How to Analyze LSD Factorial Experiments

Complementing Excel's capabilities, SmartstatXL serves as an Add-In that focuses on the analysis of experimental data. One of its specializations is in the Analysis of Variance for Factorial Latin Square Design (LSD Faktorial), which is a factorial experiment based on Latin Square Design. Although it prioritizes balanced designs (Balanced Design), SmartstatXL also offers the ability to analyze various mixed models beyond standard designs.

The features available for Factorial experiments in SmartstatXL include:

• Factorial Latin Square Design (LSD Factorial): Refers to Factorial experiments where each observational unit is measured only once.
• Factorial Latin Square Design: Sub-Sampling: Intended for repeated measurements with the capability to draw sub-samples from a single observational unit. For example, in one observational unit (treatment 3Dok1, first repetition), measurements are performed on 10 plants.
• Factorial Latin Square Design: Repeated Measure: Designed for observations carried out periodically on a single observational unit, such as every 14 days.
• Factorial Latin Square Design: Multi-Location/Season/Year: An appropriate choice for experiments conducted in different locations, seasons, or years.

When significant treatment effects are found, SmartstatXL provides various Post hoc Tests to compare treatment means. These include: Tukey, Duncan, LSD, Bonferroni, Sidak, Scheffe, REGWQ, Scott-Knott, and Dunnet.

Case Example

Fisher (1935, sections 36, 64) presented data on the pounds of potatoes harvested from a plot of land that was divided into a square consisting of 36 plots. Six treatments were randomly assigned to the plots in such a way that each treatment occurred once in every row and once in every column of the square. The treatments involved two factors, a nitrogen-based fertilizer (N) and a phosphorous-based fertilizer (P). The nitrogen fertilizer had two levels, none (n0) and a standard dose (n1). The phosphorous fertilizer had three levels, none (p0), a standard dose (p1), and double the standard dose (p2). We identify the six treatments for this 2×3 experiment as follows: A B C D E F => n0p0 n0p1 n0p2 n1p0 n1p1 n1p2.

Cited from:
Fisher (1935, sections 36, 64).

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

1. Ensure that the worksheet (Sheet) you wish to analyze is active.
2. Place the cursor on the Dataset. (For information on how to create 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 on the Factorial Menu > Factorial Latin Square Design (RBSL Faktorial).
   
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 that the Dataset is correct, press the Next Button
8. A subsequent dialog box titled Anova – Factorial Latin Square Design will appear:
   
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 choose a Factor, SmartstatXL will provide additional information about the number of levels and the names of those levels.
11. Details of the Anova STAGE 1 dialog box can be seen in the following image:
    
12. After confirming that the Dataset is correct, press 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 based on your research methodology. In this example, the Post hoc Test used is the Scott-Knott cluster test.
15. To configure additional outputs and default values for subsequent outputs, press the "Advanced Options…" button.
16. Here is the display of the Advanced Options Dialog Box:
    
17. After completing the settings, 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 mean table, ID for each Factor, and rounding of mean values. The details can be seen in the following image:
    
20. As a final step, click "OK"

Analysis Results

Analysis Information

1. Experimental Design: LSD Factorial
   • This experimental design is a factorial design involving combinations of multiple factors.
2. Post hoc Test: Scott-Knott
   • The Scott-Knott test is a method of post hoc testing used to determine groups that are significantly different in an analysis of variance.
3. Response: Response
   • This indicates the response variable or output measured in the experiment. In this case, it could be the amount of potatoes harvested, although further information is needed for confirmation.
4. Factors
   • Row (6 levels) - This indicates that there are 6 rows in the experimental design.
   • Column (6 levels) - This indicates that there are 6 columns in the experimental design.
   • Nitrogen (2 levels) - There are two levels of nitrogen treatment: no fertilizer (n0) and standard dose (n1).
   • Phosphorous (3 levels) - There are three levels of phosphorus treatment: no fertilizer (p0), standard dose (p1), and twice the standard dose (p2).

This experiment was designed using a Factorial LSD approach to examine the effects of combined nitrogen and phosphorus treatments on the measured response (e.g., amount of potatoes harvested). With this approach, researchers can understand how these two factors (nitrogen and phosphorus) influence the response both individually and in combination. In this case, there are 6 rows and 6 columns, which may represent experimental blocks or plots, and there are 6 treatment combinations resulting from the combination of 2 levels of nitrogen and 3 levels of phosphorus.

