Quantitative Methods

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Quantitative Methods

• Regression

2
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?

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Regression
Similarities to analysis of variance
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Regression
Geometry
y
Y
M
x
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Regression
Geometry
y
Y
M
x
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Regression
Geometry
y
Y
M
x
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Regression
Geometry
y
Y
M
x
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Regression
Geometry
y
Y
M
x
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Regression
Geometry
y
Y
M
x
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Regression
Geometry
y
Y
M
F1
x
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Regression
Geometry
y
Y
M
F1
x
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Regression
Geometry
y
Y
M
F1
x
Sum of squares of residuals Squared distance
from Y to F1
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Regression
Geometry
y
Y
M
x
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Regression
Geometry
y
Y
M
F1
F2
F3
x
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Regression
Geometry
y
Y
M
F1
F2
F3
x
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Regression
Geometry
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Regression
Geometry
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Regression
Minitab commands
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Regression
Minitab commands
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Regression
Minitab commands
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Regression
Minitab commands
Minitab Supplement is in a PDF file in the same
directory as the dataset.
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Regression
Regression Output
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Regression
Regression Output
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Regression
Regression Output
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Regression
Confidence intervals and t-tests
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Regression
Confidence intervals and t-tests
estimate tcrit ? Standard Error of estimate
Coef tcrit (on 29 DF) ? SECoef
1.5433 2.0452 ? 0.3839 (0.758, 2.328)
tcrit is always on Error degrees of freedom
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Regression
Confidence intervals and t-tests
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Regression
Confidence intervals and t-tests
t distance between estimate and hypothesised
value, in units of standard error
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Regression
Confidence intervals and t-tests
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Regression
Confidence intervals and t-tests
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Regression
Regression output
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Regression
Regression output
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Regression
Extreme residuals
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Regression
Outliers
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Regression
Regression output
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Regression
Four possible outcomes
Low p-value significant
High p-value non-significant
Low R-sq
High R-sq
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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

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Regression
Why linear?

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

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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.

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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.

Next week Models, parameters and GLMs Read
Chapter 3