Minimum Required Design Elements for Participation in the Global Maize Project:
EXPERIMENTAL DESIGN
Table 1. The minimum required design consists of the following factors and levels:
Fixed/random
effect
random
Factor
R
Management
system
fixed
M
Ecological intensification (EI)
Average farmer practice (FP)
m=2
Nitrogen
sequence
fixed
N
Adequate N applied every year
No N every year
n=2
Description
Replication
Levels
Variable, but 2 blocks is a minimum;
4 blocks are used as an example since
that is a common practice
No. of
levels
r=4
The experimental layout is a split-plot design with management system as the whole plot and nitrogen
sequence as a split plot in space. An example of this design is shown below:
The split plot design is used to accommodate treatment application logistics in some locations,
particularly where larger equipment, such as tillage equipment, needs to be changed between
management systems. The implication of this design upon analysis is that there is greater sensitivity to
detecting differences between the sub-plots (N) and their interaction with management system (MxN)
than between the management systems themselves (M).
Most project locations are accustomed to using 4 blocks as replicates and have incorporated this
number into their project designs. The required number is at least 2.
Over time, the same treatments are to be applied to the same experimental units for each crop. This
experiment is interested in the cumulative effects of management practices over time. It is expected
that the longer various treatments are in place, the more evident the differences will become. The levels
of the inputs comprising the EI and FP treatments may vary from crop to crop and from year to year, to
ensure that appropriate practices are used for each crop and to allow for management improvements.
However, on plots where no nitrogen is to be applied, no nitrogen should be applied to any crop in any
season. Such long-term omissions of nitrogen will provide a baseline for evaluating the cumulative
effects of the EI and FP systems on soil N supplies.
STATISTICAL ANALYSES
Statistical analyses are based on the number of years and/or locations to be considered. Four analyses
can be performed:
1. single crop season at a single location
2. single crop season across multiple locations
3. multiple crop seasons at a single location
4. multiple crop seasons at multiple locations
The unique offerings of this project are analyses No. 2 and 4 conducted at several major maize growing
areas of the world, providing a larger inference space than most other single studies. Each location in
the project has been preselected as representing a major maize production area.
Each of the above analyses can be further subdivided into two categories, based on whether or not
dependent variables are a) limited to measurements/calculations involving a single sub-plot, or b)
calculations involving more than one sub-plot. Examples of measurements or calculations involving a
single sub-plot are as follows:
 Grain dry matter and nutrient accumulation
 Whole plant dry matter and nutrient accumulation
 Calculations such as:
o Harvest index (ratio of grain dry matter to whole plant dry matter)
o Partial factor productivity (ratio of dry matter grain yield to nutrient application rate)
o Partial nutrient balance (ratio of nutrient removal to nutrient application rate)
Examples of calculations involving more than one sub-plot are:
 Agronomic efficiency of a nutrient (difference in yield between a fertilized sub-plot and an
unfertilized sub-plot, divided by the amount of nutrient applied to the fertilized sub-plot)
 Apparent crop recovery efficiency (difference in nutrient uptake between a fertilized sub-plot
and an unfertilized sub-plot, divided by the amount of nutrient applied to the fertilized sub-plot)
In the minimum required design, calculations involving both the fertilized and unfertilized sub-plots
within a whole plot result in only one observation per whole plot. In such cases, the experiment must be
analyzed as a randomized complete block design (RCBD), rather than as a split-plot design, since results
of calculations no longer exist at the sub-plot level.
Combining the spatial/temporal combinations (locations and cropping seasons) with dependent variable
scales (sub-plot or whole-plot scales of calculations) leads to 8 possible analyses that can be performed:
1. Single crop season at a single location; sub-plot scale
2. Single crop season at a single location; whole-plot scale
3. Single crop season at multiple locations; sub-plot scale
4. Single crop season at multiple locations; whole-plot scale
5. Multiple crop seasons at a single location; sub-plot scale
6. Multiple crop seasons at a single location; whole-plot scale
7. Multiple crop seasons at multiple locations; sub-plot scale
8. Multiple crop seasons at multiple locations; whole plot scale
The following sections discuss all 8 of these possibilities.
1. Single Crop Season at a Single Location; Sub-Plot Scale
The following statistical linear model is for a split plot design with management system as the whole plot
and nitrogen sequence as the split-plot. For factors and levels, see Table 1.
