Report

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 For example: Three seed-bed preparation methods Four nitrogen levels Both factors will be applied with large scale machinery S3 S1 S2 S1 N1 N2 N2 N3 N0 N1 N3 N0 S3 S2 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 Block r-1 SSR MSR A a-1 SSA MSA FA SSEA MSEA Factor A error SSB MSB FB (r-1)(b-1) SSEB MSEB Factor B error (a-1)(b-1) SSAB MSAB FAB SSEAB MSEAB Subplot error Error(a) B Error(b) AB (r-1)(a-1) b-1 Error(ab) (r-1)(a-1)(b-1) SSTot Computations There are three error terms - one for each main plot and interaction plot SSTot i j k Yijk Y rb Y SSR ab k Y ..k Y SSA SSEA i i.. Y 2 2 2 b i k Y i .k Y 2 SSA SSR SSAB Y Y a Y Y SSB SSR r Y Y SSA SSB SSEAB SSTot-SSR-SSA-SSEA-SSB-SSEB-SSAB 2 SSB ra . j. j 2 SSEB j k . jk 2 i j ij. F Ratios F ratios are computed somewhat differently because there are three errors FA = MSA/MSEA tests the sig. of the A main effect FB = MSB/MSEB tests the sig. of the B main effect FAB = MSAB/MSEAB tests the sig. of the AB interaction Standard Errors of Treatment Means Factor A Means M SE A rb Factor B Means M SEB ra Treatment AB Means M S E AB r SE of Differences for Main Effects Differences between 2 A means 2 * M SE A with (r-1)(a-1) df rb Differences between 2 B means 2 * M SEB ra with (r-1)(b-1) df SE of Differences Differences between A means at same level of B 2 * b 1 M S E A B M S E A rb Difference between B means at same level of A 2 * a 1 M S E A B M S E B ra Difference between A and B means at diff. levels 2 * a b a b M S E A B a * M S E A b * M S E B ra b For sed that are calculated from >1 MSE, t tests and 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 Potash x Block Phosphorus x Block K I II III Mean P 1 43.0 45.0 49.0 45.67 1 45.67 62.67 61.67 56.67 2 53.5 57.0 58.5 56.33 2 59.67 50.00 49.67 53.11 3 61.5 67.0 59.5 62.67 Mean 52.67 56.33 55.67 54.89 Mean 52.67 56.33 55.67 I II III Mean 54.89 SSR=6*devsq(range) Potash x Phosphorus P Main effect of Potash SSA=6*devsq(range) SSEA = 2*devsq(range) – SSR – SSA 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 Construct two-way tables Potash x Block Phosphorus x Block K I II III Mean P 1 43.0 45.0 49.0 45.67 1 45.67 62.67 61.67 56.67 2 53.5 57.0 58.5 56.33 2 59.67 50.00 49.67 53.11 3 61.5 67.0 59.5 62.67 Mean 52.67 56.33 55.67 54.89 Mean 52.67 56.33 55.67 I II III Mean 54.89 Main effect of Phosphorus SSB=9*devsq(range) Potash x Phosphorus P K1 K2 K3 Mean SSEB = 3*devsq(range) – SSR – SSB 1 46.00 58.67 65.33 56.67 2 45.33 54.00 60.00 53.11 SSAB= 3*devsq(range) – SSA – SSB Mean 45.67 56.33 62.67 54.89 ANOVA Source df SS MS Total 17 1833.78 Block 2 45.78 22.89 Potash (K) 2 885.78 442.89 Error(a) 4 78.22 19.56 Phosphorus (P) 1 56.89 56.89 Error(b) 2 693.78 346.89 KxP 2 19.11 9.56 Error(ab) 4 54.22 13.55 F 22.64** 0.16ns 0.71ns See Excel worksheet calculations Interpretation Potash None 25 kg/ha 50 kg/ha SE Mean Yield 45.67 56.33 62.67 1.80 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 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 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 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 Formal names for required assumptions 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 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 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 Covariance Structure for Residuals Y Y Y Y 2 Y ,Y 2 2 1 2 sed2 Y Y 2 2 1 2 2 1 se2 2 se2 Y ,Y 0 1 2 1 2 covariance Y Y 2 1 2 2M S E r Covariance Structure for Residuals No correlation (independence) – 4 measurements per subject 2 – All covariances = 0 0 0 0 0 0 0 2 0 0 0 2 0 1 0 0 2 0 0 2 0 0 0 1 0 0 1 0 0 0 0 0 1 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) 1 2 1 1 1 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) 1 Complex and computer intensive 12 No particular pattern for the 2 covariances is assumed 13 May have low power due to loss of df for error 14 Unstructured (UN) – – – 1 2 2 3 1 1 2 12 13 1 23 23 1 24 34 2 3 2 1 14 24 34 1 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 = (-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 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