Quantitative Methods - PowerPoint PPT Presentation

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

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

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

Sequential and Adjusted Sums of Squares

Using more than one explanatory variable

Sequential and Adjusted Sums of Squares

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

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 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

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 Greek

Known quantities that are estimates of them, in Latin

Using more than one explanatory variable

Models and parameters

Y =  + 

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

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

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

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

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

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

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 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

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

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