Summary
The article explains the concept, structure, and applications of the split-plot design in factorial experiments, emphasizing its usefulness when factors require different plot sizes. It details the randomization process, layout strategy, and ANOVA model for analyzing main plot, sub-plot, and interaction effects. A worked example using irrigation levels and fertilizer types illustrates data organization, degree of freedom calculation, and ANOVA table construction. The results show significant effects of irrigation and fertilizer on crop yield, while their interaction is non-significant. The article also discusses the advantages and limitations of split-plot design in agricultural experiments. Finally, it provides step-by-step guidance for performing split-plot analysis using the Agri Analyze online tool.
1.Introduction
The split-plot design is commonly employed in factorial experiments. This design can integrate various other designs,
such as completely randomized designs (CRD) and randomized complete block designs (RCBD).
The fundamental principle involves dividing whole plots or whole units, to which levels of one or more factors are applied (main plots).
These main-plots are then subjected to levels of one or more additional factors called sub plot.
Consequently, each whole unit functions as a block for the treatments applied to the sub-units.
In a split-plot design, the precision in estimating the main plot factor's effect is reduced to enhance the precision of the sub-plot factor's effect.
This design allows for more accurate measurement of the sub-plot factor's main effect and its interaction with the main plot factor compared to a randomized block design.
However, the precision in measuring the main plot treatments (i.e., the levels of the main plot factor) is less than that achieved with RCBD.
2. Ideal applications of this design
A split-plot design is particularly advantageous when treatments associated with one or more factors necessitate larger experimental units than treatments for other factors.
For instance, in a field experiment, factors such as methods of land preparation or irrigation application typically require large plots or experimental units.
In contrast, another factor, like crop varieties, can be evaluated using smaller plots.
The split-plot design ensures efficient resource utilization and enhances the precision of certain factor measurements, making it ideal for complex experimental setups with hierarchical treatment structures.
The split-plot design is also beneficial when incorporating an additional factor to broaden the scope of an experiment.
For example, if the primary goal is to compare the effectiveness of various fungicides in protecting against disease infection, the experiment's scope can be expanded by including several crop varieties known to differ in disease resistance.
In this setup, the varieties can be arranged in whole units, while the fungicide treatments (seed protectants) are applied to sub-units. This approach allows for a comprehensive analysis of both fungicide efficacy and varietal resistance within a single experimental framework,
optimizing resource use and experimental precision.
3. Randomization and layout strategies for split-plot experiments
In a split-plot design, there are distinct randomization procedures for the main plots and sub-plots.
Within each replication, main plot treatments are initially randomly allocated to the main plots.
Subsequently, sub-plot treatments are randomly assigned within each main plot.
This sequential randomization ensures independent and controlled assignment of treatments at both the main plot and sub-plot levels,
maintaining the integrity and statistical validity of the experimental design.
- Step 1: Partition the experimental area into "r" replications, each subdivided into "a" main plots.
- Step 2: Randomly assign the treatment levels to the main plots within each replication independently.
- Step 3: Partition each replication into "a" main plots, and within each main plot, partition into 'b' sub-plots. Randomly assign the levels of the sub-plot factors within each sub-plot.
Statistical Model for Split Plot Design
The statistical model for a split plot design is given by:
$$
Y_{ijk} = \mu + \gamma_i + A_j + \varepsilon_{ij} + B_k + (AB)_{jk} + \delta_{ijk}
$$
Where,
\( Y_{ijk} \) = The response of yield from the ith unit receiving the
jth main plot factor and kth sub-plot factor
\( \mu \) = General mean
\( \gamma_i \) = Effect of ith replication
\( A_j \) = Effect of jth main plot factor
\( \varepsilon_{ij} \) = Uncontrolled variation associated with the
jth main plot factor in the ith replication
\( B_k \) = Effect of kth sub-plot factor
\( (AB)_{jk} \) = Interaction effect between the
jth main plot factor and kth sub-plot factor
\( \delta_{ijk} \) = Uncontrolled variation associated with the
jth main plot factor and kth sub-plot factor
in the ith replication
4. Advantage of split plot design:
In a split-plot design, the effects of sub-plot treatments and their interactions with main plot treatments are tested with greater precision than the effects of the main plot treatments.
This design is more convenient for handling agricultural operations. When treatments such as irrigation, tillage, sowing dates, and other cultural practices are involved, these treatments can be assigned to the main plots.
