How to Analyze Split Plot Experiments

SmartstatXL is an Excel Add-In that can be used to simplify the analysis of experimental data, including analysis of variance RBD/CRD Split Plot (main plot using Randomized Block Design/Completely Randomized Design). Currently, only for balanced designs ( Balanced Design ). Besides being able to be used for analysis of experimental data with standard designs,

Leveraging Excel's functionality, SmartstatXL serves as an Add-In that simplifies experimental data analysis. Its expertise includes Split Plot Analysis based on RBD or CRD, where the main plots are arranged in a Randomized Block Design or Completely Randomized Design. Although the primary focus is on balanced design, SmartstatXL is also capable of handling mixed models in addition to standard designs.

Special features for CRD/RBD Split Plot experiments in SmartstatXL include:

• CRD/RBD Split Plot: Refers to Split Plot experiments where each observational unit is measured only once.
• CRD/RBD Split Plot: Sub-Sampling: Intended for repeated observations with the ability to draw sub-samples from a single observational unit. For example, in a single observational unit (treatment 3Dok1, first replication), measurements are taken on 10 plants.
• Split Plot: Repeated Measure: Specifically for observations that are conducted periodically on a single observational unit, such as every 14 days.
• Split Plot: Multi-Location/Season/Year: The ideal solution for experiments conducted in different locations, seasons, or years.

If the treatment effects are significant, SmartstatXL allows for Post hoc Test execution to compare the mean values of the treatments. Available options include: Tukey, Duncan, LSD, Bonferroni, Sidak, Scheffe, REGWQ, Scott-Knott, and Dunnet.

Case Example

D.C Arny from the University of Wisconsin compared four wheat varieties (factor A), namely Vicland 1, Vicland 2, Clinton, and Branch, which were placed as main plots, and chemical treatments for seeds (factor B) such as control, Ceresan M, Panogen, and Agrox, which were placed in sub-plots with as many as four groups. The research results are presented in the following table:

Cited from: Steel and Torrie, 1960

Steps for Analysis of Variance (ANOVA) and Post Hoc Test:

1. Make sure the worksheet (Sheet) you want 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 Menu Split Plot > RBD/CRD Split Plot.
   
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, click the Next Button
8. The following ANOVA – RBD/CRD Split Plot Dialog Box will appear:
   
9. There are three stages in this dialog. In the first stage, choose the Factor and at least one Response to be analyzed.
10. When selecting a Factor, SmartstatXL will provide additional information about the number of levels and their names. In Split Plot experiments (CRD/RBD/LSD), Replications are also included as a factor.
11. Details of the ANOVA STAGE 1 Dialog Box can be seen in the following image:
    
12. After confirming that the Dataset is correct, click the Next Button to proceed to the ANOVA Stage-2 Dialog Box
13. The dialog box for the second stage will appear.
    
14. Adjust the settings according to your research methodology. In this example, the Post Hoc Test used is the Tukey Test.
15. To set additional output and default values for subsequent outputs, press the "Advanced Options…" button.
16. Here is the display of the Advanced Options Dialog Box:
    
17. After setting it 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 prompted to specify the average table, ID for each Factor, and rounding of the average values. Details can be seen in the following image:
    
20. As a final step, click "OK"

Analysis Results

Analysis Information

From the preliminary analysis information, we can understand the basic structure of the experiment conducted:

1. Experimental Design: In this experiment, a Randomized Block Design (RBD) Split Plot is employed. This is an experimental design that involves two factors (in this case, wheat varieties as the main plot and chemical treatments as the subplot), where one factor is assigned to the main plot and the other is assigned to the subplot.
2. Post Hoc Test: Following the Analysis of Variance, if significant differences exist between groups, the Post Hoc test used to identify which groups differ is the Tukey test (BNJ).
3. Response: The response variable measured in this experiment is "Yield."
4. Factors:
   • Rep (4 levels): This refers to the four different groups in the experiment.
   • Lot (Main) (4 levels): This refers to the four different wheat varieties, namely Vicland 1, Vicland 2, Clinton, and Branch.
   • Treatment (Sub) (4 levels): This refers to the four different chemical treatments applied to the seeds, namely control, Ceresan M, Panogen, and Agrox.

Therefore, the basis of this experiment is to evaluate how four different wheat varieties respond to four different chemical treatments in terms of yield obtained.

