduced in this note. The Pearson Type I11 distribution, as standardized by Mielke, Berry, and Brier (1981) to have a mean of and a variance of 1, is described by the following probability density function: where - 2/y < Z < W, Z is the standardized statistic, y is the skewness parameter, and r(*) refers to the usual gamma function. The function converges to the standard normal probability density function when the skewness converges to zero, suggesting that a normal approximation may be used directly when the absolute value of y is less than, say,.1. Tables of critical values for the Pearson Type I11 distribution are available (Salvosa 193; Harter 1969), and these could be consulted. Alternatively, approximate probabilities can be constructed rather easily using a common subroutine for evaluating cumulative probabilities for the gamma distribution in the IMSL series available at most computer facilities. In particular, for upper-tail probabilities

3 Research Notes and Comments I 349 corresponding to an observed value of Z and positive skewness, we merely evaluate where = 4/y2 and b = (Z + 2/y)(2/y). When the user specifies and b, the integral from to b can be directly evaluated by a single call to IMSL subroutine MDGAM. A program, QAPII is available from the first author that carries out the necessary calculations. This routine calculates A, its first three moments, the skewness parameter, and the approximate probability of obtaining such a value by reference to Pearson s Type 111 distribution. In addition, the program implements a Monte Carlo test by randomly permuting the rows and columns of {Cij} against a fixed {Wi.}(or equivalently, the converse). The size of the sample is under the control of the researcher. A Numerical Illustration: Assessing Spatial Autocorrelation of Directions As an actual application, we consider the directions traveled to vandalize in the city of Minneapolis as recorded in a recent study (Frisbie et al., 1978). The data consist of sixty-six observations for which both the offense site and the site of the suspect s residence are identified by Cartesian coordinates. Our interest is in whether or not nearby offenses can be attributed to suspects who have come from similar directions. Put another way, we would like to assess the degree of spatial autocorrelation in the directions from which suspects have traveled to commit vandalism. Following Hubert et al. (1983), we define C, as cos (di - 4), where di and dj are the directions from which suspects have traveled to commit offenses in locations Zi and 4, respectively. Thus C, is equal to 1 when both suspects traveled in precisely the same direction, - 1 when they have come from opposite directions, and when the directions are at right angles to one another. Since we let Wij be the Euclidean distance between the offense sites, lj and 4, the two indexes, C, and Wij, are keyed in opposite directions. Therefore, positive spatial autocorrelation should yield a relatively low A. For this example, A is equal to , with exact mean and variance under randomization of and 3.85 x lo6, respectively. The resultant Z-statistic, given by (A - E(A))/(var(A))%, is Significance testing involves assessing the probability of observing a value of A as low or lower than by chance alone (i.e., under the randomization model). If we were to assume that the reference distribution for A was modeled adequately by the normal distribution, this probability may be obtained by referring to one of the familiar tables which display areas of the normal curve. In particular, for a Z of , the one-tailed probability is approximately.4, and we would be led to reject the null hypothesis of no spatial autocorrelation at the.5 level. In this case, however, the distribution is skewed in the negative direction (y = -.799), suggesting that this latter probability is suspect and probably too small in relation to the true distribution. Figure 1 displays a histogram of the empirical frequency distribution for A,

4 35 I Geographical Analysis, ,OOO 8 a. 6 5 = B 4 2 Z-Value FIG. 1. Frequency Histogram Based on 1, Random Permutations, and the Density Function of the Pearson Type 111 for y = generated by 1, random permutations of the original data. The standardized Pearson Type I11 distribution for y equal to also is shown in the figure. The frequency distribution is nearly mirrored by the theoretical density, and the departure from normality at this skewness level is obvious. Based on the random sample, 65 permutations yielded values of A as low or lower than , giving a Monte Carlo significance level of approximately.65 (651/11). Whereas the corresponding probability obtained by reference to the Pearson Type 111,.56, is smaller than the Monte Carlo value, neither of them would lead us to reject the null hypothesis at the.5 level. Figure 2 shows more detail in lower tails of the three cumulative distributions. Although not perfect, the Pearson Type 111, as compared to the normal, more nearly reflects the empirical distribution. - Bosed on 1, Permutations -_ Based on Penrron Type Ill Baed on Normol Z-Value FIG. 2. Cumulative Frequencies Based on 1, Random Permutations, Pearson s Type 111 Distribution, and the Normal Distribution