Due to the combination of factors within the same experiment, this design incurs very little extra cost. Conducting separate experiments for each factor would be more expensive. It saves experimental area and resources by devoting them only to the border rows in the main plot.
5. Disadvantage of split plot design
We lose precision for main plot treatments but gain precision for sub-plot treatments.
With the limitation of experimental area, the degrees of freedom for error often do not meet the minimum requirement of 12.
When missing plots occur, the analysis becomes more complicated.
Analysis of Variance (ANOVA) for Split Plot Design
| Source |
DF |
SS |
MS |
Cal F |
| Replication |
\( r-1 \) |
\[
\frac{\sum_{i=1}^{r} Y_{i..}^2}{ab}
-
\frac{(\sum Y_{ijk})^2}{rab}
\]
|
\( \dfrac{RSS}{r-1} \) |
\( \dfrac{RMS}{E(a)MS} \) |
| Main Factor (A) |
\( a-1 \) |
\[
\frac{\sum_{j=1}^{a} Y_{.j.}^2}{rb}
-
\frac{(\sum Y_{ijk})^2}{rab}
\]
|
\( \dfrac{ASS}{a-1} \) |
\( \dfrac{AMS}{E(a)MS} \) |
| Error (a) |
\( (r-1)(a-1) \) |
\[
\frac{\sum_{i=1}^{r}\sum_{j=1}^{a} Y_{ij.}^2}{b}
-
\frac{(\sum Y_{ijk})^2}{rab}
- RSS - ASS
\]
|
\( \dfrac{E(a)SS}{(r-1)(a-1)} \) |
— |
| Sub Factor (B) |
\( b-1 \) |
\[
\frac{\sum_{k=1}^{b} Y_{..k}^2}{ra}
-
\frac{(\sum Y_{ijk})^2}{rab}
\]
|
\( \dfrac{BSS}{b-1} \) |
\( \dfrac{BMS}{E(b)MS} \) |
| A × B |
\( (a-1)(b-1) \) |
\[
\frac{\sum_{j=1}^{a}\sum_{k=1}^{b} Y_{.jk}^2}{r}
-
\frac{(\sum Y_{ijk})^2}{rab}
- ASS - BSS
\]
|
\( \dfrac{ABSS}{(a-1)(b-1)} \) |
\( \dfrac{ABMS}{E(b)MS} \) |
| Error (b) |
\( a(r-1)(b-1) \) |
By subtraction |
\( \dfrac{E(b)SS}{a(r-1)(b-1)} \) |
— |
| Total |
\( rab-1 \) |
\[
\sum_{i=1}^{r}\sum_{j=1}^{a}\sum_{k=1}^{b} Y_{ijk}^2
-
\frac{(\sum Y_{ijk})^2}{rab}
\]
|
— |
— |
Calculation of Standard Error of Mean and Critical Difference (CD)
1. Standard error of mean for main factor (A):
\[
S.Em.(A) = \sqrt{\frac{E(a)MS}{rb}}
\]
2. Standard error of mean for sub factor (B):
\[
S.Em.(B) = \sqrt{\frac{E(b)MS}{ra}}
\]
3. Standard error of mean for interaction (A × B):
\[
S.Em.(A \times B) = \sqrt{\frac{E(b)MS}{r}}
\]
4. Critical difference for main factor (A):
\[
CD_{0.05} = S.Em.(A) \times \sqrt{2} \times t_{0.05,\;error(a)\;df}
\]
5. Critical difference for sub factor (B):
\[
CD_{0.05} = S.Em.(B) \times \sqrt{2} \times t_{0.05,\;error(b)\;df}
\]
6. Critical difference for interaction (A × B):
\[
CD_{0.05} = S.Em.(A \times B) \times \sqrt{2} \times t_{0.05,\;error(b)\;df}
\]
Calculation of Coefficient of Variation (CV%)
1. Coefficient of variation for main factor (A):
\[
CV\%(A) = \frac{\sqrt{E(a)MS}}{\bar{Y}} \times 100
\]
2. Coefficient of variation for sub factor (B):
\[
CV\%(B) = \frac{\sqrt{E(b)MS}}{\bar{Y}} \times 100
\]
Example for Split Plot Design
A split-plot design was used to investigate the effects of irrigation levels (main plot factor)
and fertilizer types (sub-plot factor) on the yield of a particular crop.
The experiment was conducted over four replicates
\( (R_1, R_2, R_3, R_4) \).
