Combined Experiments as Split Plots
Repeat experiment at several times
Repeat experiment at several locations
Measure same experiment several times (repeated measures)
Advantages
1. Assess performance over wide range of conditions, eg climate, soil
2. Assess consistency of performance
3. Better basis for practical recommendations
4. Can have larger number of reps
Cautions
1. Variance may not be homogeneous over time and location
2. Time is systematic in order - measurements can not be randomized to times
3. Errors are not independent in repeated measures. Successive measurements on the
same experimental unit are correlated.
Common uses
1. Variety trials
2. Comparing results from different labs or experiment stations, eg monitoring food
processing at several plants
Analysis
1. As split plot, even though experiment may be installed at CRD, RCBD or Latin
square
2. Very different variance components with measurements over space and time. Must
be partitioned from variance for individual treatments
Repeated Experiments at Different Locations and/or Times
For repeated experiments a separate randomization is done at each time or location
Time or location is the main plot
e.g.
4 Treatments Tested at 3 Different Hospitals
10 Patients per Treatment at each Hospital
ANOVA for 1 Hospital
Combined ANOVA
Source
df
Source
df
Total
39
Total
119
Treatment
3
Hospital
2
Patient within Trt (Error)
36
Rep within Hospital
27
Treatment
3
1
Treatment x Hospital
6
Error
81
Repeated Measures
For repeated measures the experiment is not re-randomized between measurements
Time is in the split plot ie time is nested in treatment
e.g.
3 Pig Feeds
4 Blocks (Breeds)
Growth Measured Weekly for 5 Weeks
ANOVA for 1 Week
Combined ANOVA for 5 Weeks
Source
df
Source
df
Total
11
Total
59
Feed
2
Feed
2
Block
3
Block
3
Error
6
Error a
6
Week
4
Feed x Week
8
Error b
36
Example: Randomized Complete Block Design
Experiment with:
4 Varieties of white clover
3 Blocks
4 Harvests
V1
V4
V3
V2
Block 1
V4
V2
V1
V3
Block 2
V2
V3
V4
V1
Block 3
Harvested (cut) 4 times a year
2
ANOVA for 1 cutting
Combined analysis for 4 harvests
Source
df
Source
df
Total
11
Total
47
Blk
2
Blk
2
Var
3
Var
3
Error
6
Err A
6
Cutting
3
CxV
9
Err B
24
Alfalfa Variety Trial (from Little and Hills)
RCBD with 4 Varieties, 5 Blocks, 4 Harvests. Data are tons/acre alfalfa
Blocks
Variety
Harvest
1
2
3
4
5
Total
Mean
1
1
2.69
2.40
3.23
2.87
3.27
14.46
2.89
2
1
2.87
3.05
3.09
2.90
2.98
14.89
2.98
3
1
3.12
3.27
3.41
3.48
3.19
16.47
3.29
4
1
3.23
3.23
3.16
3.01
3.05
15.68
3.14
Totals
11.91
11.95
12.89
12.26
12.49
61.50
1
2
2.74
1.91
3.47
2.87
3.43
14.42
2.88
2
2
2.50
2.90
3.23
2.98
3.05
14.66
2.93
3
2
2.92
2.63
3.67
2.90
3.25
15.37
3.07
4
2
3.50
2.89
3.39
2.90
3.16
15.84
3.17
Totals
11.66
10.33
13.76
11.65
12.89
60.29
1
2
3
4
5
Variety
Harvest
Total
Mean
1
3
1.67
1.22
2.29
2.18
2.30
9.66
1.93
2
3
1.47
1.85
2.03
1.82
1.51
8.68
1.74
3
3
1.67
1.42
2.81
1.51
1.76
9.17
1.83
4
3
2.60
1.92
2.36
1.92
2.14
10.94
2.19
3
Totals
7.41
6.41
9.49
7.43
7.71
38.45
1
4
1.92
1.45
1.63
1.60
1.96
8.56
1.71
2
4
2.00
2.03
1.71
1.60
1.96
9.30
1.86
3
4
2.03
1.96
1.85
1.82
2.40
10.06
2.01
4
4
2.07
1.89
1.92
1.82
1.78
9.48
1.90
Totals
8.02
7.33
7.11
6.84
8.10
37.40
Variety by Block Totals
Block
Variety
1
2
3
4
5
Total
Mean
1
9.02
6.98
10.62
9.52
10.96
47.10
9.42
2
8.84
9.83
10.06
9.30
9.50
47.53
9.51
3
9.74
9.28
11.74
9.71
10.60
51.07
10.21
4
11.40
9.93
10.83
9.65
10.13
51.94
10.39
Totals
39.00
36.02
43.25
38.18
41.19
197.64
ANOVA for each Alfalfa Harvest
Harvest 1
MS
Harvest 2
SS
MS
Harvest 3
Source
df
SS
Total
19
1.180
Blocks
4
0.165
0.041
1.725
0.431
1.256
0.314
0.311
0.078
Variety
3
0.473
0.158
0.255
0.085
0.566
0.189
0.230
0.077
12vs34
1
0.392
0.392*
0.227
0.227
0.157
0.157
0.141
0.141*
1vs2
1
0.019
0.019
0.006
0.006
0.096
0.096
0.055
0.055
3vs4
1
0.062
0.062
0.022
0.022
0.313
0.313
0.034
0.034
Error
12
0.542
0.045
1.125
0.094
1.479
0.123
0.297
0.025
3.105
SS
MS
Harvest 4
3.302
4
SS
MS
0.838
Test for Homogeneity of Variance
Compare Error variances or MS Error
F test for 2 variances
larger variance in the numerator
smaller variance in the denominator
compared to tabular values in the standard F table
use the df associated with each of the variances
Bartlett's test for more than 2 variances uses a Chi-square test for homogeneity of
variance
Example comparing MS Error for each alfalfa harvest.
Coded si2*
si
Log coded si22
Harvest
df
1
12
0.0452
4.52
0.656
2
12
0.0937
9.37
0.972
3
12
0.1233
12.33
1.091
4
12
0.0247
2.47
0.393
Total
28.69
Mean
28.69/4 = 7.1725
Log of mean
3.112
0.856
*Values coded to avoid negative logs by multiplying Column 3 by 100.
For low df,
must be adjusted by dividing by a correction factor, C.
5
The adj
value exceeds the value at the 0.05 probability, so
the null hypothesis is rejected
there is a significant difference among the variances of the 4 harvests
Harvests can be combined into a single ANOVA only with caution.
ANOVA is considered to be “robust”, i.e., not too seriously affected by small departures
from homogeneity of variances.
ANOVA for Combined Alfalfa Harvests
Source
df
SS
MS
F
F.05
F.01
Total
79
34.8690
Main plot
(19)
5.0769
Block
4
1.9386
0.4846
Variety
3
0.9014
0.3005
1.61
3.49
5.95
1+2vs3+4
1
0.8778
0.8778
4.71
4.75
9.33
1 vs 2
1
0.0046
0.0046
3 vs 4
1
0.0189
0.0189
Error a
12
2.2369
0.1864
Harvest
3(1)
26.4452
8.8151
4.49*
8.53*
Var x Har
9(3)
0.6217
0.0690
3.24*
5.29*
Error b
48(16)
2.7252
0.0568
155.2
1.21
*df in parenthesis are divided by df harvest. F test is for adjusted df.
Repeated Measures - a better approach
Calculate a growth curve for each plot, animal or experimental unit.
Compare the curves (intercept, linear and quadratic coefficients) to determine treatment
effects.
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