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## Strip-Plot Designs

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**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**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**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**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**Computations**• There are three error terms - one for each main plot and interaction plot SSTot SSR SSA SSEA SSB SSEB SSAB SSEAB SSTot-SSR-SSA-SSEA-SSB-SSEB-SSAB**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**Standard Errors of Treatment Means**• Factor A Means MSEA/rb • Factor B Means MSEB/ra • Treatment AB Means MSEAB/r**SE of Differences**• Differences between 2 A means 2MSEA/rb • Differences between 2 B means 2MSEB/ra • Differences between A means at same level of B 2[(b-1)MSEAB + MSEA]/rb • Difference between B means at same level of A 2[(a-1)MSEAB + MSEB]/ra • Differences between A and B means at diff. levels 2[(ab-a-b)MSEAB + (a)MSEA + (b)MSEB]/rab For se that are calculated from >1 MSE, df are approximated**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**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**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**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**Construct two-way tables**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 Potash 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 Potash x Phosphorus 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**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 .16ns Error(b) 2 693.78 346.89 KxP 2 19.11 9.56 .71ns Error(ab) 4 54.22 13.55**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 72 51 SSTot=devsq(range)**ANOVA**Source df SS MS F Total 17 1833.78**Construct two-way tables**Phosphorus x Block Potash 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 49.0 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 Potash x Phosphorus 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 Sums of Squares for Blocks SSR=6*devsq(range)**ANOVA**Source df SS MS F Total 17 1833.78 Block 2 45.78 22.89**Construct two-way tables**Phosphorus x Block Potash 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 49.0 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 Potash x Phosphorus 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)**ANOVA**Source df SS MS F Total 17 1833.78 Block 2 45.78 22.89 Potash 2 885.78 442.89**Construct two-way tables**Phosphorus x Block Potash 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 49.0 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 Potash x Phosphorus 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 SSEA =2*devsq(range) – SSR – SSA**ANOVA**Source df SS MS F Total 17 1833.78 Block 2 45.78 22.89 Potash 2 885.78 442.89 22.64** Error(a) 4 78.22 19.56**Construct two-way tables**Phosphorus x Block Potash 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 49.0 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 Potash x Phosphorus 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 Phosphorous SSB=9*devsq(range)**ANOVA**Source df SS MS F Total 17 1833.78 Block 2 45.78 22.89 Potash 2 885.78 442.89 22.64** Error(a) 4 78.22 19.56 Phosphorus 1 56.89 56.89**Construct two-way tables**Phosphorus x Block Potash 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 49.0 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 Potash x Phosphorus 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**ANOVA**Source df SS MS F Total 17 1833.78 Block 2 45.78 22.89 Potash 2 885.78 442.89 22.64** Error(a) 4 78.22 19.56 Phosphorus 1 56.89 56.89 .16ns Error(b) 2 693.78 346.89**Construct two-way tables**Phosphorus x Block Potash 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 49.0 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 Potash x Phosphorus 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 Interaction of P and K SSAB=3*devsq(range) – SSA – SSB**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 .16ns Error(b) 2 693.78 346.89 KxP 2 19.11 9.56 .71ns Error(ab) 4 54.22 13.55**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 Potash None 25 kg/ha 50 kg/ha SE Mean Yield 45.67 56.33 62.67 1.80**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**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**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**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.