Divisional Presentation Template

1
Modern Experimental Designs Construction and
Analysis
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Emlyn Williams CSIRO Australia
2
Topics covered

• Standard Design Types
• Completely Randomized
• Randomized Block
• Factorial
• Split-plot
• Incomplete Block Designs
• Alpha
• Row-column
• Crossover
• Spatial

3
Analysis of variance

• Completely Randomized Design
• Randomized Complete Block Design

4
Dbh (cm) means for a seed orchard (SO) and a
routine plantation (P) seedlot
Seedlot   Replication SO
P ______________________
1 30.38 28.16 2
27.91 25.62 3
28.06 26.61 4
31.42 32.59 5
30.11 28.80 6 31.52
28.19 7 31.72
31.23 8 33.53
28.34
5
GenStat output from analysis of variance of dbh
means for a Completely Randomized Design
Source of variation d.f. s.s.
m.s. v.r. F pr.   plot stratum seedlot
1 14.269 14.269 3.25
0.093 Residual 14 61.410
4.386 Total 15 75.679
Tables of means Grand mean 29.64
seedlot SO P 30.58
28.69
6
Completely Randomized Design
Model
7
Estimation - Completely Randomized Design
Overall mean
Seedlot effects
8
Residuals -Completely Randomized Design
Plot residuals vs fitted values
9
How good is the model?

• If the assumed linear model is good, then the
  residuals will be small
• We need to be able to measure how good the model
  is

10
Calculation of sums of squares
Seedlot sum of squares
Residual sum of squares
11
Check assumptions!!!!

• Use plot of residuals vs fitted values
• Look for possible outliers
• Look for any relationship between the fitted
  values and the spread of residuals

12
Example of a plot of residuals against fitted
values
I I
2 3 5 I
3 2 2 3 3 2 2
2 0.0 I 3 5 3 3 5
3 2 4 I
3 2
2 I 2
2 I I
I
-5.0 I I
I I
I
I -10.0 I
---------------------------------------------
------------ 0.0 2.0
4.0 6.0 8.0 10.0 12.0
resid v. fitted using
symbol
13
Layout of plots with seedlot labels (SO or P)
and dbh means
  Replicate   1 2 3 4
5 6 7 8
__________________________________________________
____ SO SO P SO P
P SO P 30.38 27.91 26.61 31.42
28.80 28.19 31.72 28.34     P P
SO P SO SO P SO 28.16
25.62 28.06 32.59 30.11 31.52 31.23 33.53
14
GenStat output from analysis of variance of dbh
means for a Randomized Complete Block Design
  Source of variation d.f. s.s.
m.s. v.r. F pr.   repl stratum
7 48.867 6.981 3.90   repl.plot
stratum seedlot 1 14.269
14.269 7.96 0.026 Residual
7 12.543 1.792   Total
15 75.679     Tables of means
  Grand mean 29.64   seedlot SO
P 30.58 28.69
15
Randomized Complete Block Design
Model
16
Estimation
Replicate effects
Fitted values
17
Residuals
Plot residuals vs fitted values
18
Residual sums of squares

• Completely randomized design61.410
• Randomized complete block design 12.543
• Residuals are smaller for RCB
• It is a better model

19
Testing

• We test the null hypothesis that the seedlots do
  not differ from each other
• Use the F test
• For the RCB design the F value is 7.96 on 1 and 7
  d.f. (5 value for F1,75.59)
• Reject the null hypothesis

20
Standard errors of difference (SED)

• Between two seedlot means is

where
t- value 1.89/0.67 2.82 (5 value for
t72.365)
21
Least significant difference (LSD)

• LSD t7 SED2.3650.67 1.585
• this is smaller than the difference between
  seedlot means (1.89)
• Hence seedlot means significantly different

22
Important issues

• Use of blocking structures (RCB reduces residual
  mean square from 4.386 to 1.792)
• Data needs to be presented in field order so that
  blocking structures are clear
• Check assumptions

23
Strata

• Each time we place a restriction on the
  allocation of treatments to plots, we create a
  stratum
• Each replicate contains one plot of each seedlot
  replicate stratum
• The plots within each replicate create the
  plot-within-replicate stratum

