Chapter 5 Generalized Least Squares (GLS)

5.1 Exercise 1

This first exercise will consider data collected by Eric North, an urban forester who took my class the first or second year I taught it. Some of Eric’s masters research was aimed at predicting damage to sidewalks from boulevard trees planted between the curb and sidewalk (North, Johnson, & Burk, 2015). The degree of damage largely depends on how much the trunk of the tree spreads out (measured by “trunk flare” Figure 5.1 right), which can be predicted from measures of the diameter at breast height (Figure 5.1 left).

Measurements of truck flare diameter (right photo) and diameter at breast height (DBH, left photo).

Figure 5.1: Measurements of truck flare diameter (right photo) and diameter at breast height (DBH, left photo).

We will explore linear models relating trunk flare circumference (\(Y\)) to diameter at breast height (\(X\)). As noted by Hilbert et al. (2020) and North et al. (2015), there are several potential uses of this type of model, including:

  • determining planting space needed for trees based on their potential size
  • identifying areas where there might be current or future damage to infrastructure due to tree roots
  • estimating stump size for cost estimates associated with stump removal.

The data for this set of exercises is contained in the Data4Ecologists package and can be accessed using:

library(Data4Ecologists)
data(trunkfl)
  1. Subset the data to include only Acer saccharinum (Species = 4; sugar maple).
  2. Fit a linear model relating trunk flare (\(Y\)) to diameter at breast height (\(X\)) using lm. Evaluate the assumptions of the model using residual plots.
  3. Fit a model where the variance is assumed to increase with dbh using the varPower function.
  4. Describe the fitted GLS model using a set of equations. Match the parameters in your equations to the output generated using the summary function applied to your fitted GLS model.
  5. Create a plot of Pearson residuals, \(r_i = \frac{Y_i - \hat{Y}_i}{\sqrt{\widehat{var}(Y_i)}}\) versus \(\hat{Y}_i\) (you can use plot(modelname) to accomplish this task). Comment on whether you think the GLS model is appropriate.
  6. Use AIC to compare the fit of the standard linear model and the fit of the GLS model (use AIC(model1, model2), where model1 and model2 are the names you assign to the two fitted models). Before making this comparison, refit the linear model using gls (this will ensure both models are fit using “REML”, which will be discussed in a later chapter). Which model appears to give the better fit (note: lower AIC is “better”)?
  7. Inspect the estimated regression coefficients and their standard errors. Do they change much when you account for non-constant variance?

5.2 Exercise 2

This exercise was contributed by Anu Li, who took FW8051 in the spring 2024.

The data for this exercise come from a Le Sueur River Basin sediment characterization project carried out by then-graduate student at the University of Minnesota, Anna Baker, Professor Jacques Finlay, and others from 2015-2018. We will consider a subset of the data in this data repository (Baker et al., 2019). Specifically, we will focus on three variables contained in the data set:

  • Type = Type of sediment/erosional source
  • P = Sediment phosphorus concentration, in milligrams per kilogram of sieved fine sediment
  • SRP = Water extractable soluble reactive phosphorus concentration, in milligrams per kilogram of sieved fine sediment

The latter two variables were renamed (from P (mg/kg) <63 µm and Water Extractable SRP (mg/kg) <63µm, respectively) to make the data easier to read into R. In addition, cases with missing values for either phosphorus measurement have been dropped.

Soluble reactive phosphorus (SRP) is a biologically-available form of phosphorus that causes eutrophication when found in excess in water. Phosphorus can be measured using many different methods, some of which are costly, and none of the methods are perfect. Let’s see if we can predict SRP from phosphorus in the sediment, focusing on the fine particle fraction, defined as having a diameter less than 63 microns.

To complete this exercise, you will want to load the following packages:

library(nlme) # for gls function
library(ggplot2) # for plotting

You can access the data using:

library(Data4Ecologists)
data(SoilChem)

  1. Fit a linear regression model using P (x-axis) to predict SRP (y-axes). Plot the data with the regression line overlayed. Also, use the summary function to inspect the fitted model.

  2. Check the linear model assumptions graphically and evaluate whether they appear reasonable.

  3. Fit a model in which the variance is assumed to increase with phosphorus in the sediment, P, using the varPower function. Use the summary function to inspect the model.

  4. Write out equations that describe the fitted GLS model. Match parameters to the output from the summary function.

  5. Create a plot of the Pearson residuals versus fitted values using plot(modelname). Is the variance model appropriate for the data?

  6. Sediment and soil samples were collected from different points in the basin, such as bluffs, stream banks, and agricultural sites (captured in the variable, Type). Fit a power variance model for the same response and predictor variables, except this time allow the variance parameters to depend on each level of the categorical value Type.

  7. Write out equations for calculating the variance for Type = Ag and Type = SB, replacing the parameters with their estimates from the output of the summary function.

  8. Create a plot of the Pearson residuals versus fitted values for this model. Is the variance model appropriate for the data?

  9. Compare the AIC of all three models (linear, power variance, and power variance with Type) using AIC(model 1, model 2, model 3). To do this, you must recreate the linear model using gls instead of lm. Given a low AIC indicates a better fit, which model is best for these data?

  10. How do the regression coefficients change between the 3 models?

References

Baker, A., Finlay, J., Gran, K., Rorer, M., Belo, T., Atkins, W., … Katherine, K. (2019). Le sueur river basin sediment characterization-chemical and physical properties of sediments collected 2015-2018. Data Repository for the University of Minnesota (DRUM). doi:10.13020/zbm3-9d68
Hilbert, D. R., North, E. A., Hauer, R. J., Koeser, A. K., McLean, D. C., Northrop, R. J., … Parbs, S. (2020). Predicting trunk flare diameter to prevent tree damage to infrastructure. Urban Forestry & Urban Greening, 49, 126645.
North, E. A., Johnson, G. R., & Burk, T. E. (2015). Trunk flare diameter predictions as an infrastructure planning tool to reduce tree and sidewalk conflicts. Urban Forestry & Urban Greening, 14(1), 65–71.