How to test for significant differences between 3 groups of 5 dimensional polytomous data I have asked participants in a study to categorize the meaning of messages across 5-dimensions pertaining to Happiness, Sadness, Sarcasm, Honesty, and Anger.  For each dimension there were 5 potential values they could assign to the message (a 5 point Likert-Scale response ranging from, for example, "Not at all Angry" to "Very Angry").  There are three messages that I will be comparing these responses across.  Furthermore, all three messages have an intended meaning which is also represented by ratings for the 5 dimensions.  The last column in my dataset is each response's Euclidean distance from the intended meaning.
My hypothesis is two-fold: First, that the ratings for one message will vary more than for the other messages.  In other words, certain messages are interpreted more precisely.  Second, that the distance between each of these ratings and the intended meaning is dependent upon the message.  In other words, certain messages are interpreted more accurately.
I have included a reproduction of my data set below:

And the point that represents the intended meaning (which is the same for all three messages):

My question is, what is the most responsible approach to testing my hypotheses? Is the data appropriately formatted to test this with R's multinom() function? Preferably, what method in R would allow me to run my intended tests? 
 A: It seems that the key to your question is "what is the most responsible approach".  Given the small sample size, testing the hypothesis might be difficult.  Added to the issue of small sample size, is that the same subjects did not answer all three components.  These are more delimitations rather than limitations, however and it important to focus on why the sample is composed the way it is.  The small sample size could be considered a pilot study and useful for determining suitable sample size estimates for a larger, more-rebust study.  Unfortunately the data is a picture and I couldn't grab it to demonstrate a possible solution, so bare with me.
If you load the psych package into R (1), you can use the describeBy function to provide most of the information you require.  

describeBy(dataname$Sarcastic, dataname$Condition) 

For each condition, that function will return the mean, median, standard deviation, MAD, standard error and other measures. 
Hope this helps!


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*Revelle W. psych: Procedures for Psychological, Psychometric, and Personality Research [Internet]. Evanston, Illinois: NorthWestern University; 2013. Available from: http://CRAN.R-project.org/package=psych
