Experiments in Complex Stands

Full text

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Experiments in Complex

Stands

Valerie LeMay, Craig Farnden and Peter Marshall

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March 28 to April 3, 2009 LeMay, Farnden, Marshall 2

The Challenge

• Design an experiment based on an objective • Constraints: Must have three blocks; treatment is a

cutting pattern; two or three treatments

• Variations, then:

– Variables of interest – Treatments applied

– Numbers of replicates (i.e., experimental units) – Subsampling of experimental units

• Data: 250 m X 800 m area, mixed species, uneven-aged, measured in 1985, 1993 and 2001

• Assumed: There is an interest in testing for a treatment effect as well as getting an estimate of this effect

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The Experimental Design:

Generalized RCB

Block 1

Block 2

Block 3

C

TR1

TR1

TR1

TR2

TR2

TR2

C

TR2

C

TR1

C

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March 28 to April 3, 2009 LeMay, Farnden, Marshall 4

Analysis of Variance (

n

=replicates)

Source Df MS Effect F test

Block 2 MSBL Random Treatment 2 MSTR Fixed MSTR/ MSBL X TR Block X Treatment 4 MSBL X TR Random Replicates nested in BL X TR 3 X 3 X (n-1) MSE Random Total

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Analysis of Variance (

m

=subsamples)

Source Df MS Effect F test

Block 2 MSBL Random Treatment 2 MSTR Fixed MSTR/ MSBL X TR Block X Treatment 4 MSBL X TR Random Replicates nested in BL X TR 3 X 3 X (n-1) MSE Random Samples 3 X 3 X n X (m-1) MSSE Random Total

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March 28 to April 3, 2009 LeMay, Farnden, Marshall 6

Generalized RCB: Expected Values

J

=#blocks,

n

=#replicates

T B EE

+

n

× 2 2

σ

σ

2 EE

σ

= × + − + K k k T B EE n Jn K 1 2 2 ) 1 ( 2 τ σ σ E[MSTR]= E[MSBL X TR ]= E[MSE]=

• If no Block by Treatment interaction, MSE can be used in the F-test

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E[MSTR]=

E[MSBL X TR ]= E[MSE]=

• Often experimental units are comprised of a number of individuals (e.g. trees)

• Subsamples within experimental units increase both the numerator and denominator of the F-test

Generalized RCB with

subsampling: Expected Values

J

=#blocks,

n

=#replicates,

m

=#subsamples

2 2 2 TR BLK EE SE +mσ +nmσ × σ 2 2 EE SE mσ σ +

= × + − + + K k k TR BLK EE SE m nm Jnm K 1 2 2 2 2 ) 1 ( τ σ σ σ

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March 28 to April 3, 2009 LeMay, Farnden, Marshall 8

Sizes of Experimental Units and

Numbers of Subsampling Units

• As the size of the experimental unit increases, the variance of experimental units decreases to a minimum for the treatment

• As the number of subsamples per experimental unit increases and/or the size of subsamples

increases, the variance of the experimental units decreases to the same value as when the entire experimental unit is measured

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Power Analysis

• Need a good estimate of

variance among

experimental units within treatments

and

blocks for a given size of experimental unit

• Vary the number of replicates (

n

) & size of

the effect to obtain power

• Choose an experimental unit size and

number of replicates based on power and

cost

• This choice

will likely vary among

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March 28 to April 3, 2009 LeMay, Farnden, Marshall 10

Particular Issues

with Complex Stands

• When spatial/structural complexity is increased via cutting, there will be very high variability

among small-size experimental units within a specific treatment

• For very complex stands will need large

experimental units and/or many units to obtain an acceptable level of power

• There are many possible treatments – can only include a few in the experiment – likely need to

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Simulated Experiment

Objective: To assess the results of two

treatments relative to a control in terms

of average tree size

Other variables of interest:

– Diameter and height growth rates – Average tree size

– Volume per ha

– Tree regeneration

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March 28 to April 3, 2009 LeMay, Farnden, Marshall 12

Treatments Simulated

• Control:

No cutting

• Wildlife Habitat (WH): Remove trees in

regularly spaced 20 X 20 m

patches to

permit understory vegetation regrowth to

enhance sites for wildlife

• Growth Release (GR): Remove

25% of

trees randomly

to promote growth of

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Issues in Choosing Treatments

• Wildlife Habitat (WH):

– Regular patches vs. randomly located patches?

– Patches plus removal of stems in patches to promote growth response?

• Growth Release (GR):

– What level of removal?

– Which trees? Even species & size targeted? – Recovery (value) of removed stems?

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March 28 to April 3, 2009 LeMay, Farnden, Marshall 14

Process Used to Assess Power

1. Simulate the treatment over the entire study area

2. Break each area into experimental units

3. Calculate the variance between experimental units within a treatment

4. Use this as the experimental unit variance (within a block and treatment)

5. Forecasted changes in variance discussed but not explicitly simulated, since WH requires a spatially explicit model.

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March 28 to April 3, 2009 LeMay, Farnden, Marshall 16

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March 28 to April 3, 2009 LeMay, Farnden, Marshall 18 0 10 20 30 40 50 60 0 4000 8000 12000 K function, WH r (m) K(r) border theo 0 10 20 30 40 50 60 0 2000 6000 10000 K function, 1985 r (m) K(r) border theo C or GR

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March 28 to April 3, 2009 LeMay, Farnden, Marshall 20

Post-Cutting Statistics

C WH GR DBH Mean 24.2 24.2 24.4 St. Dev. 19.5 19.7 19.7 Volume/ ha Mean 526.7 137.1 406.5 Volume/ Tree Mean 1.03 1.05 1.06

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What Variances Can We Expect?

• Divided each simulated treatment into

experimental units: 40 X 40m (N=120); 80 X 80 m (N= 30)

• Calculated the variance between all possible plots for mean dbh (i.e., variance among cell means)

• St.Dev. 40 X 40 m: 3.1 (C); 5.3 (WH); 3.2 (GR) • St.Dev. 80 X 80 m: 1.5 (C); 2.8 (WH); 1.6 (GR)

• Changes in time: might expect these to increase, particularly for WH. May become

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March 28 to April 3, 2009 LeMay, Farnden, Marshall 22

Simulated Power Analysis

• Fix α =0.05

• Size of differences for practical importance

– Mean dbh: 5 cm

• Other Variables (not done):

– Dbh Increment: 0.5 cm – Volume/ha: 20 m3/ha

– Understory vegetation: 30% increase in desirable species

• Standard deviation of experimental units in each block X treatment: 5 10 15

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n=3

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March 28 to April 3, 2009 LeMay, Farnden, Marshall 24

Overall Comments

• What about using a model to assess outcomes instead?

– Models can be used to assess a wider range of treatments

– If already available, this is a cheap option

– Downside: Really need a spatially explicit, process model?

– Models have unknown accuracy

– Still need experiments to build models for: 1) data and 2) knowledge of the processes

– Very hard to get estimates of treatments effects and accuracies of those affects

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What is important in experiments in

complex stands?

• Treatments: Should be extremes?

• Very large experimental units are needed:

– For some variables, such as spatial patterns – For spatially variable treatments (e.g., WH)

• Should do power analysis via assumed variances of experimental units

• Complexity of the design: For repeated measures, better to keep this simple

• Use covariates to reduce between experimental unit variability

• Consider using covariates instead of using blocks, since blocks do not explicitly help in

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