24
Layout of seedlings in four replicates of five
plots, each with four trees (each x refers to one
tree)
Replicate   Plot 1
2 3 4
__________________________________________ 1
x x x x x x x x x x x x x x x x   2
x x x x x x x x x x x x x x x
x   3 x x x x x x x x x x x x x
x x x   4 x x x x x x x x x x x x
x x x x 5 x x x x x x x x x x x
x x x x x     direction of trend

25
Replicate (repl) stratum
Replicate   Plot 1
2 3 4
__________________________________________ 1
x x x x x x x x x x x x x x x x   2
x x x x x x x x x x x x x x x
x   3 x x x x x x x x x x x x x
x x x   4 x x x x x x x x x x x x
x x x x 5 x x x x x x x x x x x
x x x x x     direction of trend

26
Plot-within-replicate (repl.plot) stratum
Replicate   Plot 1
2 3 4
__________________________________________ 1
x x x x   2 x x x x   3 x
x x x   4 x x x x 5 x x x
x     direction of trend

27
Tree-within-plot-within-replicate
(repl.plot.tree) stratum
Replicate   Plot 1
2 3 4
__________________________________________ 1
x x x x   2   3   4
5     direction of
trend

28
Two-dimensional layout with four rows and four
columns (each X refers to a whole plot)
Column   Row 1
2 3 4
______________________ 1 x x
x x direction 2
x x x x of trend  
3 x x x x   4
x x x x    
direction of trend
29
What are the strata?

• Row (row) stratum
• Column (column) stratum
• Plot (row.column) stratum
• Could also then have a row.column.tree stratum

30
Factorial Designs

• More than one treatment factor
• e.g. seedlot, fertilizer
• or treatments A and B

model
31
Example of interaction
Interaction caused by combined effect of A and B
32
Germination test

• Treatment A
• 6 Acacia mangium seedlots
• Treatment B
• 4 seed pre-treatments
• control
• nick
• boiling water and soak
• boiling water 1 min
• 25 seeds per dish
• 3 replicates (trays)
• 24 dishes per replicate (4 x 6 array)
• variate is percentage germination

33
Control comparison

• In GenStat we can set up an option to compare the
  control treatment against the other treatments
• This gives a test of whether the treatments
  overall are doing better than the control

34
GenStat output from analysis of variance of
percentage germination
Analysis of variance Variate v1
percent - percent(count/25)100 Source of
variation d.f. s.s. m.s. v.r.
F pr. repl stratum 2 35.11
17.56 0.18 repl.row.column
stratum contcomp 1 58542.30
58542.30 601.52 lt.001 seedlot
5 2894.44 578.89 5.95
lt.001 contcomp.treat 2 5300.15
2650.07 27.23 lt.001 contcomp.seedlot
5 1347.04 269.41 2.77
0.029 contcomp.treat.seedlot 10 961.19
96.12 0.99 0.467 Residual
46 4476.89 97.32 Total
71 73557.11
35
(continued)
Tables of means Variate v1
percent - percent(count/25)100 Grand mean
51.4 contcomp 1 2
2.0 67.9 rep. 18 54
seedlot 18265 18249 18248 18211
18212 18217 59.7 48.7
40.7 58.0 52.3 49.0 contcomp
treat control nick bws bw1min
1 2.0 2
56.9 65.8 80.9
36
Plot of residuals against fitted values from
analysis of variance of germination percentage
I I I
25.0 I I
I
I

I
I
2 0.0
I 73
2 I 2
2 I
2
I
I
I -25.0 I

---------------------------------------------
------------ -20.0 0.0
20.0 40.0 60.0 80.0 100.0
resid v. fitted using
symbol
37
Options to correct for problem with spread of
residuals

• Transformation?
• Here small fitted values and small range of
  residuals related just to the control treatment,
  so better to remove the control from the analysis
• Use the restrict option of GenStat

