Summary
The blog explains the concept and application of Strip Plot Design (SPD) for two-factor experiments where precise estimation of interaction effects is required. It describes the layout, randomization procedure, ANOVA model, and calculation of degrees of freedom and mean squares using a solved example on irrigation and fertilizer treatments. The analysis shows significant effects of both irrigation and fertilizer levels on crop yield, while their interaction was found to be non-significant. A complete ANOVA table is presented to support the findings with statistical evidence. The blog also demonstrates how to perform strip plot analysis using the Agri Analyze tool through a step-by-step procedure. Overall, it serves as a practical guide combining theory, calculations, and software-based implementation for researchers and students.
1.Introduction
The Strip Plot Design (SPD) is particularly suitable for two-factor experiments where higher precision is needed for measuring the interaction effect between the factors compared to measuring
the main effects of either factor individually. This design is also ideal when both sets of treatments require large plots.
For instance, in experiments involving spacing and ploughing treatments, cultural convenience necessitates larger plots.
Ploughing strips can be arranged in one direction, and spacing strips can be laid out perpendicular to the ploughing strips.
This arrangement is achieved using:
Vertical strip plot for the first factor (the vertical factor)
strip plot for the second factor (the horizontal factor)
Interaction plot for the interaction between the two factors.
The vertical and horizontal strip plots are always perpendicular to each other. However, their sizes are unrelated, unlike the main plot and subplot in the split plot design. The interaction plot is the smallest. In a strip plot design,
the precision of the main effects of both factors is sacrificed to improve the precision of the interaction effect.
2. Randomization and Layout Planning for Strip Plot Design
Step 1:Assign horizontal plots by dividing the experimental area into r blocks, then dividing each block into horizontal strips.
Follow the randomization procedure used in RBD, and randomly assign the levels of the first factor to the horizontal strips within each of the r blocks, separately and independently.
Step 2:Assign vertical plots by dividing each block into b vertical strips. Follow the randomization procedure used in RBD with b treatments and r replications,
and randomly assign the b levels to the vertical strips within each block, separately and independently.
3. Layout Example:
A sample layout of strip-plot design with six varieties (V1, V2, V3, V4, V5 and V6) as a horizontal factor and three nitrogen rates (N1, N2 and N3) as a vertical factor in three replications.
| Replication I |
|
Replication II |
|
Replication III |
| N1 |
N3 |
N2 |
|
N3 |
N2 |
N1 |
|
N3 |
N1 |
N2 |
| V6 |
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V4 |
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V5 |
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| V5 |
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V2 |
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V2 |
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| V3 |
