Quadrat sampling Quadrat shape Quadrat size Lab Regression and ANCOVA Review Categorical variables ANCOVA if time

Quadrat shape 1. Edge effects ? ? ? worst best

Quadrat shape 2. Variance 4 5 4 1 best

Quadrat size 1. Edge effects best worst 3/5 on edge 3/8 on edge ? ? ?

Quadrat size 1. Edge effects Density Quadrat size

Quadrat size 2. Variance High variance Low variance

(bigger quadrats take l o n g e r to sample) Quadrat size So should we always use as large a quadrat as possible? Tradeoff with cost (bigger quadrats take l o n g e r to sample)

Quadrat lab What is better quadrat shape? Square or rectangle? What is better quadrat size? 4, 9 ,16, 25 cm2 ? Does your answer differ with tree species (distribution differs)? 22cm 16 cm

Quadrat lab Use a cost (“time is money”): benefit (low variance) approach to determine the optimal quadrat design for 10 tree species. Hendrick’s method Wiegert’s method Cost: total time = time to locate quadrat + time to census quadrat Benefit: Variance Size & shape affect!

Quadrat lab Quadrats can also be used to determine spatial pattern! We will analyze our data for spatial pattern (only) in the computer lab next week (3-5 pm).

Quadrat lab: points for thought 1. You need to establish if any species shows a density gradient. How will you do this? 2. You will have a bit of time to do something extra; what would be useful? Group work fine here. 3. Rules: - if quadrat doesn’t fit on map ? - if leaves are on edge of quadrat ?

Regression Problem: to draw a straight line through the points that best explains the variance

Regression Problem: to draw a straight line through the points that best explains the variance

Regression Problem: to draw a straight line through the points that best explains the variance

Regression Test with F, just like ANOVA: Variance explained by x-variable / df Variance still unexplained / df Variance explained (change in line lengths2) Variance unexplained (residual line lengths2)

Regression Test with F, just like ANOVA: Variance explained by x-variable / df Variance still unexplained / df In regression, each x-variable will normally have 1 df

Regression Test with F, just like ANOVA: Variance explained by x-variable / df Variance still unexplained / df Essentially a cost: benefit analysis – Is the benefit in variance explained worth the cost in using up degrees of freedom?

Regression example Total variance for 32 data points is 300 units. An x-variable is then regressed against the data, accounting for 150 units of variance. What is the R2? What is the F ratio?

Regression example Total variance for 32 data points is 300 units. An x-variable is then regressed against the data, accounting for 150 units of variance. What is the R2? What is the F ratio? R2 = 150/300 = 0.5 F 1,30 = 150/1 = 30 150/30 Why is df error = 30?

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 72 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 “b” 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

ANCOVA In an Analysis of Covariance, we look at the effect of a treatment (categorical) while accounting for a covariate (continuous) Fertilized P Fertilized N Growth rate (g/day) Plant height (cm)

ANCOVA Fertilizer treatment (X1): code as 0 = N; 1 =P Plant height (X2): continuous Fertilized P Fertilized N Growth rate (g/day) Plant height (cm)

ANCOVA Fertilizer treatment (X1): code as 0 = N; 1 = P 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) ? Fertilized P Fertilized N Plant height (cm)

ANCOVA Fit full model (categorical treatment, covariate, interaction) Y=m1X1+ m2X2 +m3X1X2 +b Fertilized N+P Fertilized N 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 N fertilizer (X1= 0) Write out equation for P fertilizer (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 Fertilized N+P Fertilized N 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 intercept } m1 Fertilized N+P Fertilized N Growth rate (g/day) No interaction Intercepts differ Plant height (cm)