38
GenStat output from analysis of variance with
the control pre-treatment deleted
Analysis of variance Variate v1
percent - percent(count/25)100 Source of
variation d.f. s.s. m.s. v.r.
F pr. repl stratum 2 64.6
32.3 0.25 repl.row.column stratum treat
2 5300.1 2650.1
20.61 lt.001 seedlot 5
4148.1 829.6 6.45 lt.001 treat.seedlot
10 961.2 96.1 0.75
0.676 Residual 34 4372.7
128.6 Total 53 14846.8
39
Interpretation of analysis

• Seedlots significantly different (). Best
  germination is seedlot 18265 (79.1)LSD10.6
• Control treatment removed from the analysis
• Other pre-treatments significantly different
  (). Best pre-treatment is boiling water for
  one minute (80.9)LSD7.67
• Interaction not significant

40
Split-plot designs

• More than one treatment factor
• Strata normally called replicates,
  main-plots-within-replicates and
  sub-plots-within-main-plots-within-replicates
• Treatment factors on different strata
• e.g. basal nitrogen at the main-plots level,
  seedlots at the sub-plots level

41
Split-plot model
42
Irrigation / Fertilizer trial

• Main-plot treatment factors
• irrigation (yes or no)
• fertilizer (yes or no)
• Sub-plot treatment factor
• 4 Eucalyptus grandis seedlots
• 2 replicates
• 42 trees per plot (7 x 6)
• Variate is height at 34 months

43
Layout of plots with seedlot numbers and height
means

Replicate
1
2    irrigation none none plus plus
plus plus none none fertilizer none
plus plus none none plus none
plus   _______________________________
_______________________
4 2 1 3 2
1 4 3 4.71 16.36 14.38
4.66 5.41 14.60 4.32 14.98
3 1 2
4 3 4 1 2
6.23 15.29 16.89 4.95 5.73 12.21 4.16
15.98 2 3 4
1 4 2 3 1
7.46 13.99 11.25 5.81 5.80 14.84 5.02
14.40 1 4 3
2 1 3 2 4
6.39 11.08 15.58 7.50 6.39 15.00 6.79
11.98    Seedlots 1
Bulahdelah 2
Coffs Harbour seed orchard
3 Pomona plantation
4 Atherton
44
Plot of residuals against fitted values from
analysis of variance of height means
1.2 I I
I
I

I
I
0.0 I
I
I
2
I
I
I
-1.2 I
---------------------------------------------
------------ 2.5 5.0
7.5 10.0 12.5 15.0 17.5
resid v. fitted using
symbol
45
GenStat output from analysis of variance of
height means
Source of variation d.f. s.s.
m.s. v.r. F pr. repl stratum
1 0.7564 0.7564 1.08 repl.mainpl
stratum irrig 1 0.1081
0.1081 0.15 0.721 fert
1 590.6485 590.6485 841.11
lt.001 irrig.fert 1 0.0072
0.0072 0.01 0.926 Residual
3 2.1067 0.7022 1.05
repl.mainpl.subpl stratum seedlot
3 39.6538 13.2179 19.68
lt.001 irrig.seedlot 3 1.1098
0.3699 0.55 0.657 fert.seedlot
3 9.9503 3.3168 4.94
0.018 irrig.fert.seedlot 3 1.7360
0.5787 0.86 0.487 Residual
12 8.0596 0.6716 Total
31 654.1364
46
(continued)
Tables of means Grand mean 10.00
irrig none plus 9.95
10.06 fert none plus
5.71 14.30 seedlot Bulahdelah Coffs SO
Pomona pltn Atherton 10.18
11.40 10.15 8.29
fert seedlot Bulahdelah Coffs SO
Pomona pltn Atherton none
5.69 6.79 5.41 4.95 plus
14.67 16.02 14.89
11.63
47
Interpretation of analysis

• Fertilizer significant ()
• Irrigation not significant
• Irrigation by fertilizer interaction not
  significant
• Seedlots significant ()
• Coffs harbour seed orchard best (11.40m)
• LSD 0.89
• Seedlot by fertilizer interaction significant ()
• due to different behaviour of Atherton seedlot

