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Strip-Plot Designs

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. S3 S1 S2.

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Strip-Plot Designs

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

  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 Source df SS MS F Total rab-1 SSTot Block r-1 SSR MSR A a-1 SSA MSA FA Error(a) (r-1)(a-1) SSEA MSEAFactor A error B b-1 SSB MSB FB Error(b) (r-1)(b-1) SSEB MSEBFactor B error AB (a-1)(b-1) SSAB MSAB FAB Error(ab) (r-1)(a-1)(b-1) SSEAB MSEABSubplot error

  5. Computations • There are three error terms - one for each main plot and interaction plot SSTot SSR SSA SSEA SSB SSEB SSAB SSEABSSTot-SSR-SSA-SSEA-SSB-SSEB-SSAB

  6. F Ratios • F ratios are computed somewhat differently because there are three errors • FA = MSA/MSEAtests the sig. of the A main effect • FB = MSB/MSEBtests the sig. of the B main effect • FAB = MSAB/MSEABtests the sig. of the AB interaction

  7. Standard Errors of Treatment Means • Factor A Means • Factor B Means • Treatment AB Means

  8. SE of Differences for Main Effects • Differences between 2 A means with (r-1)(a-1) df • Differences between 2 B means with (r-1)(b-1) df

  9. SE of Differences • Differences between A means at same level of B • Difference between B means at same level of A • Difference between A and B means at diff. levels For se that are calculated from >1 MSE, t tests and dfare approximated

  10. 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

  11. 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

  12. K3 K1 K2 P1 56 32 49 P2 67 54 58 K1 K3 K2 P2 38 62 50 P1 52 72 64 K2 K1 K3 P2 54 44 51 P1 63 54 68

  13. Raw data - dry matter yields Treatment I II III P1K1 32 52 54 P1K2 49 64 63 P1K3 56 72 68 P2K1 54 38 44 P2K2 58 50 54 P2K3 67 62 51

  14. Construct two-way tables Potash x Block Phosphorus x Block P I II III Mean 1 45.67 62.67 61.67 56.67 2 59.67 50.00 49.67 53.11 Mean 52.67 56.33 55.67 54.89 K I II III Mean 1 43.0 45.0 49.045.67 2 53.5 57.0 58.556.33 3 61.5 67.0 59.562.67 Mean 52.67 56.33 55.67 54.89 Potash x Phosphorus SSR=6*devsq(range) P K1 K2 K3 Mean 1 46.00 58.67 65.33 56.67 2 45.33 54.00 60.00 53.11 Mean 45.67 56.33 62.67 54.89 Main effect of Potash SSA=6*devsq(range) SSEA =2*devsq(range) – SSR – SSA

  15. Construct two-way tables Potash x Block Phosphorus x Block K I II III Mean 1 43.0 45.0 49.0 45.67 2 53.5 57.0 58.5 56.33 3 61.5 67.0 59.5 62.67 Mean 52.67 56.33 55.67 54.89 P I II III Mean 1 45.67 62.67 61.6756.67 2 59.67 50.00 49.6753.11 Mean 52.67 56.33 55.67 54.89 Potash x Phosphorus Main effect of Phosphorus SSB=9*devsq(range) P K1 K2 K3 Mean 1 46.00 58.67 65.33 56.67 2 45.33 54.00 60.00 53.11 Mean 45.67 56.33 62.67 54.89 SSEB =3*devsq(range) – SSR – SSB SSAB=3*devsq(range) – SSA – SSB

  16. ANOVA Source df SS MS F Total 17 1833.78 Block 2 45.78 22.89 Potash (K) 2 885.78 442.89 22.64** Error(a) 4 78.22 19.56 Phosphorus (P) 1 56.89 56.89 0.16ns Error(b) 2 693.78 346.89 KxP 2 19.11 9.56 0.71ns Error(ab) 4 54.22 13.55 See Excel worksheet calculations

  17. Interpretation • Only potash had a significant effect on barley dry matter production • Each increment of added potash resulted in an increase in the yield of dry matter (~340 g/plot per kg increase in potash • The increase took place regardless of the level of phosphorus Potash None 25 kg/ha 50 kg/ha SE Mean Yield 45.67 56.33 62.67 1.80

  18. 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 harvests of a fruit or vegetable crop during a season • Annual yield of a perennial crop • 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 • Violates the assumption that errors (residuals) are independent

  19. Analysis of repeated measurements • 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 a split-plot • Univariate adjustments can be made • Multivariate procedures can be used to adjust for the correlations among sampling periods

  20. Split-plot in time • In a sense, a split-plot is a specific case of repeated measures, where sub-plots represent repeated measurements on a common main plot • Analysis as a split-plot is valid only if all pairs of sub-plots in each main plot can be assumed to be equally correlated • Compound symmetry • Sphericity • When time is a sub-plot, correlations may be greatest for samples taken at short time intervals and less for distant sampling periods, so assumptions may not be valid • Not a problem when there are only two sampling periods Formal names for required assumptions

  21. 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

  22. Multivariate adjustments for repeated measures • Stage one: estimate covariance structure for residuals • Stage two: • include appropriate 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

  23. Covariance Structure for Residuals sed2 se2 se2 covariance

  24. Covariance Structure for Residuals • No correlation (independence) • 4 measurements per subject • All covariances = 0 • Compound symmetry (CS) • All covariances (off-diagonal elements) are the same • Often applies for split-plot designs (sub-plots within main plots are equally correlated) • Autoregressive (AR) • Applies to time series analyses • For a first-order AR(1) structure, the within subject correlations drop off exponentially as the number of time lags between measurements increases (assuming time lags are all the same) • Unstructured (UN) • Complex and computer intensive • No particular pattern for the covariances is assumed

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