Jonathan D. Bakker and Margaret M. Moore. 2007. Controls on vegetation structure in southwestern ponderosa pine forests, 1941 and 2004. Ecology 88:2305–2319.

Appendix A. Summary of meta-analytical techniques applied in this paper.

Here, we provide a detailed description of the meta-analytical techniques used in this paper. We focus first on the calculation of grazing effects, and then on the modifications required for calculation of temporal effects. We follow the notation of Gurevitch and Hedges (2001). Formulae not provided here are reported in Hedges and Olkin (1985), Gurevitch and Hedges (2001), and Lipsey and Wilson (2001).

Grazing Effects

For each variable, the grazing effect size (d_ij) of site j in class i (1941 or 2004) was calculated as:

(A.1)

where and are the mean values inside and outside the exclosure, s_ij is the pooled standard deviation of the two groups, and J is a correction factor for small sample sizes. Positive and negative grazing effect sizes indicate larger responses inside and outside of exclosures, respectively. Effect sizes are in standard deviation units and are commonly interpreted as follows: 0.2 is small, 0.5 is medium, 0.8 is large, and >1 is very large (Gurevitch and Hedges 2001).

Effect sizes from the five sites were combined using a mixed effects model, which assumes random variation in effect size among and within sites (Becker 1988, Gurevitch and Hedges 2001). Grazing effect sizes were weighted by the inverse of their variance (w*_ij ; combination of sampling and effect size variances), thus weighting intensely sampled sites more heavily than less intensely sampled sites. The formulae for the cumulated grazing effect size (d*_i+ ) and its standard deviation (s*_i+) are:

	(A.2)
	(A.3)

where k is the number of sites in class i (Gurevitch and Hedges 2001). The cumulated grazing effect size was assessed for significance by dividing it by its standard deviation to form a z-statistic, which was then compared to a Z-distribution (Lipsey and Wilson 2001).

Effect sizes were converted into correlation coefficients (r_ij) using the formula:

(A.4)

where N is the total sample size, and and are the sample sizes of each grazing treatment. The correlation coefficient for the grazing effect size at each site was converted to a z-score (z_ij), and the weighted average z-score (z_i+) was calculated as the sum of the product of each z_ij and its weight ():

	(A.5)
	(A.6)
	(A.7)

The common correlation coefficient across sites was estimated by converting z_i+ back to a correlation coefficient (r_i+):

(A.8)

Temporal Dynamics

The meta-analytic techniques described above were also used to calculate temporal effect sizes, except that: i ) mean values inside and outside exclosures were replaced with mean values in 1941 and 2004, and ii) sampling variances were calculated using a formula that accounted for the correlation between 1941 and 2004 data (Becker 1988, Lipsey and Wilson 2001). All else being equal, a site with larger correlation between data, either positively or negatively, will have a smaller variance and therefore a larger weight. Negative temporal effect sizes indicate a decline in the response variable between 1941 and 2004 and positive values indicate an increase between 1941 and 2004. Cumulated temporal effect sizes and standard deviations were calculated for each grazing treatment using formulae (2) and (3).

Cumulated temporal effect sizes from the two grazing treatments were tested for equality using the homogeneity statistic . Between-class homogeneity () was calculated as:

(A.9)

where w*_ij is the weight of the jth site in the ith class (grazing treatment), d*_i+ is defined in formula (2), and d*₊₊ is the grand cumulated temporal effect size across both grazing treatments. is distributed as a -statistic with one degree of freedom, since there are two classes being compared.

LITERATURE CITED

Becker, B. J. 1988. Synthesizing standardized mean-change measures. British Journal of Mathematical and Statistical Psychology 41:257–278.

Gurevitch, J., and L. V. Hedges. 2001. Meta-analysis: combining the results of independent experiments. Pages 347–369 in S. M. Scheiner and J. Gurevitch, editors. Design and analysis of ecological experiments. Oxford University Press, New York, New York, USA.

Hedges, L. V., and I. Olkin. 1985. Statistical methods for meta-analysis. Academic Press, Orlando, Florida, USA.

Lipsey, M. W., and D. B. Wilson. 2001. Practical meta-analysis. SAGE Publications, Thousand Oaks, California, USA.