Quantitative Methods

Using more thanone explanatory variable

Using more than one explanatory variable

Why use more than one?

• Intervening or “3rd” variables (schoolchildren’s maths)• Reducing error variation (saplings)• There is more than one interesting predictor (trees)

Using more than one explanatory variable

Statistical elimination

Using more than one explanatory variable

Statistical elimination

Using more than one explanatory variable

Statistical elimination

Using more than one explanatory variable

Statistical elimination

Using more than one explanatory variable

Statistical elimination

Using more than one explanatory variable

Sequential and Adjusted Sums of Squares

Using more than one explanatory variable

Sequential and Adjusted Sums of Squares

Using more than one explanatory variable

Sequential and Adjusted Sums of Squares

2761.1

Using more than one explanatory variable

Sequential and Adjusted Sums of Squares

Using more than one explanatory variable

Sequential and Adjusted Sums of Squares

Using more than one explanatory variable

Sequential and Adjusted Sums of Squares

Using more than one explanatory variable

Sequential and Adjusted Sums of Squares

Using more than one explanatory variable

Sequential and Adjusted Sums of Squares

MTB > glm lvol=lhgt;SUBC> covar lhgt.

Source DF Seq SS Adj SS Adj MS F PLHGT 1 3.5042 3.5042 3.5042 21.14 0.000Error 29 4.8080 4.8080 0.1658Total 30 8.3122

MTB > glm lvol=lhgt+ldiam;SUBC> covar lhgt ldiam.

Source DF Seq SS Adj SS Adj MS F PLHGT 1 3.5042 0.1987 0.1987 30.14 0.000LDIAM 1 4.6234 4.6234 4.6234 701.33 0.000Error 28 0.1846 0.1846 0.0066Total 30 8.3122

Using more than one explanatory variable

Models and parameters

Using more than one explanatory variable

Models and parameters

Y = +

Unknown quantities we would like to know, in Known quantities that are estimates of them, in Latin

Using more than one explanatory variable

Models and parameters

Y = +

YIELD=μ +

Fertil Coeff

A α1

B α2

C −α1 −α2

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥

+ε

FINALHT=μ +β ×INITHT+

Water Coeff

A α1

B α2

C α3

D −α1 −α2 −α3

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

+ε

AMA=μ +β ×YEARS+γ ×HGHT+ε

LVOL=μ +β ×LDIAM +γ ×LHGT+ε

WGHT=μ +β ×LLEG+γ ×RLEG+ε

Using more than one explanatory variable

Models and parameters

LVOL=μ +β ×LDIAM +γ ×LHGT+ε

MTB > glm lvol=ldiam+lhgt;SUBC> covar ldiam lhgt.

Analysis of Variance for LVOL, using Adjusted SS for Tests

Source DF Seq SS Adj SS Adj MS F PLDIAM 1 7.9289 4.6234 4.6234 701.33 0.000LHGT 1 0.1987 0.1987 0.1987 30.14 0.000Error 28 0.1846 0.1846 0.0066Total 30 8.3122

Term Coef SE Coef T PConstant -6.6467 0.7983 -8.33 0.000LDIAM 1.98306 0.07488 26.48 0.000LHGT 1.1203 0.2041 5.49 0.000

Using more than one explanatory variable

Models and parametersMTB > glm lvol=ldiam;SUBC> covariate ldiam.

Analysis of Variance for LVOL

Source DF Seq SS Adj SS Adj MS F PLDIAM 1 7.9254 7.9254 7.9254 599.72 0.000Error 29 0.3832 0.3832 0.0132Total 30 8.3087

Using more than one explanatory variable

Models and parametersMTB > glm lvol=ldiam;SUBC> covariate ldiam.

Analysis of Variance for LVOL

Source DF Seq SS Adj SS Adj MS F PLDIAM 1 7.9254 7.9254 7.9254 599.72 0.000Error 29 0.3832 0.3832 0.0132Total 30 8.3087

QuickTime™ and aAnimation decompressor

are needed to see this picture.

QuickTime™ and aAnimation decompressor

are needed to see this picture.

QuickTime™ and aAnimation decompressor

are needed to see this picture.

QuickTime™ and aAnimation decompressor

are needed to see this picture.

QuickTime™ and aAnimation decompressor

are needed to see this picture.

QuickTime™ and aAnimation decompressor

are needed to see this picture.

QuickTime™ and aAnimation decompressor

are needed to see this picture.

Source DF Seq SS Adj SS Adj MS F PLDIAM 1 7.9254 7.9254 7.9254 599.72 0.000Error 29 0.3832 0.3832 0.0132Total 30 8.3087

Source DF Seq SS Adj SS Adj MS F PLDIAM 1 7.9254 4.6275 4.6275 698.63 0.000LHEIGHT 1 0.1978 0.1978 0.1978 29.86 0.000Error 28 0.1855 0.1855 0.0066Total 30 8.3087

Using more than one explanatory variable

Models and parameters

Using more than one explanatory variable

Geometry in 3-D

Using more than one explanatory variable

Geometry in 3-D

Source DF Seq SS Adj SS Adj MS F PLHGT 1 3.5042 0.1987 0.1987 30.14 0.000LDIAM 1 4.6234 4.6234 4.6234 701.33 0.000Error 28 0.1846 0.1846 0.0066Total 30 8.3122

Source DF Seq SS Adj SS Adj MS F PLDIAM 1 7.9289 4.6234 4.6234 701.33 0.000LHGT 1 0.1987 0.1987 0.1987 30.14 0.000Error 28 0.1846 0.1846 0.0066Total 30 8.3122

Using more than one explanatory variable

Geometry in 3-D

Using more than one explanatory variable

Geometry in 1-D

Using more than one explanatory variable

Next week: Designing experiments

Read Chapter 5

• Two or more x-variables are often useful and often necessary, and are easy to fit

• Statistical elimination, Seq and Adj SS, plug-in parts• Two variables may duplicate each others’

information (right and left legs)...• ... or they may unmask it (poets’ dates)

Last words…