Quantitative Methods Using more than one explanatory variable
Quantitative Methods Using more than one explanatory variable
Using more than one explanatory variable Why use more than one? • Intervening or “ 3 rd” 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 Why use more than one? • Intervening or “ 3 rd” 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 “ 3 rd” 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 LHGT Error Total DF 1 29 30 Seq SS 3. 5042 4. 8080 8. 3122 Adj SS 3. 5042 4. 8080 Adj MS 3. 5042 0. 1658 F 21. 14 P 0. 000 Adj SS 0. 1987 4. 6234 0. 1846 Adj MS 0. 1987 4. 6234 0. 0066 F 30. 14 701. 33 P 0. 000 MTB > glm lvol=lhgt+ldiam; SUBC> covar lhgt ldiam. Source LHGT LDIAM Error Total DF 1 1 28 30 Seq SS 3. 5042 4. 6234 0. 1846 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 Gr k 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 LDIAM LHGT Error Total DF 1 1 28 30 Term Constant LDIAM LHGT Coef -6. 6467 1. 98306 1. 1203 Seq SS 7. 9289 0. 1987 0. 1846 8. 3122 SE Coef 0. 7983 0. 07488 0. 2041 Adj SS 4. 6234 0. 1987 0. 1846 T -8. 33 26. 48 5. 49 Adj MS 4. 6234 0. 1987 0. 0066 P 0. 000 F 701. 33 30. 14 P 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 LDIAM LHGT Error Total DF 1 1 28 30 Term Constant LDIAM LHGT Coef -6. 6467 1. 98306 1. 1203 Seq SS 7. 9289 0. 1987 0. 1846 8. 3122 SE Coef 0. 7983 0. 07488 0. 2041 Adj SS 4. 6234 0. 1987 0. 1846 T -8. 33 26. 48 5. 49 Adj MS 4. 6234 0. 1987 0. 0066 F 701. 33 30. 14 P 0. 000 Fitted LVOL = -6. 6467 + 1. 98306*LDIAM + 1. 1203*LHGT P 0. 000
Using more than one explanatory variable Models and parameters Model Formula Best Fit Equation lvol=ldiam+lhgt 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 LDIAM Error Total DF 1 29 30 Seq SS 7. 9254 0. 3832 8. 3087 Adj SS 7. 9254 0. 3832 Adj MS F P 7. 9254 599. 72 0. 000 0. 0132
Using more than one explanatory variable Models and parameters MTB > glm lvol=ldiam; SUBC> covariate ldiam. Analysis of Variance for LVOL Source LDIAM Error Total DF 1 29 30 Seq SS 7. 9254 0. 3832 8. 3087 Adj SS 7. 9254 0. 3832 Adj MS F P 7. 9254 599. 72 0. 000 0. 0132
Using more than one explanatory variable Models and parameters Source LDIAM Error Total DF 1 29 30 Seq SS 7. 9254 0. 3832 8. 3087 Adj SS 7. 9254 0. 3832 Adj MS F P 7. 9254 599. 72 0. 000 0. 0132 Source LDIAM LHEIGHT Error Total DF 1 1 28 30 Seq SS 7. 9254 0. 1978 0. 1855 8. 3087 Adj SS 4. 6275 0. 1978 0. 1855 Adj MS F P 4. 6275 698. 63 0. 000 0. 1978 29. 86 0. 000 0. 0066
Using more than one explanatory variable Geometry in 3 -D
Using more than one explanatory variable Geometry in 3 -D Source LHGT LDIAM Error Total DF 1 1 28 30 Seq SS 3. 5042 4. 6234 0. 1846 8. 3122 Adj SS 0. 1987 4. 6234 0. 1846 Adj MS 0. 1987 4. 6234 0. 0066 F 30. 14 701. 33 P 0. 000 Source LDIAM LHGT Error Total DF 1 1 28 30 Seq SS 7. 9289 0. 1987 0. 1846 8. 3122 Adj SS 4. 6234 0. 1987 0. 1846 Adj MS 4. 6234 0. 1987 0. 0066 F 701. 33 30. 14 P 0. 000
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
- Slides: 31