1. Part III The General Linear Model. Multiple Explanatory Variables Chapter 12 Multiple Regression

2. Introduction

3. GLM | Multiple Regression Pcorn example…again Example 9.3.1 from Snedecor and Cochran (1989) Interested in the relationship between: – Phosphorus content of corn and phosphorus (organic and inorganic) levels in soil samples.

4. 1. Construct Model Verbal: Plant available phosphorus depends on the amount of both organic and inorganic soil phosphorus Graphical:

5. 1. Construct Model Verbal: Plant available phosphorus depends on the amount of both organic and inorganic soil phosphorus Graphical:

6. 1. Construct Model Formal: Start with individual explanatory variables:

7. 1. Construct Model Formal: Now we construct a model with both explanatory variables

8. 1. Construct Model

9. Partial Regression

10. 1. Construct Model Formal: Now we construct a model with both explanatory variables

11. 1. Construct Model Formal:

12. 1. Construct Model Formal: Finally, we add an interaction term Investigate potential interactive effects on the response variable

13. 2. Execute analysis mr <- lm(Pcorn~ioP+oP+ioP*oP, data=corn) ioPoPPcorn 12.65851 4.72454 1.93654 0.42360 0.63461 0.45364 3.11971 10.93776 1.76577 23.15077 9.44481 10.13193 11.62993 21.64493 23.15695 23.14696 29.95199

14. 3. Evaluate Model □ Straight line model ok? □ Errors homogeneous? □ Errors normal? □ Errors independent?

15. 4.State the population and whether the sample is representative. 5.Decide on mode of inference. Is hypothesis testing appropriate? 6.State H A / H o pair, test statistic, distribution, tolerance for Type I error. – Separate statement for each explanatory variable

16. 4.State the population and whether the sample is representative. 5.Decide on mode of inference. Is hypothesis testing appropriate? 6.State H A / H o pair, test statistic, distribution, tolerance for Type I error. – Separate statement for each explanatory variable var

17. 7. ANOVA n = 17

19. 8. Recompute p-value if necessary. Assumptions met, skip 9. Declare decision about model terms.

20. Present parameter estimates along with CL – Pcorn = 45.92 + 0.3278 oP + 5.304 ioP + 0.0830 ioP*oP Organic and inorganic soil phosphorus have interactive effects on phosphorus content of corn. If we wish to look at the effects of soil phosphorus on corn phosphorus content we need to know both organic and inorganic concentrations in the soil. We need to use the interaction term to compute the expected levels of corn phosphorus. 10. Report and interpret parameters of biological interest.