Feb 1 2006

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Lecture 10

• Feb 1 2006

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Added-variable

• Added variable plots give you a visual sense of
  whether x2 is a useful addition to the model
• E(yx1) a b x1

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Steps to making one

• Regress y on x1
• Compute residuals of y on x1 y "-"x1 (remove x1
  from y)
• Regress x2 on x1
• Compute residuals of x1 on x2 x2 "-"x1
• (remove x1 from x2)
• Plot y"-"x1 vs. x2 "-"x1

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Interpret

• If there is a "significant" slope, then x2 is
  useful.
• Slope of the added variable plot is the same as
  the coefficient if you fit
• E(yx1, x2) B0 B1 x1 B2 x2

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Significance Tests

• Find the slope of an added variable plot, and do
  a t-test to see if the slope is significant. The
  value of the t-stat is almost the same as the
  value of the t-stat for the "full" model.
• The p-values will differ because the degrees of
  freedom are different n-2 for added-variable
  slope, n-3 for full model

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Mussel Beds
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Is density related to food levels?
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Is density related to human use?
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If we know the human use level, do we need to
know food leve?
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Added-Variable
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Summary of added variable plot

• slope 10.87
• additional amount of food worth 10.87 mm of
  thickness, on avg., controlling for human use
• t 4.742, p1.45e-05

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Summary of lm(thicknessfoodhuman.use)

• E(thicknessfood, human)
• 6318.87food - 6.294 human.use
• t_food 4.70, p 1.84e-05

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Testing one variable Plan 1

• H0 Beta1 0, Beta0, Beta2, Beta3, etc.
  "arbitrary"
• Ha Beta1 ltgt 0, others arbitrary
• Fit full model y B0B1x1B2x2B3x3 etc.
• Fit reduced modely B0 B2x2B3x3
• Compare RSS

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partial F-test

• Compare RSS (full) with RSS (reduced)
• Note RSS(reduced) will be ???? than RSS(full)
• F ((RSS(red) - RSS(full))/1)/RSS(full)/(n-p)
• Note The denominator is just ????

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Formula for partial F

• (Reduction in RSS)/DFnum divided byRSS/DFden
• DF of a model is n - of parameters estimated.
• DFnum is DF(full)-DF(reduced)
• DFden is n - p

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In R (long)

• full lt- lm(y x1 x2 x3 x4)
• red lt- lm(yx2 x3 x4)
• anova(full) anova(red)
• compute by hand from output

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R(short)

• full lt- lm(y x1x2x3)
• anova(full)
• Read output

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Plan 2

• t2 F
• full lt- lm(yx1x2x3)
• summary(full)
• look at t-statistic