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

2T

0i

i

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

Fixed and random models for the CRD

Fixed Model(Model I)

Random Model(Model II)

Yij = µ + i + ij

Expected Source df Mean Square

Treatment t -1

Error tr -t

2T

2e rs+s2e

s

Expected

+Te r 22s

Source df Mean Square

Treatment t -1

Error tr -t

2e

s

2 2t i

i(t 1)

variance among fixed treatment effects

Models for the RBD

Fixed Model Random Model

Yij = µ + i +j + ij

2 2T j

j

2 2B i

i

(t 1)

(r 1)

Source dfExpectedMean Square

Block r-1

Treatment t-1

Error (r-1)(t-1)

Source

Treatment

Block2 2e Bt 2 2e Tr 2e

Source dfExpectedMean Square

Block r-1

Treatment t-1

Error (r-1)(t-1)

Source

Treatment

Block2 2e Bt 2 2e Tr 2e

Source dfExpectedMean Square

Block r-1

Treatment t-1

Error (r-1)(t-1)

Source

Treatment

Block2 2e Bt 2 2e Tr 2e

Mixed Model

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

Nested vs. Cross-Classified Factors

Cross-classified

A1

A2

A3

B1 B2

X X

X X

X X

All possible combinations of

A and B

Nested

A1 A2 A3

B1 B2 B3 B4 B5 B6

Each unit of B is uniqueto each unit of A

General form for degrees of freedom

B nested in A a(b-1) A*B (a-1)(b-1)

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

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.

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

2 21

2

t sn

d

For Example

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.

Subsample6.2 mean 6.507.4 variance 0.455.8 t (0.05, 4 df) 2.78

7 d 0.506.1 n 13.88

Suppose we were measuring grain protein content and we wanted to increase the precision with which we were measuring each replicate of a treatment.

2 2 21

2 2

t s 2.78 * 0.45n 13.88

d 0.5

Linear model with sub-sampling

For a CRD Yijk= + i + ij + ijk

= mean effecti = ith treatment effectij = random errorijk= sampling error

For an RBD Yijk= + i + j + ij + ijk

= mean effectβi = ith block effectj = jth treatment effectij = treatment x block interaction, treated as errorijk= sampling error

Source df Expected Mean Square

Block r-1 2 2 2s e bσ nσ tnσ

Treatment t-1 2 2 2s e tn rn

Error (r-1)(t-1) 2 2s en

Sampling Error rt(n-1) 2s

Expected Mean Squares – RBD with subsampling

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

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

ijk ijkY Y

2

iitn Y Y

2

jjrn Y Y

2

kkn Y Y SSB SST

Significance Tests

MSS estimates – the variation among samples

MSE estimates – the variation among samples plus– the variation among plots treated

alike

MST estimates– the variation among samples plus– 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

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

Ys MSE rn

iiL MSE rntY

1 2Y Ys 2MSE rn

1 21 2L 2MSE rntY Y

1 2Y Yt2MSE rn

2 2 2 22 s e s eY

nMSEs

rn rn rn r

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

2e2

2s1

c

cn

rrn

2e

2s2

y

See Kuehl pg 163 for an example