How to Analyze Strip Plot (Split Block) Experiments

As an extension of Excel, SmartstatXL is designed to facilitate the analysis of experimental data, specifically in the Strip Plot or Split Block analysis of variance. Although the main focus is on balanced designs (Balanced Design), SmartstatXL also offers flexibility in analyzing various mixed models beyond standard designs.

The specific features available for Strip Plot/Split Block experiments in SmartstatXL include:

• Strip Plot/Split Block: Refers to experiments where each observational unit is measured only once.
• Strip Plot/Split Block: Sub-Sampling: Intended for repeated observations with the ability to draw sub-samples from a single observational unit. For example, in one observational unit (treatment 3Dok1, replicate 1), there are 10 plants measured.

If there are significant treatment effects, SmartstatXL allows for the execution of Post Hoc Tests to compare the average treatment values. These include: Tukey, Duncan, LSD, Bonferroni, Sidak, Scheffe, REGWQ, Scott-Knott, and Dunnet.

Case Example

Data on the yield of six rice varieties, sown directly with three levels of nitrogen in a Strip Plot design with three replicates. The Rice 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. Translated by Endang Sjamsuddin and Justika S. Baharsjah. Second Edition. Jakarta: UI-Press, 1995. ISBN: 979-456-139-8. p. 114.

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 determine the appropriate dataset.
4. Activate the SmartstatXL Tab
5. Click the Strip Plot Menu.
   
6. SmartstatXL will display a dialog box to confirm whether the Dataset is correct or not (usually, the cell address for the Dataset is automatically selected correctly).
   
7. After confirming the Dataset is correct, press the Next Button
8. A dialog box titled Anova – Strip Plot will appear next:
   
9. There are three stages in this dialog. In the first stage, select the Factor and at least one Response to be analyzed.
10. When you select a Factor, SmartstatXL will provide additional information about the number of levels and their names. In a Strip Plot experiment, Replicates are included as factors.
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 proceed to the Anova Stage-2 Dialog Box
13. A dialog box for the second stage will appear.
    
14. Adjust the settings based on your research method. In this example, the Post Hoc Test used is Tukey's Test.
15. To set up additional output and default values for subsequent output, press the "Advanced Options..." button.
16. Here is the appearance of the Advanced Options Dialog Box:
    
17. After completing the setup, 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 average values. The details can be seen in the following image:
    
20. As the final step, click "OK"

Analysis Results

Analysis Information

From the analysis information, it is evident that the Analysis of Variance (ANOVA) was performed on rice yield data from six rice varieties, planted with three levels of nitrogen and three replicates. The experimental design is Strip Plot, and the Post Hoc Test to be used is Tukey (BNJ).

Below are some points that will be the focus of interpretation and discussion:

1. Replicate Effect: With three replicates, the extent to which variation within replicates influences rice yield will be identified.
2. Variety Effect: Six rice varieties will be analyzed to determine which variety yields the best rice output.
3. Nitrogen Effect: Three levels of nitrogen will be analyzed to understand how nitrogen intake affects the rice yield for each variety.
4. Interaction among Factors: In addition to the single effects of each factor, it's also important to examine how these factors interact with each other in influencing rice yield.
5. Post Hoc Test: Tukey (BNJ) will be used to determine significant differences between sample groups.

Analysis of Variance

Interpretation of Analysis of Variance Results

1. Replicate Effect (U)
   • F-Value: 3.090
   • P-Value: 0.090
   • F-0.05: 4.103
   • Conclusion: Not significantly different (ns)
   • The replicate effect on rice yield shows no significant difference (P-Value > 0.05). This indicates that the replicates in this experiment are fairly consistent and do not significantly affect the outcome.
2. Variety Effect (V)
   • There is a significant difference in rice yield between varieties (P-Value < 0.01). This indicates that the rice variety significantly influences the rice yield.
3. Nitrogen Effect (N)
   • Nitrogen intake also shows a very significant effect on rice yield (P-Value < 0.01). This indicates that different levels of nitrogen affect the rice yield.
4. Interaction between Variety and Nitrogen (V x N)
   • There is a significant interaction between variety and nitrogen in influencing rice yield (P-Value < 0.01). This indicates that the effect of nitrogen on rice yield varies depending on the rice variety used.