𝑌𝑖𝑗𝑘 = 𝜇 + 𝑅𝑖 + 𝑀𝑗 + 𝑎(𝑖𝑗) + 𝑁𝑘 + (𝑀𝑁)𝑗𝑘 + 𝜀(𝑖𝑗𝑘)
In this model, there are two error terms, defined as follows:
 𝑎(𝑖𝑗) = (𝑅𝑀)𝑖𝑗 , the block x management system interaction;

𝜀(𝑖𝑗𝑘) = (𝑅𝑁)𝑖𝑘 + (𝑅𝑀𝑁)𝑖𝑗𝑘 , the sum of the block x nitrogen sequence interaction and the
block x management system x nitrogen sequence interaction.
The first error term, 𝑎(𝑖𝑗) , arises from restricting the random assignment of nitrogen sequence sub-plots
to whole plot units.
In the F-tests for determining the statistical significance of main effects and interactions, appropriate
error terms must be used in the denominator. Table 2 contains the expected mean squares (EMS) of the
various terms in the linear model given above. Interpretations of the EMS in this table are:
 Error 𝑎(𝑖𝑗) is used as the denominator in the F-test determining statistical significance of the
effect of blocks, 𝑅𝑖 , and the management practice main effect, 𝑀𝑗 .

The experimental error, 𝜀(𝑖𝑗𝑘) , is used as the denominator in the F-test for determining
statistical significance of the nitrogen sequence main effect, 𝑁𝑘 , and the interaction of nitrogen
sequence with management system, (𝑀𝑁)𝑗𝑘 .
Table 2. Expected mean squares (EMS) for the analysis of variance (ANOVA) of a balanced, split-plot
design with no missing data for one year at one location and observations taken at the sub-plot level.
Source
Degrees of freedom (df)
Expected mean squares (EMS)
𝑅𝑖
(𝑟 − 1)
𝜎𝜀2 + 𝑛𝜎𝑎2 + 𝑚𝑛𝜎𝑅2
𝑀𝑗
(𝑚 − 1)
𝜎𝜀2
𝑎(𝑖𝑗)
(𝑟 − 1)(𝑚 − 1)
𝜎𝜀2 + 𝑛𝜎𝑎2
𝑁𝑘
(𝑛 − 1)
𝜎𝜀2 + 𝑟𝑚
(𝑀𝑁)𝑗𝑘
(𝑚 − 1)(𝑛 − 1)
𝜎𝜀2
𝜀(𝑖𝑗𝑘)
𝑚(𝑟 − 1)(𝑛 − 1)
𝜎𝜀2
Total
𝑟𝑚𝑛 − 1
+
𝑛𝜎𝑎2
+ 𝑟𝑛
∑ 𝑀𝑗2
(𝑚 − 1)
∑ 𝑁𝑘2
(𝑛 − 1)
∑ ∑(𝑀𝑁)2𝑖𝑗
+𝑟
(𝑚 − 1)(𝑛 − 1)
2. Single Crop Season at a Single Location; Whole Plot Scale
The factors and levels corresponding to the analysis of a RCBD are provided in Table 3.
Table 3. Factors and levels for the RCBD for a single crop season and single location with observations
taken at the whole plot level.
Description
Replication
Management
system
Fixed/random
effect
random
Factor
R
fixed
M
Levels
Variable, but 2 blocks is a minimum;
4 blocks are used as an example since
that is a common practice
Ecological intensification (EI)
Average farmer practice (FP)
No. of
levels
r=4
m=2
The following statistical linear model is for a RCBD examining management system as the main effect:
𝑌𝑖𝑗 = 𝜇 + 𝑅𝑖 + 𝑀𝑗 + 𝜀(𝑖𝑗)
There is only one error term: 𝜀(𝑖𝑗) = (𝑅𝑀)𝑖𝑗 , which is the block x management system interaction. It is
used as the denominator in the F-tests for determining statistical significance of all main effects and
interactions. Table 4 contains the expected mean squares (EMS) of the various terms in the linear model
given above.
Table 4. EMS for the ANOVA of a balanced RCBD with no missing data for one crop season at one
location and observations taken at the whole-plot level.
