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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
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Regression
Geometry
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Regression
Geometry
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Regression
Geometry
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Regression
Geometry
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Regression
Geometry
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Regression
Geometry
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F1
Regression
Geometry
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F1
Regression
Geometry
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Sum of squares of residuals = Squared distance from Y to F1
F1
Regression
Geometry
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Regression
Geometry
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F1 F2F3x
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Regression
Geometry
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F1 F2F3x
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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.