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 N1 N2 N0 N3 S1 S3 S2

Advantages --- Disadvantages 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 MSEA Factor A error B b-1 SSB MSB FB Error(b) (r-1)(b-1) SSEB MSEB Factor B error AB (a-1)(b-1) SSAB MSAB FAB Error(ab) (r-1)(a-1)(b-1) SSEAB MSEAB Subplot 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/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 Factor B Means Treatment AB Means

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

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

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 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 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 Phosphorus x Block 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 Potash x Phosphorus SSR=6*devsq(range) SSA=6*devsq(range) Main effect of Potash SSEA = 2*devsq(range) – SSR – SSA

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 61.67 56.67 2 59.67 50.00 49.67 53.11 Mean 52.67 56.33 55.67 54.89 Potash x Phosphorus SSB=9*devsq(range) Main effect of 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 SSAB= 3*devsq(range) – SSA – SSB

See Excel worksheet calculations 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

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

Formal names for required assumptions 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 Compound symmetry – covariances within subjects are equal Sphericity – variances of differences within subjects are equal You usually have sphericity when compound symmetry occurs. The two assumptions are usually met at the same time. Formal names for required assumptions

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   sed2 se2 se2 covariance

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) GLM repeated option can handle cases of CS PROC MIXED or GLIMMIX needed for more complex covariance structures

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 GLM repeated option can handle cases of CS PROC MIXED or GLIMMIX needed for more complex covariance structures

Mixed Model adjustment for error structure Stage one: estimate covariance structure for residuals Determine which covariance structures would make sense for the experimental design and type of data that is collected Use graphical methods to examine covariance patterns over time Likelihood ratio tests of more complex vs simpler models 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

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