Abstract
Spatial dependence or spatial autocorrelation often occurs in ecological data and can be a serious problem in analysis, affecting the significance rates of statistical tests, making them too liberal when the dependence is positive. Ecological phenomena often are patchy and give data with a wave structure, producing autocorrelation that cycles between positive and negative with increasing distance, further complicating the situation. This article describes the essentials of dealing with this problem as commonly encountered in analyzing ecological data for two variables. We investigated two related approaches to correcting statistical tests for data with spatial autocorrelation from one-dimensional sampling schemes like the transects used in plant ecology, the example of interest here. Both approaches estimate the “effective sample size” based on the observed autocorrelation structures of the variables. We examined tests of correlation and bivariate goodness-of-fit tests, as well as extensions beyond both of these test classes. The correction methods prove to be robust for a wide range of spatial autocorrelation structures in one-dimensional data and provide reliable corrections in most cases. They fail only when the data have strong and consistent waves that cause persistent cycles in the autocorrelation as a function of distance. By examining the spatial autocorrelation structure of the ecological data, we can predict the likelihood of successful correction for these bivariate tests.
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References
Alpargu, G., and Dutilleul, P. (2003), “To Be or Not To Be Valid in Testing the Significance of the Slope in Simple Quantitative Linear Models With Autocorrelated Errors,” Journal of Statistical Computation and Simulation, 73, 165–180.
— (2006), “Stepwise Regression in Mixed Quantitative Linear Models With Autocorrelated Errors,” Communications in Statistics—Simulation and Computation, 35, 79–104.
Armitage, P. (1955), “Tests for Linear Trends in Proportions and Frequencies,” Biometrics, 11, 375–386.
Bartlett, M. S. (1935), “Some Aspects of the Time-Correlation Problem in Regard to Tests of Significance,” Journal of the Royal Statistical Society, 98, 536–543.
Cerioli, A. (1997), “Modified Tests of Independence in 2×2 Tables With Spatial Data,” Biometrics, 53, 619–628.
— (2002), “Testing Mutual Independence Between Two Discrete-Valued Spatial Processes: A Correction to Pearson Chi-Squared,” Biometrics, 58, 888–897.
Cerioli, A. (2003), “The Cochran#x2014;Armitage Trend Test Under Spatial Autocorrelation,” in Proceedings of the Conference “Complex Models and Computational Methods for Estimation and Prediction,” Treviso, Italy.
Chatfield, C. (1975), The Analysis of Time Series: Theory and Practice, London: Chapman & Hall.
Cliff, A. D., and Ord, J. K. (1981), Spatial Processes: Models and Applications, London: Pion.
Clifford, P., and Richardson, S. (1985), “Testing the Association Between Two Spatial Processes,” Statistics and Decisions, Supp. 2, 155–160.
Clifford, P., Richardson, S., and Hémon, D. (1989), “Assessing the Significance of Correlation Between Two Spatial Processes,” Biometrics, 45, 123–134.
— (1993), “Response: Reader Reaction. Modifying the t Test for Assessing the Correlation Between Two Spatial Processes,” Biometrics, 49, 305–314.
Cochran, W. G. (1954), “Some Methods for Strengthening the Common χ2 Tests,” Biometrics, 10, 417–451.
Cressie, N. A. (1991), Statistics for Spatial Data, New York: Wiley.
Dale, M. R. T. (1999), Spatial Pattern Analysis in Plant Ecology, Cambridge: Cambridge University Press.
Dale, M. R. T., and Fortin, M.-J. (2002), “Spatial Autocorrelation and Statistical Tests in Ecology,” Écoscience, 9, 162–167.
Dale, M. R. T., Henry, G. H. R., and Young, C. (1993), “Markov Models of Spatial Dependence in Vegetation,” Coenoses, 8, 21–24.
Dutilleul, P. (1993), “Modifying the t Test for Assessing the Correlation Between Two Spatial Processes,” Biometrics, 49, 305–314.
Fortin, M.-J., and Dale, M. R. T. (2005), Spatial Analysis: A Guide for Ecologists, Cambridge: Cambridge University Press.
Franco, M., and Harper, J. (1988), “Competition and the Formation of Spatial Pattern in Spatial Gradients: An Example Using Kochia scoparia,” Journal of Ecology, 76, 959–974.
Getis, A., and Boots, B. (1978), Models of Spatial Processes, Cambridge: Cambridge University Press.
Haining, R. P. (1978), “The Moving Average Model for Spatial Interaction,” Transactions of the Institute of British Geographers, 3, 202–225.
Legendre, P., and Legendre, L. (1998), Numerical Ecology (2nd English ed.), Amsterdam: Elsevier.
Legendre, P., Dale, M. R. T., Fortin, M.-J., Gurevitch, J., Hohn, M., and Myers, D. (2002), “The Consequences of Spatial Structure for the Design and Analysis of Ecological Field Surveys,” Ecography, 25, 601–615.
Lomnicki, Z. A., and Zaremba, S. K. (1955), “Some Applications of Zero-One Processes,” Journal of the Royal Statistical Society, Ser. B, 17, 243–255.
Marriott, F. H. C., and Pope, J. A. (1954), “Bias in the Estimation of Autocorrelation,” Biometrika, 41, 390–402.
Ord, J. K. (1979), “Time Series and Spatial Patterns in Ecology,” in Spatial and Temporal Analysis in Ecology, eds. R. M. Cormack and J. K. Ord, Fairland, MD: International Co-Operative Publishing House.
Ripley, B. D. (1981), Spatial Statistics, New York: Wiley.
Tavaré, S. (1983), “Serial Dependence in Contingency Tables,” Journal of the Royal Statistical Society, Ser. B, 45, 100–106.
Tavaré, S., and Altham, P.M. E. (1983), “Serial Dependence of Observations Leading to Contingency Tables, and Corrections to Chi-Squared Statistics,” Biometrika, 70, 139–144.
Upton, G. J. G., and Fingleton, B. (1989), Spatial Data Analysis by Example, Vol. II. Categorical and Directional Data, New York: Wiley.
Whittle, P. (1954), “On Stationary Process in the Plane,” Biometrika, 41, 434–449.
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Dale, M.R.T., Fortin, MJ. Spatial autocorrelation and statistical tests: Some solutions. JABES 14, 188–206 (2009). https://doi.org/10.1198/jabes.2009.0012
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DOI: https://doi.org/10.1198/jabes.2009.0012