Ecological Archives E086-095-A1

Camilo Mora and D. Ross Robertson. 2005. Causes of latitudinal gradients in species richness: a test with fishes of the tropical eastern Pacific. Ecology 86:1771–1782.

Appendix A. A description of the methods used to account for spatial autocorrelation.

We used two methods that accommodate spatial autocorrelation in data: Trend Surface Analysis and Spatial Regression Models (both of these methods are described in detail by Legendre and Legendre [1998], Kaluzny et al. [1998], Lichstein et al. [2002]).

In brief, Trend Surface Analysis is an ordinary regression model including the spatial structure of the data (i.e., the spatial trend) as variable. This trend is quantified with a third order polynomial of the latitudinal coordinates of the bands. Specifically, the trend is quantified as:

Y = X + X2 + X3

where Y is the number of species and X the latitudinal position of the bands. Because in the Tropical Eastern Pacific most latitudinal bands are approximately at the same longitudinal band, longitude of the bands was omitted in the model we used. No parameter exists for the intercept because, prior to the analysis, X was centered on its mean (by subtracting the mean latitude from the value of each latitudinal band). This was done to reduce collinearity between first and higher order terms (Legendre and Legendre 1998). Monomials that were non-significant were removed.

With the addition of the trend we can assess specific relationships between diversity patterns and predictors in terms of their spatial structure. To exemplify, imagine the area enclosed by the square in Fig. A1 as the variance in the dependent variable, the area enclosed by the circle (A) as the variance explained by the predictor, (B) as the variance explained by the trend and (a+b+c) as the variance explained by the trend and the predictor jointly (These variances are obtained from R2-values of regression models containing the predictor alone, the trend alone and the trend and the predictor combined). The different components of the Trend Surface Analysis are calculated by subtraction of those R2-values. Specifically, the component

Non-spatial environmental (a in Fig. A1) = R2(predictor and trend combined) R2(trend alone).
 
Spatially-structured environmental (b in Fig. A1) = R2(predictor r alone) + R2(trend alone)R2(predictor and trend combined).
 
Non-environmental spatial (c in Fig. A1) = R2(predictor and trend combined)R2(predictor  alone).
 
Unexplained variation (d in Fig. A1) = 1 – R2(predictor and trend combined).

The component (a) refers to the variance that could be accounted by the predictor independently of any spatial signal (e.g., spatial autocorrelation in the response), (b) refers to the spatial signal contained in the variation explained by the predictor, (c) refers to the spatial signal in the response, and (d) is the variance that is not explained by the predictor nor the spatial signal.

In addition we performed Spatial Regression analyses using Conditional Autoregressive models (Kaluzni et al. 1998). This tool accounts for autocorrelation at fine spatial scales, which often is not the case with Trend Surface Analysis (Lichstein et al. 2002).  For the spatial regressions, we defined as the spatial neighbors the bands immediately above and below each latitudinal band, with only one neighbor for the southernmost and northernmost bands.  We used Splus Spatial Stats software for these analyses (Kaluzni et al. 1998).

 
   FIG. A1. Partitioning of the response variable into the four components of Trends Surface Analysis. The square represents 100% of the variance in the response. (A) is the variation explained by the predictor and (B) the variation explained by the trend independently. (a+b+c) is the variance explained by the predictor and the trend combined. The names of the different components are: (a) the nonspatial environmental, (b) the spatially structured environmental, (c) the non-environmental spatial, and (d) the unexplained variance.

 

LITERATURE CITED

Kaluzny, S. P., S. C. Vega, T. P. Cardoso, and A. A. Shelly. 1998. S+ Spatial stats user’s manual for Windows and Unix. Springer-Verlag, New York, New York, USA.

Legendre, P., and P. Legendre. 1998. Numerical ecology. Second English edition. Elsevier Science, Amsterdam, The Netherlands.

Lichstein, J. W., T. R. Simons, S. A. Shriner, and K. E. Franzreb. 2002. Spatial autocorrelation and autoregressive models in ecology. Ecological Monographs 72:445–463.



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