## Experiments in Complex

## Stands

Valerie LeMay, Craig Farnden and Peter Marshall

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

### The Experimental Design:

### Generalized RCB

### Block 1

### Block 2

### Block 3

### C

### TR1

### TR1

### TR1

### TR2

### TR2

### TR2

### C

### TR2

### C

### TR1

### C

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

### 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

### 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

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[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

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* +*m*σ +*nm*σ ×
σ
2
2
*EE*
*SE* *m*σ
σ +

### ∑

= × + − + +*K*

*k*

*k*

*TR*

*BLK*

*EE*

*SE*

*m*

*nm*

*Jnm*

*K*1 2 2 2 2 ) 1 ( τ σ σ σ

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

## 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

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

## 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

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

## 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?

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.

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

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**

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### 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

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

*n*=3

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

### 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