Evidence of Tree Species’ Range Shifts in a Complex Landscape

Climate change is expected to change the distribution of species. For long-lived, sessile species such as trees, tracking the warming climate depends on seedling colonization of newly favorable areas. We compare the distribution of seedlings and mature trees for all but the rarest tree species in California, Oregon and Washington, United States of America, a large, environmentally diverse region. Across 46 species, the mean annual temperature of the range of seedlings was 0.120°C colder than that of the range of trees (95% confidence interval from 0.096 to 0.144°C). The extremes of the seedling distributions also shifted towards colder temperature than those of mature trees, but the change was less pronounced. Although the mean elevation and mean latitude of the range of seedlings was higher than and north of those of the range of mature trees, elevational and latitudinal shifts run in opposite directions for the majority of the species, reflecting the lack of a direct biological relationship between species’ distributions and those variables. The broad scale, environmental diversity and variety of disturbance regimes and land uses of the study area, the large number and exhaustive sampling of tree species, and the direct causal relationship between the temperature response and a warming climate, provide strong evidence to attribute the observed shifts to climate change.


S2 Appendix. Details of the statistical analysis
Statistical analysis follows standard survey sampling procedures [1], but from a continuous population perspective [2]. For each species, an approximate design unbiased estimator of the mean elevation, latitude, and annual temperature of the range of seedlings or mature trees (ˆk d  ) is given by the weighted domain sample mean (after [1], section 5.8):       A are the land areas of California, Oregon, and Washington, respectively, and the denominator, proper probability density function. Then, for a sample of size n, the inclusion density function is The estimator in eq. (1) reduces to a weighted average of the latitude, elevation or temperature of the plots that contain either seedlings or trees of the species of interest. The weight is 1 for plots in California and Oregon, and 10/9 for plots in Washington, thus accounting for the different sampling intensity. In this analysis, nonresponse is treated as if it were missing at random within the range of the species ( [4], p. 41). However, the bias introduced by non-response is likely to be negligible. The non-response rate is relatively low. More importantly, the parameter of interest is the difference between the means of the range of seedlings and mature trees. A bias in the estimator of the difference would require a different non-response process affecting the range of seedlings and trees, which is unlikely given the large overlap between those ranges.
An approximate estimator of the variance of this ratio estimator is ([1], eq. 5.6.10): These estimators and the associated confidence intervals are obtained by treating the sample as if it had been selected using independent random sampling, instead of a spatially balanced design.
Because a balanced design is more efficient than an independent sampling design in the presence of spatial correlation, those variance estimators would tend to be conservative and overstate the sampling variance [5].
For each species, we estimated the difference in the mean elevation, latitude or annual temperature for the range of seedlings minus that for the range of mature trees as the difference between their respective domain ratio estimators: The approximate variance of ˆk  , using a Taylor linearization method ( [6], eq. 6.9.1), is: We estimated a 95% confidence interval as: where 0.975 z is the 97.5 percentile of the normal distribution.
For each species, we estimated the 5th and 95th percentiles of the distribution of temperature for seedlings and mature trees. First, we estimated the population distribution function ( [7], p.69), and then calculated its inverse evaluated at 0.05 and 0.95. We computed the difference between the estimator for seedlings minus that for trees and obtained 95% confidence intervals by the percentile bootstrap method ( [8], Chapter 6).

Estimation of mean differences across all species
The variance of the estimator for the individual species effect differed widely among species, mostly due to large differences in sample sizes. Further, because the estimators were calculated from the same sample, they are correlated. To deal with the unequal variance and lack of independence, we estimated the mean difference across all species using a generalized least squares estimator: where δ is a vector of the individual species differences, ˆk  , c is a vector of 1s of the same length as δ (46) and Σ is the covariance matrix of δ. The variance of this estimator is: We estimated the variance Σ from the sample. An analytical expression for Σ would be cumbersome and difficult to obtain, so we used the bootstrap (8, Chapter 6). We took a sample with replacement from the 33,674 plots and computed δ (eq. (7)) and the estimates of the difference in the 5th and 95th percentiles. We repeated this process 20,000 times and calculated the covariance matrices of the bootstrap replications.