48
Cross-over model
49
2 x2 drug trial(Jones and Kenward Ex 2.1)

• 56 patients
• Drug (A) vs Placebo (B)
• 27 in AB group, 29 in BA group
• Variate is exploratory flow rate (response)

50
Part of the data file for 2 x 2 drug trial
51
GenStat analysis of variance of 2 x 2 drug trial
Source of variation d.f. s.s.
m.s. v.r. F pr. patient stratum group
1 10572.7 10572.7 0.90
0.347 Residual 54 634865.6
11756.8 36.04 period stratum treat
1 479.6 479.6
patient.period stratum treat
1 3026.1 3026.1 9.28 0.004 Residual
54 17617.1 326.2 Total
111 666561.1
52
(continued)
Tables of means Variate response
Grand mean 232.5 group AB BA
242.5 223.1 rep. 54
58 treat Drug(A) Placebo(B)
237.7 227.3 Standard errors of
differences of means Table
group treat rep. unequal
56 d.f. 54
54 s.e.d. 20.50 3.42
53
Plot of residuals against fitted values from
analysis of variance of response
I 40.0 I I
I
I
I
2 2 I
3 24
0.0 I 2
322 2 I
323
I 3 2
I
I
I
-40.0 I ------------------
----------------------------------------
0.0 80.0 160.0 240.0
320.0 400.0 480.0
resid v. fitted using symbol
54
SAS analysis of variance of 2 x 2 drug trial

Sum of Source DF
Squares Mean Square F Value Pr gt F  
Model 57 648943.9852 11384.9822
34.90 lt.0001   Error 54
17617.1338 326.2432   Corrected Total
111 666561.1190     Source
DF Type I SS Mean Square F Value Pr
gt F   GROUP 1 10572.6799
10572.6799 32.41 lt.0001 PATIENT(GROUP)
54 634865.5790 11756.7700 36.04
lt.0001 PERIOD 1 479.6066
479.6066 1.47 0.2306 TREAT
1 3026.1198 3026.1198 9.28
0.0036  
55
(continued)

H0LSMean1
RESPONSE LSMean2
TREAT LSMEAN Pr gt
t   Drug(A)
238.000215 0.0036
Placebo(B) 227.597632  

Dependent Variable RESPONSE response
Standard

Parameter
Estimate Error t Value Pr gt t  
Trt2 -Trt1 -10.4025830 3.41561476
-3.05 0.0036
56
Incomplete block designs

• Explanation
• Model
• Choice of incomplete block design
• Alpha designs
• Row-column designs
• Latinized row-column designs
• CycDesigN

57
An RCB design for three replicates of 42
seedlots
42 12 15 37 17 22 4
21 26 25 18 28 36 16 10
35 6 29 11 8 2 Replicate 1
23 38 41 3 31 24 30 39
1 27 40 14 34 7 19 20 33
5 9 13 32   25 16 5 6
29 40 24 23 33 21 42 22 19
41 12 36 8 34 32 15 39
Replicate 2 2 20 28 35 26 13
3 9 14 1 7 10 4 30
18 37 27 17 31 38 11   33
16 21 7 1 41 37 17 28 40
29 24 36 35 4 26 34 11 10
18 39 Replicate 3 22 19 9
13 14 31 25 5 6 20 12 2
38 32 23 3 30 27 15 42 8
Equal variance between any two plots
58
Problems with RCB assumptions

• Equal pairwise plot variances within replicates?
• E.g. in example 42 plots of 5 x 5 trees at 2m x
  2m spacing 0.43 ha per replicate
• too big to assume uniformity
• Better to break the area up into smaller units
• E.g. Incomplete blocks

59
Replicate 1 subdivided to illustrate the use of
incomplete blocks to adjust for trend 

Block   1 2 3 4 5
6 7 __________________________
______ 42 12 15 37
17 22 4 21 26 25 18
28 36 16 10 35 6 29
11 8 2 23 38 41 3
31 24 30 39 1 27 40
14 34 7 19 20 33 5
9 13 32     Low
High ground
ground
  direction of trend

