Model allows us to estimate response distribution at any X
Bivariate data sets contain response measurements corresponding to a few specific values of X, whereas a normal linear model provides a response distribution for all X. By fitting a normal linear model to the data, it is therefore possible to estimate the response distribution at x-values for which we do not have data.
Tree diameter and timber volume
The value of hardwood trees depends on the volume of timber that can be obtained when the trees are harvested. However the volume of timber cannot be easily measured when the tree is standing, so volume is usually estimated from measurements that are easier to make, such as the tree diameter 4.5 feet above ground level. The diagram below plots the cross-sectional area at this height against the volume of timber for 31 black cherry trees that were harvested in the Allegheny National Forest in Pennsylvania.
The relationship seems reasonably linear, so we will try to fit a normal linear model to the data. The least squares line is shown in the diagram below with the grey band representing ± twice the estimate of σ.
Drag the slider to display the estimated normal distribution of the volume of timber obtained from a tree of any cross-sectional area.
The mean of this estimated distribution (i.e. the least squares line) provides a prediction of the volume of timber for a different black cherry tree of any cross-sectional area.