Regression designs Y X1 Plant size 1 2 3 4 5 6 7 8 9 Growth rate 1 10 1 2 3 4 5 6 7 8 9 Growth rate Y 1 10 Plant size X1 X Y 1 1.5 2 3.3 4 4.0 6 4.5 8 5.2 10 72

Regression designs Y Y X1 X1 Plant size Plant size 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 Growth rate Y 1 2 3 4 5 6 7 8 9 Growth rate Y 10 Plant size X1 1 10 Plant size X1 X Y 1 1.5 2 3.3 4 4.0 6 4.5 8 5.2 10 72 X Y 1 0.8 1 1.7 1 3.0 10 5.2 10 7.0 10 8.5

Regression designs Y Y X1 Y X1 X1 Code 0=small, 1=large Plant size 1 2 3 4 5 6 7 8 9 Growth rate Y 1 2 3 4 5 6 7 8 9 Growth rate Y 10 Plant size X1 1 2 3 4 5 6 7 8 9 Growth rate Y Plant size X1 1 10 Plant size X1 X Y 1 1.5 2 3.3 4 4.0 6 4.5 8 5.2 10 7.2 X Y 1 0.8 1 1.7 1 3.0 10 5.2 10 7.0 10 8.5 X Y 0 0.8 0 1.7 0 3.0 1 5.2 1 7.0 1 8.5

Growth = m*Size + b Y X1 Questions on the general equation above: Code 0=small, 1=large Growth = m*Size + b 1 2 3 4 5 6 7 8 9 Growth rate Y Plant size X1 Questions on the general equation above: 1. What parameter predicts the growth of a small plant? 2. Write an equation to predict the growth of a large plant. 3. Based on the above, what does “m” represent? X Y 0 0.8 0 1.7 0 3.0 1 5.2 1 7.0 1 8.5

Growth = m*Size + b If small Y Growth = m*0 + b If large X1 Difference in growth Growth of small Code 0=small, 1=large Growth = m*Size + b 1 2 3 4 5 6 7 8 9 Growth rate Y Plant size X1 If small Growth = m*0 + b If large Growth = m*1 + b X Y 0 0.8 0 1.7 0 3.0 1 5.2 1 7.0 1 8.5 Large - small = m

Nitrogen What about “covariates”… - looking at the effect of salmon on tree growth rates Nitrogen

Compare tree growth around 2 streams, one with and one without salmon No Salmon Growth rate (g/day) t(9) = 0.06, p = 0.64

ANCOVA In an Analysis of Covariance, we look at the effect of a treatment (categorical) while accounting for a covariate (continuous) Salmon No Salmon Growth rate (g/day) Plant height (cm)

ANCOVA Fertilizer treatment (X1): code as 0 = No Salmon; 1 = Salmon Plant height (X2): continuous Salmon No Salmon Growth rate (g/day) Plant height (cm)

ANCOVA Fertilizer treatment (X1): code as 0 = No Salmon; 1 = Salmon Plant height (X2): continuous ? X1 X2 Y 0 1 1.1 0 2 4.0 : : : 1 1 3.1 1 2 5.2 1 5 11.3 X1*X2 : 1 2 5 Growth rate (g/day) ? Salmon No Salmon Plant height (cm)

ANCOVA Fit full model (categorical treatment, covariate, interaction) Y=m1X1+ m2X2 +m3X1X2 +b Salmon No Salmon Growth rate (g/day) Plant height (cm)

ANCOVA Fit full model (categorical treatment, covariate, interaction) Y=m1X1+ m2X2 +m3X1X2 +b Questions: Write out equation for No Salmon (X1= 0) Write out equation for Salmon (X1 = 1) What differs between two equations? If no interaction (i.e. m3 = 0) what differs between eqns?

ANCOVA Fit full model (categorical treatment, covariate, interaction) Y=m1X1+ m2X2 +m3X1X2 +b If X1=0: Y=m1X1+ m2X2 +m3X1X2 +b If X1=1: Y=m1 + m2X2 +m3X2 +b Difference: m1 +m3X2 Difference if no interaction: m1 +m3X2

Difference between categories…. Constant, doesn’t depend on covariate Depends on covariate = m1 + m3X2 (interaction) = m1 (no interaction) 12 10 8 Growth rate (g/day) Growth rate (g/day) 6 4 2 2 4 6 Plant height (cm) Plant height (cm)

ANCOVA Fit full model (categorical treatment, covariate, interaction) Test for interaction (if significant- stop!) If no interaction, the lines will be parallel Salmon No Salmon Growth rate (g/day) Plant height (cm)

ANCOVA Fit full model (categorical treatment, covariate, interaction) Test for interaction (if significant- stop!) Test for differences in intercepts between lines = m1 } m1 Growth rate (g/day) No interaction Intercepts differ Plant height (cm)

Multiple X variables: Both categorical …………... ANOVA One categorical, one continuous……………...ANCOVA Both continuous …………....Multiple Regression

Regression’s deep dark secret: Order matters! Input: height p=0.001 weight p=0.34 age p=0.07 age p=0.04 weight p=0.88 Why? In the first order, even though weight wasn’t significant, it explained some of the variation before age was tested. Common when x-variables are correlated with each other.