A Spatially Adjusted ANOVA Model, by Daniel A. GrifSzth. yi,( i = 1,2,...,q;

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1 A Spatially Adjusted ANOVA Model, by Daniel A. GrifSzth lntroduction Adaptation of classical statistical theory for the purpose of geographical analysis has uncovered the general problem of parameter estimation in statistical models involving spatially autocorrelated error terms. Consequences attributable to this problem have already been documented, especially for the classical linear regression model (see, for example [2, 4, 6, 9, 14, 18, 19, 22, 23, 24, 25, 29, 30, 31, 36, 37]), and have been briefly explored in terms of factor analysis [28]. Furthermore, Cliff and Ord [9] have derived a solution for the linear regression model whose errors are spatially autocorrelated. The objective of this essay is twofold: (1) to present the solution for a one-way univariate analysis of a variance model (hereafter referred to as a one-way ANOVA model) involving spatially autocorrelated error terms; and (2) to show that any linear statistical model that can be rewritten as a linear regression model may be solved by the Cliff-Ord technique. Geographical Applications of the ANOVA Model An introduction to the geographical application of the ANOVA technique has been adequately described in, for example, [ 17, 26, 27, 34, 381. In addition, a brief bibliography of its application to spatial problems has been provided in [IS], with additional examples including [3, 7, 20, 321. The general question addressed in these writings will now be outlined. Consider some planar surface that has been partitioned into a set of M mutually exclusive and collectively exhaustive regions, with A$ ( i = 1,2,..., M ) distinct areal units in the corresponding jth region, wherec;i, = n. Let each of these n areal units have associated with it a given value for some random variate Y, denoting this measure in the jth region for the ith areal unit as i = 1,2,..., M). One question that could be asked about these yli values is whether or not their respective regional averages 1/, ( i = I$,..., M ) come from the same parent population. One-way ANOVA furnishes a standard method for assessing dfferences in yi,( i = 1,2,...,q; the set of regional means r/, ( i = 1,2,..., M), given their respective regional variances sy, ( i = 1,2,,.., M ) and some probability of observed differences occur- I should like to acknowledge the helpful comments of Dr. L. Curry of the Geography Department at the University of Toronto, and Dr. E. Sheppard of the Geo aphy Department at the University of Minnesota on an earlier version of this paper. Of course, I gne am responsible for any errors or omissions. Daniel A. Griffith is professor of geogrcipliy, Ryerson Polytechnical Institute. C~ /78/ $00.50/ Ohio State University Press GEOGRAPHICAL ANALYSIS, vol. X, no. 3 (July 1978)

2 ring. This method utilizes the matrix model where Y = n x 1 vector of measures for some characteristic py = n X 1 mean-score vector Research Notes and Comments / 297 Y = py + t + E, (1) t = n x 1 vector of differences between the mean-score vector entry and the mean of region i( j =1,2,..., M ) to which areal unit k( k = 1,2,..., n) belongs E = n x 1 vector whose entries are normally distributed, random, uncorrelated error terms having a mean of zero and a constant variance. Draper and Smith [13] have shown that equation (1) may be rewritten as the following regression equation: Y=XP+E, (2) where Y and E are defined as in equation (l), p is an M X 1 vector of regression parameters, and X is an nxm matrix of dummy variables whose M column vectors are defined such that' xkl-l (k=1,2,..., n),and 1 if areal unit k belongs to region I, 'kl= { 0 otherwise. (k=1,2,..., n; 1=2,3,..., M ). The one-way ANOVA summary table accompanying equation (2) appears in Table 1. TABLE 1 ANOVA a la Regression Source Sim of Squares Degrees of Mean Freedom Square Between regions b'xy - ny2 M- 1 4 Within regions YY - bxy n-m 4 Overall mean ny2 1 Total YY n The use of dummy variables in the case of xkl = 1 (k = 1,2,3,..., n) requires that any arbitrary X, variate be deleted from the n x M matrix X. Thus, there will be M- 1 dummy variables in addition to XI. A model such as equation (2) can be fitted using a zero-intercept linear regression computer program.