Main Plot Factor (A – Irrigation Levels)
- \( A_1 \): Low Irrigation
- \( A_2 \): Medium Irrigation
- \( A_3 \): High Irrigation
Sub-Plot Factor (B – Fertilizer Types)
- \( B_1 \): Organic Fertilizer
- \( B_2 \): Inorganic Fertilizer
- \( B_3 \): Mixed Fertilizer
Observed Data
| Treatment |
R1 |
R2 |
R3 |
R4 |
| A1B1 |
386 |
396 |
298 |
387 |
| A1B2 |
496 |
549 |
469 |
513 |
| A1B3 |
476 |
492 |
436 |
476 |
| A2B1 |
376 |
406 |
280 |
347 |
| A2B2 |
480 |
540 |
436 |
500 |
| A2B3 |
455 |
512 |
398 |
468 |
| A3B1 |
355 |
388 |
201 |
337 |
| A3B2 |
446 |
533 |
413 |
482 |
| A3B3 |
433 |
482 |
334 |
435 |
Solution
| Treatment |
R1 |
R2 |
R3 |
R4 |
Treatment Total |
Treatment Mean |
| A1B1 |
386 |
396 |
298 |
387 |
1467 |
366.75 |
| A1B2 |
496 |
549 |
469 |
513 |
2027 |
506.75 |
| A1B3 |
476 |
492 |
436 |
476 |
1880 |
470.00 |
| A2B1 |
376 |
406 |
280 |
347 |
1409 |
352.25 |
| A2B2 |
480 |
540 |
436 |
500 |
1956 |
489.00 |
| A2B3 |
455 |
512 |
398 |
468 |
1833 |
458.25 |
| A3B1 |
355 |
388 |
201 |
337 |
1281 |
320.25 |
| A3B2 |
446 |
533 |
413 |
482 |
1874 |
468.50 |
| A3B3 |
433 |
482 |
334 |
435 |
1684 |
421.00 |
| Total |
3903 |
4298 |
3265 |
3945 |
15411 |
— |
1. Correction Factor (CF)
$$
CF = \frac{(\sum Y_{ijk})^2}{rab}
$$
$$
CF = \frac{(15411)^2}{4 \times 3 \times 3} = 6{,}597{,}192
$$
2. General Mean (\(\bar{Y}\))
$$
\bar{Y} = \frac{15411}{4 \times 3 \times 3} = 428.08
$$
3. Replication Sum of Squares (RSS)
$$
RSS = \frac{\sum_{i=1}^{r} Y_{i..}^2}{ab}
- \frac{(\sum Y_{ijk})^2}{rab}
$$
$$
= \frac{3903^2 + 4298^2 + \cdots + 3945^2}{3 \times 3}
- 6{,}597{,}192
= 61{,}636.97
$$
A × B Table
|
B1 |
B2 |
B3 |
Total A |
| A1 |
1467 |
2027 |
1880 |
5374 |
| A2 |
1409 |
1956 |
1833 |
5198 |
| A3 |
1281 |
1874 |
1684 |
4839 |
| Total B |
4157 |
5857 |
5397 |
|
4. Main Factor Sum of Squares (ASS)
$$
ASS = \frac{\sum_{j=1}^{a} Y_{.j.}^2}{rb}
- \frac{(\sum Y_{ijk})^2}{rab}
$$
$$
= \frac{5374^2 + 5198^2 + 4839^2}{4 \times 3}
- 6{,}597{,}192
= 12{,}391.17
$$
Main Factor (A) × Replication Table
|
R1 |
R2 |
R3 |
R4 |
| A1 |
1358 |
1437 |
1203 |
1376 |
| A2 |
1311 |
1458 |
1114 |
1315 |
| A3 |
1234 |
1403 |
948 |
1254 |
5. Error Associated with Main Factor (\(E(a)SS\))
$$
E(a)SS = \frac{\sum_{i=1}^{r} \sum_{j=1}^{a} Y_{ij.}^2}{b}
- \frac{(\sum Y_{ijk})^2}{rab}
- RSS - ASS
$$
$$
= \frac{1358^2 + 1437^2 + \cdots + 1254^2}{3}
- 6{,}597{,}192
- 61{,}636.97
- 12{,}391.17
$$
$$
= 4{,}382.61
$$
6. Sub-Factor Sum of Squares (BSS)
$$
BSS = \frac{\sum_{k=1}^{k} Y_{..k}^2}{ra}
- \frac{(\sum Y_{ijk})^2}{rab}
$$
$$
= \frac{4157^2 + 5857^2 + 5397^2}{4 \times 3}
- 6{,}597{,}192
= 128{,}866.70
$$
7. A × B SS (ABSS)
$$
ABSS = \frac{\sum_{j=1}^{j} \sum_{k=1}^{k} Y_{.jk}^2}{r}
- \frac{(\sum Y_{ijk})^2}{rab}
- ASS - BSS
$$
$$
= \frac{1467^2 + 2027^2 + \cdots + 1684^2}{4}
- 6{,}597{,}192
- 12{,}391.17
- 128{,}866.70
$$
$$
= 304.16
$$
8. Total Sum of Squares (TSS)
$$
TSS = \sum_{i=1}^{i} \sum_{j=1}^{j} \sum_{k=1}^{k} Y_{ijk}^2
- \frac{(\sum Y_{ijk})^2}{rab}
$$
$$
= 386^2 + 396^2 + 298^2 + \cdots + 435^2
- 6{,}597{,}192
