Figure 1.
Probability density functions for beta distributions.
Probability density functions for beta distributions with (left) and
(right).
Figure 2.
Distribution of lakes that were sampled for the 2007 U.S. National Lakes Assessment.
Figure 3.
Normal quantile-quantile plots.
Normal quantile-quantile plots of arcsine-square-root-transformed (“arcsine”), logit-transformed (“logit”) and untransformed (“raw lm”) EPHEptax values (panels (a) - (c)). Panel (d) shows a beta quantile-quantile plot using the untransformed EPHEptax values. It is seen that EPHEptax is best approximated by a beta distributed random variable.
Figure 4.
Analysis of the NLA Data. The figure contains boxplots of values obtained from the 100 bootstrap samples (left panel) and from the 100 sets of out-of-bootstrap observations (right panel).
Figure 5.
Number of selected predictor variables.
Analysis of the NLA Data. The two panels contain the number of selected predictor variables (averaged over 100 bootstrap samples) for various modeling approaches. Dark grey bars represent linear effects, light grey bars represent non-linear effects. In case of beta regression with fixed precision parameter (“beta fix”), the precision model contains only one predictor (namely, the intercept).
Figure 6.
Function estimates for the proportion of developed land in catchment.
Analysis of the NLA Data. The five panels contain the function estimates for the proportion of developed land in catchment (computed from 100 bootstrap samples). In case of beta regression, estimates present the effects of the proportion of developed land in catchment on the mean parameter . Black lines correspond to the mean and the 0.05 and 0.95 quantiles of the function estimates. For reasons of interpretability, the range of the x-axes was restricted to the lower
of the sample values.
Figure 7.
Function estimates for the site elevation.
Analysis of the NLA Data. The five panels contain the function estimates for the site elevation (computed from 100 bootstrap samples). In case of beta regression, estimates present the effects of the site elevation on the mean parameter . Black lines correspond to the mean and the 0.05 and 0.95 quantiles of the function estimates. For reasons of interpretability, the range of the x-axes was restricted to the lower
of the sample values.
Figure 8.
Function estimates for the chlorophyll- a concentration.
Analysis of the NLA Data. The five panels contain the function estimates for the chlorophyll- a concentration (computed from 100 bootstrap samples). In case of beta regression, estimates present the effects of the chlorophyll- a concentration on the mean parameter . Black lines correspond to the mean and the 0.05 and 0.95 quantiles of the function estimates. For reasons of interpretability, the range of the x-axes was restricted to the lower
of the sample values.
Figure 9.
Function estimates for the total nitrogen concentration.
Analysis of the NLA Data. The five panels contain the function estimates for the total nitrogen concentration (computed from 100 bootstrap samples). In case of beta regression, estimates present the effects of the total nitrogen concentration on the mean parameter . Black lines correspond to the mean and the 0.05 and 0.95 quantiles of the function estimates. For reasons of interpretability, the range of the x-axes was restricted to the lower
of the sample values.
Figure 10.
Function estimates for mean site depth.
Analysis of the NLA Data. The five panels contain the function estimates for the mean depth at the sites (computed from 100 bootstrap samples). In case of beta regression, estimates present the effects of the depth on the mean parameter . Black lines correspond to the mean and the 0.05 and 0.95 quantiles of the function estimates. For reasons of interpretability, the range of the x-axes was restricted to the lower
of the sample values.
Figure 11.
Effects of basin-climate regions.
Estimated effects of basin-climate regions on EPHEptax, as obtained from applying boosted beta regression to 100 bootstrap samples of the NLA data. The figure presents median effect estimates for the mean parameter computed from the 100 model fits (LM = Lower Missouri).
Figure 12.
Estimated spatial surface function.
Estimated spatial surface function for the mean parameter
in boosted beta regression. The figure presents the median spatial surface obtained from the 100 bootstrap samples.
Figure 13.
Effect of chlorophyll- a concentration on the precision parameter.
Analysis of the NLA Data. The figure contains the estimated effect of the chlorophyll- a concentration on the logarithm of the precision parameter (computed from 100 bootstrap samples). Black lines correspond to the mean and the 0.05 and 0.95 quantiles of the function estimates. For reasons of interpretability, the range of the x-axis was restricted to the lower
of the sample values.