[latex]{\chi}^2= \frac{(18)\left[6(2.450) -14.217 \right]}{1+\frac{6+1}{3(6)(18)}}=8.521[/latex].

[latex]\textrm{Equation 3}[/latex] Example calculation of Bartlett’s Test for Homogeneity of Variance.

Ex. 1: Evaluating the Variances

Let us try this using R and a different set of data.

The data in the QM-mod12-ex1data.xls worksheet are from a growth chamber experiment in which a number of treatments were applied to eastern gamagrass seed in an attempt to break dormancy and thereby increase germination percentage. The treatments consisted of:

• control (no treatment)
• wet chilling for 2 weeks
• wet chilling for 4 weeks
• wet chilling for 2 weeks with scarification, and
• wet chilling for 4 weeks with scarification

The experimental design was a CRD with five replications.

The data are expressed as percentages. This should lead us to suspect that there may be a potential for problems with the assumptions for the ANOVA.

In this exercise, we will use Excel to create a table of treatment means and variances which we will examine to see if they conform to the homogeneity assumption.

Ex. 1: Creating a Pivot Table

1. Using the mouse, select all the data in the worksheet. Be sure to select the top row, which contains the data labels.

2. Select PivotTable from the Insert menu.

3. A dialog box will open, and the data you have selected will automatically appear in the box next to Select a table or range.

4. Click the circle next to New Worksheet and then OK.

5. The next screen will show an empty table with a panel on the right side titled PivotTable Field List, which is used to format the table.

6. Drag the Treatment field into the Row Labels box in the panel.

7. Drag the Germination button into the Values box in the panel.

8. Click or Double-click on the Sum of Germination field and select Value Field Settings… from the popup menu that appears.

9. Select Average from the list of options that appear, then click OK to calculate and display the five treatment means.

10. Once again, drag the Germination button into the Values box in the panel.

11. Click on the Sum of Germination field and select Value Field Settings… from the popup menu that appears.

12. This time, select Var from the list of options that appear, then click OK to calculate and display the five treatment variances.

Your completed pivot table should look like Fig. 3.

Ex. 1: Examining Homogeneity Assumption

In looking at the variances associated with the five treatments, there appears to be a potential problem with the homogeneity assumption (Fig. 4). The variance of Treatment 1 is 1.093 compared with 24.773 for Treatment 3. The ratio of these two variances is 22.67, which is relatively large. This situation should lead us to examine more thoroughly the assumption of homogeneity.

Note: Later in this unit, we will learn how to “transform” data so that the variances are more equal. Sometimes it will help us to decide what transformation function to use if we look at how both variance and standard deviation vary with treatment means.

To calculate the standard deviation for treatment means, simply choose “Std Dev” instead of “Var” in Step 12.

Ex. 1: Plotting the Variance against Means

A good way to visualize what is going on with the data is to plot the variances against their associated means.

In this exercise, we will use Excel to create a plot of the means and variances. The most expedient way to accomplish this is to create a Pivot Chart using the Pivot Table we just made in Exercise 1.

Create an XY graph of the means and variances using the following steps:

1. Using your mouse, place the cursor anywhere within in the Pivot Table you created for the last exercise.
2. At the very top of the Excel window to the right, a menu item labeled PivotTable Tools will appear. Select this menu item and then click on the PivotChart icon that appears.
3. Select the first chart style that appears in the Column template.
4. The graph that is created should look something like (Fig. 5).

Our assumption for the ANOVA is that the variances of the five treatments are equal. By looking at the graph, we see that the variances for Treatments 3 and 4 are considerably higher than the others, which should lead us to explore our assumption further.

Note that you can produce the same style chart if you have produced a pivot table with standard deviations instead of variances.

Sometimes, it is easier to visualize the relationship between the variance and treatment means if we plot them in a scatter plot. This is a little more tricky than the PivotChart you just made, but it allows us to better see how variance increases with treatment means. In addition, to choose which transformation function to use (see Table 12.5), sometimes you need to compare the relationship between variance and mean to that between standard deviation and mean to see which relationship is more linear.

Ex. 1: Creating a Scatterplot

To produce a scatterplot with variances on the Y-axis and treatment means on the X-axis:

1. Select the “Average” and “Var” columns from your PivotChart (Fig. 6). Do not select the treatment column or the Grand Total row. Copy and paste these below your original chart.

2. Highlight your new table. Then choose Insert from the menu bar, then Scatter, then the first chart style (Fig. 7).

Your chart should look like Fig. 8. Again, the treatment means are on the X-axis, and the variances are on the Y-axis.

In this case, there does not seem to be a linear relationship between variance and treatment mean. However, this chart again shows that the variances associated with two of the treatments are many times greater than those associated with the other. Data transformation may be required so that the variances meet the assumptions of the Analysis of Variance.

