Quantitative Methods Regression

Examples for linear regression Do more brightly coloured birds have more parasites? 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?

Regression Similarities to analysis of variance

x y M Y Regression Geometry

x y M Y Regression Geometry

x y M Y Regression Geometry

x y M Y Regression Geometry

x y M Y Regression Geometry

x y M Y Regression Geometry

x y M Y F1F1 Regression Geometry

x y M Y F1F1 Regression Geometry

x y M Y Sum of squares of residuals = Squared distance from Y to F 1 F1F1 Regression Geometry

x y M Y Regression Geometry

M Y F1F1 F2F2 F3F3 x y Regression Geometry

M Y F1F1 F2F2 F3F3 x y Regression Geometry

Regression Geometry

Regression Geometry

Regression Minitab commands

Regression Minitab commands

Regression Minitab commands

Regression Minitab commands Minitab Supplement is in a PDF file in the same directory as the dataset.

Regression Regression Output

Regression Regression Output

Regression Regression Output

Regression Confidence intervals and t-tests

Regression estimate ± t crit  Standard Error of estimate Coef ± t crit (on 29 DF)  SECoef ±  = (0.758, 2.328) Confidence intervals and t-tests t crit is always on Error degrees of freedom

Regression Confidence intervals and t-tests

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

Regression Confidence intervals and t-tests

Regression Confidence intervals and t-tests

Regression Regression output

Regression Regression output

Regression Extreme residuals

Regression Outliers

Regression Regression output

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

Regression Difference from analysis of variance Continuous vs Categorical Continuously varying Values have meaning as numbers Values are ordered Interpolation makes sense Examples: –height –concentration –duration Discrete values Values are just “names” that define subsets Values are unordered Interpolation is meaningless Examples –drug –breed of sheep –sex

Regression Not because relationships are linear Good simple starting point - cf recipes Approximation to a smoothly varying curve Why linear?

Regression Last words… Regression is a powerful and simple tool, very commonly used in biology Regression and ANOVA have deep similarities Learn the numerical skills of calculating confidence intervals and testing for non-zero slopes.

Regression Last words… Next week: Models, parameters and GLMs Read Chapter 3 Regression is a powerful and simple tool, very commonly used in biology Regression and ANOVA have deep similarities Learn the numerical skills of calculating confidence intervals and testing for non-zero slopes.