3 208 / Geographical Analysis A Sputially Adjusted ANOVA Model Cliff and Ord have demonstrated the biases introduced by spatially autocorrelated values ek (k = 1,2,.,., n) in a general linear regression situation [9], and yk (k = 1,2,..., n) in a difference of means test based upon student's t-distribution [8]. Unfortunately neither equation (1) nor equation (2) accommodates these situations. Assuming the presence of autocorrelation in the error terms implies that E may be partitioned such that (after [12, 15, 22, 29, 30, 31, 33, 36]), where E=pCE+V (3) p =some parameter of the existing spatial autocorrelation level C = first-order Markov matrix depicting the planar arrangement of those areal units constituting observations E = error terms for equation (I) V = true random error component. Equation (1) may now be mathematically manipulated in the following manner: pcy = pcp,, + pct + pce (4) Y - pcy=p, - pcp,, + t - pct +E- pce (I - pc)y = (I - pc) p,, + (I - pc)t + pce +V- pce = (I - pc) ( py + t) + v :. Y = p,, + t + (I- pc) -'v. A similar sequence of algebraic operations on equation (2) yields (5) (6) ( 7) (8) Y = XP + (I - pc)- 'v, or (9) (I-pC)Y=(I-pC)xp+V. (10) Cliff and Ord have developed an iterative procedure for estimating p from equation (10) [9, pp The one-way ANOVA summary table accompanying equation (10) appears in Table 2. TABLE 2 ANOVA a la Regression Adjusted for the Presence of Spatial Autocorrelation Sollrca Sum of Sqware\ Degrees of Mean Freedom Square Between regions b[(i - pc)x]'(i- pc)y - ny2 M-1 d Within regions [(I-pC)Y]'(I-pC)Y- n-m 4v b"(i - pc)x]'(i - pc)y Overall mean n r2 1 Total [(I- PC)YI'(I - PC)Y n

4 Research Notes and Comments / 299 Extensions The foregoing methodology has at least two immediate extensions. Consider the case of two groups in discriminant function analysis. This particular problem reduces to a multiple regression analysis where Y is a binary dummy variable indicating to which group each areal unit belongs, and X is an n X p attribute matrix. Consequently, the solution to this discriminant function problem follows that for equation (9). Another extension involves estimating p for a single variate Y. This problem is a special case of equation (2) in which x,, =O (k = 1,2,...,n; Z=2,3,...,M). In other words, following equation (10) (I - pc)y = (z - pc)x/3, +v, (11) where x, E 1 (k = 1,2,..., n). Once again the Cliff-Ord solution technique can be used to estimate p and Po. The value obtained for p should be proportional to Moran coefficient and Geary ratio measures associated with variate Y. Concluding Comments In this paijer a one-way ANOVA model was developed, which takes into account the presence of spatial autocorrelation in its error terms. This ANOVA model complements the one formulated by Moellering and Tobler [32] for treating geographical scale problems. However, subsequent research that ultimately synthesizes these two versions is needed. Two extensions to the methodology used in deriving equation (8) have been noted. These extensions provide solutions to the two-group discriminant function analysis problem, as well as estimating p of the spatial operator (I - pc) for a single variable. Finally, some additional research needed to complement findings reported on here includes: 1. The extension of model (8) to Q-way schemes as well as multivariate situations; 2. The generalization of model (8) so that successively higher orders of spatial autocorrelation can be accommodated [24, 251; 3. A treatment of the other possible autocorrelation situations in ANOVA problems, such as the regional means being spatially dependent (see [ 191); 4. The uncovering of the sampling distribution for si and s& estimates obtained from autocorrelated data; 5. The development of appropriate tests of significance for parameter estimates based on autocorrelated data (some preliminary findings have already been published in [I, 5, 3.51); 6. A reassessment of the discipline s previous analysis of variance contributions. LITERATURE CITED 1. Bar-Shalom, Y. On the Asymptotic Properties of the Maximum-Likelihood Estimates Obtained from Dependent Observations. Journal of the Royal Statistical Society, 33B (1971),