The same method can be used to create a plot with Standard Deviation on the Y-axis and Treatment means on the X-axis. Simply create a pivot table with treatment means and standard deviation, and then follow the two steps above.

Ex. 2: Introduction

When conducting an analysis of data, there are numerous assumptions that are made about the data. Sadly, in the real world, these assumptions are often violated, leading to poor interpretation of the results. In this activity, we will explore ways to deal with these assumption violations when it comes to performing an analysis of variance (ANOVA).

R Code Used In This Exercise

Packages:

• plyr
• reshape2
• ggplot2
• agricolae

Code:

• setwd()
• detach()
• names()
• log()
• read.csv()
• as.factor()
• cbind()
• sqrt()
• head()
• aov()
• melt()
• asin()
• str()
• summary()
• qplot()
• LSD.test()
• attach()
• aggregate()
• bartlett.test()

Ex. 2: Review of 3 ANOVA’s Main Assumptions

Review – These Are The 3 Main Assumptions Of The Anova

The error terms are normally, independently, and randomly distributed.
This means that the error terms follow a normal distribution, although this is difficult in smaller sample sizes. There should also be independence between the size of the error of a treatment group and the experimental units to which it is allocated.

The variances are homogenous and not correlated with the means of different levels.
The error variances should be constant. Remember that the difference between all levels of a treatment is tested using one error term (pooled), and if this assumption is violated, then the error term used to test the differences between treatments will be too large for comparing treatment levels with small variances and too small for comparing treatment levels with large variances.

The main effects and interactions are additive.
This means that we assume that the linear additive model we are using to analyze the data holds true. Sometimes this assumption is violated because there is an interaction between the plots and the treatments, but this data can be transformed using the log transformation.

In this lesson, we will explore how to determine if your data should be transformed and, if so, what kind of transformation is appropriate.

Ex. 2: Exercise Introduction

You are a student, and you wish to study the effects of 5 treatments on breaking gamagrass seed dormancy. The treatments were control (no treatment), wet chilling for 2 weeks, wet chilling for 4 weeks, wet chilling for 2 weeks plus scarification, and wet chilling for 4 weeks plus scarification. You decide to test the treatments on a single, common variety using a completely random design with 5 replications for each of the treatments. The data for these treatments are expressed as the percent germination in the file Set1.csv. In this activity, we will use R to check the second and third assumptions of the ANOVA by creating a table of treatment means and variances which we will examine to see if the homogeneity of the variances assumption was violated. Then, if the assumption is violated, we will explore ways to transform the data for a more accurate analysis.

Our first step is to start visualizing our data, and one way to do this is to create a table of means and variances of the means. First, you will need to read in the data and ensure that it was read in properly as well as make sure that the “Treatment” column is considered to be a factor.

Ex. 2: Read the Data

Ex. 2: Visualize the Data

To help visualize the data, we will start by making a table of the treatment means and variances, and standard deviations. This can be done with the following code, which analyzes the data set using the specified function (mean, variance, and standard deviation, respectively) based on a specified variable in the data set (here, it is Treatment). The code also renames the new column that is made to better reflect which function was used. Otherwise, without renaming, the column head would just read “Germination,” regardless of what you have actually calculated. This first code calculates the treatment means:

Ex. 2: Combine into a Single Table

Ex. 2: Graph Means and Variance

Ex. 2: Reshape Data

Ex. 2: Reshape GUI

reshapeGUI

Ex. 2: Preview Result

You can hit the preview button to see what your data will look like. The Execute button will actually run the code you have generated, but if you want to save the code itself, you can copy and paste it into a script editor.

Now we can start plotting our means and variances. The package “ggplot2” has a lot of versatility in visualizing data, so we will use this package:

library(reshapeGUI)

Now use this code to create a bar plot that includes both the treatment means and the treatment variances and also includes some nice colors and a graph title:

means.var.varplot <- qplot(x-Treatment, y=value, fill=variable, data=Summary.melt, geom=”bar”, stat=”identity”, position=”dodge”, main=”Barplot of Means and Variance of germination for 5 treatments”)

means.var.barplot

This gives us this graph (Fig. 9).

Remember how one of the assumptions of the ANOVA is equal variances across the treatments? Looking at this graph here, we can clearly see that this is not true of this data set. The variances for treatments 3 and 4 are much higher than the other treatments, and this should prompt you to explore this further.

Ex. 2: Scatter Plot to Visualize Data

Another way to visualize this data is to plot the variances against the means in a scatterplot which can be done using ggplot2 with the following code:

means.var.scatterplot <- qplot(data=Summary, x=GermMean, y=GermVar,
+                             main=”Scatterplot of germination means against germination variances”) +
+                             geom_point(size = 5)

means.var.scatterplot

And the graph looks like this (Fig. 10):

Ex. 2: Bartlett’s Test

As we saw with the bar graph, treatments 3 and 4 clearly have greater variances than the other treatments in the data set. However, this is not enough evidence to prompt data transformation, and a different test should be run to be sure.