Source
Degrees of freedom (df)
Expected mean squares (EMS)
𝑅𝑖
(𝑟 − 1)
𝜎𝜀2 + 𝑚𝜎𝑅2
𝑀𝑗
(𝑚 − 1)
𝜎𝜀2
𝜀(𝑖𝑗)
(𝑟 − 1)(𝑚 − 1)
𝜎𝜀2
Total
𝑟𝑚 − 1
+𝑟
∑ 𝑀𝑗2
(𝑚 − 1)
3. Single Crop Season across Multiple Locations; Sub-Plot Scale
In a single season, multiple locations can be considered in an analysis to determine their differences in
response. In this experiment, locations were chosen for their relevance to global maize production.
Because they were pre-selected based on many considerations, the location effect is considered fixed,
not random, as shown in Table 5.
Table 5. Factors and levels considered in an analysis examining multiple locations for a single crop
season, employing the minimum required experimental design.
Description
Location
Fixed/random
effect
fixed
Factor
L
Replication
random
R
Levels
14 locations are planned
Variable, but 2 blocks is a minimum;
4 blocks are used as an example since
that is a common practice
No. of
levels
l = 14
r=4
Management
system
fixed
M
Ecological intensification (EI)
Average farmer practice (FP)
m=2
Nitrogen
sequence
fixed
N
Adequate N applied every year
No N every year
n=2
The following statistical linear model is for a split plot design with management system as the whole
plot, nitrogen sequence as the split-plot, fixed locations, and replications nested within locations.
𝑌𝑖𝑗𝑘𝑙 = 𝜇 + 𝐿𝑖 + 𝑅(𝑖)𝑗 + 𝑀𝑘 + (𝐿𝑀)𝑖𝑘 + 𝑎(𝑖𝑗𝑘) + 𝑁𝑙 + (𝐿𝑁)𝑖𝑙 + (𝑀𝑁)𝑘𝑙 + (𝐿𝑀𝑁)𝑖𝑘𝑙 + 𝜀(𝑖𝑗𝑘𝑙)
The model contains two error terms:
 𝑎(𝑖𝑗𝑘) = (𝑅𝑀)(𝑖)𝑗𝑘 , the block x management system interaction;

𝜀(𝑖𝑗𝑘𝑙) = (𝑅𝑁)(𝑖)𝑗𝑙 + (𝑅𝑀𝑁)(𝑖)𝑗𝑘𝑙 , the sum of the block x nitrogen sequence interaction and the
block x management system x nitrogen sequence interaction.
The following interpretations follow from the EMS listed in Table 6:
 The replication effect, 𝑅(𝑖)𝑗 , is the appropriate denominator in the F-test determining statistical
significance of the location main effect, 𝐿𝑖 .
 The error arising from randomization restrictions for the sub-plots, 𝑎(𝑖𝑗𝑘) , is the denominator for
testing the statistical significance of management system, 𝑀𝑘 , and the location by management
system interaction (𝐿𝑀)𝑖𝑘 .
 The experimental error term, 𝜀(𝑖𝑗𝑘𝑙) , is used as the denominator in the F-tests for determining
statistical significance of all effects (main effect and interactions) involving nitrogen sequence:
𝑁𝑙, , (𝐿𝑁)𝑖𝑙 , (𝑀𝑁)𝑘𝑙 , (𝐿𝑀𝑁)𝑖𝑘𝑙 .
Table 6. EMS for the ANOVA of a balanced, split-plot design with no missing data for one crop season at
many locations, with observations taken at the sub-plot level.