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V6 |
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V3 |
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| V2 |
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V3 |
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V4 |
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| V4 |
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V1 |
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V6 |
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| V1 |
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V5 |
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V1 |
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Statistical Model of Strip Plot Design
$$
Y_{ijk} = \mu + \gamma_i + A_j + \varepsilon_{ij} + B_k + \beta_{ik} + (AB)_{jk} + \delta_{ijk}
$$
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
Analysis of Variance for Strip Plot Design
| Source |
DF |
SS |
MS |
Cal F |
| Replication |
r-1 |
$$
\sum_{i=1}^{r} Y_{i..}^2 - \frac{(\sum Y_{ijk})^2}{rab}
$$
|
$$
\frac{RSS}{r-1}
$$
|
$$
\frac{RMS}{E(a)MS}
$$
|
| Horizontal Factor (A) |
a-1 |
$$
\sum_{j=1}^{a} Y_{.j.}^2 - \frac{(\sum Y_{ijk})^2}{rab}
$$
|
$$
\frac{ASS}{a-1}
$$
|
$$
\frac{AMS}{E(a)MS}
$$
|
| Error (a) |
(r-1)(a-1) |
$$
\sum_{i=1}^{r}\sum_{j=1}^{a} \frac{Y_{ij.}^2}{b}
- \frac{(\sum Y_{ijk})^2}{rab}
- RSS - ASS
$$
|
$$
\frac{E(a)SS}{(r-1)(a-1)}
$$
|
– |
| Vertical Factor (B) |
b-1 |
$$
\sum_{k=1}^{b} Y_{..k}^2 - \frac{(\sum Y_{ijk})^2}{rab}
$$
|
$$
\frac{BSS}{b-1}
$$
|
$$
\frac{BMS}{E(b)MS}
$$
|
| Error (b) |
(r-1)(b-1) |
$$
\sum_{i=1}^{r}\sum_{k=1}^{b} \frac{Y_{i.k}^2}{a}
- \frac{(\sum Y_{ijk})^2}{rab}
- RSS - BSS
$$
|
$$
\frac{E(b)SS}{(r-1)(b-1)}
$$
|
– |
| A × B |
(a-1)(b-1) |
$$
\sum_{j=1}^{a}\sum_{k=1}^{b} \frac{Y_{.jk}^2}{r}
- \frac{(\sum Y_{ijk})^2}{rab}
- ASS - BSS
$$
|
$$
\frac{ABSS}{(a-1)(b-1)}
$$
|
$$
\frac{ABMS}{E(c)MS}
$$
|
| Error (c) |
(r-1)(a-1)(b-1) |
By subtraction |
$$
\frac{E(c)SS}{(r-1)(a-1)(b-1)}
$$
|
– |
| 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 Horizontal factor (A)
$$
S.Em.(A) = \sqrt{\frac{E(a)MS}{rb}}
$$
2. Standard error of mean for Vertical factor (B)
$$
S.Em.(B) = \sqrt{\frac{E(b)MS}{ra}}
$$
3. Standard error of mean for interaction (A × B)
$$
S.Em.(A \times B) = \sqrt{\frac{E(c)MS}{r}}
$$
4. Critical difference for main factor (A)
$$
CD_{0.05}(A) = S.Em.(A) \times \sqrt{2} \times t_{0.05,ne(a)}
$$
5. Critical difference for sub factor (B)
$$
CD_{0.05}(B) = S.Em.(B) \times \sqrt{2} \times t_{0.05,ne(b)}
$$
6. Critical difference for interaction (A × B)
$$
CD_{0.05}(A \times B) = S.Em.(A \times B) \times \sqrt{2} \times t_{0.05,ne(c)}
$$
Calculation of Coefficient of Variation (CV%)
1. CV for Horizontal factor (A)
$$
CV\%(A) = \frac{\sqrt{E(a)MS}}{\bar{Y}} \times 100
$$
2. CV for Vertical factor (B)
$$
CV\%(B) = \frac{\sqrt{E(b)MS}}{\bar{Y}} \times 100
$$
3. CV for interaction (A × B)
$$
CV\%(AB) = \frac{\sqrt{E(c)MS}}{\bar{Y}} \times 100
$$
Example of Strip Plot Design
In the previous chapter, this dataset was used for a split-plot design and now the same dataset will be used to illustrate a strip plot design.