Analysis of Variance

The Analysis of Variance table provides statistical information about the effects of the two factors (wheat varieties and chemical treatments) and their interaction on the response variable, "Yield." Below is the interpretation and discussion of the provided table:

Interpretation:

1. Rep (R):
   • F-Value (13.794) is much greater than F-0.05 (3.863) and F-0.01 (6.992), with a P-Value of 0.001. This indicates that there is a significant difference between the four groups at the 1% significance level.
2. Lot (Main) (L):
   • F-Value (13.819) is also much greater than F-0.05 (3.863) and F-0.01 (6.992), with a P-Value of 0.001. This indicates that there is a significant difference between the four wheat varieties at the 1% significance level.
3. Treatment (Sub) (T):
   • F-Value (2.799) is close to F-0.05 (2.866) but smaller than it, with a P-Value of 0.054. This indicates that the chemical treatments do not significantly influence yield at the 5% significance level. However, it is close to the significance threshold.
4. L x T (Interaction between Lot and Treatment):
   • F-Value (3.208) is greater than F-0.05 (2.153) but smaller than F-0.01 (2.946), with a P-Value of 0.006. This indicates that there is a significant interaction between the wheat varieties and the chemical treatments at the 1% significance level.
5. CV(a) and CV(b):
   • CV(a) and CV(b) are coefficients of variation. CV(a) is 15.70% for error a and CV(b) is 8.53% for error b. These provide a measure of variability in the data that is not explained by the model.

Discussion:

• The results show that there is a significant difference between the four groups and between the four wheat varieties in terms of yield. This means that both the group and wheat varieties influence the yield obtained.
• Chemical treatments do not significantly influence yield at the 5% significance level. However, it is important to note that the effect is close to the significance threshold, so more data or further experiments may be needed to determine the actual effect of the chemical treatments.
• The significant interaction between wheat varieties and chemical treatments indicates that the effect of chemical treatments on yield varies depending on the wheat variety used. In other words, some wheat varieties may be more responsive to certain chemical treatments compared to others.
• The coefficients of variation indicate that there is still variability in the data that is not explained by the model. This could be due to other factors not included in the analysis or natural variation within the experiment.

Post hoc Test

Based on the Analysis of Variance, there is an interaction effect between Lot and Treatment on Yield. Although the single effects are significant, the discussion should focus on the interaction effect between these two factors.

Single Effect

1. Single Effect:

Lot (Main) (L):

Among the wheat varieties (Lot), Vicland 1 has a lower average yield compared to the other varieties. Vicland 2, Clinton, and Branch show no significant difference in yields but all have higher yields than Vicland 1.

Treatment (Sub) (T):

There is no significant difference between different chemical treatments on seed yield.

2. Interaction Effect

Two presentation formats are available for the interaction effect's mean table. You can choose either one or both. The first format is a one-way table, where treatment levels are combined, and the layout is similar to the single effect table. The second format tests the simple effects and is presented in a two-way table. The choice of mean table and graph presentation can be adjusted through Advanced Options (refer back to step 15 of Analysis of Variance Steps).

First Format: Lot x Treatment Effect

The First Format displays combinations of wheat varieties and chemical treatments. It gives a clear picture of how specific combinations affect yield. Some combinations, like Vicland 1 with Check and Agrox, have lower yields compared to other combinations. Meanwhile, Branch with Ceresan M shows the highest yield among all combinations.

Second Format: Simple Effect of Lot x Treatment

The Second Format breaks down the interaction into simple effects for the wheat varieties and chemical treatments.

Vertically:

Vicland 1 with all chemical treatments has lower yields compared to other wheat varieties with the same chemical treatments. Branch, on the other hand, tends to have the highest yields among all wheat varieties with the same chemical treatments.

Horizontally:

There is no significant difference between different chemical treatments for each wheat variety. However, for Vicland 1, the treatment with Ceresan M shows higher yields compared to Check and Agrox.

Discussion:

The results indicate that wheat varieties have a significant influence on yield. Vicland 1, in particular, has lower yields compared to other varieties. Chemical treatments, on the other hand, show no significant difference in yields.

However, when considering the interaction between wheat varieties and chemical treatments, we see some interesting patterns. Some combinations of wheat varieties and chemical treatments show higher yields compared to others. This suggests that while wheat varieties and chemical treatments may not have significant single effects, their combination can be crucial.

Specifically, Vicland 1 appears to be less responsive to chemical treatments compared to other wheat varieties. Branch, on the other hand, shows a better response to some chemical treatments.

It's essential to consider these interactions when selecting wheat varieties and chemical treatments to achieve the best yield. Further experiments may be needed to better understand the mechanisms behind these interactions.

Assumptions Check in ANOVA

Formal Approach (Statistical Tests)

The assumptions of homogeneity of variance and data normality are two main assumptions in Analysis of Variance (ANOVA). If these assumptions are violated, the ANOVA results may not be valid.

1. Levene's Test for Homogeneity of Variance:
   • Null hypothesis (H0): Variance is the same across all groups.
   • Alternative hypothesis (H1): At least one group has a variance different from the other groups.
   • With a p-value greater than 0.05 (0.431 > 0.05), we fail to reject the null hypothesis. This indicates that there is insufficient evidence to state that variances differ between groups. Therefore, the assumption of homogeneity of variance is met.
2. Normality Test:
   • Null hypothesis (H0): Residuals are normally distributed.
   • Alternative hypothesis (H1): Residuals are not normally distributed.
   • All tests show p-values greater than 0.05, meaning we fail to reject the null hypothesis that residuals are normally distributed. Therefore, the assumption of normality is met.