Discussion

1. Replicate Consistency: Results indicate that the replicates in this experiment are fairly consistent, validating the design and execution of the experiment.
2. Importance of Variety Selection: Choosing the right variety is crucial as it significantly affects rice yield. Further research may need to be conducted to identify which variety is most efficient under specific conditions.
3. Nitrogen Effect: Due to the significant effect of nitrogen, nitrogen fertilization strategies may need to be optimized to maximize rice yield.
4. Variety-Nitrogen Interaction: The interaction between variety and nitrogen means that nitrogen application strategies need to be tailored to the rice variety used.

Overall, the right choice of variety and nitrogen level is crucial for enhancing rice yield. Next, it will be useful to look at the results of the Tukey Post Hoc Test to understand the more detailed differences between factors.

Post Hoc Test

Based on the ANOVA results conducted, it was found that the Single Effects of both variety and nitrogen on rice yield are significant. Furthermore, the interaction between variety and nitrogen also showed high significance. This indicates that the combined effects of variety and nitrogen levels are not simple but affect rice yield in a more complex way.

Since the interaction between variety and nitrogen is significant, the next focus will be on interaction effects. This means that it's not just about choosing the right variety or nitrogen level, but also important to consider how the combination of both can optimize rice yield.

Further analysis will be conducted to identify the most effective combination of variety and nitrogen levels to improve rice yield. This will provide valuable information for farmers and researchers in developing more efficient and effective rice cultivation strategies.

Single Effect of Variety

Single Effect of Nitrogen
Interaction Effect of Variety and Nitrogen

There are two formats for presenting the average table for interaction effects. You can choose one or both. The first format is in a one-way table layout, where treatment levels are combined and laid out like a Single Effect table. The second format tests simple effects and is presented in a two-way table layout. The choice of average table and graph display can be set through Advanced Options (refer back to step 15 from the ANOVA Steps).

First Format: Variety x Nitrogen Effect

Second Format: Simple Effect of Variety x Nitrogen

Explanation of the Two Presentation Formats

• First Format: In this format, each combination of variety and nitrogen is presented as a separate entity with its own average value and confidence interval (CI). Each combination is also labeled with letters indicating whether the differences are significant or not, according to the Tukey Post Hoc Test at the 0.05 significance level.
• Second Format: This format provides more detailed information on how each variety behaves under different nitrogen conditions, and vice versa. It compares between two varieties at the same nitrogen level (lowercase letters) and between two nitrogen levels at the same variety (uppercase letters).

Interpretation and Discussion of the Second Format

Critical Values

• Between 2 Varieties: Standard Error = 0.5072, Tukey (BNJ) 0.05 = 2.4073
• Between 2 Nitrogen Levels: Standard Error = 0.3945, Tukey (BNJ) 0.05 = 1.5648

These critical values are used as a threshold to determine whether the difference between group averages is significant or not. The larger the critical value, the greater the difference that must exist between two groups to be considered significant.

Table Interpretation

• Variety v1: Shows a significant increase in rice yield from nitrogen level 1 to level 3 (read vertically, from "a" to "b").
• Variety v2: No significant difference between nitrogen levels 2 and 3, but there is a significant difference with level 1 (read vertically, from "a" to "b").
• Variety v3: Shows a significant increase in rice yield from nitrogen level 1 to level 3 (read vertically, from "A" to "C").
• Nitrogen n1: Variety v6 shows a lower rice yield and is not significantly different from other varieties (read horizontally, all "A").
• Nitrogen n2: Varieties v1, v3, and v2 show higher rice yields and are significantly different from other varieties (read horizontally, from "A" to "B").
• Nitrogen n3: Variety v3 shows the highest rice yield and is significantly different (read horizontally, from "A" to "C").

Discussion

• Variability among Varieties: There is significant variation in the response to nitrogen among different varieties, highlighting the importance of variety selection in the context of nitrogen management.
• Nitrogen Effect: Nitrogen levels have different effects on each variety, reinforcing the importance of considering the interaction between variety and nitrogen in management strategies.
• Statistical Significance: The critical values from Tukey (BNJ) provide a clear boundary for assessing significant differences, validating the need to consider the interaction between varieties and nitrogen in rice cultivation strategies.

Overall, the data indicate that both variety and nitrogen level significantly affect rice yield, but their effects are highly dependent on each other. This underscores the importance of a more holistic approach in the management of rice cultivation.

ANOVA Assumption Checks

Formal Approach (Statistical Tests)

Levene's Test for Homogeneity of Variances

Levene's Test is used to test the homogeneity of variances across groups. In this case, the P-value is 0.640, which is greater than 0.05, indicating that the assumption of homogeneity of variances is met. In other words, the variances of rice yields across all groups are considered homogeneous, validating the application of Analysis of Variance (ANOVA) to this data.