A strip plot design was used to investigate the effects of irrigation levels (Horizontal factor) and fertilizer types (Vertical factor) on the yield of a particular crop. The experiment was conducted over four replicates (R1, R2, R3, R4).
Factors:
Horizontal Factor (A - Irrigation Levels):
A1: Low Irrigation
A2: Medium Irrigation
A3: High Irrigation
Vertical Factor (B - Fertilizer Types):
B1: Organic Fertilizer
B2: Inorganic Fertilizer
B3: Mixed Fertilizer
Treatments
| 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:
| Treatments |
R1 |
R2 |
R3 |
R4 |
Treatment total |
Treatment means |
| 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):
$$
CF = \frac{(\sum Y_{ijk})^2}{rab}
$$
$$
CF = \frac{(15411)^2}{4 \times 3 \times 3} = 6597192
$$
2. General mean ($\bar{Y}$):
$$
\bar{Y} = \frac{15411}{4 \times 3 \times 3} = 428.08
$$
3. Replication SS (RSS):
$$
RSS = \frac{\sum Y_{i..}^2}{ab} - \frac{(\sum Y_{ijk})^2}{rab}
$$
$$
= \frac{3903^2 + 4298^2 + 3265^2 + 3945^2}{3 \times 3} - 6597192
$$
$$
= 61636.97
$$
Horizontal factor (A) × Replication table:
|
R1 |
R2 |
R3 |
R4 |
| A1 |
1358 |
1437 |
1203 |
1376 |
| A2 |
1311 |
1458 |
1114 |
1315 |
| A3 |
1234 |
1403 |
948 |
1254 |
4. Horizontal factor SS (ASS):
$$
ASS = \frac{\sum Y_{.j.}^2}{rb} - \frac{(\sum Y_{ijk})^2}{rab}
$$
$$
= \frac{5374^2 + 5198^2 + 4839^2}{4 \times 3} - 6597192
$$
$$
= 12391.17
$$
5. Error associated with horizontal factor ($E(a)SS$):
$$
E(a)SS = \frac{\sum \sum Y_{ij.}^2}{b} - \frac{(\sum Y_{ijk})^2}{rab} - RSS - ASS
$$
$$
= 4382.61
$$
6. Vertical factor SS (BSS):
$$
BSS = \frac{\sum Y_{..k}^2}{ra} - \frac{(\sum Y_{ijk})^2}{rab}
$$
$$
= 128866.70
$$
Vertical factor (B) × Replication table:
|
R1 |
R2 |
R3 |
R4 |
| B1 |
1117 |
1190 |
779 |
1071 |
| B2 |
1422 |
1622 |
1318 |
1495 |
| B3 |
1364 |
1486 |
1168 |
1379 |
7. Error associated with vertical factor ($E(b)SS$):
$$
E(b)SS = \frac{\sum \sum Y_{i.k}^2}{a} - \frac{(\sum Y_{ijk})^2}{rab} - RSS - BSS
$$
$$
= 4752.44
$$
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 |
|
8. A × B SS (ABSS):
$$
ABSS = \frac{\sum \sum Y_{.jk}^2}{r} - \frac{(\sum Y_{ijk})^2}{rab} - ASS - BSS
$$
$$
= 304.16
$$
9. Total SS (TSS):
$$
TSS = \sum \sum \sum Y_{ijk}^2 - \frac{(\sum Y_{ijk})^2}{rab}
$$
$$
= 213796.80
$$
10. Main Error ($E(c)SS$):
$$
E(c)SS = TSS - RSS - ASS - E(a)SS - E(b)SS - BSS - ABSS
$$
$$
= 1462.72
$$
Final ANOVA Table for Crop Yield Analysis Using Strip Plot Design with Irrigation and Fertilizer Treatments
TABLE F
| SV |
DF |
SS |
MS |
CAL F |
5% |
1% |
| Replication |
3 |
61636.97 |
20545.66 |
28.12 |
3.49 |
10.80 |
| Horizontal plot (A) |
2 |
12391.17 |
6195.58 |
8.48 |
5.14 |
10.92 |
| Error (A) |
6 |
4382.61 |
730.44 |
|
|
|
| Vertical plot (B) |
2 |
128866.67 |
64433.33 |
81.35 |
5.14 |
10.92 |
| Error (B) |
6 |
4752.44 |
792.07 |
|
|
|
| $A \times B$ |
4 |
304.17 |
76.04 |
0.62 |
3.26 |
5.41 |
| Error (C) |
12 |
1462.72 |
121.89 |
|
|
|
| Total |
35 |
213796.75 |
|
|
|
|
Calculation of degrees of freedom:
Replication DF: r-1 = 4-1=3
Main plot (A): a-1=3-1=2
Error (A):(r-1)*(a-1)=3*2=6
Main plot (B): b-1=3-1=2
Error (B): (r-1)*(b-1)=3*2=6
$A \times B$: (a-1)*(b-1)=2*2=4
Error (C): (r-1)*(a-1)*(b-1)=3*2*2=12
Total: rab-1=4*3*3-1=35
Calculation of MS:
Replication: 61636.97/3=20545.66
Main plot (A): 12391.17/2=6195.58
Error (A): 4382.61/6=730.44
Main plot (B): 128866.67/2=64433.33
Error (B): 4752.44/6=792.07.