$$
$$
= 213{,}796.80
$$
9. Main Error Sum of Squares \(E(b)SS\)
$$
E(b)SS = TSS - RSS - ASS - E(a)SS - BSS - ABSS
$$
$$
= 213{,}796.80 - 61{,}636.97 - 12{,}391.17
- 4{,}382.61 - 128{,}866.70 - 304.16
$$
$$
= 6{,}215.16
$$
Calculation of Degrees of Freedom
Replication DF = r – 1 = 4 – 1 = 3
Main Plot DF = A - 1 = 3 - 1 = 2
Error a DF = (r-1)*(A-1) = 6
Sub Plot DF = B – 1 = 3 – 1 = 2
Interaction DF = (A-1) * (B-1) = 4
Error b DF = A*(r-1)*(B-1) = 18
Total DF = A*B*r – 1 = 35
The Mean Square for different component is obtained by dividing SS with DF for respective component
Calculated F value for different ANOVA components
Replication Cal F = Replication MS / Error a MS = 28.12
Main Plot A Cal F = Main Plot A MS / Error a MS = 8.48
Sub Plot B Cal F = Sub Plot B MS / Error b MS = 186.61
Interaction Cal F = Interaction MS / Error b MS = 0.22
Final ANOVA Table for Crop Yield Analysis Using Split-Plot Design
(Irrigation as Main Plot Factor and Fertilizer as Sub Plot Factor)
| SV |
DF |
SS |
MS |
Cal F |
Table F (5%) |
Table F (1%) |
| Replication |
3 |
61636.97 |
20545.66 |
28.12 |
3.16 |
8.49 |
| Main Plot A |
2 |
12391.17 |
6195.58 |
8.48 |
5.14 |
10.92 |
| Error (a) |
6 |
4382.61 |
730.44 |
– |
– |
– |
| Sub Plot B |
2 |
128866.67 |
64433.33 |
186.61 |
3.55 |
6.01 |
| Interaction (A × B) |
4 |
304.17 |
76.04 |
0.22 |
2.93 |
4.58 |
| Error (b) |
18 |
6215.17 |
345.29 |
– |
– |
– |
| Total |
35 |
213796.75 |
– |
– |
– |
– |
Calculation of Standard Error of Mean and Critical Difference (CD)
1. Standard Error of Mean for Main Factor (A)
$$
\text{S.Em.(A)} = \sqrt{\frac{E(a)\text{MS}}{rb}}
$$
$$
= \sqrt{\frac{730.44}{4 \times 3}} = 7.80
$$
2. Standard Error of Mean for Sub Factor (B)
$$
\text{S.Em.(B)} = \sqrt{\frac{E(b)\text{MS}}{ra}}
$$
$$
= \sqrt{\frac{345.19}{4 \times 3}} = 5.36
$$
3. Standard Error of Mean for Interaction (A × B)
$$
\text{S.Em.(A × B)} = \sqrt{\frac{E(b)\text{MS}}{r}}
$$
$$
= \sqrt{\frac{345.19}{4}} = 9.29
$$
4. Critical Difference (CD) for Main Factor (A)
$$
\text{CD}_{0.05} = \text{S.Em.(A)} \times \sqrt{2} \times t_{0.05,\;ne(a)}
$$
$$
= 7.80 \times \sqrt{2} \times 2.44 = 26.99
$$
5. Critical Difference (CD) for Sub Factor (B)
$$
\text{CD}_{0.05} = \text{S.Em.(B)} \times \sqrt{2} \times t_{0.05,\;ne(b)}
$$
$$
= 5.36 \times \sqrt{2} \times 2.10 = 15.93
$$
6. Critical Difference (CD) for Interaction (A × B)
$$
\text{CD}_{0.05} = \text{S.Em.(A × B)} \times \sqrt{2} \times t_{0.05,\;ne(b)}
$$
$$
= 9.29 \times \sqrt{2} \times 2.10 = 27.60
$$
Calculation of Coefficient of Variation (CV%)
1. Coefficient of Variation for Main Factor (A)
$$
\text{CV % (A)} = \frac{\sqrt{E(a)\text{MS}}}{\bar{Y}} \times 100
$$
$$
= \frac{\sqrt{730.44}}{428.08} \times 100 = 6.31
$$
2. Coefficient of Variation for Sub Factor (B)
$$
\text{CV % (B)} = \frac{\sqrt{E(b)\text{MS}}}{\bar{Y}} \times 100
$$
$$
= \frac{\sqrt{345.29}}{428.08} \times 100 = 4.34
$$
6. Conclusion:
The calculated F-value (28.12) is much greater than the critical F-values at both 5% (3.16) and 1% (8.49) significance levels.