60
Model for an incomplete block design
61
Choice of incomplete block design

• Many ways to organize seedlots in incomplete
  block designs
• Which is the best??
• We need a measure of how good a design is
  relative to other designs with the same
  parameters r, v and k
• We use the average efficiency factor, E

62
Two possible arrangements for an incomplete
block design with r 2, v 9 and k 3
  Replicate 1 Replicate
2   Block 1 2 3 1 2
3 ____________
___________   1 4 7
1 2 3 2 5 8
4 5 6 3 6 9 7
8 9       Replicate 1
Replicate 2   Block 1 2 3
1 2 3 ___________
___________   1 4 7
1 5 4 2 5 8 2
8 6 3 6 9 3
9 7
63
Alpha designs

• They are a type of incomplete block design
• One-dimensional blocking structure
• Available for all parameter sets
• Good for nested treatment stuctures
• Construct designs in CycDesigN

64
Unrandomized alpha design for r 3, s 5 and k
4
Block  
1 2 3 4 5
_____________________   1 2
3 4 5 6 7 8 9 10
Replicate 1 11 12 13
14 15 16 17 18 19
20       1
2 3 4 5 7 8 9
10 6 Replicate 2 13
14 15 11 12 19 20 16
17 18     1 2 3 4 5
8 9 10 6 7
Replicate 3 15 11 12 13
14 17 18 19 20 16
Randomization?
Nested treatment structure?
65
Alpha design 1D - blocking v19, k4,5, r3
66
Row-column designs

• Two-dimensional blocking structures
• Strata
• replicates
• rows within replicates
• columns within replicates
• plots
• Better field control than incomplete blocks
• Latinized designs if replicates are together

67
Latinized row-column design for v20, s5, k4,
r3
68
2-Latinized row-column design for v18, s6, k3,
r3
69
Partially-latinized design for v9, s3, k3, r4
70
Software - CycDesigN

• Windows 95 to XP
• Visual C
• Resolvable / non-resolvable
• Block / row-column
• One / two stage
• Cyclic / alpha / other
• Factorial / nested treatments
• t-Latinized / partially-latinized
• Unequal block sizes
• http//www.ffp.csiro.au/software/

71
Analysis of Incomplete Block Designs

• Fixed-Effects model
• Mixed-Effects model
• Summary of Analysis options
• Nested treatment structure

72
Model for an Incomplete Block Design (Fixed
Effects)
73
Incomplete Block Design Matrix form of Model
distributed as
74
Replicate 1
Replicate 2   Block 1 2 3 1
2 3 ___________
___________   1 4 7 1
2 3 2 5 8 4
5 6 3 6 9 7
8 9
75
Treatment comparisons in an RCB design
4 seedlots 2 replicates
Replicate 1 2 1 1 2 2 3 3 4 4
All pairwise comparisons available in each
replicate
76
Treatment comparisons in an Incomplete Block
Design
4 treatments 2 replicates 2 plots per
incomplete block
Replicate 1
2 Block 1 2 1 2 1 3 1 2 2 4 3 4
Direct comparison
Indirect comparison
Block comparison
Average Efficiency Factor
77
Incomplete Block Designs - Tradeoffs

• Extra blocking structures help to reduce the
  residual mean square
• Smaller block sizes
• better control of field variation
• less treatment information within incomplete
  blocks
• smaller average efficiency factor
• Decrease in residual mean square needs to
  outweigh loss of treatment information within
  blocks

78
Mixed-effects Model
where
is distributed as
and independent of
Can write the model as
where
79
Analysis of incomplete block designs

• Mixed-model analysis
• blocks random
• variance between two plots
• within a block
• between blocks
• estimated treatment means can be thought of as a
  weighted combination of information within and
  between blocks

80
Combination of treatment information
Special Cases
81
Analysis of a Latinized Row-column Design

• Weipa casuarina equisetifolia trial
• 4 replicates, 6 rows and 10 columns
• 60 families
• originally 64 but 4 did not germinate
• nested treatment structure for regions
• Inoculation treatment at repls level