5 . 300 / Geographical Analysis 2. Bartlett, M. The Statistical Analysis of Spatial Patterns. New York: Wiley, Berry, B., and W. Garrison. The Functional Bases of the Central Place Hierarchy. Economic Geography, 34 (1958), Besag, J. The Statistical Analysis of Non-Lattice Data. The Statistician, 24 (1975), Bhat, B. On the Method of Maximum-Wtelihood for Dependent Observations. JOUTIIU~ of the Royal Statistical Society, 36B (1974), Bodson, P., and D. Peeters. Estimation of the Coefficients of a Linear Regression in the Presence of Spatial Autocorrelation: An Application to a Belgian Labour-Demand Function. Enuironment and Planning, 7A (1975), Brunn, S. Changes in the Service Structure of Rural Trade Centres. Rural Sociology, 33 (1968), Cliff, A., and J. Ord. The Comparison of Means When Samples Consist of Spatially Autocorrelated Observations. Enuironment and Planning, 7A (1975), Spatial Autocorrelation. London: Pion (Monographs in Spatial and Environmental Systems Analysis), Curry, L. A Bivariate Spatial Regression Operator. The Canadian Geographer, 16 (1972), ~ A Note on Spatial Association. The Professional Geographer, 18 (1966), Dacey, M. A Review of Measures of Contiguity for Two and K-Color Maps. In Spatial Analysis: A Reader in Statistical Geography, edited by B. Berry and D. Marble, pp Englewood Cliffs: Prentice-Hall, Draper, N., and H. Smith. Applied Regression Analysis. New York: Wiley, Fisher, W. Econometric Estimation with Spatial Dependence. Regional and Urban Ecm- ics, 1 (1971), Geary, R. The Contiguity Ratio and Statistical Mapping. The Incurporated Statistician, 5 (1954), Greer-Wootten, B. A Bibliography of Statistical Applications in Geography. Washington, D.C.: A. A. G. Commission on College Geography, Technical Paper No. 9, Gregory, S. Statistical Methods and the Geographer. London: Longmans, Green and Co., Griffith, D. A Note on Spatial Autocorrelation. The Professional Geographer, 27 (1975), , Spatial Autocorrelation Problems: Some Preliminary Sketches of a Structural Taxonomy. The East Lakes Geographer, 11 (1976), Haggett, P. Regional and Local Components in the Distribution of Forested Areas in South East Brazil: A Multivariate Approach. Geographical Journal, 130 (1964), Haining, R. Model Specification in Stationary Random Fields. Geographical Analysis, 9 (1977), Hepple, L. A Maximum Likelihood Model for Econometric Estimation With Spatial Series, In Theory and Practice in Regiunul Science, edited by I. Masser, pp London: Pion, Hordijk, L. Spatial Correlation in the Disturbances of a Linear Interregional Model. Regional and Urban Economics, 4 (1974), Hordijk, L., and J. Paelinck. Spatial Econometrics: Some Contributions. Foundations of Empirical Economic Research Series, No. 1974/6. Netherlands Economic Institute, Spatial Econometrics: Some Further Results. Foundations of Empirical Economic Research Series, No. 1975/5. Netherlands Economic Institute, King, L. Statistical Analysis in Geography. Englewood Cliffs: Prentice-Hall, Krumbein, W., and F. Graybill. An Introduction to statistical Models in Geology. New York: McGraw-Hill, 19%. 28. Lebart, L. Analyse statistique de la contiguite. Publications de l institute de statistique de l uniuersite de Paris, 18 (1969), Martin, R. On Spatial Dependence, Bias, and the Use of First Spatial Differences in Regression Analysis. Area, 6 (1974),

6 Research Notes and Cmmenis / Miron, J. A Note on the Estimation of a Spatially Autoregressive Model. proceedings, 22nd IGU Congress, 2 (1972), On the Estimation of a Partial Adjustment Model with Autocorrelated Errors. International Economic Review, 14 (1973), Moellering, H., and W. Tobler. Geographical Variances. Geogmphical Adysis, 4 (1972), Moran, P. The Interpretation of Statistical Maps. Journal of the Royal Statistical Society, 10B (1948), Norcliffe, G. Inferential Statistics for Geogmphas. London: Hutchinson, Silvey, S. A Note on Maximum Likelihood in the Case of Dependent Random Variables. Joumal of the Royal Statistical Society, 23B (1961), Smith, T. On the Specification of Autocorrelation Errors. RSRI Discussion Paper No. 67. Philadelphia: RSFU, Unwin, D., and L. Hepple. The Statistical Analysis of Spatial Series. The Statistician, 23 (1974), Yeates, M. An Introduction to Quantitative Analysis in Human Geography. New York: Mdhw-Hill, 1973.