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

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Introduction

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

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

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

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

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

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1. Construct Model

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Partial Regression

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

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1. Construct Model Formal:

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

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2. Execute analysis mr <- lm(Pcorn~ioP+oP+ioP*oP, data=corn) ioPoPPcorn

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

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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

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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

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7. ANOVA n = 17

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

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Present parameter estimates along with CL – Pcorn = oP ioP 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.

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