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

2. 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.

3. Stein’s Sample Estimate Where t 1 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

4. 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.

5. Linear model with sub-sampling For a CRDY ijk =  +  i +  ij +  ijk  =mean effect  i = i th treatment effect  ij = random error  ijk =sampling error For an RBDY ijk =  +  i +  j +  ij +  ijk  =mean effect β i = i th block effect  j = j th treatment effect  ij = treatment x block interaction, treated as error  ijk =sampling error

6. 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

7. The RBD ANOVA with Subsampling SourcedfSSMSF Totalrtn-1SSTot = Blockr-1SSB=SSB/(r-1) Trtmtt-1SST =SST/(t-1)F T = MST/MSE Error(r-1)(t-1)SSE = SSE/(r-1)(t-1)F E = MSE/MSS Sample Err.rt(n-1)SSS =SSS/rt(n-1) SSTot-SSB-SST-SSE

8. 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

9. 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 n Therefore: n F E – tests the significance of the variation among plots treated alike n F T – tests the significance of the differences among the treatment means

10. Allocating resources – reps vs samples Cost function C = c 1 r + c 2 rn –c 1 = cost of an experimental unit –c 2 = 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