Running the Bartlett’s test for homogeneity of variances:

This test lets us test to see if the variances between treatments are significantly different enough to warrant a data transformation. It uses the formula in Equation 2 above.

Ex. 2: Conclusions

This tests the null hypothesis that all treatment variances are equal, and the calculated chi-square value is compared to the critical value, which is based on the degrees of freedom. When it comes to interpreting the test, there are several possible outcomes based on the p-value:

• If the p-value is >0.01, then we accept the null hypothesis and don’t transform the data
• If P<0.001, the null is rejected, and we need to transform the data
• If 0.01>P>0.001, then you must try to figure out whether there is any reason why the data would not meet the requirement of heterogeneity of variances; otherwise, we don’t transform the data.

Performing Bartlett’s test in R is very easy and can be done with one line of code

bartlett.test(Germination~Treatment, germdata)

Bartlett test of homogeneity of variances

data: Germination by Treatment

Bartlett’s K-squared = 13.4583, df = 4, p-value = 0.009241

Looking at these results, our p-value falls between 0.01 and 0.001, which is the range in which you must decide if there is a theoretical basis for transforming the data or not. Since our experimental data is expressed in percentages (which often leads to unequal variances), we can conclude that the variances are not homogenous, and we do have a reason for transforming this data set.

Generally, the most appropriate data transformation for percentage data is the angular or arcsine transformation. In this exercise, we will transform the germination data using the angular transformation in Excel.

The angular transformation is more complicated to calculate than appears on the surface. Although we will calculate it using a single equation, the calculation actually involves the following steps:

1. The percentage data are converted to proportions by dividing by 100.
2. The square root of the proportion is taken using the SQRT function.
3. The arcsine of the square root is taken using the ASIN function.
4. Convert from radians to degrees by multiplying by (180/PI). This last step is not required. Either degrees or radians give the same results. However, converting back to degrees makes the transformed data look more like the original data - only compressed.

Ex. 3: Natural Log Transformation

Once you have decided how to transform the data, this can be done quickly in Excel. Using the germination percentages found in the Exercise 12.1 data, follow these steps. In each case, you will need to create a new column heading. Since you will likely be importing these data into R, it is easiest to give them a one-word heading. Don’t worry about originality — just select something simple that will be easy to use in R code.

The Natural Log Transformation
The natural log is sometimes referred to as “Ln”, so change the title of column D to “LnGerm.” Below that, type in “=ln(C2)”. Select that cell, then double click on the tiny box in the right hand corner to copy down the rest of the data. Your table should look like the one below (Fig. 18).

Ex. 3: Square Root Transformation

The Square Root Transformation
Change the title of column E to “SqRtGerm” (real original, see?). Below that, type in “=sqrt(C2)”. Select that cell, then double click on the tiny box in the right hand corner to copy down the rest of the data. Your table should look like this (Fig. 19):

Ex. 3: Arc Sin (Angular) Transformation

The Arc Sin (Angular) Transformation
The arc sin transformation is somehow legitimate statistical voodoo. There is a whole lot going on, converting percentages to proportions, taking the square root and arcsin, and converting from radians to degrees.

Change the title of column F to “AsinGerm. Below that, type in “=asin(sqrt(C2/100))*180/PI()”. Select that cell, then double-click on the tiny box in the right-hand corner to copy down the rest of the data. Your table should look like Fig. 20, below.

Ex. 3: Data Transformation using R

So, what kind of data transformation should we perform? There are many different ways, and each type of transformation is appropriate in different circumstances. We will explore only 3 of them in this activity.

The Natural Log Transformation

This transformation is good for when the standard deviation of the treatments is more or less proportional to the means of the treatments and where the effects seem to be multiplicative instead of additive.

R code:

1nGerm<-log(Germination)

The Square Root Transformation

This is most often used when the data consists of counting rare events and tends to follow a Poisson distribution, not a normal distribution. In this instance, the variance tends to be proportional to the mean. R code:

sqrtGerm<-sqrt(Germination)

The Arcsine (Angular) Transformation

This is typically used on data that is expressed in percentages or proportions because they are more likely to have a binomial distribution where the variance tends to be greater at the center of the distribution. Calculating the arcsine transformation is a bit tricky and involves several steps:

• Percentage data is divided by 100
• Then the square root is taken
• Then the arcsine is taken
• The last step of converting from radians to degrees is optional and is done by multiplying by 180/pi

R code:

asingerm<-asin(sqrt(Germination/100))*180/pi

Ex. 3: Combine All Transformations in One Table

Ex. 3: Bartlett's Test and ANOVA

Ex. 3: LSD and Conclusions