Therefore, there is strong evidence to suggest that there are significant differences between the replicates.
The calculated F-value (8.48) for main factor exceeds the critical F-value 5% (5.14) significance levels.
This indicates that there are significant differences among the irrigation level.
The calculated F-value (186.61) for sub factor exceeds the critical F-value at 1% (6.01) significance level.
This indicates that there is highly significant variation among level of fertilizer.
The calculated F-value (0.22) for interaction between main factor and sub factor (A x B) which is less than critical F-value at 5% (2.93) significance level.
This indicate that there is non-significant interaction between irrigation and fertilizer.
For the irrigation, highest yield was observed for A1 and A2 were found statistically at par with it based on critical difference.
For the fertilizer, highest yield was observed for B2 and none of the level of fertilizer at par with it based on critical difference.
For the interaction (A x B), highest yield was observed for A1 x B2 and A2 x B2 were found statistically at par with it.
Steps to perform analysis of split plot design in Agri Analyze
Step 1: To create a CSV file with columns for replication, main factor (A), sub factor (B) and Yield.
| Main Plot |
Sub Plot |
Replication |
Yield |
| A1 |
B1 |
R1 |
386 |
| A1 |
B2 |
R1 |
496 |
| A1 |
B3 |
R1 |
476 |
| A2 |
B1 |
R1 |
376 |
| A2 |
B2 |
R1 |
480 |
| A2 |
B3 |
R1 |
455 |
| A3 |
B1 |
R1 |
355 |
| A3 |
B2 |
R1 |
446 |
| A3 |
B3 |
R1 |
433 |
| A1 |
B1 |
R2 |
396 |
| A1 |
B2 |
R2 |
549 |
| A1 |
B3 |
R2 |
492 |
| A2 |
B1 |
R2 |
406 |
| A2 |
B2 |
R2 |
540 |
| A2 |
B3 |
R2 |
512 |
| A3 |
B1 |
R2 |
388 |
| A3 |
B2 |
R2 |
533 |
| A3 |
B3 |
R2 |
482 |
| A1 |
B1 |
R3 |
298 |
| A1 |
B2 |
R3 |
469 |
| A1 |
B3 |
R3 |
436 |
Step 2: Click on ANALYTICAL TOOL ->DESIGN OF EXPERIMENT ->SPLIT PLOT DESIGN ANALYSIS ->SPLIT PLOT 1,1 (SPLIT PLOT) ANALYSIS
Step 3: Open link https://www.agrianalyze.com/SplitPlotOneOne.aspx (For first time users free registration is mandatory)
Step 4: Link Here to download sample file Sample File Download
Step 5: Select replication, main factor (A), sub factor (B) and dependent variable (Yield).
Step 6: Select a test for multiple comparisons, such as the Least Significant Difference (LSD) test,
to determine significant differences among groups. Same as for Duncan’s New Multiple Range Test (DNMRT), Tukey’s HSD Test.
Step 7: Click submit, pay a nominal fee, and download the output report with detailed interpretation.
Output Report:
Link of the output report
Video Tutorial:
Link of the Youtube Tutorial
Quiz :
Split Plot Quiz
Reference
Gomez, K. A., & Gomez, A. A. (1984). Statistical Procedures for Agricultural Research. John wiley & sons. 50-67.
The blog is written by:
Darshan Kothiya, PhD Scholar, Department of Agricultural Statistics, BACA, AAU, Anand