Discussion:

Based on the results of the Levene test and various normality tests, the assumptions of variance homogeneity and data normality for analysis of variance (ANOVA) are met. This means that the results from the previously conducted ANOVA are valid and reliable. Meeting these assumptions is crucial to ensure that the conclusions drawn from the ANOVA are valid.

Visual Approach (Plot Charts)

1. Normal P-Plot of Residual Data
   • The Normal P-Plot is used to check the assumption of residual normality. If the points closely follow the diagonal line, this indicates that the data is approximately normally distributed. In this chart, the points seem to follow the diagonal line relatively well, although there are some deviations at the ends. This suggests that the assumption of normality is largely met.
2. Residual Data Histogram
   • The histogram is used to assess data distribution. In this case, we examine the distribution of the residuals. A bell-shaped histogram indicates a normal distribution. The histogram appears symmetric and approaches a bell shape, albeit with some deviations. This supports the findings from the Normal P-Plot that the assumption of normality is largely met.
3. Residual vs. Predicted Plot
   • This plot is used to examine the assumption of homoscedasticity (constant variance) of the residuals. If the points are randomly scattered without any specific pattern, this assumption is met. In this chart, there's no specific pattern evident, although there are some deviations. Therefore, the assumption of homoscedasticity appears to be met.
4. Standard Deviation vs. Mean
   • This chart is used to check the variability within the data. If the points are randomly scattered without a specific pattern, this indicates that the variance is constant across levels of the predictor variable. In this chart, the points seem to scatter randomly without a specific pattern, indicating variance homogeneity.

Conclusion:

Based on the graphical examination of the charts, it appears that the primary assumptions of ANOVA are met. This reaffirms the earlier findings from statistical tests and ensures that your ANOVA results are valid and reliable.

Box-Cox Transformation and Residual Analysis

1. Box-Cox Transformation:

The Box-Cox transformation is used to address violations of assumptions in regression and ANOVA, such as homoscedasticity and normality. The obtained lambda value is 0.964. As this value is close to 1, no transformation is required, meaning your data already meet the basic assumptions of ANOVA.

2. Residual Values and Outlier Examination:

The "Residual" column shows the difference between the actual observation values ("Yield") and the predicted values. Large residuals could indicate the presence of outliers.

Some observations have been labeled as "Outliers" in the "Diagnostic" column, meaning they have large residuals and are considered outliers. For instance, observations for Vicland 2 with the Check treatment and Clinton with the Panogen treatment have significant residuals, indicating that their response values differ significantly from what is predicted by the model.

Discussion:

• Outlier: An outlier is an observation that differs significantly from the majority of other observations. In this context, outliers are identified based on the magnitude of their residuals. An observation is considered an outlier if its residual is significantly larger or smaller than the average residual.
• Leverage: Leverage measures how far the predictor values (X) deviate from the average of other predictors. Observations with high leverage can have a substantial impact on the regression results and may affect model assumptions.

Recommendation:

• For observations identified as outliers, it is advised to revisit the raw data. The aim is to evaluate whether there are recording errors or other factors that can explain the unusual values. If no errors are found, consider removing the outliers or replacing them with other values.
• If model assumptions are still not met even after removing outliers, attempting a Box-Cox transformation could be an option. In this case, the recommended lambda value can be used to correct for assumption violations. However, it appears that no transformation is needed as the lambda value is close to 1.
• It is crucial to make decisions based on a deep understanding of the data and the experimental context. Any actions taken should be clearly documented in the analysis report.

Conclusion

1. Analysis of Variance:
   • There is a significant difference among the four groups and among the four wheat varieties in terms of yield. Vicland 1 has lower yield compared to other wheat varieties.
   • Chemical treatment on the seeds does not show a significant effect on the yield at the 5% level. However, there are indications that its effect is approaching the threshold of significance.
   • There is a significant interaction between wheat varieties and chemical treatment. This means that the effect of chemical treatment on the yield varies depending on the wheat variety used.
2. Assumption Checks:
   • The Levene's test and various normality tests indicate that the assumptions of homogeneity of variances and data normality are met. This ensures that the results from the Analysis of Variance (ANOVA) are valid and reliable.
   • Assumption check graphs, including Normal P-Plot, Histogram, Residual vs. Predicted, and Standard Deviation vs. Mean, all indicate that the data meet the basic assumptions of ANOVA.
3. Box-Cox Transformation:
   • The Box-Cox transformation indicates that the data already meet the basic assumptions of ANOVA, with a lambda value approaching 1, so no further transformation is needed.
4. Outlier Data Examination:
   • Some observations are identified as outliers based on the magnitude of their residuals. Although these outliers may impact the model, further investigation is needed before making any decisions about them.

Taking into account all the above findings, it can be concluded that wheat varieties have a significant influence on yield, while chemical treatment tends to have no significant effect. However, the combination of both—the wheat variety and the chemical treatment—shows a significant interaction, suggesting that the effect of the chemical treatment may vary depending on the wheat variety used. Additionally, the data meet the basic assumptions required for ANOVA, making this analysis valid and reliable.