82
Country of origin of the provenances
Country
Provenance Number Name
Number _____________________________
_______________ 1 Australia
1,2,3,4 2 Benin
5 3 China
6,7,63 4 Egypt
8,10,11 5 Fiji
12,13,56 6 Guam
14 7 India
15,16,17,18,19,20 8 Kenya
21,22,23,24,25,26,27,28 9
Malaysia 29,30,31,32,33,34,35,36,57
10 Mauritius 58
11 PNG 37 12
Philippines 38,39,40 13
Puerto Rico 59 14 Solomon
Is. 41,42 15 Sri Lanka
60,61,62 16 Thailand
45,46,47,48 17 Vanuatu
50,64 18 Vietnam
51,52,53,54,55
83
Field layout

Column   1 2 3 4 5 6
7 8 9 10 Row
_____________________________________  
1 20 51 8 27 13 56 16 25 54 45
2 36 11 32 38 2 37 29 48
17 23 3 21 41 58 52 15 60
26 4 30 14 Replicate 1 4
24 64 35 53 22 61 10 6 42 12
5 59 55 18 28 31 33 40 62 63 46
6 34 3 50 19 7 57 47 39
5 1 1 4 63 31 40 36 20
14 47 25 6 2 19 15 24 54
28 30 55 3 41 61 3 29 57
42 18 32 26 17 59 58 34 Replicate 2
4 45 53 16 23 56 2 22 60 7
13 5 5 21 46 51 1 38 48
52 12 10 6 50 39 11 37 62
35 27 8 33 64 1 31 13 20
7 37 51 8 56 26 16 2 38
17 28 48 29 59 1 64 3 5 3
58 35 33 21 12 27 42 54 14 32
Replicate 3 4 62 23 63 50 30
15 57 22 36 2 5 55 47 40
10 11 46 19 45 18 41 6 61
4 6 60 39 25 34 24 53 52 1
48 56 2 42 26 23 20 12 32 55
2 40 16 29 62 58 63 61 5 13
47 3 15 8 34 22 6 36 4
10 19 39 Replicate 4 4 52
30 25 33 17 64 41 38 46 24 5
11 27 14 3 45 31 54 57 21 18
6 1 7 60 35 59 50 53 51 37
28
84
Part of the GenStat output from the RCB analysis
Analysis of variance Variate v1
dbh - dbhsqrt(dbh1dbh1 dbh2dbh2
dbh3dbh3) Source of variation d.f.
s.s. m.s. v.r. F pr. repl
stratum inoc 1 11.5415
11.5415 11.46 0.077 Residual
2 2.0142 1.0071 1.66
repl.row.column stratum country
17 54.6185 3.2129 5.30
lt.001 inoc.country 17 10.0724
0.5925 0.98 0.487 country.prov
41 18.6057 0.4538 0.75
0.854 inoc.country.prov 41 21.4625
0.5235 0.86 0.698 Residual
116 70.2557 0.6057 Total
235 188.5705
85
Plot of residuals against fitted values from RCB
analysis of variance of dbh means
I I I 1.5 I
I
I 2
I 2
I 2 23 2222
I 2 2 3 233
22422223 0.0 I
2 22 232 2 2 42 I
2 2 2 2433 6232 3
I 2 3 4 22 2 2
I 2
I 2 2
I -1.5 I
------- -----
---------- ------- -------- --------- -----
1.6 2.4 3.2 4.0
4.8 5.6 6.4
resid v. fitted using symbol
86
Part of the GenStat output from the
fixed-effects analysis
Regression Analysis Response
variate v1 dbh - dbhsqrt(dbh1dbh1
dbh2dbh2
dbh3dbh3) Fitted terms Constant inoc
repl column repl.row
repl.column prov inoc.prov Accumulated
analysis of variance Change d.f.
s.s. m.s. v.r. F pr. inoc
1 11.5415 11.5415
48.05 lt.001 repl 2 2.0142
1.0071 4.19 0.020 column
9 65.2421 7.2491 30.18 lt.001
repl.row 20 16.5932 0.8297
3.45 lt.001 repl.column 27 16.4051
0.6076 2.53 0.001 prov
58 53.8939 0.9292 3.87 lt.001
inoc.prov 58 8.4698 0.1460
0.61 0.971 Residual 60 14.4107
0.2402 Total 235
188.5705 0.8024
87
GenStat output from the mixed-effects analysis
REML Variance Components Analysis
Response Variate v1 dbh -
dbhsqrt(dbh1dbh1 dbh2dbh2 dbh3dbh3)
Fixed model Constantreplcolumnprov Rand
om model repl.rowrepl.column  
Estimated Variance Components Random term
Component S.e. repl.row
0.0640 0.0294 repl.column
0.0459 0.0262 units
0.1951 0.0253 Wald tests
for fixed effects Fixed term
Wald statistic d.f. repl
12.8 3 column
144.9 9 prov
284.1 58
88
Table of estimated seedlot means from analyses of
the RCB and mixed-effects models
Seedlot RCB Mixed 1
1.85 2.42 2 3.24 3.14
3 2.94 2.86 4
2.50 2.26 5 3.34 3.72
6 3.38 3.54 7
3.98 3.95 8 2.64 2.75
10 2.47 2.65 11
2.40 2.05 12 2.39 2.75
13 2.58 2.98 14
2.34 2.65 15 3.77 3.45
16 3.60 3.92 17
3.45 3.43 18 3.26 3.36
19 3.83 3.65 SED
0.55 0.35
89
Rows within replicates BLUPs
90
Estimated long column means
91
Summary of Analysis Options

