When analyzing a questionnaire, one often wants to view the correlation between two or more Likert questionnaire item’s (for example: two ordered categorical vectors ranging from 1 to 5).

When dealing with several such Likert variable’s, a clear presentation of all the pairwise relation’s between our variable can be achieved by inspecting the (Spearman) correlation matrix (easily achieved in R by using the “cor.test” command on a matrix of variables). Yet, a challenge appears once we wish to plot this correlation matrix. The challenge stems from the fact that the classic presentation for a correlation matrix is a scatter plot matrix – but scatter plots don’t (usually) work well for ordered categorical vectors since the dots on the scatter plot often overlap each other.

There are four solution for the point-overlap problem that I know of:

Jitter the data a bit to give a sense of the “density” of the points

Use a color spectrum to represent when a point actually represent “many points”

Use different points sizes to represent when there are “many points” in the location of that point

Add a LOWESS (or LOESS) line to the scatter plot – to show the trend of the data

In this post I will offer the code for the a solution that uses solution 3-4 (and possibly 2, please read this post comments). Here is the output (click to see a larger image):

Daniel Malter just shared on the R mailing list (link to the thread) his code for performing the Siegel-Tukey (Nonparametric) test for equality in variability. Excited about the find, I contacted Daniel asking if I could republish his code here, and he kindly replied “yes”. From here on I copy his note at full.

(The R code for Barnard’s exact test is at the end of the article, and you could also just download it from here, or from github)

About Barnard’s exact test

About half a year ago, I was studying various statistical methods to employ on contingency tables. I came across a promising method for 2×2 contingency tables called “Barnard’s exact test“. Barnard’s test is a non-parametric alternative to Fisher’s exact test which can be more powerful (for 2×2 tables) but is also more time-consuming to compute (References can be found in the Wikipedia article on the subject).

When comparing Fisher’s and Barnard’s exact tests, the loss of power due to the greater discreteness of the Fisher statistic is somewhat offset by the requirement that Barnard’s exact test must maximize over all possible p-values, by choice of the nuisance parameter, π. For 2 × 2 tables the loss of power due to the discreteness dominates over the loss of power due to the maximization, resulting in greater power for Barnard’s exact test. But as the number of rows and columns of the observed table increase, the maximizing factor will tend to dominate, and Fisher’s exact test will achieve greater power than Barnard’s.

About the R implementation of Barnard’s exact test

After finding about Barnard’s test I was sad to discover that (at the time) there had been no R implementation of it. But last week, I received a surprising e-mail with good news. The sender, Peter Calhoun, currently a graduate student at the University of Florida, had implemented the algorithm in R. Peter had found my posting on the R mailing list (from almost half a year ago) and was so kind as to share with me (and the rest of the R community) his R code for computing Barnard’s exact test. Here is some of what Peter wrote to me about his code:

On a side note, I believe there are more efficient codes than this one. For example, I’ve seen codes in Matlab that run faster and display nicer-looking graphs. However, this code will still provide accurate results and a plot that gives the p-value based on the nuisance parameter. I did not come up with the idea of this code, I simply translated Matlab code into R, occasionally using different methods to get the same result. The code was translated from:

Trujillo-Ortiz, A., R. Hernandez-Walls, A. Castro-Perez, L. Rodriguez-Cardozo. Probability Test. A MATLAB file. URL

My goal was to make this test accessible to everyone. Although there are many ways to run this test through Matlab, I hadn’t seen any code to implement this test in R. I hope it is useful for you, and if you have any questions or ways to improve this code, please contact me at calhoun.peter@gmail.com