Appendix E. Expanded methods for statistical analyses and spatial autocorrelation.

*Logistic Regression*

Because the dependent variable is binary and the independent variables are continuous, we used logistic regression to test the null hypothesis that tree establishment is randomly distributed and not influenced by local site conditions. We did this for the entire study site and the zonal analysis. Traditional forward and/or backward logistic regressions are susceptible to strongly correlated independent variables, which frequently occur in treeline environments. Using such techniques may provide unreliable rankings of the independent variables because highly significant variables may erroneously elevate the importance of other highly correlated variables (Mac Nally 2000, 2002). Thus, we opted to use the 'all.regs' and 'hier.part' functions in *R* 2.4.1 (2006) to perform logistic regression within the hierarchical partitioning framework for the multivariate analyses (Chevan and Sutherland 1991, Mac Nally 1996). With this approach, one can achieve incremental improvements of traditional regression models for a given variable by averaging its effects throughout every possible combination with the other variables (Mac Nally 1996). The hierarchical partitioning approach mitigates the effects of multicollinearity between the independent variables and provides higher confidence in the relative rankings of the independent variables.

The independent variables displayed substantial skewness (up to 4.8) and kurtosis (up to 8.2). Thus, we removed outliers and employed the use of several data transformation techniques (i.e. square root, logarithmic, cubic, and quadratic) to lower the skewness and kurtosis values to just above 0.5 or lower. We only applied logistic regression to the transformed data and used the 'rand.hp' function in *R* 2.4.1 (2006) to calculate Z-scores at the 95% and 99% confidence levels (Mac Nally 2002). We reported the independent contributions of the local site conditions because this is a reliable method of ranking them. However, these values do not include any joint contributions, which can significantly lower the overall variance explained by the model (Mac Nally 2002).

*Spatial autocorrelation*

The effects of spatial autocorrelation (i.e. spatial interdependence) may alter the ranked importance of independent variables and produce erroneous interpretations when utilizing multivariate statistical analyses in ecological research (González-Megías et al. 2005, Griffith and Peres-Neto 2006). We used *R* 2.4.1 (2006) to mitigate the effects of spatial autocorrelation by creating a spatially weighted landscape matrix for our sampled points with the modified principal coordinates of neighbour matrices approach discussed by Dray et al. (2006). The approach is based off eigenvectors and distance, producing positively (i.e., similar) and negatively (i.e. dissimilar) clustered neighbors with associated eigen values. Data-driven Akaike information criterion rankings selected the Delaunay triangulation for creating the spatially weighted matrix. The resultant continuous data corresponded well with the well-known Moran’s I measure of spatial autocorrelation and were included as an additional independent variable in the regression analyses (Griffith and Peres-Neto 2006).

LITERATURE CITATIONS

Chevan, A., and M. Sutherland. 1991. Hierarchical partitioning. American Statistician 45:90–96.

Dray, S., P. Ledendre, and P. R. Peres-Neto. 2006. Spatial modelling: a comprehensive framework for principal coordinate analysis of neighbour matrices (PCNM). Ecological Modelling 196:483–493.

González-Megías, A., J. M. Gómez, and F. Sánchez-Piñero. 2005. Consequences of spatial autocorrelation for the analysis of metapopulation dynamics. Ecology 86(12):3264–3271.

Griffith, D. A., and P. R. Peres-Neto. 2006. Spatial modeling in ecology: the flexibility of eigenfunction spatial analyses. Ecology 87:2603–2613.

Mac Nally, R. 1996. Hierarchical partitioning as an interpretative tool in multivariate inference. Australian Journal of Ecology21:224–228.

Mac Nally, R. 2000. Regression and model-building in conservation biology, biogeography and ecology: the distinction between - and reconciliation of - 'predictive' and 'explanatory' models. Biodiversity and Conservation 9:655–671.

Mac Nally, R. 2002. Multiple regression and inference in ecology and conservation biology: further comments on identifying important predictor variables. Biodiversity and Conservation 11:1397–1401.