# Analysing the correlation of two ranked scales

So I am researching if self-esteem has an effect on hearing aid satisfaction in patients. The patients have filled out two questionnaires, the first measuring self esteem, the second measuring their hearing aid satisfaction. My hypothesis is that a higher self esteem score will correlate to a higher hearing aid satisfaction.

The self esteem questionnaire is a ten part likert scale which asks the patients to ring whether they strongly agree, agree, disagree or strongly disagree with a statement.

The hearing aid satisfaction questionnaire is similar but uses tick boxes for the patients to choose which option best describes their use.

So 20 people have done the questionnaires and so I have 20 scores for self esteem and hearing aid satisfaction for each patient. I want to see if the two scores are correlated.

What statistical test do I use?

(I literally have no experience in stats at all)

From your description, it seems that you already have two summary scores for each individual (that is, each individual has one score for each test).

If you don't have these summary scores yet (that is, you have 10 + 10 = 20 items for each individual), then you will need to determine a way to create a summary score for each set of items. In a very brief way, these are your options: (1) create a standardized summed scale and report their reliability by using a measure such as Cronbach's alpha (see alpha from the psych package in R); (2) assuming that the scales are uni-dimensional, then use factor or principal components analysis to extract a component or factor for each set of items (see fa in the psych package for a start on how to do this).

However, keep in mind that if your scale isn't uni-dimensional (that is, it "loads" on more than one factor or component), then you'll need to use the factor or principal component analysis to determine the number of clusters.

Once you have these two continuous scores for each individual, you can run a basic correlation between the set of items. There's a standard interpretation of correlation, which ranges from -1 to 1 (with values closer to -1 or 1 indicating stronger negative or positive relationships, accordingly). For examining correlations again see the psych package (which I mention here because of the excellent documentation). F

If you have other predictors (such as the age or sex of the respondent) you could also run a multiple regression, in which you estimate the mean score of one scale using (for example) sex, age, and the score on the other scale as predictors. Conceptually regression is a way of estimating a mean conditional on values of a set of inputs. For regression analysis see lm in the stats package in R.

If you simply want a measure and test of correlation between satisfaction and esteem, you (or your statistical software) could calculate a quantity called Spearman's rank correlation coefficient, $r_{\text{S}}$—a monotonic measure of correlation. A monotonic correlation assumes only that satisfaction always either changes in the same direction (i.e. either increases or decreases) or remains constant as esteem increases.

You could then, while bearing in mind Peter Flom's caveat about your low sample size, test whether $r_{\text{S}}$ is significantly different than 0 (i.e. there is some correlation, rather than none) using one of several methods detailed in the Wikipedia article. If you have statistical software, it is likely that it will perform such a test for you. If you need help performing such a test (e.g. how to perform a t test) by hand, that is probably a good candidate for another search through the CrossValidated archives.

First, you can't analyze whether one thing has an effect on another in a study like this, you can only analyze whether the two things are related.

Second, with only 20 people, the relationship will have to be quite strong to be detected.

Finally, to your question: If I understand, there are 10 questions in each scale and you have added those 10 to form 2 scales. If that's right, then you ought to be able to use regression with satisfaction as the dependent variable and self-esteem as the independent variable. You could start with ordinary least squares regression and check the assumptions; if they are violated, then please write again about how they are violated.

If the satisfaction scale is very skew, or limited to only a few different results, you might need a different form of regression.

• Why would the skewness of satisfaction matter, if the residuals turned out to be distributed normally, with mean 0? – Alexis Apr 27 '14 at 15:02
• If it is very skew, then there are likely to be ceiling effects. – Peter Flom - Reinstate Monica Apr 27 '14 at 19:23
• Can you amplify that a bit? I don't recall an assumption of "no ceiling effects" with respect to OLS. Do you mean to say that a very skew dependent variable likely means a non-linear functional relationship between the dependent variable and the predictor? Just trying to understand your terms and meaning. – Alexis Apr 27 '14 at 19:44
• Thinking about it some more, I think a problematic ceiling effect would reveal itself in non-normal residuals. – Peter Flom - Reinstate Monica Apr 27 '14 at 21:00