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Quantitative Methods. Using more than one 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 ) .

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quantitative methods

Quantitative Methods

Using more thanone explanatory variable

slide2

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)
slide8

Using more than one explanatory variable

Sequential and Adjusted Sums of Squares

slide9

Using more than one explanatory variable

Sequential and Adjusted Sums of Squares

slide10

2761.1

Using more than one explanatory variable

Sequential and Adjusted Sums of Squares

slide11

Using more than one explanatory variable

Sequential and Adjusted Sums of Squares

slide12

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)
slide13

Using more than one explanatory variable

Sequential and Adjusted Sums of Squares

slide14

Using more than one explanatory variable

Sequential and Adjusted Sums of Squares

slide15

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)
slide16

Using more than one explanatory variable

Sequential and Adjusted Sums of Squares

slide17

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 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 > glm lvol=lhgt+ldiam;

SUBC> 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

slide19

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

slide20

Using more than one explanatory variable

Models and parameters

Y =  + 

slide21

Using more than one explanatory variable

Models and parameters

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 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

slide22

Using more than one explanatory variable

Models and parameters

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 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 + 1.98306*LDIAM + 1.1203*LHGT

slide23

Using more than one explanatory variable

Models and parameters

Model

Model Formula

lvol=ldiam+lhgt

Best Fit Equation

Fitted LVOL = -6.6467 + 1.98306*LDIAM + 1.1203*LHGT

slide24

Using more than one explanatory variable

Models and parameters

MTB > glm lvol=ldiam;

SUBC> 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

slide25

Using more than one explanatory variable

Models and parameters

MTB > glm lvol=ldiam;

SUBC> 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

slide26

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

slide28

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

slide31

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