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
FA
SSEA
MSEA
Factor A error
SSB
MSB
FB
(r-1)(b-1)
SSEB
MSEB
Factor B error
(a-1)(b-1)
SSAB
MSAB
FAB
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
Computations
 There are three error terms - one for each main plot and
interaction plot
SSTot

 i  j  k Yijk  Y

rb   Y
SSR
ab  k Y ..k  Y
SSA

SSEA
i
i..
Y


2
2
2

b  i  k Y i .k  Y

2
 SSA  SSR
SSAB
Y  Y 
a    Y  Y   SSB  SSR
r   Y  Y   SSA  SSB
SSEAB
SSTot-SSR-SSA-SSEA-SSB-SSEB-SSAB
2
SSB
ra 
. j.
j
2
SSEB
j
k
. jk
2
i
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
M SE A
rb
 Factor B Means
M SEB
ra
 Treatment AB Means
M S E AB
r
SE of Differences for Main Effects
 Differences between 2 A means
2 * M SE A
with (r-1)(a-1) df
rb
 Differences between 2 B means
2 * M SEB
ra
with (r-1)(b-1) df
SE of Differences
 Differences between A means at same level of B
2 *   b  1  M S E A B  M S E A 
rb
 Difference between B means at same level of A
2 *   a  1  M S E A B  M S E B 
ra
 Difference between A and B means at diff. levels
2 *   a b  a  b  M S E A B  a * M S E A  b * M S E B 
ra b
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
Potash x Block
Phosphorus x Block
K
I
II
III
Mean
P
1
43.0
45.0
49.0
45.67
1
45.67 62.67 61.67
56.67
2
53.5
57.0
58.5
56.33
2
59.67 50.00 49.67
53.11
3
61.5
67.0
59.5
62.67
Mean 52.67 56.33 55.67
54.89
Mean 52.67 56.33 55.67
I
II
III
Mean
54.89
SSR=6*devsq(range)
Potash x Phosphorus
P
Main effect of Potash
SSA=6*devsq(range)
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
Construct two-way tables
Potash x Block
Phosphorus x Block
K
I
II
III
Mean
P
1
43.0
45.0
49.0
45.67
1
45.67 62.67 61.67
56.67
2
53.5
57.0
58.5
56.33
2
59.67 50.00 49.67
53.11
3
61.5
67.0
59.5
62.67
Mean 52.67 56.33 55.67
54.89
Mean 52.67 56.33 55.67
I
II
III
Mean
54.89
Main effect of Phosphorus
SSB=9*devsq(range)
Potash x Phosphorus
P
K1
K2
K3
Mean
SSEB =
3*devsq(range) – SSR – SSB
1
46.00 58.67 65.33
56.67
2
45.33 54.00 60.00
53.11
SSAB=
3*devsq(range) – SSA – SSB
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**
0.16ns
0.71ns
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
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
Formal names for required assumptions
 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
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

 Y  Y   Y   Y  2  Y ,Y
2
2
1
2
sed2
Y  Y
2
2
1
2
2
1
se2
2
se2
 Y ,Y  0
1
2
1
2
covariance
 Y Y 
2
1
2
2M S E
r
Covariance Structure for Residuals
 No correlation (independence)
– 4 measurements per subject
2

– All covariances = 0
0
  
 0

 0
0
0

0
2
0

0
0
2
0 
1


0
0
  2 
0
0 

2
 
0
0
0
1
0
0
1
0
0
0

0

0

1
 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)
1


2

  





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)
 1
Complex and computer intensive

 12
No particular pattern for the
2
 
covariances is assumed
  13
May have low power due to loss of df

for error
  14
 Unstructured (UN)
–
–
–
1

2  
 
 2
 3



1


1


2
 12
 13
1
 23
 23
1
 24
 34
2
3
 
2




1
 14 

 24

 34 

1 
Mixed Model adjustment for error structure
 Stage one: estimate covariance structure for residuals
1. 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
= (-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