How to Analyze Split Plot Experiments (CRD/RCBD) across Multiple Locations, Years, Seasons

In agricultural research and other applied sciences, more complex experimental designs such as Split Plot are often required to understand the interaction between treatments under various environmental conditions. SmartstatXL provides an optimal solution for researchers looking to analyze Split Plot experimental data, with main plots based on CRD (Completely Randomized Design), RBD (Randomized Block Design), or LSD (Latin Square Design), conducted across different locations, seasons, or years.

The strength of SmartstatXL lies in its ease of accommodating Balanced Designs, ensuring data analysis integrity under various conditions. If a significant treatment effect is found, this tool allows researchers to delve deeper through various Post Hoc Tests. Available options include: Tukey, Duncan, LSD, Bonferroni, Sidak, Scheffe, REGWQ, Scott-Knott, and Dunnet. With these advanced features, researchers can ensure that their data interpretation is based on thorough and accurate statistical analysis.

Case Example

Yield of two rice varieties (ton/ha) tested with six levels of nitrogen at three locations in an RBD-based Split Plot, with three replications. The Yield Data (ton/ha) is presented in the following table:

Cited from:
Gomez, Kwanchai A. and Gomez, Arturo A. 1995. Statistical Procedures for Agricultural Research. [trans.] Endang Sjamsuddin and Justika S. Baharsjah. Second Edition. Jakarta: UI-Press, 1995. ISBN: 979-456-139-8. p. 350.

Steps for Analysis of Variance (Anova) and Post Hoc Tests:

1. Ensure the worksheet (Sheet) you wish to analyze is active.
2. Place the cursor on the Dataset. (For information on creating a Dataset, please refer to the 'Data Preparation' guide).
3. If the active cell is not on the dataset, SmartstatXL will automatically detect and select the appropriate dataset.
4. Activate the SmartstatXL Tab
5. Click on the Menu Split Plot > Multi Location/Year/Season.
   
6. SmartstatXL will display a dialog box to confirm whether the Dataset is correct (usually the cell address for the Dataset is automatically selected correctly).
   
7. After confirming that the Dataset is correct, press the Next Button
8. A dialog box titled Anova – Split Plot (repeated at multiple locations) will appear next:
   
   The Factorial Experimental Model is analyzed by including an additional Factor for Season/Location/Year and is analyzed collectively (sometimes referred to as Mixed Design or Split Split Plot in Time).
   In the Split Split Plot in Time model, for both CRD and RBD, the Replication Factor must be included in the model!
   Compare with the Factorial Experimental Model that is partially analyzed at each location, as shown in the following image:
   
   In the above standard Split Plot Experimental Model, there is no Location Factor.
9. There are 3 stages. In the first stage, select the Factors and at least one Response to be analyzed (as shown in the image above)!
10. When you select Factors, SmartstatXL will provide additional information about the number of levels and their names.
11. Details of the Anova STAGE 1 dialog box can be seen in the following image:
    
12. After confirming the Dataset is correct, press the Next Button to enter the Anova Stage-2 Dialog Box
13. The dialog box for the second stage will appear.
    
14. Adjust the settings according to your research method. In this example, the Post Hoc Test used is the Tukey Test.
15. To adjust additional outputs and default values for subsequent outputs, press the "Advanced Options…" button.
16. Here is the appearance of the Advanced Options Dialog Box:
    
17. Once you've finished setting up, close the "Advanced Options" dialog box.
18. Next, in the Anova Stage 2 Dialog Box, click the Next Button.
19. In the Anova Stage 3 Dialog Box, you will be asked to specify the average table, ID for each Factor, and rounding of average values. The details can be seen in the following image:
    
20. As the final step, click "OK"

Analysis Results

Analysis Information

Experimental Design

The design used in this experiment is an RBD (Randomized Block Design) Split Plot, repeated across various locations, seasons, or years. This is a fairly complex design often used in agricultural research to minimize the effects of land heterogeneity or experimental conditions.

Post Hoc Tests

The post hoc test to be used is Tukey (HSD), which is one of the statistical methods for comparing effects between groups after Analysis of Variance (ANOVA) shows a significant difference.

