Estimating Effects of Limiting Factors With Regression Quantiles

Abstract

In a recent Concepts paper in Ecology, Thomson et al. emphasized that assumptions of conventional correlation and regression analyses fundamentally conflict with the ecological concept of limiting factors, and they called for new statistical procedures to address this problem. The analytical issue is that unmeasured factors may be the active limiting constraint and may induce a pattern of unequal variation in the biological response variable through an interaction with the measured factors. Consequently, changes near the maxima, rather than at the center of response distributions, are better estimates of the effects expected when the observed factor is the active limiting constraint. Regression quantiles provide estimates for linear models fit to any part of a response distribution, including near the upper bounds, and require minimal assumptions about the form of the error distribution. Regression quantiles extend the concept of one-sample quantiles to the linear model by solving an optimization problem of minimizing an asymmetric function of absolute errors. Rank-score tests for regression quantiles provide tests of hypotheses and confidence intervals for parameters in linear models with heteroscedastic errors, conditions likely to occur in models of limiting ecological relations. We used selected regression quantiles (e.g., 5th, 10th, . . . , 95th) and confidence intervals to test hypotheses that parameters equal zero for estimated changes in average annual acorn biomass due to forest canopy cover of oak (Quercus spp.) and oak species diversity. Regression quantiles also were used to estimate changes in glacier lily (Erythronium grandiflorum) seedling numbers as a function of lily flower numbers, rockiness, and pocket gopher (Thomomys talpoides fossor) activity, data that motivated the query by Thomson et al. for new statistical procedures. Both example applications showed that effects of limiting factors estimated by changes in some upper regression quantile (e.g., 90-95th) were greater than if effects were estimated by changes in the means from standard linear model procedures. Estimating a range of regression quantiles (e.g., 5-95th) provides a comprehensive description of biological response patterns for exploratory and inferential analyses in observational studies of limiting factors, especially when sampling large spatial and temporal scales.

The Speed of Invasion: Rates of Spread for Thirteen Exotic Forest Insects and Diseases

“…Quantile regression can fit a model to portray the change in the 50th (median) or any other quantile of the distribution of distance as a function of time. While the 50th quantile provides a measure of the central trend of the data, the upper quantiles, i.e., the 90th or 95th, can be considered the unconstrained rate of spread [34,35].…”

Trait Mindfulness as a Limiting Factor for Residual Depressive Symptoms: An Explorative Study Using Quantile Regression

“…upper quantiles. This is because, if the predictor has an effect on the response variable, the highest values in the distribution of the response along the predictor gradient will be constrained by the predictor itself whereas all the remaining lower values, scattered underneath, will be limited by a combination of hidden factors [30], [31].…”

Analysis of Limiting Factors Across the Life Cycle of Delta Smelt (Hypomesus transpacificus)

“…The concept was derived from observations that applying more of a nutrient that was not limiting did not improve crop yields. The concept of limiting factors more recently has been applied to explain ecological phenomena (McNamara and Houston 1987; Klein 1991; Messier 1991; Thomson et al 1996; Cade et al 1999; Rettie and Messier 2000; Dunham et al 2002; Kaiser et al 1994). Recognition of the role of limiting factors in determining population responses in imperiled species is relevant for two reasons.…”

“…The presence and manifestation of limiting factors can be difficult to detect because they do not always control the response variable. When they do not control the observed response, those data points should not be given weight in a factor analysis (Kaiser et al 1994; Cade et al 1999); yet conventional additive approaches do just that (Augspurger 1996; Thomson et al 1996, and see Appendix A). …”