Transcript Strip-Plot Designs

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
SSEA
MSEA Factor A error
SSB
MSB
(r-1)(b-1)
SSEB
MSEB Factor B error
(a-1)(b-1)
SSAB
MSAB
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
FA
FB
FAB
Computations
 There are three error terms - one for each main plot and
interaction plot
SSTot
SSR
SSA

rb  Y
ab k Y..k  Y
i
SSEA
SSB
SSEB
SSAB
i..

Y


2

2
bi  k Yi.k  Y

ra  j Y. j.  Y

r   Y

2
a  j  k Y. jk  Y


2

2
2
 SSA  SSR
 SSB  SSR
 Y  SSA  SSB
SSTot-SSR-SSA-SSEA-SSB-SSEB-SSAB
i
SSEAB

i  j  k Yijk  Y
2
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
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 means
2MSEB/ra
 Differences between A means at same level of B
2[(b-1)MSEAB + MSEA]/rb
 Difference between B means at same level of A
2[(a-1)MSEAB + MSEB]/ra
 Differences between A and B means at diff. levels
2[(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
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
Phosphorus x Block
P
Potash x Block
K
1
I
43.0
II
45.0
III
49.0
Mean
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
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
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
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**
.16ns
.71ns
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
72
51
SSTot=devsq(range)
ANOVA
Source
df
SS
Total
17
1833.78
MS
F
Construct two-way tables
Phosphorus x Block
Potash x Block
P
I
II
III
Mean
1
45.67 62.67 61.67
56.67
45.67
2
59.67 50.00 49.67
53.11
58.5
56.33
Mean 52.67 56.33 55.67
54.89
59.5
62.67
K
I
II
III
1
43.0
45.0
49.0
2
53.5
57.0
3
61.5
67.0
Mean 52.67 56.33 55.67
Mean
54.89
Potash x Phosphorus
P
Sums of Squares for Blocks
SSR=6*devsq(range)
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
ANOVA
Source
df
SS
Total
17
1833.78
Block
2
45.78
MS
22.89
F
Construct two-way tables
Phosphorus x Block
Potash x Block
P
I
II
III
Mean
1
45.67 62.67 61.67
56.67
45.67
2
59.67 50.00 49.67
53.11
58.5
56.33
Mean 52.67 56.33 55.67
54.89
59.5
62.67
K
I
II
III
1
43.0
45.0
49.0
2
53.5
57.0
3
61.5
67.0
Mean 52.67 56.33 55.67
Mean
54.89
Potash x Phosphorus
P
Main effect of Potash
SSA=6*devsq(range)
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
ANOVA
Source
df
SS
MS
Total
17
1833.78
Block
2
45.78
22.89
Potash
2
885.78
442.89
F
Construct two-way tables
Phosphorus x Block
Potash x Block
P
I
II
III
Mean
1
45.67 62.67 61.67
56.67
45.67
2
59.67 50.00 49.67
53.11
58.5
56.33
Mean 52.67 56.33 55.67
54.89
59.5
62.67
K
I
II
III
1
43.0
45.0
49.0
2
53.5
57.0
3
61.5
67.0
Mean 52.67 56.33 55.67
Mean
54.89
Potash x Phosphorus
P
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
ANOVA
Source
df
SS
MS
Total
17
1833.78
Block
2
45.78
22.89
Potash
2
885.78
442.89
Error(a)
4
78.22
19.56
F
22.64**
Construct two-way tables
Phosphorus x Block
Potash x Block
P
I
II
III
Mean
1
45.67 62.67 61.67
56.67
45.67
2
59.67 50.00 49.67
53.11
58.5
56.33
Mean 52.67 56.33 55.67
54.89
59.5
62.67
K
I
II
III
1
43.0
45.0
49.0
2
53.5
57.0
3
61.5
67.0
Mean 52.67 56.33 55.67
Mean
54.89
Potash x Phosphorus
P
Main effect of Phosphorous
SSB=9*devsq(range)
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
ANOVA
Source
df
SS
MS
Total
17
1833.78
Block
2
45.78
22.89
Potash
2
885.78
442.89
Error(a)
4
78.22
19.56
Phosphorus
1
56.89
56.89
F
22.64**
Construct two-way tables
Phosphorus x Block
Potash x Block
K
I
II
III
1
43.0
45.0
49.0
2
53.5
57.0
3
61.5
67.0
Mean
P
I
II
III
Mean
1
45.67 62.67 61.67
56.67
45.67
2
59.67 50.00 49.67
53.11
58.5
56.33
Mean 52.67 56.33 55.67
54.89
59.5
62.67
Mean 52.67 56.33 55.67
54.89
Potash x Phosphorus
P
SSEB =
3*devsq(range) – SSR – SSB
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
ANOVA
Source
df
SS
MS
Total
17
1833.78
Block
2
45.78
22.89
Potash
2
885.78
442.89
Error(a)
4
78.22
19.56
Phosphorus
1
56.89
56.89
Error(b)
2
693.78
346.89
F
22.64**
.16ns
Construct two-way tables
Phosphorus x Block
Potash x Block
P
I
II
III
Mean
1
45.67 62.67 61.67
56.67
45.67
2
59.67 50.00 49.67
53.11
58.5
56.33
Mean 52.67 56.33 55.67
54.89
59.5
62.67
K
I
II
III
1
43.0
45.0
49.0
2
53.5
57.0
3
61.5
67.0
Mean 52.67 56.33 55.67
Mean
54.89
Potash x Phosphorus
P
K1
K2
K3
Mean
Interaction of P and K
1
46.00 58.67 65.33
56.67
SSAB=
3*devsq(range) – SSA – SSB
2
45.33 54.00 60.00
53.11
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**
.16ns
.71ns
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 splitplot
– 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.