1. Quantitative Methods Using more than one explanatory variable

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

3. Using more than one explanatory variable Statistical elimination

4. Using more than one explanatory variable Statistical elimination

5. Using more than one explanatory variable Statistical elimination

6. Using more than one explanatory variable Statistical elimination

7. Using more than one explanatory variable Statistical elimination

8. Using more than one explanatory variable Sequential and Adjusted Sums of Squares

9. Using more than one explanatory variable Sequential and Adjusted Sums of Squares

10. Using more than one explanatory variable Sequential and Adjusted Sums of Squares 2761.1

11. Using more than one explanatory variable Sequential and Adjusted Sums of Squares

12. Using more than one explanatory variable Sequential and Adjusted Sums of Squares

13. Using more than one explanatory variable Sequential and Adjusted Sums of Squares

14. Using more than one explanatory variable Sequential and Adjusted Sums of Squares

15. 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

16. Using more than one explanatory variable Models and parameters

17. 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

18. Using more than one explanatory variable Models and parameters Y =  + 

19. 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

20. 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

21. 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

22. 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 Models and parameters

23. Using more than one explanatory variable Geometry in 3-D

24. 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

25. Using more than one explanatory variable Geometry in 3-D

26. Using more than one explanatory variable Geometry in 1-D

27. 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…