Quadrat sampling

1

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

2
Quadrat shape
1. Edge effects
?
best
worst
3
Quadrat shape
2. Variance
4
5
4
1
best
4
Quadrat size
1. Edge effects
?
?
?
?
?
?
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Quadrat size
1. Edge effects
Density
Quadrat size
6
Quadrat size
2. Variance
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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)
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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
9

• Quadrat lab
• Use a cost (time is money) benefit (low
  variance) approach to determine the optimal
  quadrat design for 10 tree species.
• Hendricks method
• Wiegerts method
• Cost
• total time time to locate quadrat time to
  census quadrat
• Benefit
• Variance

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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).
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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 doesnt fit on map ? - if leaves are on
edge of quadrat ?
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Regression

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

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Regression

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

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Regression

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

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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)
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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
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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?
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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?

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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?
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Regression designs
1
10
Plant size
X1
X Y 1 1.5 2 3.3 4 4.0 6 4.5 8 5.2 10 72
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Regression designs
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
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Regression designs
Code 0small, 1large
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
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Code 0small, 1large
Growth mSize 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?
X Y 0 0.8 0 1.7 0 3.0 1 5.2 1 7.0 1 8.5
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Code 0small, 1large
Growth mSize b
If small Growth m0 b
If large Growth m1 b
X Y 0 0.8 0 1.7 0 3.0 1 5.2 1 7.0 1 8.5
Large - small m
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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)
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ANCOVA

• Fertilizer treatment (X1) code as 0 N 1 P
• Plant height (X2) continuous

Growth rate (g/day)
Plant height (cm)
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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
X1X2 0 0 1 2 5
Growth rate (g/day)
Plant height (cm)
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ANCOVA

• Fit full model (categorical treatment, covariate,
  interaction)
• Ym1X1 m2X2 m3X1X2 b

Fertilized NP
Fertilized N
Growth rate (g/day)
Plant height (cm)
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ANCOVA

• Fit full model (categorical treatment, covariate,
  interaction)
• Ym1X1 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?

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ANCOVA

• Fit full model (categorical treatment, covariate,
  interaction)
• Ym1X1 m2X2 m3X1X2 b

If X11 Ym1 m2X2 m3X2 b
Difference m1 m3X2
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Difference between categories.
Constant, doesnt depend on covariate
Depends on covariate
m1 m3X2 (interaction)
m1 (no interaction)
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10
8
Growth rate (g/day)
Growth rate (g/day)
6
4
2
0
0
2
4
6
Plant height (cm)
Plant height (cm)
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ANCOVA

1. Fit full model (categorical treatment, covariate,
   interaction)
2. Test for interaction (if significant- stop!)

If no interaction, the lines will be parallel
Fertilized NP
Fertilized N
Growth rate (g/day)
Plant height (cm)
33
ANCOVA

1. Fit full model (categorical treatment, covariate,
   interaction)
2. Test for interaction (if significant- stop!)
3. Test for differences in intercept

m1
Fertilized NP
Fertilized N
Growth rate (g/day)
No interaction Intercepts differ
Plant height (cm)