Source
Degrees of freedom (df)
Expected mean squares (EMS)
𝐿𝑖
(𝑙 − 1)
𝜎𝜀2
𝑅(𝑖)𝑗
𝑙(𝑟 − 1)
𝜎𝜀2 + 𝑛𝜎𝑎2 + 𝑚𝑛𝜎𝑅2
𝑀𝑘
(𝑚 − 1)
𝜎𝜀2
(𝐿𝑀)𝑖𝑘
(𝑙 − 1)(𝑚 − 1)
𝜎𝜀2
+
𝑛𝜎𝑎2
+
𝑚𝑛𝜎𝑅2
∑ 𝐿2𝑖
+ 𝑟𝑚𝑛
(𝑙 − 1)
+
𝑛𝜎𝑎2
∑ 𝑀𝑘2
+ 𝑙𝑟𝑛
(𝑚 − 1)
+
𝑛𝜎𝑎2
∑ ∑(𝐿𝑀)2𝑖𝑘
+𝑛
(𝑙 − 1)(𝑚 − 1)
𝑎(𝑖𝑗𝑘)
𝑙(𝑟 − 1)(𝑚 − 1)
𝜎𝜀2 + 𝑛𝜎𝑎2
𝑁𝑙
(𝑛 − 1)
𝜎𝜀2 + 𝑙𝑟𝑚
(𝐿𝑁)𝑖𝑙
(𝑙 − 1)(𝑛 − 1)
𝜎𝜀2 + 𝑚
∑ ∑(𝐿𝑁)2𝑖𝑙
(𝑙 − 1)(𝑛 − 1)
(𝑀𝑁)𝑘𝑙
(𝑚 − 1)(𝑛 − 1)
𝜎𝜀2 + 𝑙𝑟
∑ ∑(𝑀𝑁)2𝑘𝑙
(𝑚 − 1)(𝑛 − 1)
(𝐿𝑀𝑁)𝑖𝑘𝑙
(𝑙 − 1)(𝑚 − 1)(𝑛 − 1)
𝜎𝜀2 +
𝜀(𝑖𝑗𝑘𝑙)
𝑙𝑚(𝑟 − 1)(𝑛 − 1)
𝜎𝜀2
Total
𝑙𝑟𝑚𝑛 − 1
∑ 𝑁𝑙2
(𝑛 − 1)
∑ ∑ ∑(𝐿𝑀𝑁)2𝑖𝑘𝑙
(𝑙 − 1)(𝑚 − 1)(𝑛 − 1)
4. Single Crop Season across Multiple Locations; Whole Plot Scale
Table 7 provides the factors and levels considered in the linear model for this scenario.
Table 7. Factors and levels for the RCBD resulting from an analysis of multiple locations within a single
crop season, with observations taken at the whole plot level.
Description
Location
Fixed/random
effect
fixed
Factor
L
Replication
random
R
Variable, but 2 blocks is a minimum;
4 blocks are used as an example since
that is a common practice
r=4
fixed
M
Ecological intensification (EI)
Average farmer practice (FP)
m=2
Management
system
Levels
14 locations are planned
No. of
levels
l = 14
The following statistical linear model is for a RCBD based on the factors in Table 7 and considering
replications nested within locations.
𝑌𝑖𝑗𝑘 = 𝜇 + 𝐿𝑖 + 𝑅(𝑖)𝑗 + 𝑀𝑘 + (𝐿𝑀)𝑖𝑘 + 𝜀(𝑖𝑗𝑘)
There is one experimental error term, 𝜀(𝑖𝑗𝑘) = (𝑅𝑀)(𝑖)𝑗𝑘 , which is the block x management system
interaction.
The following interpretations follow from the EMS listed in Table 8:
 The replication effect, 𝑅(𝑖)𝑗 , is the appropriate denominator in the F-test determining statistical
significance of the location main effect, 𝐿𝑖 .
 The experimental error term, 𝜀(𝑖𝑗𝑘) , is used as the denominator in the F-tests for determining
statistical significance of all remaining main effects and interactions.
Table 8. EMS for the ANOVA of a balanced, RCBD with no missing data for one year at many locations
and observations taken at the whole plot level.
Source
Degrees of freedom (df)
Expected mean squares (EMS)
𝐿𝑖
(𝑙 − 1)
𝜎𝜀2 + 𝑚𝜎𝑅2 + 𝑟𝑚
𝑅(𝑖)𝑗
𝑙(𝑟 − 1)
𝜎𝜀2 + 𝑚𝜎𝑅2
𝑀𝑘
(𝑚 − 1)
𝜎𝜀2 + 𝑙𝑟
(𝐿𝑀)𝑖𝑘
(𝑙 − 1)(𝑚 − 1)
𝜎𝜀2 + 𝑟
𝜀(𝑖𝑗𝑘)
𝑙(𝑟 − 1)(𝑚 − 1)
𝜎𝜀2
Total
𝑟𝑙𝑚 − 1
∑ 𝐿2𝑖
(𝑙 − 1)
∑ 𝑀𝑘2
(𝑚 − 1)
∑ ∑(𝐿𝑀)2𝑖𝑘
(𝑙 − 1)(𝑚 − 1)