5 Perspective Research Notes and Comments I 351 We have examined the viability of alternative significance testing strategies for a very general cross-product statistic. Since many statistics, and particularly those that are used for assessing spatial autocorrelation, can be viewed as special cases of this general measure, our findings have widespread applicability. In this paper, the superiority of the Pearson Type I11 approximation over the Normal approximation has been demonstrated with respect to a single analysis. A more thorough examination is presently being carried out as part of the first author s doctoral dissertation, and that work as well as other studies which we have conducted strongly support these findings. The Type I11 approximation is still not perfect, as one can see from the figures. The discrepancy is most obvious in the upper tail of the distribution shown in Figure 1, but it should be noted that the skewed tail is extremely close and that our experience as well as Mielke s (1981) has been that one is almost invariably concerned with that tail and not the other. Our final comment concerns the utility of performing the Type I11 significance test in conjunction with a Monte Carlo test, as provided for by the program QAPII. The need is predicated by the fact that both procedures are subject to error. Although the Type I11 distribution is generally more appropriate than the Normal, the test still is prone to approximation error. On the other hand, a Monte Carlo test may suffer from sampling error. After all, one does not ordinarily perform 1, permutations as we have done for this analysis; sample sizes this large are rather expensive for the purpose of routine significance testing. Therefore, it is beneficial to perform both tests so that the results of each can be used to check on the other. It is not necessary to choose a huge sample of permutations for the Monte Carlo test in that case-1 permutations is probably sufficient for most applications. The two tests complement each other in this way, and together they provide a much better way to infer significance than has been hitherto available. LITERATURE CITED Ascher, S., and J. Bailar (1982). Moments of the Mantel-Valand Procedure. ]ournal ofstatistica1 and Computational Simulation, 14, Cliff, A,, and J. Ord (1981). Spatial Processes: Models and Applications. London: Pion. Frisbie, D., G. Fishbine, R. Hintz, M. Joelson, and J. Nutter (1978). Crime in Minneapolis. Minneapolis: Minnesota Crime Prevention Center. Geary, R. (1954). The Contiguity Ratio and, Statistical Mapping. The Incorporated Statistician, 5, Glick, B. (1979). Tests for Space-Time Clustering Used in Cancer Research. Geographical Analysis, 11, Harter, H. (1969). A New Table of Percentage Points of the Pearson Type 111 Distribution. Technometrics, 11, Hubert, L., R. Colledge, and C. Costanzo (1981). Generalized Procedures for Evaluating Spatial Autocorrelation. Geographical Analysis, 13, Hubert, L., R. Golledge, C. Costanzo, N. Gale, and W. Halperin (1983). Non arametric Tests for Directional Data. In Recent Developments in Sb)atial, Analysis: Methodoigy, Measurement, Models, edited by G. Bahrenberg, M. Fischer, an P. Nijkamp Aldershot, U.K.: Gower. Mielke, P. (1979). On Asymptotic Non-Normality of Null Distributions of MRPP Statistics. Communications in Statistics, AS, (errata, A1 (1981). p. 1795; All (1982), p. 847). Mielke, P., K. Berry, and G. Brier (1981). Ap lication of Multi-res onse Permutation Procedures for Examining Seasonal Changes in Montby Mean Sea-Lever Pressure Patterns. Monthly Weather Review, 19, 12e26. Mielke, P., K. Berry, and E. Johnson (1976). Multi-response Permutation Procedures for a priori Classifications. Communications in Statistics, A5, Royaltey, H., E. Astrachan, and R. Sokal (1975). Tests for Patterns in Geographic Variation. Geographical Analysis, 7, Salvosa, L. (193). Tables of Pearson s Type 111 Function. Annals of Mathematical Statistics, 1,