1. Strip-Plot Designs Sometimes called split-block design For experiments involving factors that are difficult to apply to small plots Three sizes of plots so there are three experimental errors The interaction is measured with greater precision than the main effects

2. For example: Three seed-bed preparation methods Four nitrogen levels Both factors will be applied with large scale machinery S3 S1 S2 N1 N2 N0 N3 S1 S3 S2 N2 N3 N1 N0

3. Advantages --- Disadvantages Advantages –Permits efficient application of factors that would be difficult to apply to small plots Disadvantages –Differential precision in the estimation of interaction and the main effects –Complicated statistical analysis

4. Strip-Plot Analysis of Variance SourcedfSSMSF Totalrab-1SSTot Blockr-1SSRMSR Aa-1SSAMSAF A Error(a)(r-1)(a-1)SSEAMSE A Factor A error Bb-1SSBMSBF B Error(b)(r-1)(b-1)SSEBMSE B Factor B error AB(a-1)(b-1)SSABMSABF AB Error(ab)(r-1)(a-1)(b-1)SSEABMSE AB Subplot error

5. Computations SSTot SSR SSA SSE A SSB SSE B SSAB SSE AB SSTot-SSR-SSA-SSE A -SSB-SSE B -SSAB There are three error terms - one for each main plot and interaction plot

6. F Ratios F ratios are computed somewhat differently because there are three errors F A = MSA/MSE A tests the sig. of the A main effect F B = MSB/MSE B tests the sig. of the B main effect F AB = MSAB/MSE AB tests the sig. of the AB interaction

7. Standard Errors of Treatment Means Factor A MeansMSE A /rb Factor B MeansMSE B /ra Treatment AB MeansMSE AB /r

8. SE of Differences Differences between 2 A means 2MSE A /rb Differences between 2 B means 2MSE B /ra Differences between A means at same level of B 2[(b-1)MSE AB + MSE A ]/rb Difference between B means at same level of A 2[(a-1)MSE AB + MSE B ]/ra Differences between A and B means at diff. levels 2[(ab-a-b)MSE AB + (a)MSE A + (b)MSE B ]/rab For se that are calculated from >1 MSE, df are approximated

9. Interpretation Much the same as a two-factor factorial: First test the AB interaction –If it is significant, the main effects have no meaning even if they test significant –Summarize in a two-way table of AB means If AB interaction is not significant –Look at the significance of the main effects –Summarize in one-way tables of means for factors with significant main effects

10. Numerical Example A pasture specialist wanted to determine the effect of phosphorus and potash fertilizers on the dry matter production of barley to be used as a forage –Potash: K1=none, K2=25kg/ha, K3=50kg/ha –Phosphorus: P1=25kg/ha, P2=50kg/ha –Three blocks –Farm scale fertilization equipment

11. K3K1K2 K1K3K2 K2K1K3 P1 P2 P1 P2 P1 563249 675458 386250 527264 544451 635468

12. Raw data - dry matter yields TreatmentIIIIII P1K1325254 P1K2496463 P1K3567268 P2K1543844 P2K2585054 P2K3676251

13. Construct two-way tables KIIIIIIMean 143.045.049.045.67 253.557.058.556.33 361.567.059.562.67 Mean52.6756.3355.6754.89 PIIIIII Mean 145.6762.6761.6756.67 259.6750.0049.6753.11 Mean52.6756.3355.6754.89 PK1K2K3 Mean 146.0058.6765.3356.67 245.3354.0060.0053.11 Mean45.6756.3362.6754.89 Potash x Block Phosphorus x Block Potash x Phosphorus

14. ANOVA Source dfSSMS F Total 171833.78 Block 245.7822.89 Potash (K) 2885.78442.8922.64** Error(a) 478.2219.56 Phosphorus (P) 156.8956.89.16ns Error(b) 2693.78346.89 KxP 219.119.56.71ns Error(ab) 454.2213.55

15. TreatmentIIIIII P1K1325254 P1K2496463 P1K3567268 P2K1543844 P2K2585054 P2K3677251 Raw data - dry matter yields SSTot=devsq(range)

16. ANOVA SourcedfSSMSF Total171833.78

17. Construct two-way tables KIIIIIIMean 143.045.049.045.67 253.557.058.556.33 361.567.059.562.67 Mean52.6756.3355.6754.89 PIIIIII Mean 145.6762.6761.6756.67 259.6750.0049.6753.11 Mean52.6756.3355.6754.89 PK1K2K3 Mean 146.0058.6765.3356.67 245.3354.0060.0053.11 Mean45.6756.3362.6754.89 Potash x Block Phosphorus x Block Potash x Phosphorus SSR=6*devsq(range) Sums of Squares for Blocks

18. ANOVA SourcedfSSMSF Total171833.78 Block245.7822.89

19. Construct two-way tables KIIIIIIMean 143.045.049.045.67 253.557.058.556.33 361.567.059.562.67 Mean52.6756.3355.6754.89 PIIIIII Mean 145.6762.6761.6756.67 259.6750.0049.6753.11 Mean52.6756.3355.6754.89 PK1K2K3 Mean 146.0058.6765.3356.67 245.3354.0060.0053.11 Mean45.6756.3362.6754.89 Potash x Block Phosphorus x Block Potash x Phosphorus SSA=6*devsq(range) Main effect of Potash

