strip-plot designs
DESCRIPTION
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. S3 S1 S2. - PowerPoint PPT PresentationTRANSCRIPT
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
Factor A Means MSEA/rb
Factor B Means MSEB/ra
Treatment AB Means MSEAB/r
SE of Differences Differences between 2 A means
2MSEA/rb
Differences between 2 B means2MSEB/ra
Differences between A means at same level of B2[(b-1)MSEAB + MSEA]/rb
Difference between B means at same level of A2[(a-1)MSEAB + MSEB]/ra
Differences between A and B means at diff. levels2[(ab-a-b)MSEAB + (a)MSEA + (b)MSEB]/rab
For se that are calculated from >1 MSE, 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
K1 K3 K2
K2 K1 K3
P1
P2
P2
P1
P2
P1
56 32 49
67 54 58
38 62 50
52 72 64
54 44 51
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 49.045.67
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
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 .16ns
Error(b) 2 693.78 346.89
KxP 2 19.11 9.56 .71ns
Error(ab) 4 54.22 13.55
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 72 51
Raw data - dry matter yields
SSTot=devsq(range)
ANOVA
Source df SS MS F
Total 17 1833.78
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 BlockPhosphorus x Block
Potash x Phosphorus
SSR=6*devsq(range)
Sums of Squares for Blocks
ANOVA
Source df SS MS F
Total 17 1833.78
Block 2 45.78 22.89
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 BlockPhosphorus x Block
Potash x Phosphorus
SSA=6*devsq(range)
Main effect of Potash
ANOVA
Source df SS MS F
Total 17 1833.78
Block 2 45.78 22.89
Potash 2 885.78 442.89
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 BlockPhosphorus x Block
Potash x Phosphorus
SSEA =2*devsq(range) – SSR – SSA
ANOVA
Source df SS MS F
Total 17 1833.78
Block 2 45.78 22.89
Potash 2 885.78 442.89 22.64**
Error(a) 4 78.22 19.56
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 BlockPhosphorus x Block
Potash x Phosphorus
SSB=9*devsq(range)
Main effect of Phosphorous
ANOVA
Source df SS MS F
Total 17 1833.78
Block 2 45.78 22.89
Potash 2 885.78 442.89 22.64**
Error(a) 4 78.22 19.56
Phosphorus 1 56.89 56.89
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 BlockPhosphorus x Block
Potash x Phosphorus
SSEB =3*devsq(range) – SSR – SSB
ANOVA
Source df SS MS F
Total 17 1833.78
Block 2 45.78 22.89
Potash 2 885.78 442.89 22.64**
Error(a) 4 78.22 19.56
Phosphorus 1 56.89 56.89 .16ns
Error(b) 2 693.78 346.89
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 BlockPhosphorus x Block
Potash x Phosphorus
SSAB=3*devsq(range) – SSA – SSB
Interaction of P and K
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 .16ns
Error(b) 2 693.78 346.89
KxP 2 19.11 9.56 .71ns
Error(ab) 4 54.22 13.55
Interpretation Only potash had a significant effect
Each increment of added potash resulted in an increase in the yield of dry matter
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 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
– Correlations are greatest for samples taken at short time intervals and less for distant sampling periods
Repeated measurements over time 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 split-
plot
– This is valid only if all pairs of sub-plots in each main plot can be assumed to be equally correlated.
• Compound symmetry
• Sphericity
Univariate adjustments can be made Multivariate procedures can be used to adjust for the
correlations among sampling periods
Univariate adjustments for repeated measures
Reduce df for subplots, interactions, and subplot error terms to obtain more conservative F tests
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– Can be done with the REPEATED statement in PROC GLM
Multivariate adjustments for repeated measures
Stage one: estimate covariance structure for residuals Stage two:
– include 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
Reference: Littell et al., 2002. SAS for Linear Models, Chapter 8.