Type 2 Modified Augmented Design:
Rows, Columns, Whole Plots, Subplots Primary Check
The field is divided into rows and columns, which form a grid of whole plots (i.e.,
incomplete blocks). The experimental units (plots) within each whole plot are referred to as
subplots. At the center of each whole plot, there is a subplot of the primary check cultivar.
Column
1
1
2
4
3
5
6
WholePlot
(wp) 1
wp2
wp3
wp4
wp5
wp6
wp7
wp8
wp9
wp10
wp11
wp12
wp13
wp14
wp15
wp16
wp17
wp18
wp19
wp20
wp21
wp22
wp23
wp24
Row
2
3
4
= Primary Check
(Check1)
Type 2 Modified Augmented Design:
Secondary Checks
An arbitrary number of secondary checks is selected (must be >1); here, I selected 2. An
arbitrary number of whole plots is selected to be assigned secondary checks; here, I
selected 4. One subplot of each secondary check is randomly located within each of the
selected whole plots.
Column
1
1
2
4
3
5
6
WholePlot
(wp) 1
wp2
wp3
wp4
wp5
wp6
wp7
wp8
wp9
wp10
wp11
wp12
wp13
wp14
wp15
wp16
wp17
wp18
wp19
wp20
wp21
wp22
wp23
wp24
Row
2
3
4
= Primary Check
(Check1)
= Secondary Check A
(Check2)
= Secondary Check B
(Check3)
Type 2 Modified Augmented Design:
Adjustments – Method 1
There are two ways to make adjustments, Method 1 and Method 3. Method 1 ignores the
secondary checks and adjusts experimental lines based on row and column averages for
the primary check alone. To obtain Method 1 adjustments, first run an ANOVA with
Check1 plots as the dependent variable and Row and Column as class effects.
Column
1
1
2
4
3
5
6
WholePlot
(wp) 1
wp2
wp3
wp4
wp5
wp6
wp7
wp8
wp9
wp10
wp11
wp12
wp13
wp14
wp15
wp16
wp17
wp18
wp19
wp20
wp21
wp22
wp23
wp24
Row
2
3
4
= Primary Check
(Check1)
= Secondary Check A
(Check2)
= Secondary Check B
(Check3)
Type 2 Modified Augmented Design:
1/18/13
Adjustments - Method 1 - ANOVA
SAS Output
Method 1 ANOVA -­ Spike Length (cm)
The GLM Procedure
Dependent Variable: SpkLngth
Source
DF
Sum of Squares
Mean Square
F Value
Pr > F
8
1.39740741
0.17467593
0.79
0.6212
Error
15
3.32685185
0.22179012
Corrected Total
23
4.72425926
Model
R-­Square
Coeff Var
Root MSE
SpkLngth Mean
0.295794
8.619245
0.470946
5.463889
DF
Type I SS
Mean Square
F Value
Pr > F
Row
3
0.74537037
0.24845679
1.12
0.3721
Column
5
0.65203704
0.13040741
0.59
0.7093
Source
DF
Type III SS
Mean Square
F Value
Pr > F
Row
3
0.74537037
0.24845679
1.12
0.3721
Column
5
0.65203704
0.13040741
0.59
0.7093
Source
“whole plot error”=
1) error for comparing
plots in different
whole plots
2) Error used to
calculate standard
error for T3
Use F test results for
Row and Column
effects to help you
determine Method 1
appropriateness
Method1AdjustedLine102Row(R)2 Column(C)3 = RawLine102R2C3 – (Check1AvgR2 – CheckAvg) - (Check1AvgC3 – Check1Avg)
Line “MALT-102” from data in
corresponding Excel file
= RawLine102R2C3 – Check1AvgR2 – Check1AvgC3 + 2*Check1Avg
= 8.47 – 5.86 – 5.58 + 2*5.46
= 7.95
Type 2 Modified Augmented Design:
Adjustments – Method 3
Method 3 uses data from all checks to calculate adjustments. Adjustments are based on
the slope from regressing the average of all secondary check subplots within each whole
plot upon the primary check subplot within each whole plot. Thus, the adjustment
parameter is estimated using only check subplots in wholeplots with secondary checks.