Normality Test

All normality tests show P-values greater than 0.05, indicating that the residual data is normally distributed. This is another critical assumption that must be met for the validity of the Analysis of Variance.

Discussion

• Homogeneity of Variances: The homogeneity of variances is crucial to ensure that the conclusions drawn from the Analysis of Variance are valid. In this case, Levene's Test validates the homogeneity of variances, allowing for further analysis.
• Data Normality: The assumption of normality is critical as many statistical methods, including ANOVA, rely on the normal distribution of data or residuals. In this context, the normality tests validate this assumption.
• Validity of Analysis: Both of these assumptions are key to the validity of the Analysis of Variance. Meeting these assumptions increases the confidence in the ANOVA results and the conclusions drawn from it.

Overall, the assumption checks validate the applicability of Analysis of Variance to this data, thereby strengthening the integrity and confidence in the analytical outcomes.

Visual Approach (Graphical Plots)

Interpretation and Discussion of Assumption Checks Graphically

1. Normal P-Plot of Residual Data

The Normal P-Plot is commonly used to check whether a dataset is normally distributed. In this graph, the data points tend to follow the diagonal line, indicating that the residual data approximates a normal distribution. This supports the results from the normality statistical tests and validates the normality assumption in the Analysis of Variance.

2. Histogram of Residual Data

The histogram also provides information about the data distribution. From the shape of the histogram, it appears that the residual data is spread around the center and forms a pattern approximating a bell shape, again indicating a near-normal distribution.

3. Residual vs. Predicted Plot

This plot is used to assess homoscedasticity, which is the homogeneity of the residual variances across the range of predictor values. If the plot shows a random pattern, then the homoscedasticity assumption is met. From this plot, no specific pattern stands out, indicating the homogeneity of residual variances, thus fulfilling the ANOVA assumption.

4. Standard Deviation vs. Mean

This graph shows the relationship between the standard deviation and the mean for each group. In the context of ANOVA, we want to see how much the standard deviation changes as the mean changes. From this graph, it appears that there is no specific pattern or trend showing that the standard deviation changes significantly as the mean changes, affirming the assumption of homogeneity of variances.

Conclusion

All the above graphs support the validity of the assumptions required for performing Analysis of Variance, namely the homogeneity of variances and normality of the residual data. This strengthens the confidence in the Analysis of Variance results and validates the conclusions drawn from the analysis.

Box-Cox Transformation and Residual Analysis

Interpretation and Discussion of Analysis Results

Box-Cox Transformation

• Lambda: 1.056
• Transformation: No transformation (No Transformation: \( Y^1 \))

The Box-Cox method is used to stabilize the variances and make the data more closely approximate a normal distribution. A lambda value of 1.056 and the decision not to perform any transformation indicate that the data is already sufficiently close to a normal distribution, thus not requiring any transformation.

Residual Values and Outlier Inspection

• Replication, Variety, Nitrogen: These are the factors considered in the experiment.
• Grain Yield: This is the response variable being measured.
• Predicted: This is the expected or predicted value from the model.
• Residual: This is the difference between the observation and the prediction.
• Leverage: This measures how much a data point has the potential to influence the model estimation.
• Studentized Residual and Studentized Deleted Residual: These are normalized residuals used for assessing the presence of outliers.
• Cook's Distance and DFITS: These are measures assessing the influence of an observation on the overall model.
• Diagnostic: This provides information on whether a data point is an outlier or not.
• Box-Cox Data: These are the values resulting from the Box-Cox transformation.

In the table, there is one observation identified as an outlier with high Cook's Distance and high Studentized Deleted Residual. This outlier has the potential to affect the model's validity and may require further investigation.

Discussion

• Box-Cox Transformation: The absence of a need for data transformation suggests that the ANOVA assumptions are met, validating the analysis results.
• Outlier Data: Identifying outliers in the data is an important step in ensuring the accuracy of the model. These outliers might represent rare events or measurement errors and need to be carefully addressed in further analysis.
• Residuals and Diagnostics: Residuals and other diagnostic statistics provide insights into the quality of the model. In this case, the presence of outliers and high leverage values for some observations require further attention.

Overall, the analysis results validate most of the model's assumptions and indicate a relatively good model quality but also highlight areas that may require further investigation, such as the presence of outliers.