Analysis of Variance

Interpretation and Discussion:

1. Row (R)
   • Degrees of Freedom (DF): 5
   • Sum of Squares (SS): 54198.5833
   • Mean Square (MS): 10839.7167
   • F-Value: 7.098
   • P-Value: 0.001
   • The row factor has a significant effect on the response at the 1% level (as P-Value < 0.01 and F-Value > F-0.01). This suggests that there is a significant difference between the outcomes in different rows.
2. Column (C)
   • The column factor has a significant effect on the response at the 5% level (as P-Value < 0.05 and F-Value > F-0.05). This suggests that there is a significant difference between the outcomes in different columns.
3. Nitrogen (N)
   • The nitrogen factor has a highly significant effect on the response at the 1% level. This indicates that the type of nitrogen fertilizer used affects the potato yield.
4. Phosphorous (P)
   • The phosphorus factor also has a highly significant effect on the response at the 1% level. This indicates that the phosphorus fertilizer dosage affects the potato yield.
5. Nitrogen x Phosphorous Interaction (N x P)
   • The interaction between nitrogen and phosphorus does not have a significant effect on the response (as P-Value > 0.05). This means that the combination of nitrogen and phosphorus dosages does not significantly affect the potato yield.
6. Coefficient of Variation (CV): 8.44%
   • The coefficient of variation indicates the relative variation of the obtained data. With a value of 8.44%, the data variation is considered low, indicating that the experimental data are relatively consistent.

Conclusion:

• Individual factors of row, column, nitrogen, and phosphorus have a significant impact on potato yield.
• However, the interaction between nitrogen and phosphorus does not show a significant impact on the yield.
• The experimental data show relatively low variation, indicating consistency in data collection.

Post hoc Test

Based on the Analysis of Variance, only the single effect has a significant impact. Below is a summary of the post hoc test results using the Scott-Knott test.

Single Effect Nitrogen

Interpretation and Discussion:

1. Partition:
   • Bo-Max: 4288.4090
   • Lambda: 21.709
   • db: 1.754
   • Chi-sqr: 5.506
   • p-value: 0.000
   • From the p-value less than 0.01, it is evident that there is a significant difference at the 1% level between the two nitrogen levels (n0 vs n1) based on the Scott-Knott test.
2. Table of Mean Response Values:
   • Nitrogen n1: The average response is 509.06 with a confidence interval ±50.36. This is labeled as "b".
   • Nitrogen n0: The average response is 416.44 with a confidence interval ±39.24. This is labeled as "a".
   • Based on labels "a" and "b", both nitrogen levels show a significant difference at the 5% level based on the Scott-Knott post hoc test. This means that the nitrogen dosage affects the potato yield.

Conclusion:

• There is a significant difference between the two nitrogen levels (n0 and n1) with respect to the response (the number of potatoes harvested). In other words, treatment with nitrogen dosage (n1) results in a higher potato yield compared to no nitrogen dosage (n0).
• This underscores the importance of nitrogen dosage in enhancing potato yield productivity. In the context of agricultural practice, this can be valuable information for farmers in considering the appropriate nitrogen dosage to improve their harvest.

Single Effect Phosphorous

Interpretation and Discussion:

1. Partition:
   • (p0) vs (p1;p2):
     • Bo-Max: 11528.1667
     • Lambda: 22.424
     • db: 2.632
     • Chi-sqr: 7.166
     • p-value: 0.000
     • Based on the p-value less than 0.01, there is a significant difference at the 1% level between level p0 and the combination of levels p1 and p2.
   • (p1) vs (p2):
     • Chi-sqr: 5.506
     • p-value: 0.001
     • From the p-value less than 0.01, there is a significant difference at the 1% level between levels p1 and p2.
2. Table of Mean Response Values:
   • Phosphorous p1: The average response is 473.33 with a confidence interval ±55.36. This is labeled as "b".
   • Phosphorous p0: The average response is 375.08 with a confidence interval ±40.49. This is labeled as "a".
   • Phosphorous p2: The average response is 539.83 with a confidence interval ±48.59. This is labeled as "c".
   • Based on labels "a," "b," and "c," all three phosphorous levels show a significant difference at the 5% level based on the Scott-Knott post hoc test. This means that the phosphorous dosage affects the potato yield.

Conclusion:

• There is a significant difference between the three phosphorous levels (p0, p1, and p2) with respect to the response (the number of potatoes harvested). In other words, as the phosphorous dosage increases, the potato yield also increases.
• Specifically, treatment with double the standard phosphorous dosage (p2) results in the highest potato yield, followed by the standard dosage (p1), and no phosphorous dosage (p0).
• This conclusion emphasizes the importance of phosphorous dosage in enhancing potato yield productivity. This information is highly useful for farmers in considering the appropriate phosphorous dosage to improve their harvest.

Interaction Effects

There are two formats for displaying average tables for interaction effects. You can choose one or both. The first format is in a one-way table, where treatment levels are combined, and the layout is similar to the table for single effects. The second format tests simple effects and is presented in a two-way table format. The display settings for average tables and graphs can be adjusted via Advanced Options (refer back to step 15 of the Analysis of Variance Steps).

