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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
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
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 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
SSTot
SSR
SSA
SSEA
SSB
SSEB
SSAB
SSEAB SSTot-SSR-SSA-SSEA-SSB-SSEB-SSAB
There are three error terms - one for each main plot and interaction plot
2i j k ijkY Y
2..kkab Y Y
2i..irb Y Y
2i.ki kb Y Y SSA SSR
2. j.jra Y Y
2. jkj ka Y Y SSB SSR
2ij.i jr Y Y SSA SSB
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
AMSE
rb
BMSE
ra
ABMSE
r
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
2 A* MSE
rb
2 B* MSE
ra
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
2 1 AB A* b MSE MSE
rb
2 1 AB B* a MSE MSE
ra
2 AB A B* ab a b MSE a * MSE b * MSE
rab
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
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 49.045.67
2 53.5 57.0 58.556.33
3 61.5 67.0 59.562.67
Mean 52.67 56.33 55.6754.89
Potash x BlockP I II III Mean
1 45.67 62.67 61.6756.67
2 59.67 50.00 49.6753.11
Mean 52.67 56.33 55.6754.89
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
SSEA =2*devsq(range) – SSR – SSA
SSR=6*devsq(range)
SSA=6*devsq(range)
Main effect of Potash
Construct two-way tables
K I II III Mean
1 43.0 45.0 49.045.67
2 53.5 57.0 58.556.33
3 61.5 67.0 59.562.67
Mean 52.67 56.33 55.6754.89
P I II III Mean
1 45.67 62.67 61.6756.67
2 59.67 50.00 49.6753.11
Mean 52.67 56.33 55.6754.89
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 Block Phosphorus x Block
Potash x Phosphorus
SSEB =3*devsq(range) – SSR – SSB
SSB=9*devsq(range)Main effect of Phosphorus
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 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
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
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 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
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
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
1 2 1 2 1 2
2 2 2Y Y Y Y Y ,Y
2 sed2 se2 se2 covariance
1 2
2 2Y Y
1 2Y ,Y
0 1 2
2Y Y
2MSE
r
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)
2
22
2
2
1 0 0 00 0 0
0 1 0 00 0 0
0 0 1 00 0 0
0 0 0 10 0 0
2
1
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)
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
2 3
22
2
3 2
1
1
1
1
12 13 14
12 23 242
13 23 34
14 24 34
1
1
1
1
Mixed Model adjustment for error structure
Stage one: estimate covariance structure for residuals1. 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 contentadjust for loss in power due to loss of df in more complex models
Null model - no adjustment for correlated errors
2