20. ANOVA SourcedfSSMSF Total171833.78 Block245.7822.89 Potash2885.78442.89

21. Construct two-way tables KIIIIIIMean 143.045.049.045.67 253.557.058.556.33 361.567.059.562.67 Mean52.6756.3355.6754.89 PIIIIII Mean 145.6762.6761.6756.67 259.6750.0049.6753.11 Mean52.6756.3355.6754.89 PK1K2K3 Mean 146.0058.6765.3356.67 245.3354.0060.0053.11 Mean45.6756.3362.6754.89 Potash x Block Phosphorus x Block Potash x Phosphorus SSE A = 2*devsq(range) – SSR – SSA

22. ANOVA SourcedfSSMSF Total171833.78 Block245.7822.89 Potash2885.78442.8922.64** Error(a)478.2219.56

23. Construct two-way tables KIIIIIIMean 143.045.049.045.67 253.557.058.556.33 361.567.059.562.67 Mean52.6756.3355.6754.89 PIIIIII Mean 145.6762.6761.6756.67 259.6750.0049.6753.11 Mean52.6756.3355.6754.89 PK1K2K3 Mean 146.0058.6765.3356.67 245.3354.0060.0053.11 Mean45.6756.3362.6754.89 Potash x Block Phosphorus x Block Potash x Phosphorus SSB=9*devsq(range) Main effect of Phosphorous

24. ANOVA SourcedfSSMSF Total171833.78 Block245.7822.89 Potash2885.78442.8922.64** Error(a)478.2219.56 Phosphorus156.8956.89

25. Construct two-way tables KIIIIIIMean 143.045.049.045.67 253.557.058.556.33 361.567.059.562.67 Mean52.6756.3355.6754.89 PIIIIII Mean 145.6762.6761.6756.67 259.6750.0049.6753.11 Mean52.6756.3355.6754.89 PK1K2K3 Mean 146.0058.6765.3356.67 245.3354.0060.0053.11 Mean45.6756.3362.6754.89 Potash x Block Phosphorus x Block Potash x Phosphorus SSE B = 3*devsq(range) – SSR – SSB

26. ANOVA SourcedfSSMSF Total171833.78 Block245.7822.89 Potash2885.78442.8922.64** Error(a)478.2219.56 Phosphorus156.8956.89.16ns Error(b)2693.78346.89

27. Construct two-way tables KIIIIIIMean 143.045.049.045.67 253.557.058.556.33 361.567.059.562.67 Mean52.6756.3355.6754.89 PIIIIII Mean 145.6762.6761.6756.67 259.6750.0049.6753.11 Mean52.6756.3355.6754.89 PK1K2K3 Mean 146.0058.6765.3356.67 245.3354.0060.0053.11 Mean45.6756.3362.6754.89 Potash x Block Phosphorus x Block Potash x Phosphorus SSAB= 3*devsq(range) – SSA – SSB Interaction of P and K

28. ANOVA Source dfSS MS F Total 171833.78 Block 245.7822.89 Potash (K) 2885.78442.8922.64** Error(a) 478.2219.56 Phosphorus (P) 156.8956.89.16ns Error(b) 2693.78346.89 KxP 219.119.56.71ns Error(ab) 454.2213.55

29. Interpretation Only potash had a significant effect Each increment of added potash resulted in an increase in the yield of dry matter The increase took place regardless of the level of phosphorus PotashNone25 kg/ha50 kg/haSE Mean Yield45.6756.3362.671.80

30. Repeated measurements over time We often wish to take repeated measures on experimental units to observe trends in response over time. –repeated cuttings of a pasture –multiple observations on the same animal (developmental responses) Often provides more efficient use of resources than using different experimental units for each time period May also provide more precise estimation of time trends by reducing random error among experimental units – effect is similar to blocking Problem: observations over time are not assigned at random to experimental units. –Observations on the same plot will tend to be positively correlated –Correlations are greatest for samples taken at short time intervals and less for distant sampling periods

31. Repeated measurements over time The simplest approach is to treat sampling times as sub- plots in a split-plot experiment. –Some references recommend use of strip-plot rather than split- plot –This is valid only if all pairs of sub-plots in each main plot can be assumed to be equally correlated. Compound symmetry Sphericity Univariate adjustments can be made Multivariate procedures can be used to adjust for the correlations among sampling periods

32. Univariate adjustments for repeated measures Reduce df for subplots, interactions, and subplot error terms to obtain more conservative F tests Fit a smooth curve to the time trends and analyze a derived variable –average –maximum response –area under curve –time to reach the maximum Use polynomial contrasts to evaluate trends over time (linear, quadratic responses) and compare responses for each treatment –Can be done with the REPEATED statement in PROC GLM

33. Multivariate adjustments for repeated measures Stage one: estimate covariance structure for residuals Stage two: –include covariance structure in the model –use generalized least squares methodology to evaluate treatment and time effects Computer intensive –use PROC MIXED or GLIMMIX in SAS Reference: Littell et al., 2002. SAS for Linear Models, Chapter 8.