Comparison of Methods to Estimate Reineke's Maximum Size-Density Relationship Species Boundary Line Slope

Abstract

Abstract Maximum size-density relationships (MSDR) provide natural resource managers useful information about the relationship between tree density and average tree size. Obtaining a valid estimate of how maximum tree density changes with changes in average tree size is necessary to describe these biological relationships accurately. This article examines three methods to estimate the slope (b) of the MSDR species boundary line across a range of planting densities: ordinary least-squares (OLS), first-difference model, and the linear mixed-effects model. For this article, stability refers to the extent to which parameter estimates do not change when the range of planting densities in the fitting data set changes. When using data from a planting density trial consisting of planting densities ranging from 6,727 to 747 seedlings per hectare, mixed-effect models produced the most stable estimates of b while OLS resulted in the least stable estimates. MSDR boundaries have been defined as either (1) those that describe the boundaries of individual stands (MSDR dynamic thinning line) or (2) those that describe MSDRs common across all sites for a particular species in a certain geographical region (MSDR species boundary line). A further refinement of the MSDR species boundary line is proposed by defining two MSDR species boundary lines, labeled here as I and II. Although both MSDR species boundary lines are positioned above all observations, one MSDR species boundary line, II, has a slope that can be considered the population average of all MSDR dynamic thinning lines; the other species boundary line (I) has a slope that results from positioning the boundary above all observations without accounting for self-thinning patterns of individual stands. A mixed-effects analysis was used to estimate the slope of MSDR species boundary line II.