Quantitative Methods

Regression

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

F1

Regression

Geometry

x

y

M

Y

F1

Regression

Geometry

x

y

M

Y

Sum of squares of residuals = Squared distance from Y to F1

F1

Regression

Geometry

x

y

M

Y

Regression

Geometry

M

Y

F1 F2F3x

y

Regression

Geometry

M

Y

F1 F2F3x

y

Regression

Geometry

Regression

Geometry

Regression

x F0 F1 F2 F314.9 32.5 32.0 31.6 31.014.3 32.5 31.8 31.1 30.219.3 32.5 33.8 35.1 36.814.7 32.5 31.9 31.4 30.712.4 32.5 31.1 29.6 27.814.7 32.5 31.9 31.4 30.712.5 32.5 31.1 29.7 27.917.9 32.5 33.2 34.0 34.916.5 32.5 32.7 32.9 33.118.6 32.5 33.5 34.5 35.817.5 32.5 33.1 33.6 34.419.5 32.5 33.9 35.3 37.017.5 32.5 33.1 33.6 34.314.4 32.5 31.8 31.1 30.3

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 ± tcrit Standard Error of estimate

Coef ± tcrit (on 29 DF) SECoef

1.5433 ± 2.0452 0.3839 = (0.758, 2.328)

Confidence intervals and t-tests

tcrit 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

t=Coef−βSECoef

vs tcrit

CI =Coef±tcrit×SECoef

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.