Column
1
1
2
4
3
5
6
WholePlot
(wp) 1
wp2
wp3
wp4
wp5
wp6
wp7
wp8
wp9
wp10
wp11
wp12
wp13
wp14
wp15
wp16
wp17
wp18
wp19
wp20
wp21
wp22
wp23
wp24
Row
2
3
4
= Primary Check
(Check1)
= Secondary Check A
(Check2)
= Secondary Check B
(Check3)
Type 2 Modified Augmented Design:
Adjustments – Method 3 - Regression
Row
2
3
3
4
Column
1
2
4
4
WholePlot
7
14
16
22
Check1
5.50
5.47
5.73
4.53
Check2
4.83
4.83
5.00
5.17
Check3
8.17
7.83
10.00
7.33
AvgCheck2,3 vs Check1
8
y = 0.7026x + 2.916
R² = 0.41406
7
6
Check23Avg
6.50
6.33
7.50
6.25
Use slope of
regression to adjust
experimental lines
5
4
4
5
6
7
8
Method3AdjustedLine102WholePlot(wp)9 = RawLine102wp9 – slope*(Check1wp9 – Check1Avg)
Line “MALT-102” from data in
corresponding Excel file
= 8.47 – 0.7026*(5.5 – 5.46)
= 8.44
NOTE: Method1AdjustedEntryX = 7.95
Type 2 Modified Augmented Design:
Adjustments – Method 3 - Regression
NOTE: if the data is missing for any of the checks in a whole plot, then you must exclude the
remaining check subplots in that whole plot from the regression. However, you may not want to
exclude those other subplots from the dataset as a whole, because they can be used in other
calculations, just not calculation of the regression coefficient.
Row
2
3
3
4
Column
1
2
4
4
WholePlot
7
14
16
22
Check1
5.50
5.47
5.73
4.53
Check2
4.83
4.83
5.00
.
Check3
8.17
7.83
10.00
7.33
Check23Avg
6.50
6.33
7.50
.
Type 2 Modified Augmented Design:
Adjustments – Method 1 vs Method 3 - Relative Efficiency
Prior to adjusting the values for the experimental subplots, you will probably want to
decide whether you want to use a Method 1 adjustment, a Method 3 adjustment, or no
adjustment.
The first step in comparing the adjustment methods is to look at the relative efficiencies
for Method 1 and Method 3. As you can see by the equation below, a higher relative
efficiency means a greater reduction in experimental error after adjustments:
Relative Efficiency =
Intra-WholePlotErrorWithoutAdjustment
Intra-WholePlotErrorWithAdjustment
x 100%
Type 2 Modified Augmented Design:
Adjustments – Method 1 vs Method 3 - Relative Efficiency
The relative efficiency of Method 1 or Method 3 compares the experimental error
within whole plots for unadjusted versus adjusted data. The intra-whole plot errors
used to calculate relative efficiency are based on secondary check subplots only.
Column
1
1
2
4
3
5
6
WholePlot
(wp) 1
wp2
wp3
wp4
wp5
wp6
wp7
wp8
wp9
wp10
wp11
wp12
wp13
wp14
wp15
wp16
wp17
wp18
wp19
wp20
wp21
wp22
wp23
wp24
Row
2
3
4
= Primary Check
(Check1)
= Secondary Check A
(Check2)
= Secondary Check B
(Check3)
Type 2 Modified Augmented Design:
Adjustments – Method 1 vs Method 3 - Relative Efficiency
Relative Efficiency =
Intra-WholePlotErrorWithoutAdjustment
Intra-WholePlotErrorWithAdjustment
i
Intra-WholePlotErrorWithoutAdjustment =
ΣΣ(Yini -Yi)2
i
Σ(ni -1)
i
Intra-WholePlotErrorWithAdjustment =
ni
ni
ΣΣ(Yin Adj-YiAdj)2
i
i
Σ(ni -1)
x 100%
i = number of secondary checks
ni = number of subplots of each
secondary check
Yini = value of subplot of secondary
check i in subplot ni
Yi = mean of secondary check i
across all subplots ni
NOTE: The ni term in the numerator may be a bit confusing, as the design demands planting the same number of subplots for each
secondary check (i.e., n instead of ni). However, you may have missing data for some secondary check plots, which is fine. The
important concept here is that the Intra-WholePlotError is basically a weighted average variance of the secondary checks. When there
is no missing data from the secondary checks, then the Intra-WholePlot Error is simply the average variance of the secondary checks.