Nitrogen x Phosphorous Effects

Simple Effects of Nitrogen x Phosphorous

This situation exemplifies the complexity in statistical analysis. At times, the results from the Analysis of Variance and post hoc tests may appear contradictory. The ANOVA results indicate no interaction between Nitrogen and Phosphorous, while the Scott-Knott test shows differences among some treatments (Lambda value > Chi-square or P-Value < 0.05). However, it's essential to understand that both analyses examine the same thing in different ways, each with its strengths and limitations. In this case, SmartstatXL recommends displaying the average table according to the conclusions drawn from the Analysis of Variance. Therefore, no notations are made on the post hoc test results.

Here are some points that may help you draw conclusions:

Analysis of Variance (ANOVA):

• It is a statistical method used to test differences between two or more means.
• In this context, ANOVA assesses whether there is a significant interaction effect between Nitrogen and Phosphorous.
• The ANOVA results indicate that this interaction is not significant. This means that, overall, there is no strong evidence that the combination of Nitrogen and Phosphorous jointly affects the response.

Scott-Knott Post hoc Test:

• It is a more specific method that compares differences between specific treatment combinations.
• In this context, the Scott-Knott test indicates that even if there may be no significant overall interaction between Nitrogen and Phosphorous, some specific treatment combinations may affect the outcome.
• This result suggests that, although the interaction effect may not be strong overall, there are specific situations where the combination of Nitrogen and Phosphorous could be important.

Conclusion:

Based on the above analyses, it is advised that even if there is no strong evidence from the ANOVA that the overall interaction between Nitrogen and Phosphorous affects the outcome, the post hoc test indicates that there are some specific treatment combinations that could be significant. Therefore, when considering practical recommendations based on these results, it may be useful to consider those specific treatment combinations that show differences in the post hoc tests, while noting that the overall interaction effect is likely not strong.

In other words, although the overall interaction may not be significant, there are specific situations where the combination of Nitrogen and Phosphorous could affect the outcome. As a researcher, you should consider both when making recommendations or decisions based on these results.

Various options for displaying average tables and their post hoc tests can be adjusted from Advanced Options (refer back to step 15 of the Analysis of Variance Steps). For example, in this case, we trust the results based on the Scott-Knott post hoc test. Follow the steps as shown in the following image:

Here are the post hoc test results for the interaction between Nitrogen and Phosphorous after setting the advanced options:

From the table above, regarding the simple effects, we can observe that:

1. For each level of Phosphorous (P), the average response for Nitrogen n1 is always higher compared to n0. This is indicated by lowercase notation (b higher than a), read vertically.
2. Similarly, for each level of Nitrogen (N), the average response for Phosphorous p2 is always higher compared to p1 and p0. This is indicated by uppercase notation (C higher than B and A), read horizontally.

From these observations, we can conclude that:

• Nitrogen (N) has a single effect on the response, where n1 consistently produces higher responses compared to n0, regardless of the level of Phosphorous.
• Phosphorous (P) also has a single effect on the response, where p2 consistently produces higher responses compared to p1 and p0, regardless of the level of Nitrogen, followed by p1 which is consistently higher compared to p0.

When the two factors (in this case, Nitrogen and Phosphorous) have consistent single effects and there is no variation in their combined responses, it suggests that there is likely no significant interaction between the two factors.

In other words, the effects of Nitrogen and Phosphorous are additive and do not depend on each other. Therefore, although the Scott-Knott post hoc test indicates differences between specific treatment combinations, the consistent pattern of the single effects of both factors supports the conclusion that their interaction is likely not significant, as indicated by the Analysis of Variance.

Therefore, although there are differences between specific treatment combinations, the response pattern indicates that there is likely no significant interaction between Nitrogen and Phosphorous.

Assumption Checks for ANOVA

Formal Approach (Statistical Tests)

1. Levene's Test for Homogeneity of Variance:
   • The assumption of ANOVA is that all groups have the same variance, known as homogeneity of variance.
   • The F-value for Levene's Test is 1.729 with a P-value of 0.158.
   • Since the P-value > 0.05, we fail to reject the null hypothesis and conclude that the variance across all groups is homogeneous. Therefore, the assumption of homogeneity of variance is met.
2. Normality Tests:
   • Another assumption of ANOVA is that the data are normally distributed.
   • There are several methods used to test for normality, and you have presented results from various normality tests.
   • Shapiro-Wilk's: P-value = 0.302
   • Anderson Darling: P-value = 0.265
   • D'Agostino Pearson: P-value = 0.410
   • Liliefors: P-value > 0.20
   • Kolmogorov-Smirnov: P-value > 0.20
   • All p-values from the above tests are greater than 0.05, indicating that we fail to reject the null hypothesis and conclude that the data are normally distributed. Therefore, the assumption of normality is met.

