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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

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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

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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

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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

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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

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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

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Standard Errors of Treatment Means Factor A Means Factor B Means Treatment AB Means

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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

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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 sed that are calculated from >1 MSE, t tests and df are approximated

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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

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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

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

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

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Construct two-way tables KIIIIII Mean 143.045.049.045.67 253.557.058.556.33 361.567.059.562.67 Mean52.6756.3355.6754.89 Potash x Block PIIIIII Mean 145.6762.6761.6756.67 259.6750.0049.6753.11 Mean52.6756.3355.6754.89 Phosphorus x Block PK1K2K3 Mean 146.0058.6765.3356.67 245.3354.0060.0053.11 Mean45.6756.3362.6754.89 Potash x Phosphorus SSE A = 2*devsq(range) – SSR – SSA SSR=6*devsq(range) SSA=6*devsq(range) Main effect of Potash

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Construct two-way tables KIIIIII Mean 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 BlockPhosphorus x Block Potash x Phosphorus SSE B = 3*devsq(range) – SSR – SSB SSB=9*devsq(range) Main effect of Phosphorus SSAB= 3*devsq(range) – SSA – SSB

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ANOVA Source dfSSMS F Total 171833.78 Block 245.7822.89 Potash (K) 2885.78442.89 22.64** Error(a) 478.2219.56 Phosphorus (P) 156.8956.890.16ns Error(b) 2693.78346.89 KxP 219.119.560.71ns Error(ab) 454.2213.55 See Excel worksheet calculations

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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 PotashNone25 kg/ha50 kg/haSE Mean Yield45.6756.3362.671.80

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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

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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 Mixed Model approaches can be used to adjust for the correlations among sampling periods

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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

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Univariate adjustments for repeated measures 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 Reduce df for subplots, interactions, and subplot error terms to obtain more conservative F tests

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Multivariate adjustments for repeated measures In PROC GLM, each repeated measure is treated like an additional variable in a multivariate analysis: model yield1 yield2 yield3 yield4=variety/nouni; repeated harvest / printe; MANOVA approach is very conservative –Effectively controls Type I error –Power may be low Many parameters are estimated so df for error may be too low Missing values result in an unnecessary loss of available information No real benefit compared to a Mixed Model approach

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Covariance Structure for Residuals sed 2 se 2 se 2 covariance

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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)

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Covariance Structure for Residuals 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 –May have low power due to loss of df for error

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Mixed Model adjustment for error structure Stage one: estimate covariance structure for residuals 1.Determine which covariance structures would make sense for the experimental design and type of data that is collected 2.Use graphical methods to examine covariance patterns over time 3.Likelihood ratio tests of more complex vs simpler models 4.Information content = (-2 res log likelihood) simple model minus (-2 res log likelihood) complex model df = difference in # parameters estimated AIC, AICC, BIC – information content adjust for loss in power due to loss of df in more complex models Null model - no adjustment for correlated errors

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Mixed Model adjustment for error structure 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

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