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Ordinal response variables show up often in biology, but I'm not sure how they are best analysed. Some examples are qualitative assessments (little, many, a lot) or risk assessments (low risk, medium risk, high risk).

My study is about correlation between pain and depression levels. We are measuring pain on an ordinal Likert scale. This is a number scale running from 10 (severe pain) to 0 (no pain).

We will also measure depression on a similar scale (scale: 4 - 0).

What statistical test should I use?

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    $\begingroup$ This is perhaps better suited to math SE. $\endgroup$ – canadianer Dec 19 '14 at 0:48
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    $\begingroup$ @canadianer This isn't suited to Math.SE, but Cross Validated (stats.stackexchange.com) $\endgroup$ – Luigi Dec 19 '14 at 2:04
  • $\begingroup$ This is suited I think, as Likert scales are (ab)used often in biomedical research. $\endgroup$ – AliceD Dec 19 '14 at 2:05
  • $\begingroup$ @ChrisStronks the question is not about biology but the application of a statistical method. Your answer is excellent and would make a great addition to Cross Validated but I believe it is off-topic here. $\endgroup$ – Luigi Dec 19 '14 at 2:07
  • $\begingroup$ @Luigi No wonder I couldn't find a statistics SE… $\endgroup$ – canadianer Dec 19 '14 at 2:14
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You can use ordinal multinomial regression (also known as ordered logit) if the response is ordered. These methods are basically extensions of logistic regressions, but using e.g. a cumulative logit instead of the logit. However, there are a number of different assumptions you need to consider. For instance, are you going to use a proportional-odds assumption (which is commonly used), which means that there is an equal probability of going from e.g. class 1 -> 2 and class 5 -> 6? You can also evaluate the proportional odds assumption using plots or a score test. If the response cannot be ordered, there are multinomial nominal methods that you can use, and you can also evaluate the results from an ordered analysis by comparing the predictions with ones from a multinomial nominal analysis. I have used these kinds of methods for the analysis is Red list classifications, which are clearly ordered but cannot simply be transformed into a numerical response (similar to your situation).

The book Analysis of Ordinal Categorical Data (Agresti. 2010) is a really good starting point.

In R you can look at the packages polr and vgam for ways to perform different types of analyses on ordinal data. The author of the book above has also published some examples for categorical data analysis in R, using the packages I mentioned: Examples of Using R for Modeling Ordinal Data. In SAS similar analyses can be done using Proc Genmod and Proc Logistic.

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The testing of an ordinal scale requires non-parametric statistical tests. Mean and standard deviation are invalid parameters for descriptive statistics whenever data are on ordinal scales, as are any parametric analyses based on the normal distribution.

The report of Allen & Seaman, 2007 describes a number of possible tests:

Nonparametric procedures—based on the rank, median or range—are appropriate for analyzing these data, as are chi-squared statistics.

Notably, Kruskall-Wallis models can be used to replace a standard parametric analysis of variance, as it is based on the ranks and not the means of the responses. Given these scales are representative of an underlying continuous measure, one recommendation is to analyze them as interval data as a pilot prior to gathering the continuous measure.

However, non-parametric tests are notoriously low in statistical power. There is a way of making an ordinal Likert scale like the ones you use continuous by using a ruler or slider, see the following figure (taken from Allen & Seaman, 2007):

slider

This trick makes it continuous and normal parametric tests can be used, ramping up statistical power substantially. It is a lot more work though to analyze the data. Especially when the subjects indicate their perceived pain/depression by physically putting a mark on a paper ruler, as you have to manually measure the responses. A digital slider can make your life easier. If you are planning to do hundreds of subjects you should carefully think about the possible options.

Good luck!

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    $\begingroup$ There are parametric tests for ordinal response variables, which are basically extentions of logistic regressions. $\endgroup$ – fileunderwater Dec 19 '14 at 13:01

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