Response Variable

The response variable in this experiment is grain yield (ton/ha). This is a crucial quantitative variable in the context of rice production.

Factors

• Replication: There are three levels of replication, serving to estimate experimental error.
• Location: The experiment is conducted at three different locations, serving as an indicator of environmental variability.
• Nitrogen (Main): Six levels of nitrogen are tested as the main factor in the experiment.
• Variety (Sub): Two different rice varieties are tested as the subplot factor.

Assumption Violations

In this analysis, there appears to be indications of assumption violations. Common assumptions checked in ANOVA include data normality, variance homogeneity, and observation independence. Violation of these assumptions can impact the validity of the analysis results.

Anova Assumption Checks

Formal Approach (Statistical Tests)

Levene's Test for Homogeneity of Variances

Levene's test is conducted to assess whether the variability (variance) across all experimental groups is the same. In this case, the F-value is 2.32 with a P-value of 0.001. Since the P-value is less than 0.05, this result indicates that the variances between groups differ significantly, violating the assumption of variance homogeneity in ANOVA.

This assumption violation requires further action, such as data transformation or the use of more robust analysis methods against heteroskedasticity (for example, statistical tests that do not require the assumption of variance homogeneity).

Normality Test

Several tests have been conducted to assess whether the residuals are normally distributed, including Shapiro-Wilk's, Anderson-Darling, D'Agostino-Pearson, Liliefors, and Kolmogorov-Smirnov. All tests show p-values above 0.05, indicating that the data meet the normal distribution assumption. Therefore, the normality assumption is met in this analysis.

Conclusion

1. The homogeneity of variance assumption is violated, indicating the need for corrective measures before proceeding with ANOVA or using alternative statistical methods.
2. The normality assumption is met, thus requiring no further action related to this aspect.

The violation of the homogeneity of variance assumption is a serious issue that needs to be addressed to ensure the validity of the analysis results. Subsequently, data transformation or the use of more robust alternative analysis methods may be necessary to address this assumption violation.

Visual Approach (Plot Graphics)

1. Normal P-Plot of Residual Data

The Normal P-Plot is used to check whether the data is normally distributed. In this graph, the points seem to follow the diagonal line quite well. This suggests that the residual data is likely normally distributed, consistent with the results of the previous normality tests.

2. Residual Data Histogram

The histogram is also used to check data distribution. The shape of the histogram shows a relatively symmetrical distribution and follows the shape of the normal curve. This provides additional evidence supporting that the data meets the normal distribution assumption.

3. Residual vs. Predicted Plot

This plot is used to check for homoscedasticity—that is, whether the residual variance is constant across the levels of predictors. In this graph, the points seem to be randomly scattered without a specific pattern, which is generally considered an indication of homoscedasticity. However, this result slightly contradicts the Levene's test, which indicated heteroscedasticity. Therefore, further analysis may be needed to resolve this inconsistency.

4. Standard Deviation vs. Mean

This graph is used to check for homogeneity of variances between groups. Ideally, points should be randomly scattered around the mean value. If there's a pattern (e.g., funnel shape), it could indicate heteroscedasticity. In this graph, there appears to be some variation in the standard deviations across different mean levels, which again supports Levene's test results regarding the violation of the homogeneity of variance assumption.

Conclusion

• The normal distribution assumption appears to be met based on the Normal P-Plot and Histogram.
• There are some inconsistencies between Levene's test and the graphic plots indicating homoscedasticity. This requires further attention.
• The violation of the homogeneity of variance assumption is also supported by the Standard Deviation vs. Mean graph.

Overall, the graphical assumption checks provide additional useful insights, although there are some areas requiring further clarification.

Overall, visual checks of assumptions provide additional useful insights, although there are some areas that require further clarification.

Challenges in Meeting Assumptions and Handling Outliers

Despite various efforts to meet the assumption of homogeneity of variances and to handle outlier issues, a satisfactory solution has not yet been found. The following points are worth noting:

Homogeneity of Variances

Although Box-Cox transformations and various other types of transformations have been applied, the assumption of homogeneity of variances has not yet been met. This becomes a critical issue as the violation of this assumption can affect the validity of the entire analysis. In situations like this, it may be necessary to consider more robust statistical methods against assumption violations, such as using generalized linear models or non-parametric tests.

