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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

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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?

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Scatterplot of tree volume vs height

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Minitab commands

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Regression Output

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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

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

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Regression output

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Outliers

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Residual plots

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Confidence and prediction intervals

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

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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?

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BPS - 5th Ed. Chapter 231 Inference for Regression.

BPS - 5th Ed. Chapter 231 Inference for Regression.

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