1/18/13
SAS Output
Type 2 Modified Augmented Design:
Method 1 ANOVA -­ Spike Length (cm)
The GLM Procedure
Dependent Variable: SpkLngth
Adjustments – Method 1 vs Method 3
Source
DF
Sum of Squares
Mean Square
F Value
Pr > F
8
1.39740741
0.17467593
0.79
0.6212
Error
15
3.32685185
0.22179012
Corrected Total
23
4.72425926
Model
Example Dataset Results
(see corresponding Excel file for calculations):
1/18/13
R-­Square
Coeff Var
Root MSE
SpkLngth Mean
0.295794
8.619245
0.470946
SAS Output
5.463889
Source
DF
Type I SS
Mean Square
F Value
Pr > F
0.74537037
0.24845679
The GLM Procedure
0.65203704 0.13040741
1.12
0.3721
0.59
0.7093
Method 1 ANOVA -­ Spike Length (cm)
Row
3
Column
5
Dependent Variable: SpkLngth
Source
Model
Error
Source
DF
Type III SS Mean Square F Value
Pr > F
DF Sum of Squares Mean Square F Value
Pr > F
Row
Column
Corrected Total
3
8
0.74537037
0.24845679
1.39740741
0.17467593
1.12 0.3721
0.79 0.6212
5
15
0.65203704
0.13040741
3.32685185
0.22179012
0.59
23
4.72425926
Relative Efficiency
0.295794 8.619245 0.470946
R-­Square
Source
Coeff Var
Root MSE
0.7093
SpkLngth Mean
5.463889
DF
Type I SS
Mean Square
3
0.74537037
0.24845679
Column
5
0.65203704
0.13040741
Source
DF
Type III SS
Mean Square
Row
3
0.74537037
124.
F Value
Pr > F
7
0.24845679
1.12 0.3721
Column
5
0.65203704
0.13040741
Method
Row 1
95.0
F Value
Pr > F
0.59
0.7093
0.59
0.7093
Method
3
1.12 0.3721
Type 2 Modified Augmented Design:
Adjustments – Method 1 vs Method 3
It is important to note that the F-tests from the ANOVA results apply to Row and
Column effects, which can be accounted for by Method 1 adjustments. The Row and
Column effects generally are best at accounting for gradients that stretch across a
substantial portion of the field.
Method 3 does not necessarily require gradients or any other type of pattern in the
field to account for field effects; it only requires that the secondary checks are affected
by the field in a way similar to how the primary checks are affected.
That being said, it is common to find no significant effects in the ANOVA but still find
that a Method 3 adjustment is appropriate.
Type 2 Modified Augmented Design:
Adjustments – Method 1 vs Method 3
The type-2 modified augmented design intentionally makes the final selection of an
adjustment method to be somewhat subjective, based on the user’s understanding of the
biological system. Evidence on the appropriateness of each method includes:
• Relative efficiency of Method 1 vs Method 3
• ANOVA results for Row and Column effects
• Biological meaning of analysis parameters - I once had a relative efficiency of 112%
for Method 3 but a negative Method 3 regression coefficient
• Heat maps or other semi-quantitative/qualitative evaluations of field effects
• Knowledge of the field or fieldbook notes from the experiment
• Knowledge of the lines used as checks - e.g., you may have more confidence in a
marginal Method 1 relative efficiency when you know that your primary check used
to calculate Method 1 adjustments is much less sensitive to field effects than most
of the other lines in the experiment.
MAD Type 2 References
• Lin, CS and Poushinsky, G (1983). A modified augmented design for
an early stage of plant selection involving a large number of test
lines without replication. Biometrics 39:553-561.
• Lin, CS, Poushinsky, G and Jui, PY (1983). Simulation study of three
adjustment methods for the modified augmented design and
comparison with the balanced lattice square design. The Journal of
Agricultural Science 100:527-534.
• Lin, CS and Poushinsky, G (1985). A modified augmented design
(type 2) for rectangular plots. Canadian Journal of Plant Science
65:743-749.
• May, KW, Kozub, GC and Schaalje, GB (1989). Field evaluation of a
modified augmented design (type 2) for screening barley lines.
Canadian Journal of Plant Science 69:1-15.