Handling Outliers

Efforts have been made to replace outlier data with missing data calculations, but this also seems not to have solved the problem. Outliers in the data can significantly affect the analysis and interpretation, so it is crucial to find an effective way to deal with them. One approach could involve deeper analysis to understand the cause of these outliers.

Limitations in the number of varieties could be a contributing factor to the violation of the homogeneity of variance assumption. One way to address this issue is to increase the number of varieties tested, if possible. Additionally, more robust statistical methods against assumption violations can be used, such as non-parametric methods or generalized linear models.

Understanding that the research has been conducted and data has been collected, there are several steps that can be taken to maximize the utility of the existing data, despite some limitations and assumption violations.

Analysis Options:

• Alternative Statistical Methods: Choosing more robust statistical methods against the violations of assumptions of homogeneity of variances or normal distribution, such as non-parametric tests or generalized linear models, could be an option.
• Bootstrap or Resampling: This technique allows for the estimation of parameters from the existing sample, despite its limitations, and can provide confidence intervals for these parameters.
• Sensitivity Analysis: Conducting additional analyses to evaluate how sensitive the results are to assumption violations can provide insights into how robust the conclusions drawn from the data are.
• Discussion and Critical Notes: It is crucial to explicitly discuss the limitations of the analysis in the report or publication, including any assumption violations and potential biases that may have occurred.

Despite the limitations and assumption violations, the results of the analysis can still be justified as long as corrective measures and sensitivity analyses are conducted, and as long as these limitations are openly and honestly discussed. Transparency in analysis methods and interpretation will enhance the credibility of the research.

Analysis of Variance

Summary of Results

The Analysis of Variance (ANOVA) results indicate several factors that significantly affect paddy yield, while some other factors do not. Here is the interpretation of the ANOVA table:

1. Location (L): The F-value (2.905) is not significant at either the 5% or 1% level (P-Value = 0.131). This indicates that location does not have a significant effect on paddy yield.
2. Nitrogen (Main) (N): The F-value (59.309) is highly significant at the 1% level (P-Value < 0.001). This indicates that nitrogen levels have a very significant effect on paddy yield.
3. Interaction of Location and Nitrogen (L x N): The F-value (4.029) is also significant at the 1% level (P-Value = 0.001). This indicates that the interaction between location and nitrogen levels affects paddy yield.
4. Variety (Sub) (V): The F-value (25.015) is significant at the 1% level (P-Value < 0.001). This indicates that rice varieties have a significant effect on paddy yield.
5. Interaction of Location and Variety (L x V): The F-value (3.970) is significant at the 5% level (P-Value = 0.028). This indicates that the interaction between location and variety also affects paddy yield, but not as strongly as the other factors.
6. Interaction of Nitrogen and Variety (N x V) and Interaction of Location, Nitrogen, and Variety (L x N x V): Not significant (P-Value > 0.05), indicating that these interactions do not affect paddy yield.

Coefficient of Variation (CV)

• CV(a) for Error a: 14.89%
• CV(b) for Error b: 11.63%
• CV(c) for Error c: 10.45%

These coefficients of variation indicate the extent of variation in the data, and these figures are relatively low, indicating the stability of the experiment.

Conclusion

1. Nitrogen levels and Variety are the most influential factors on paddy yield, followed by the interaction between Location and Nitrogen and Location and Variety.
2. Location alone does not show a significant effect, although its interactions with Nitrogen and Variety are quite significant.
3. Despite the violation of the assumption of homogeneity of variances, these results provide valuable insights into the factors affecting paddy yield.

In this context, despite some limitations and violations of assumptions, the analysis results still provide valuable insights and can be interpreted cautiously.

