1. Using more than one explanatory variable
Quantitative Methods
Using more than one explanatory variable

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

3. Statistical elimination
Using more than one explanatory variable
Statistical elimination

4. Statistical elimination
Using more than one explanatory variable
Statistical elimination

5. Statistical elimination
Using more than one explanatory variable
Statistical elimination

6. Statistical elimination
Using more than one explanatory variable
Statistical elimination

7. Statistical elimination
Using more than one explanatory variable
Statistical elimination

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

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

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

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

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

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

15. 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
Error
Total
MTB > glm lvol=lhgt+ldiam;
SUBC> covar lhgt ldiam.
LHGT
LDIAM
Error

16. Using more than one explanatory variable
Models and parameters

17. Models and parameters Using more than one explanatory variable
Unknown quantities we would like to know, in Greek
Known quantities that are estimates of them, in Latin

18. Models and parameters Using more than one explanatory variable

19. Models and parameters Using more than one explanatory variable
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
LHGT
Error
Total
Term Coef SE Coef T P
Constant
LDIAM
LHGT

20. Models and parameters Using more than one explanatory variable
MTB > glm lvol=ldiam;
SUBC> covariate ldiam.
Analysis of Variance for LVOL
Source DF Seq SS Adj SS Adj MS F P
LDIAM
Error
Total

21. Models and parameters Using more than one explanatory variable
MTB > glm lvol=ldiam;
SUBC> covariate ldiam.
Analysis of Variance for LVOL
Source DF Seq SS Adj SS Adj MS F P
LDIAM
Error
Total

22. Models and parameters Using more than one explanatory variable
Source DF Seq SS Adj SS Adj MS F P
LDIAM
Error
Total
Source DF Seq SS Adj SS Adj MS F P
LDIAM
LHEIGHT
Error
Total

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

24. Geometry in 3-D Using more than one explanatory variable
Source DF Seq SS Adj SS Adj MS F P
LHGT
LDIAM
Error
Total
LDIAM
LHGT

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

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

27. Next week: Designing experiments
Using more than one explanatory variable
Last words…
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)
Next week: Designing experiments
Read Chapter 5