Quadrat sampling Quadrat shape Quadrat size Lab Regression
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 best 1
Quadrat size 1. Edge effects ? ? ? ? ? ? 3/5 on edge worst 3/8 on edge best
Quadrat size Density 1. Edge effects Quadrat size
Quadrat size 2. Variance High variance Low variance
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 cm 2 ? Does your answer differ with tree species (distribution differs)? 22 cm 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 Variance unexplained (change in line lengths 2) (residual line lengths 2)
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. 1. What is the R 2? 2. 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. 1. What is the R 2? 2. What is the F ratio? R 2 = 150/300 = 0. 5 Why is df error = 30? F 1, 30 = 150/1 = 30 150/30
Y Growth rate Regression designs 9 8 7 6 5 4 3 2 1 0 1 10 Plant size X 1 2 4 6 8 10 Y 1. 5 3. 3 4. 0 4. 5 5. 2 72
9 8 7 6 5 4 3 2 1 0 Y 1 10 Plant size Growth rate Y Growth rate Regression designs 9 8 7 6 5 4 3 2 1 0 1 Y 1. 5 3. 3 4. 0 4. 5 5. 2 72 Plant size X 1 X 1 2 4 6 8 10 10 X 1 10 10 10 Y 0. 8 1. 7 3. 0 5. 2 7. 0 8. 5
Regression designs Y 1 10 Plant size 9 8 7 6 5 4 3 2 1 0 Y 1. 5 3. 3 4. 0 4. 5 5. 2 72 Plant size 9 8 7 6 5 4 3 2 1 0 0 1 Plant size X 1 X 1 2 4 6 8 10 10 Growth rate 9 8 7 6 5 4 3 2 1 0 Growth rate Y Growth rate Code 0=small, 1=large X 1 10 10 10 Y 0. 8 1. 7 3. 0 5. 2 7. 0 8. 5 X 1 X 0 0 0 1 1 1 Y 0. 8 1. 7 3. 0 5. 2 7. 0 8. 5
Growth = m*Size + b 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? Y Growth rate Code 0=small, 1=large 9 8 7 6 5 4 3 2 1 0 0 1 Plant size X 1 X 0 0 0 1 1 1 Y 0. 8 1. 7 3. 0 5. 2 7. 0 8. 5
Difference in growth Growth of small Growth = m*Size + b If small Growth = m*0 + b Y Growth rate Code 0=small, 1=large 9 8 7 6 5 4 3 2 1 0 0 1 Plant size If large Growth = m*1 + b Large - small = m X 1 X 0 0 0 1 1 1 Y 0. 8 1. 7 3. 0 5. 2 7. 0 8. 5
ANCOVA Growth rate (g/day) In an Analysis of Covariance, we look at the effect of a treatment (categorical) while accounting for a covariate (continuous) Fertilized P Fertilized N Plant height (cm)
ANCOVA Growth rate (g/day) Fertilizer treatment (X 1): code as 0 = N; 1 =P Plant height (X 2): continuous Fertilized P Fertilized N Plant height (cm)
ANCOVA Fertilizer treatment (X 1): code as 0 = N; 1 = P Plant height (X 2): continuous Growth rate (g/day) ? ? Plant height (cm) X 1 0 0 : 1 1 : 1 X 2 1 2 : 5 X 1*X 2 Y 0 1. 1 0 4. 0 : : 1 3. 1 2 5. 2 : 5 11. 3 Fertilized P Fertilized N
ANCOVA 1. Fit full model (categorical treatment, covariate, interaction) Growth rate (g/day) Y=m 1 X 1+ m 2 X 2 +m 3 X 1 X 2 +b Fertilized N+P Fertilized N Plant height (cm)
ANCOVA 1. Fit full model (categorical treatment, covariate, interaction) 2. Y=m 1 X 1+ m 2 X 2 +m 3 X 1 X 2 +b Questions: • Write out equation for N fertilizer (X 1= 0) • Write out equation for P fertilizer (X 1 = 1) • What differs between two equations? • If no interaction (i. e. m 3 = 0) what differs between eqns?
ANCOVA 1. Fit full model (categorical treatment, covariate, interaction) 2. Y=m 1 X 1+ m 2 X 2 +m 3 X 1 X 2 +b If X 1=0: Y=m 1 X 1+ m 2 X 2 +m 3 X 1 X 2 +b If X 1=1: Y=m 1 Difference: m 1 Difference if no interaction: m 1 + m 2 X 2 +m 3 X 2 +b
Difference between categories…. Constant, doesn’t depend on covariate Depends on covariate = m 1 + m 3 X 2 (interaction) Growth rate (g/day) = m 1 (no interaction) Plant height (cm) 12 10 8 6 4 2 0 0 2 4 Plant height (cm) 6
ANCOVA 1. Fit full model (categorical treatment, covariate, interaction) 2. Test for interaction (if significant- stop!) Growth rate (g/day) If no interaction, the lines will be parallel Fertilized N+P Fertilized N Plant height (cm)
ANCOVA Growth rate (g/day) 1. Fit full model (categorical treatment, covariate, interaction) 2. Test for interaction (if significant- stop!) 3. Test for differences in intercept } m 1 Fertilized N+P Fertilized N No interaction Intercepts differ Plant height (cm)
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