Simple linear models Straight line is simplest case, but key is that parameters appear linearly in the model Needs estimates of the model parameters (slope and intercept)- usually by least squares Makes a number of assumptions, usually checked graphically using residuals

Examples for linear regression How is LOI related to moisture? How should we estimate merchantable volume of wood from the height of a living tree? How is pest infestation late in the season affected by the concentration of insecticide applied early in the season?

Scatterplot of tree volume vs height

Minitab commands

Regression Output

Interpreting the output Goodness of fit (R-squared) and ANOVA table p-value? Confidence intervals and tests for the parameters Assessing assumptions (outliers and influential observations Residual plots

t = distance between estimate and hypothesised value, in units of standard error vs Confidence intervals and t-tests

Regression output

Outliers

Residual plots

Confidence and prediction intervals

Low R-sq High R-sq Low p-value: significant High p-value: non-significant Four possible outcomes

Not because relationships are linear Transformations can often help linearise Good simple starting point – results are well understood Approximation to a smoothly varying curve Why linear?