Using instrumental variables for assessing the overall bias of size-number relationships

Studies of relationships between number of trees (or other plants) per unit area and their average size have been haunted by doubts about their statistical validity. These doubts are based on the fact that both sides of the size-number equations share the same variable-number of trees. As a result, the error term may be correlated with the predicting variable which would bias the regression. Because of this bias, called artificial correlation, it has been recommended to avoid size-number relationships and use biomass-number relationships instead. This recommendation may work if the shared variable was the only source of bias. Actually, there are several other sources (error of independent variable and constrained variability), which affect both types of relationships. To estimate their overall influence, the variables (stand biomass, average diameter, and number of trees) were smoothed using age of the investigated stands. Using these smoothed variables, it was found that the overall bias of biomass-number relationships was about six times larger and of the opposite sign than that of size-number relationships. This application of the two-stage least squares (2SLS) suggests that we can relate size and number of trees without transgressing any statistical assumptions. © 2007 by The Haworth Press, Inc. All rights reserved.