Dealing with ordinal data

Back in Chapter 1, data were described as interval (measurements on a regular scale), ordinal (measurements on a scale of undefined steps) and nominal (classifications). We have dealt extensively with two of these, but ordinal data have thus far been ignored. 

1 Why ordinal data are generally analysed by non-parametric methods 

Ordinal data typically include things like patients subjective descriptions of their condition. A score may be allocated, ranging from 1 to 4, where  

Ordinal data tend not to form normal distributions. For a start, it is often recorded on scales with a very limited number of possible values. Scales of four, five or six points are frequently seen. In such cases, it is impossible for the data to form the sort of smooth, bell-shaped distribution that constitutes a true normal distribution. However, then the problem is further exacerbated. Although there is no necessary reason for it, anybody who has worked with real-world, ordinal data knows that it is frequently hideously non-normal. Offered a scale of possible scores, people will quite frequently do bizarre things like only using the extreme upper and lower values but not the middle ones, or else they will produce a completely flat distribution, with no peak frequencies anywhere. No amount of mathematical transformation is going to convert that sort of mess into anything remotely resembling a normal distribution. 

It is theoretically possible for ordinal scale data to approximate a normal distribution, but marked non-normality is all too common. 

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An example of dealing with ordinal scale data - applying the Mann-Whitney test to the effectiveness of an analgesic... 

The calculation of mean and standard deviation only really makes sense when we are dealing with continuous, score or count data. These quantities have little relevance when we are looking at binary or ordinal data. In these situations we would tend to use proportions in the various categories as our summary statistics and population parameters of interest. 

In this section we will discuss the extension of the t-tests for continuous data and the chi-square tests for binary, categorical and ordinal data to deal with more than two treatment arms. 

The statistical methods discussed up to now have required certain assumptions about the populations from which the samples were obtained. Among these was that the population could be approximated by a normal distribution and that, when dealing with several populations, these have the same variance. There are many situations where these assumptions cannot be met, and methods have been developed that are not concerned with specific population parameters or the distribution of the population. These are referred to as non-parametric or distribution-free methods. They are the appropriate methods for ordinal data and for interval data where the requirements of normality cannot be assumed. A disadvantage of these methods is that they are less efficient than parametric methods. By less efficient is meant... 

Another important feature of mathematical modeling techniques is the nature of the response data that they are capable of handling. Some methods are designed to work with data that are measured on a nominal or ordinal scale this means the results are divided into two or more classes that may bear some relation to one another. Male and female, dead and alive, and aromatic and nonaromatic, are all classifications (dichotomous in this case) based on a nominal scale. Toxic, slightly toxic, and non-toxic are classifications based on an ordinal scale since they can be written as toxic > slightly toxic > non-toxic. The rest of this section is divided into three parts methods that deal with classified responses, methods that handle continuous data, and artificial neural networks that can be used for both. 

At the NEA Data Bank the responsibility for the overall co-ordination of the Project was with Federico Mompean who until his departure in September 2007, was in charge of the preparation of the successive drafts to the semifinal version, and updating the NEA thermodynamic database. Myriam Illemassene (until December 2006) and Jane Perrone, (who also oversaw the production of the final version) were responsible, most capably, for the editing of the numerous drafts. The present volume owes a great deal to the invaluable efforts of these NEA staff... 

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