11 Analysis Of Variance (ANOVA)
11.1 One-Way ANOVA: Post-Hoc Tests and Review
11.1.1 Reminders
ANOVA’s are used when the sample data consists of a dependent measure and more than two indpendent variables.
The sample data you have will have a dependent variable, and a grouping variable. This grouping variable needs to be made into a factor.
In order to do this, we need to use the factor() function.
When R imports a csv file, there is an option to automatically import any string data as a factor.
Computing the ANOVA in R is very similar to computing a linear regression:
You should notice that the two outputs share certain values. The F statistic is the same in both cases, as is the p-value. Additionally, the linear regression has the same two degrees of freedom: 3 and 36.
When you are creating your ANOVA output, it is good practice to name the ANOVA as being a something.mod. This is helpful because you will be able to use this model name in other instances.
11.1.2 Post-Hoc Tests
Post-Hoc tests are useful because they allow us to see where the difference is in our data. If we have a One-Way ANOVA with five groups and we have a significant F value, we have no idea which groups are different.
We will be using two Post-Hoc tests: the TukeyHSD and the SNK.Test (Student-Newman-Keuls).
We will use the same data from before and try to find out where the difference is, or rather, which combinations of data are significantly different.
11.1.2.1 Tukey HSD
The TukeyHSD test stands for “Honestly-Significantly-Different”.
From this output we can see that the p adj value corresponds to whether or not the specific comparison is significant.
We can see that there is a difference between the following groups:
Treatment 2 and Control, Treatment 3 and Control, Treatment 2 and Treatment 1, Treatment 3 and Treatment 1, and Treatment 3 and Treatment 2.
There is not a significant difference between Treatment 1 and Control.
If you look at the columns labeled lwr and upr, these are the confidence intervals for this specific comparison. If the confidence intervals include 0, the comparison will not be significant.
To test this, if you notice, the comparison between Treatment and Control is the only comparison with a negative lower interval. The range of -2.32 -> 11.36 includes 0 so we can determine it is not significant.
These results can also be visualized in the form of a plot.
11.1.2.2 Plotting the Tukey Test
In order to plot the Tukey Test results you will need to pass the output of the Tukey Test into the plot() function.
As you can see from the graph, there are dashed lines to demarcate the line at 0. The only interval that crosses this range is the Treatment 1 and Control.
11.1.3 Student Newman Keuls
The Student Newman Keuls test is another post-hoc test that is used to make comparisons of differences between groups.
The Student Newman Keuls test is available through a package called agricolae.
First, we will install the package using install.packages("PackageName").
Second, we will load the package through the use of library(agricolae).
The arguments in the Student Newman Keuls test are as follows:
y - This is where you will put your model, (this is why it is so important to store your models!) trt - This is the name of your grouping variable, it will need to appear in quotes. main - This is what the title of your output will be. alpha - This is where you will specify what alpha level you will be testing your comparisons against. console - This is the option that asks whether you want to see the output (you should always make it equal to TRUE)
In full:
The output from the Newman Keuls test does not return p-values, however, the same ideas about the confidence intervals apply.
You will notice that Control and Treatment have ranges that include 0, (-6.01-10) and (-3.31-14.79) respectively.
The output also prints out a helpful tool: Means with the same letter are not significantly different.
We can see that Treatment 3 Treatment 2 are both different and Treatment 1 and Control are the same letter, indicating they are not significantly different.