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

3. 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

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 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

5. 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

6. 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

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 sed that are calculated from >1 MSE, t tests and df are 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 P2
K1 K3 K2 P2 P1
K2 K1 K3 P2 P1

13. Raw data - dry matter yields
Treatment I II III
P1K
P1K
P1K
P2K
P2K
P2K

14. Construct two-way tables
K I II III Mean
Mean
Potash x Block
P I II III Mean
Mean
Phosphorus x Block
P K1 K2 K3 Mean
Mean
Potash x Phosphorus
SSR=6*devsq(range)
SSA=6*devsq(range)
Main effect of Potash
SSEA = 2*devsq(range) – SSR – SSA

15. Construct two-way tables
Potash x Block
Phosphorus x Block
K I II III Mean
Mean
P I II III Mean
Mean
Potash x Phosphorus
SSB=9*devsq(range)
Main effect of Phosphorus
P K1 K2 K3 Mean
Mean
SSEB = 3*devsq(range) – SSR – SSB
SSAB= 3*devsq(range) – SSA – SSB

16. See Excel worksheet calculations
ANOVA
Source df SS MS F
Total
Block
Potash (K) **
Error(a)
Phosphorus (P) ns
Error(b)
KxP ns
Error(ab)
See Excel worksheet calculations

17. Interpretation
Potash None 25 kg/ha 50 kg/ha SE
Mean Yield
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

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

20. 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

21. 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

22. 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

23. Covariance Structure for Residuals
sed se se 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)
GLM repeated option can handle cases of CS
PROC MIXED or GLIMMIX needed for more complex covariance structures

25. 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

26. 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

27. 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