Experiments in Complex Stands

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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

TR2

C

TR2

C

TR1

C

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Analysis of Variance (

n

=replicates)

Source Df MS Effect F test

Block 2 MS_BL _Random Treatment 2 MS_TR _Fixed _MS_TR_/ MS_{BL X TR} Block X Treatment 4 MS_{BL X TR} _Random Replicates nested in BL X TR 3 X 3 X (n-1) MS_E _Random Total

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Analysis of Variance (

m

=subsamples)

Source Df MS Effect F test

Block 2 MS_BL Random Treatment 2 MS_TR Fixed _MS_TR_/ MS_{BL X TR} Block X Treatment 4 MS_{BL X TR} Random Replicates nested in BL X TR 3 X 3 X (n-1) MS_E Random Samples 3 X 3 X n X (m-1) MS_SE Random Total

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Generalized RCB: Expected Values

J

=#blocks,

n

=#replicates

T B EE

+

n

× 2 2

σ

2 EE

σ

∑

= × + − + K k k T B EE n Jn K 1 2 2 ) 1 ( 2 τ σ σ E[MS_TR]= E[MS_{BL X TR} ]= E[MS_E]=

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

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E[MS_TR]=

E[MS_{BL X TR} ]= E[MS_E]=

• 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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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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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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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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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 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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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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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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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