A x B: 304.17/4=76.04
Error (C): 1462.72/12=121.89
Calculation of Standard Error of Mean and Critical Difference (CD)
-
Standard error of mean for horizontal factor (A)
\[
\text{S.Em. (A)} = \sqrt{\frac{E(a)\,MS}{rb}}
= \sqrt{\frac{730.44}{4 \times 3}} = 7.80
\]
-
Standard error of mean for vertical factor (B)
\[
\text{S.Em. (B)} = \sqrt{\frac{E(b)\,MS}{ra}}
= \sqrt{\frac{792.07}{4 \times 3}} = 8.12
\]
-
Standard error of mean for interaction (A × B)
\[
\text{S.Em. (A \times B)} = \sqrt{\frac{E(b)\,MS}{r}}
= \sqrt{\frac{121.89}{4}} = 5.52
\]
-
Critical difference of mean for horizontal factor (A)
\[
\text{CD}_{0.05} = \text{S.Em. (A)} \times \sqrt{2} \times t_{0.05,\;ne(a)}
\]
\[
= 7.80 \times \sqrt{2} \times 2.44 = 26.99
\]
-
Critical difference of mean for vertical factor (B)
\[
\text{CD}_{0.05} = \text{S.Em. (B)} \times \sqrt{2} \times t_{0.05,\;ne(b)}
\]
\[
= 8.12 \times \sqrt{2} \times 2.44 = 28.11
\]
-
Critical difference of mean for interaction (A × B)
\[
\text{CD}_{0.05} = \text{S.Em. (A \times B)} \times \sqrt{2} \times t_{0.05,\;ne(c)}
\]
\[
= 5.52 \times \sqrt{2} \times 2.17 = 17.00
\]
Calculation of Coefficient of Variation (CV%)
-
Coefficient of variation for horizontal factor (A)
\[
\text{CV % (A)} = \frac{\sqrt{E(a)\,MS}}{\bar{Y}} \times 100
= \frac{\sqrt{730.44}}{428.08} \times 100 = 6.31
\]
-
Coefficient of variation for vertical factor (B)
\[
\text{CV % (B)} = \frac{\sqrt{E(b)\,MS}}{\bar{Y}} \times 100
= \frac{\sqrt{792.07}}{428.08} \times 100 = 6.57
\]
-
Coefficient of variation (Interaction)
\[
\text{CV % (C)} = \frac{\sqrt{E(c)\,MS}}{\bar{Y}} \times 100
= \frac{\sqrt{121.89}}{428.08} \times 100 = 2.57
\]
Conclusion
- The calculated F-value (28.12) is much greater than the critical F-values at both
5% (3.49) and 1% (10.80) significance levels. Therefore, there is strong evidence
to suggest that there are significant differences between the replicates.
- The calculated F-value (8.48) for the horizontal factor exceeds the critical
F-value at the 5% (5.14) significance level. This indicates that there are
significant differences among the irrigation levels.
- The calculated F-value (81.35) for the vertical factor exceeds the critical
F-value at the 1% (10.92) significance level. This indicates that there is
highly significant variation among fertilizer levels.
- The calculated F-value (0.62) for the interaction between the main factor and
sub-factor (A × B) is less than the critical F-value at the 5% (2.93)
significance level. This indicates that there is a non-significant interaction
between irrigation and fertilizer.
- For irrigation, the highest yield was observed for A1, and A2
was found to be statistically at par with it based on the critical difference.
- For fertilizer, the highest yield was observed for B2, and none of the
fertilizer levels were found to be at par with it based on the critical difference.
- For the interaction (A × B), the highest yield was observed for A1 ×
B2, and none of the combinations of the two factors were found to be
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.
| Horizontal |
Vertical |
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 |
Step 2: Click on ANALYTICAL TOOL ->DESIGN OF EXPERIMENT ->STRIP PLOT DESIGN ANALYSIS
Step 3: Open link https://www.agrianalyze.com/StripPlot.aspx (For first time users free registration is mandatory)
Step 4: Link Here to download sample file Sample File Download
Step 5: Select Replication, Horizontal factor (A), Vertical factor (B) and Dependent variable (Yield)
Step 6: Select a test for multiple comparisons, such as Least Significant Difference (LSD) test or Tuckey's test or Duncan's New Multiple Range Test (DNMRT test) for grouping of treatment means.
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
Reference
Gomez, K. A., & Gomez, A. A. (1984). Statistical Procedures for Agricultural Research. John wiley & sons. 108-120.
The blog is written by:
Darshan Kothiya, PhD Scholar, Department of Agricultural Statistics, BACA, AAU, Anand