Quantitative Methods

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Quantitative Methods

• Using more thanone explanatory variable

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Using more than one explanatory variable
Why use more than one?

• Intervening or 3rd variables (schoolchildrens
  maths)
• Reducing error variation (saplings)
• There is more than one interesting predictor
  (trees)

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Using more than one explanatory variable
Statistical elimination
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Using more than one explanatory variable
Statistical elimination
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Using more than one explanatory variable
Statistical elimination
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Using more than one explanatory variable
Statistical elimination
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Using more than one explanatory variable
Statistical elimination
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Using more than one explanatory variable
Sequential and Adjusted Sums of Squares
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Using more than one explanatory variable
Sequential and Adjusted Sums of Squares
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Using more than one explanatory variable
Sequential and Adjusted Sums of Squares
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Using more than one explanatory variable
Sequential and Adjusted Sums of Squares
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Using more than one explanatory variable
Why use more than one?

• Intervening or 3rd variables (schoolchildrens
  maths)
• Reducing error variation (saplings)
• There is more than one interesting predictor
  (trees)

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Using more than one explanatory variable
Sequential and Adjusted Sums of Squares
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Using more than one explanatory variable
Sequential and Adjusted Sums of Squares
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Using more than one explanatory variable
Why use more than one?

• Intervening or 3rd variables (schoolchildrens
  maths)
• Reducing error variation (saplings)
• There is more than one interesting predictor
  (trees)

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Using more than one explanatory variable
Sequential and Adjusted Sums of Squares
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Using more than one explanatory variable
Sequential and Adjusted Sums of Squares
MTB gt glm lvollhgt SUBCgt covar lhgt. Source
DF Seq SS Adj SS Adj MS F
P LHGT 1 3.5042 3.5042 3.5042
21.14 0.000 Error 29 4.8080 4.8080
0.1658 Total 30 8.3122 MTB gt glm
lvollhgtldiam SUBCgt covar lhgt ldiam. Source
DF Seq SS Adj SS Adj MS F
P LHGT 1 3.5042 0.1987
0.1987 30.14 0.000 LDIAM 1 4.6234
4.6234 4.6234 701.33 0.000 Error 28
0.1846 0.1846 0.0066 Total 30
8.3122
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Using more than one explanatory variable
Models and parameters
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Using more than one explanatory variable
Models and parameters
Y ? ?
Unknown quantities we would like to know, in
Greek Known quantities that are estimates of
them, in Latin
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Using more than one explanatory variable
Models and parameters
Y ? ?
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Using more than one explanatory variable
Models and parameters
MTB gt glm lvolldiamlhgt SUBCgt covar ldiam
lhgt. Analysis of Variance for LVOL, using
Adjusted SS for Tests Source DF Seq SS
Adj SS Adj MS F P LDIAM 1
7.9289 4.6234 4.6234 701.33
0.000 LHGT 1 0.1987 0.1987
0.1987 30.14 0.000 Error 28 0.1846
0.1846 0.0066 Total 30 8.3122
Term Coef SE Coef T
P Constant -6.6467 0.7983 -8.33
0.000 LDIAM 1.98306 0.07488 26.48
0.000 LHGT 1.1203 0.2041 5.49
0.000
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Using more than one explanatory variable
Models and parameters
MTB gt glm lvolldiamlhgt SUBCgt covar ldiam
lhgt. Analysis of Variance for LVOL, using
Adjusted SS for Tests Source DF Seq SS
Adj SS Adj MS F P LDIAM 1
7.9289 4.6234 4.6234 701.33
0.000 LHGT 1 0.1987 0.1987
0.1987 30.14 0.000 Error 28 0.1846
0.1846 0.0066 Total 30 8.3122
Term Coef SE Coef T
P Constant -6.6467 0.7983 -8.33
0.000 LDIAM 1.98306 0.07488 26.48
0.000 LHGT 1.1203 0.2041 5.49 0.000
Fitted LVOL -6.6467 1.98306LDIAM
1.1203LHGT
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Using more than one explanatory variable
Models and parameters
Model
Model Formula
lvolldiamlhgt
Best Fit Equation
Fitted LVOL -6.6467 1.98306LDIAM
1.1203LHGT
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Using more than one explanatory variable
Models and parameters
MTB gt glm lvolldiam SUBCgt covariate
ldiam. Analysis of Variance for LVOL Source
DF Seq SS Adj SS Adj MS F P LDIAM
1 7.9254 7.9254 7.9254 599.72 0.000 Error 29
0.3832 0.3832 0.0132 Total 30 8.3087
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Using more than one explanatory variable
Models and parameters
MTB gt glm lvolldiam SUBCgt covariate
ldiam. Analysis of Variance for LVOL Source
DF Seq SS Adj SS Adj MS F P LDIAM
1 7.9254 7.9254 7.9254 599.72 0.000 Error 29
0.3832 0.3832 0.0132 Total 30 8.3087
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Using more than one explanatory variable
Models and parameters
Source DF Seq SS Adj SS Adj MS F
P LDIAM 1 7.9254 7.9254 7.9254 599.72
0.000 Error 29 0.3832 0.3832
0.0132 Total 30 8.3087
Source DF Seq SS Adj SS Adj MS F
P LDIAM 1 7.9254 4.6275 4.6275 698.63
0.000 LHEIGHT 1 0.1978 0.1978 0.1978
29.86 0.000 Error 28 0.1855 0.1855
0.0066 Total 30 8.3087
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Using more than one explanatory variable
Geometry in 3-D
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Using more than one explanatory variable
Geometry in 3-D
Source DF Seq SS Adj SS Adj MS
F P LHGT 1 3.5042 0.1987
0.1987 30.14 0.000 LDIAM 1 4.6234
4.6234 4.6234 701.33 0.000 Error 28
0.1846 0.1846 0.0066 Total 30
8.3122 Source DF Seq SS Adj SS
Adj MS F P LDIAM 1 7.9289
4.6234 4.6234 701.33 0.000 LHGT 1
0.1987 0.1987 0.1987 30.14
0.000 Error 28 0.1846 0.1846
0.0066 Total 30 8.3122
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Using more than one explanatory variable
Geometry in 3-D
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Using more than one explanatory variable
Geometry in 1-D
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Using more than one explanatory variable
Last words

• Two or more x-variables are often useful and
  often necessary, and are easy to fit
• Two variables may duplicate or mask each others
  information
• Seq and Adj SS, plug-in parts, statistical
  elimination
• Model, model formula, and best fit equation

Next week Designing experiments Read Chapter 5