Conclusion:

Based on the results of Levene's Test and the normality tests, the two primary assumptions of ANOVA (homogeneity of variance and data normality) are met. This means you can be confident in the validity of your ANOVA results and the conclusions you draw from the analysis.

Visual Approach (Plotting Graphs)

1. Normal P-Plot of Residual Data:
   • In a normal P-Plot, the points should follow the diagonal line when the data are normally distributed.
   • From the graph you've uploaded, the points appear to follow the diagonal line fairly closely, indicating that the residuals are normally distributed.
2. Residual Data Histogram:
   • The histogram is used to check the data distribution.
   • From the residual histogram you've uploaded, the data appear to have a shape approximating a normal distribution with a slight positive skew.
3. Residual vs. Predicted Plot:
   • The Residual vs. Predicted plot is used to check for homogeneity of variance.
   • Ideally, the points should be randomly scattered with no particular pattern.
   • From the plot you've uploaded, there appears to be no specific pattern or clear structure, indicating that the residual variance is constant across all predicted values.
4. Standard Deviation vs. Mean:
   • This plot is used to check for homogeneity of variance between groups.
   • Ideally, the points should be horizontally scattered with no particular pattern.
   • From the plot you've uploaded, the points appear to be scattered without any specific pattern, indicating that the variance across all groups is homogeneous.

Conclusion:

Based on the graphs you've uploaded, it appears that the two main assumptions for ANOVA (normality and homogeneity of variance) are met. This supports the numerical results you mentioned earlier and adds confidence to the validity of your ANOVA results.

Box-Cox and Residual Analysis

Box-Cox Transformation:

• The Lambda value found is 0.736.
• The recommended transformation is the Square Root Transformation, indicating that your response data will more closely adhere to the assumption of normality when square-rooted.

Outlier Data Examination:

• Residuals: The difference between the actual observed values and the values predicted by the model.
• Leverage: Measures the extent to which an observation influences the model's estimate. Points with high leverage can potentially be influential points.
• Studentized Residual: Residuals that have been normalized by the residual variance. Absolute values larger than 2 or smaller than -2 often indicate outliers.
• Cook's Distance: Measures the influence of omitting a particular data point. High values indicate influential data points.
• DFITS: Another statistic measuring the influence of omitting a particular data point.
• Diagnostic: Based on the data you provided, it appears that there is one data point identified as an "Outlier."

Data Interpretation:

• Most data points do not show signs of being outliers or influential points, based on the diagnostic values you've provided.
• However, there is one data point (with the combination 3,6,n1,p2) identified as an "Outlier" based on the Studentized Deleted Residual. This means that this data point has a significantly large residual compared to other data points after considering its influence on the model.

Conclusion:

• A square root transformation may be needed to ensure the response data meets the assumption of normality.
• Most data points seem to be unproblematic, but there is one data point that may be an outlier and needs further examination. Outliers can influence parameter estimation and model quality, so it is important to consider appropriate action, such as further investigation into the cause of the outlier or considering its removal from the analysis (although this decision should be made carefully and consider the research context).

Overall Conclusion

Based on the entire analysis you have provided, here are the general conclusions:

1. Single Effect of Nitrogen and Phosphorous:
   • There is a significant difference in response based on Nitrogen levels (n0 and n1) and Phosphorous levels (p0, p1, p2). The n1 level of Nitrogen and p2 level of Phosphorous consistently show higher responses compared to other levels.
2. Interaction Between Nitrogen and Phosphorous:
   • Although the initial Analysis of Variance indicated no significant interaction between Nitrogen and Phosphorous, the post hoc Scott-Knott test showed differences between specific treatment combinations. However, further observation indicates that both factors seem to have consistent single effects, suggesting that the interaction may not be significant.
3. Assumption Examination:
   • The assumptions of variance homogeneity and data normality, which are primary prerequisites for ANOVA, are met based on statistical tests and graphical visualization.
4. Data Transformation:
   • The Box-Cox transformation suggests that a square root transformation could be applied to the response data to ensure compliance with the assumption of normality.
5. Outlier Data Examination:
   • Most data points do not appear to be outliers or influential points. However, there is one data point identified as an outlier based on diagnostic statistics. This data point may influence model estimation and the quality of the analysis, requiring further consideration.

Final Conclusion:

Treatment with Nitrogen level n1 and Phosphorous level p2 yields the highest response. Although there are indications from post hoc tests about a possible interaction between Nitrogen and Phosphorous, further observation indicates that both might have consistent single effects. The basic assumptions for ANOVA are met, but data transformation may be needed for further analysis. Additionally, special attention should be given to the data point identified as an outlier.