Post hoc Tests

For significant factors and interactions, post hoc tests are usually conducted to understand in greater detail how these factors or combinations of factors affect the response variable. In this case, some of the post hoc tests that could be conducted are:

• Post hoc Test for Nitrogen (N): Given the high significance of this factor, a post hoc test can be used to determine which nitrogen levels yield the best paddy results.
• Post hoc Test for Interaction of Location and Nitrogen (L x N): This test will help understand how the effects of nitrogen levels differ among different locations.
• Post hoc Test for Variety (V): To identify the variety that yields the best paddy results.
• Post hoc Test for Interaction of Location and Variety (L x V): This test will provide insights into how different varieties are effective in different locations.

Single Effect of Nitrogen

The post hoc test results using the Tukey method (BNJ) for the single effect of Nitrogen (N) show significant variations in paddy yield across different Nitrogen levels.

Interpretation

• Nitrogen level N1 yields the lowest paddy output and is significantly different from other Nitrogen levels.
• There is a significant increase in paddy yield from N1 to N2.
• Nitrogen level N4 yields the highest paddy output and is significantly different from levels N1 and N2.
• Nitrogen levels N3, N5, and N6 yield similar paddy outputs and are not significantly different from each other.

Practical Implications

• Higher Nitrogen levels generally result in better paddy yields, although there is a point where increasing Nitrogen levels no longer provides significant benefits (such as between N3, N5, and N6).
• Conclusion
• Nitrogen levels have a significant effect on paddy yield, with certain levels (such as N4) yielding better results compared to others.
• These findings provide valuable insights for agricultural practitioners on how to optimize the use of Nitrogen to improve paddy yield.

Effect of the Interaction between Location and Nitrogen

The post hoc test results show significant variations in paddy yield among different combinations of Location and Nitrogen. For example, at Location 1 (L1), the use of Nitrogen N4 yields the highest paddy output (7,385.17 ± 935.46 ton/ha) included in the 'f' group. Meanwhile, at Location 2 (L2), the use of Nitrogen N5 yields the highest paddy output (7,218.83 ± 806.09 ton/ha) included in the 'ef' group.

Simple Effect of the Interaction between Location and Nitrogen

In this analysis, lowercase letters are used to compare between two locations at the same Nitrogen level, whereas uppercase letters are used to compare between two Nitrogen levels at the same location.

• At Location 1 (L1): Nitrogen N4 and N5 yield similar and higher results compared to other Nitrogen levels (group 'C').
• At Location 2 (L2): Nitrogen N5 yields the highest paddy output (group 'B').
• At Location 3 (L3): There is no significant difference in paddy yield among different Nitrogen levels (all in group 'B').

Interpretation

• Variability Across Locations: There is variability in paddy yield responses to different Nitrogen levels across different locations. For example, N4 works best in L1, while N5 works best in L2.
• Simple Effect of Nitrogen at Different Locations: The effect of Nitrogen appears more homogeneous at Location 3 compared to Locations 1 and 2, indicating that responses to Nitrogen may be influenced by other factors related to the location.

Conclusion

• The interaction between Location and Nitrogen has a significant impact on paddy yield. This means that Nitrogen usage strategies must be adjusted based on location.
• These results facilitate more accurate recommendations on how to optimize the use of Nitrogen to improve paddy yields in different locations.

Single Effect of Varieties

The post hoc test results using the Tukey method (BNJ) for the single effect of Varieties (Sub) (V) show significant variations in paddy yield between two varieties.

• Variety V2 yields higher paddy than V1, and this difference is significant at a 95% confidence level (significance level 0.05).

Practical Implications

• Given the significant difference between the two varieties, farmers may consider using variety V2 to improve paddy yield.

This emphasizes the importance of selecting the right variety as one of the strategies to improve paddy yield. Moreover, these results can potentially be combined with other findings (such as the effects of Nitrogen and Location) to create more comprehensive and effective recommendations.

Format 1: Effect of the Interaction between Location and Varieties

The post hoc test results show significant variations in paddy yield among different combinations of Location and Varieties. For example, at Location 1 (L1), Variety V2 yields higher paddy (6,345.06 ± 832.44 ton/ha) compared to Variety V1 (5,435.94 ± 876.97 ton/ha).

