1. Fixed vs. Random Effects
Fixed effect
we are interested in the effects of the treatments (or blocks) per se
if the experiment were repeated, the levels would be the same
conclusions apply to the treatment (or block) levels that were tested
treatment (or block) effects sum to zero
Random effect
represents a sample from a larger reference population
the specific levels used are not of particular interest
conclusions apply to the reference population
inference space may be broad (all possible random effects)
or narrow (just the random effects in the experiment)
goal is generally to estimate the variance among treatments
(or other groups)
Need to know which effects are fixed or random to determine appropriate F tests in ANOVA

2. Fixed or Random?
lambs born from common parents (same ram and ewe) are given different formulations of a vitamin supplement
comparison of new herbicides for potential licensing
comparison of herbicides used in different decades (1980’s, 1990’s, 2000’s)
nitrogen fertilizer treatments at rates of 0, 50, 100, and 150 kg N/ha
years of evaluation of new canola varieties (2008, 2009, 2010)
location of a crop rotation experiment that is conducted on three farmers’ fields in the Willamette valley (Junction City, Albany, Woodburn)
species of trees in an old growth forest
tree species could be a response variable or a design factor

3. Fixed and random models for the CRD
Yij = µ + i + ij
variance among fixed treatment effects
Expected + T e  r 2 s
Source df
Mean Square
Treatment t - 1
Error tr
Fixed Model
(Model I)
Random model may be more characteristic of observational studies
Expected
Random Model
(Model II)
Source df
Mean Square
Treatment t - 1
Error tr 2 T e r s +

4. Models for the RBD Yij = µ + i +j + ij Fixed Model Random Model
Random model may be more characteristic of observational studies
Mixed Model

5. Nested (Hierarchical) Designs
Levels of one factor (B) occur within the levels of another factor (A)
Levels of B are unique to each level of A
Factor B is nested within A
Factor A = the pigs (sows)
Factor B = the piglets
Nested factors are usually random effects

6. Nested vs. Cross-Classified Factors
A A A3
B1 B B3 B B5 B6
Each unit of B is unique
to each unit of A
Cross-classified A1 A2 A3
B1 B2
X X
All possible
combinations of
A and B
General form for degrees of freedom
B nested in A  a(b-1)
A*B  (a-1)(b-1)

7. Sub - Sampling
It may be necessary or convenient to measure a treatment response on subsamples of a plot
several soil cores within a plot
duplicate laboratory analyses to estimate grain protein
Introduces a complication into the analysis that can be handled in one of two ways:
compute the average for each plot and analyze normally
subject the subsamples themselves to an analysis
The second choice gives an additional source of variation in the ANOVA – often called the sampling error

8. Use Sampling to Gain Precision
When making lab measurements, you will have better results if you analyze several samples to get a truer estimate of the mean.
It is often useful to determine the number of samples that would be required for your chosen level of precision.
Sampling will reduce the variability within a treatment across replications.

9. Stein’s Sample Estimate
Where
t1 is the tabular t value for the desired confidence level and the degrees of freedom of the initial sample
d is the half-width of the desired confidence interval
s is the standard deviation of the initial sample
Kuehls’ book (pg 163) also provides a formula for determining optimum allocation of reps and subsamples, considering both the size of the variances and relative costs of reps vs. subsamples.

10. For Example
Suppose we were measuring grain protein content and we wanted to increase the precision with which we were measuring each replicate of a treatment.
If we collected and ran five samples from the same block and same treatment, we might obtain data like that above. We decide that an alpha level of 5% is acceptable and we would like to be able to get within .5 units of the true mean.
The formula indicates that to gain that type of precision, we would need to run 14 samples per block per treatment.

11. Linear model with sub-sampling
For a CRD Yijk= + i + ij + ijk
 = mean effect
i = ith treatment effect
ij = random error
ijk= sampling error
For an RBD Yijk= + i + j + ij + ijk
βi = ith block effect
j = jth treatment effect
ij = treatment x block interaction, treated as error

12. Expected Mean Squares – RBD with subsampling
Students are not required to know forms of expected mean squares. This information is presented to clarify the reason for using specific mean squares to form F ratios.
In this example, treatments are fixed and blocks are random effects
This is a mixed model because it includes both fixed and random effects
Appropriate F tests can be determined from the Expected Mean Squares

13. The RBD ANOVA with Subsampling
Source df SS MS F
Total rtn-1 SSTot =
Block r-1 SSB= SSB/(r-1)
Trtmt t-1 SST = SST/(t-1) FT = MST/MSE
Error (r-1)(t-1) SSE = SSE/(r-1)(t-1) FE = MSE/MSS
Sampling Error SSS = SSS/rt(n-1)
rt(n-1) SSTot-SSB-SST-SSE

14. Significance Tests Therefore: FE FT MSS estimates
the variation among samples
MSE estimates
the variation among samples plus
the variation among plots treated alike
MST estimates
the variation among plots treated alike plus
the variation among treatment means
Therefore: FE
tests the significance of the variation among plots treated alike FT
tests the significance of the differences among the treatment means

15. Means and Standard Errors
Standard Error of a treatment mean
Confidence interval estimate
Standard Error of a difference
Confidence interval estimate
t to test difference between
two means

16. Allocating resources – reps vs samples
Cost function
C = c1r + c2rn
c1 = cost of an experimental unit
c2 = cost of a sampling unit
If your goal is to minimize variance for a fixed cost, use the estimate of n to solve for r in the cost function
If your goal is to minimize cost for a fixed variance, use the estimate of n to solve for r using the formula for a variance of a treatment mean
See Kuehl pg 163 for an example