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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

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.

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

Linear model with sub-sampling

- For a CRDYijk= + i + ij + ijk
=mean effect

i = ith treatment effect

ij = random error

ijk=sampling error

- For an RBDYijk= + i + j + ij + ijk
=mean effect

βi = ith block effect

j = jth treatment effect

ij = treatment x block interaction, treated as error

ijk=sampling error

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

SourcedfSSMSF

Totalrtn-1SSTot =

Blockr-1SSB=SSB/(r-1)

Trtmtt-1SST =SST/(t-1)FT = MST/MSE

Error(r-1)(t-1)SSE = SSE/(r-1)(t-1)FE = MSE/MSS

SampleErr.rt(n-1)SSS =SSS/rt(n-1)

SSTot-SSB-SST-SSE

Standard Error of a treatment mean

Confidence interval estimate

Standard Error of a difference

Confidence interval estimate

T to test difference between

two means

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

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