Interpretation

• L1_V1 vs L1_V2: At Location 1, Variety V2 yields higher paddy and is significantly different from Variety V1.
• L2_V1 vs L2_V2: At Location 2, there is no significant difference in paddy yield between Varieties V1 and V2.
• L3_V1 vs L3_V2: At Location 3, Variety V2 yields higher paddy but is not significantly different from Variety V1.

Practical Implications

• The interaction between Location and Varieties has a significant impact on paddy yield. This means that the strategy for selecting varieties must be adjusted based on location.
• For instance, at Location 1, farmers might achieve higher paddy yields using Variety V2, while at Location 3, the choice between Varieties V1 and V2 may not significantly impact paddy yield.

Conclusion

• There is significant variation in the effectiveness of rice varieties in different locations. This highlights the importance of considering specific locations when selecting varieties to plant.

This adds another layer of complexity to decisions about which varieties to plant but also offers opportunities for greater optimization through selecting the varieties best suited to specific location conditions.

Format 2: Simple Effects of Location and Varieties

Simple effects provide more specific information about how varieties perform in different locations and how locations affect yield for different varieties. This information is important for farmers or researchers who want to make more informed decisions about what varieties to plant and where.

In this analysis, lowercase letters are used to compare between two locations for the same variety, while uppercase letters are used to compare between two varieties at the same location.

• Based on Location: At Location 1 and Location 2, Variety V2 appears to be more optimal. In Location 2, both varieties have similar performance.
• Based on Variety: For Variety V1, Location 2 seems to yield the best results. Meanwhile, for Variety V2, there is no difference in yield among the 3 locations.

Overall, this information is very useful for farmers in making decisions about which variety to plant in a specific location to maximize paddy yield.

Choice Between Presenting Interaction Effects and Simple Effects

• Interaction Effects: More appropriate if the goal is to understand how location and varieties interact in affecting paddy yield.
• Simple Effects: More appropriate if the goal is to provide specific practical recommendations based on location or variety.

In this context, Simple Effects may be more informative because they provide more specific information about how varieties behave in each location. This allows for more targeted recommendations. Moreover, simple effects also facilitate interpretation and decision-making, especially for farmers or other stakeholders who need clear and easily applicable information.

Three-way Interaction Effect, Location x Nitrogen x Varieties

Further test results are not discussed here, as their effects are not significant.

Box-Cox and Residual Analysis

Box-Cox Transformation

The Box-Cox transformation is used to satisfy the assumptions of homogeneity of variance and normal distribution of the data. In this case, the obtained Lambda value is 2,000, indicating that the square transformation (Y²) is most appropriate for this data. This transformation is expected to help satisfy the assumptions required for Analysis of Variance.

Residual Analysis and Outlier Examination

• Residual: The difference between the observed value and the predicted value. Small residuals indicate that the model predicts the data well.
• Leverage: Measures how much an observation influences the model estimation. High values may indicate the presence of outliers.
• Studentized Residual: Normalized residuals. Absolute values above 2 or 3 are often considered as indications of outliers.
• Studentized Deleted Residual: Similar to Studentized Residual but calculated by removing the observation from the model.
• Cooks Distance: Measures the influence of an observation on all parameter estimates. High values indicate significant influence.
• DFITS: A diagnostic that measures how much an observation influences the estimated model.

Based on the residual analysis:

• The first observation (Rep 1, Location L1, Nitrogen N1, Variety V1) is identified as an outlier with high Studentized Residual and Studentized Deleted Residual, as well as significant Cooks Distance.
• Other observations do not appear to show signs of being outliers based on the provided diagnostic measures.

Conclusion

1. The Box-Cox transformation was successfully carried out with a Lambda value of 2, indicating that a square transformation was applied. This is expected to aid in satisfying the homogeneity of variance assumption.
2. The residual analysis indicates the presence of one outlier that could potentially affect the model. This needs to be handled carefully, either by conducting further analysis or by removing the observation from the analysis.

The Box-Cox transformation and residual analysis are crucial steps in validating model fit and the reliability of the results. The presence of outliers needs attention in the further interpretation of the analysis results.