• Replicates usually fixed
• physical interpretation
• (no) treatment information
• computation advantage
• Two-dimensional blocking structures
• Latinized designs for contiguous replicates
• long columns usually fixed

92
Nested Treatment Structure

• Number treatments 1 to v
• Better to initially estimate treatment effects
  without the nested structure
• Then analyse estimated treatment using the nested
  structure

93
GenStat output from the analysis of nested
seedlot structure
Analysis of variance Variate v1
dbh - dbhsqrt(dbh1dbh1 dbh2dbh2
dbh3dbh3) Weight variate trepl Source of
variation d.f. s.s. m.s. v.r.
F pr. country 17 47.8213
2.8130 14.42 lt.001 country.prov
41 13.6472 0.3329 1.71 0.013 residual
119 0.1951
94
13 x 4 dose-response study(Jones and Kenward Ex
5.1)

• Investigation of respiratory failure
• 13 subjects (babies)
• 4 doses of nitric oxide
• Variate is post-ductal arterial oxygen tension
  (pco2resp)

95
Design of dose-response study
Treatments (Doses of nitric oxide) A
5ppm B 10ppm C 20ppm D 40ppm
96
Plot of residuals against fitted values from the
analysis of pco2resp
  2
- I I
I 3.0 I I

I
I
I
I
2 0.0 I
2
I 2
I
I
I
I

-3.0 I ------------------
----------------------------------------
0.0 2.5 5.0 7.5
10.0 12.5 15.0
resid v. fitted using symbol
97
GenStat REML output for analysis of 13x4
dose-response study
Residual variance model Term
Factor Model(order) Parameter
Estimate S.e. Residual
Identity Sigma2 2.961
0.806 Wald tests for fixed effects
Fixed term Wald statistic
d.f. Wald/d.f. Chi-sq prob Sequentially
adding terms to fixed model subject
128.75 12 10.73
lt0.001 period 3.70
3 1.23 0.296 carry1
4.12 3 1.37
0.249 dose 0.67
3 0.22 0.879  
98
SAS mixed output for analysis of 13x4
dose-response study
Covariance Parameter
Estimates  
Cov Parm Estimate  
Residual 2.9614    
Type 1 Tests of Fixed Effects  
Num Den
Effect DF DF F Value
Pr gt F   SUBJECT
12 27 10.73 lt.0001
PERIOD 3 27 1.23
0.3172 DOSE 3
27 0.53 0.6636
CARRY1 3 27 1.06
0.3804  
99
Spatial Model
Alternative form for the incomplete block mixed
model
where
Spatial specification
Linear Variance
Williams et al. Biometrics (to appear)