Ordinal data
Ordinal data is a categorical, statistical data type where the variables have natural, ordered categories and the distances between the categories are not known. These data exist on an ordinal scale, one of four levels of measurement described by S. S. Stevens in 1946. The ordinal scale is distinguished from the nominal scale by having a "ranking". It also differs from the interval scale and ratio scale by not having category widths that represent equal increments of the underlying attribute. Examples of ordinal data. A well-known example of ordinal data is the Likert scale. An example of  a Likert scale is: Examples of ordinal data are often found in questionnaires: for example, the survey question "Is your general health poor, reasonable, good, or excellent?" may have those answers coded respectively as 1, 2, 3, and 4. Sometimes data on an interval scale or ratio scale are grouped onto an ordinal scale: for example, individuals whose income is known might be grouped into the income categories $0–$19,999, $20,000–$39,999, $40,000–$59,999, ..., which then might be coded as 1, 2, 3, 4, ... Other examples of ordinal data include socioeconomic status, military ranks, and letter grades for coursework. Ways  to analyse ordinal data. Ordinal data analysis requires a different set of analyses than other qualitative variables. These methods incorporate the natural ordering of the variables in order to avoid loss of power. Computing the mean of a sample of ordinal data is discouraged; other measures of central tendency, including the median or mode, are generally more appropriate. General. Stevens (1946) argued that, because the assumption of equal distance between categories does not hold for ordinal data, the use of means and standard deviations for description of ordinal distributions and of inferential statistics based on means and standard deviations was  not appropriate. Instead, positional measures like the median and percentiles, in addition to descriptive statistics appropriate for nominal data (number of cases, mode, contingency correlation), should be used. Nonparametric methods have been proposed as the most appropriate procedures for inferential statistics involving ordinal data, especially those developed for the analysis of ranked measurements. However, use of parametric statistics for ordinal data may be permissible with certain caveats to take advantage of the greater range of available statistical procedures. Univariate statistics. In place of means and standard deviations, univariate statistics appropriate for ordinal data include the median, other percentiles (such as  quartiles and deciles), and the quartile deviation. One-sample tests for ordinal data include the Kolmogorov-Smirnov one-sample test, the one-sample runs test, and the change-point test. Bivariate statistics. In lieu of testing differences in means with "t"-tests, differences in distributions of ordinal data from two independent samples can be tested with Mann-Whitney, runs, Smirnov, and signed-ranks tests. Test for two related or matched samples include the sign test and the Wilcoxon signed ranks test. Analysis of variance with ranks and the Jonckheere test for ordered alternatives can be conducted with ordinal data in place  of independent samples ANOVA. Tests for more than two related samples include the Friedman two-way analysis of variance by ranks and the Page test for ordered alternatives. Correlation measures appropriate for two ordinal-scaled variables include Kendall's tau, gamma, "r", and "d/d". Regression applications. Ordinal data can be considered as a quantitative variable. In logistic regression, the equation is the model and c takes on the assigned levels of the categorical scale. In regression analysis, outcomes (dependent variables) that are ordinal variables can be predicted using a variant of ordinal regression, such as ordered logit or ordered  probit. In multiple regression/correlation analysis, ordinal data can be accommodated using power polynomials and through normalization of scores and ranks. Linear trends. Linear trends are also used to find associations between ordinal data and other categorical variables, normally in a contingency tables. A correlation "r" is found between the variables where "r" lies between -1 and 1. To test the trend, a test statistic: is used where "n" is the sample size. "R" can be found by letting formula_3 be the row scores and formula_4 be the column scores. Let formula_5 be the mean of the row scores while  formula_6. Then formula_7 is the marginal row probability and formula_8 is the marginal column probability. "R" is calculated by: Classification methods. Classification methods have also been developed for ordinal data. The data are divided into different categories such that each observation is similar to others. Dispersion is measured and minimized in each group to maximize classification results. The dispersion function is used in information theory. Statistical models for ordinal data. There are several different models that can be used to describe the structure of ordinal data. Four major classes of model are described below, each defined for a random variable  formula_10, with levels indexed by formula_11. Note that in the model definitions below, the values of formula_12 and formula_13 will not be the same for all the models for the same set of data, but the notation is used to compare the structure of the different models. Proportional odds model. The most commonly-used model for ordinal data is the proportional odds model, defined by formula_14 where the parameters formula_12 describe the base distribution of the ordinal data, formula_16 are the covariates and formula_13 are the coefficients describing the effects of the covariates. This model can be generalized by defining  the model using formula_18 instead of formula_19, and this would make the model suitable for nominal data (in which the categories have no natural ordering) as well as ordinal data. However, this generalization can make it much more difficult to fit the model to the data. Baseline category logit model. The baseline category model is defined by formula_20 This model does not impose an ordering on the categories and so can be applied to nominal data as well as ordinal data. Ordered stereotype model. The ordered stereotype model is defined by formula_21 where the score parameters are constrained such that  formula_22. This is a more parsimonious, and more specialised, model than the baseline category logit model: formula_23 can be thought of as similar to formula_24. The non-ordered stereotype model has the same form as the ordered stereotype model, but without the ordering imposed on formula_25. This model can be applied to nominal data. Note that the fitted scores, formula_26, indicate how easy it is to distinguish between the different levels of formula_10. If formula_28 then that indicates that the current set of data for the covariates formula_16 do not provide much information to distinguish between levels formula_30 and formula_31 , but that does not necessarily imply that the actual values formula_30 and formula_31 are far apart. And if the values of the covariates change, then for that new data the fitted scores formula_26 and formula_35 might then be far apart. Adjacent categories logit model. The adjacent categories model is defined by formula_36 although the most common form, referred to in Agresti (2010) as the "proportional odds form" is defined by formula_37 This model can only be applied to ordinal data, since modelling the probabilities of shifts from one category to the next category implies that an ordering of those categories  exists. The adjacent categories logit model can be thought of as a special case of the baseline category logit model, where formula_38. The adjacent categories logit model can also be thought of as a special case of the ordered stereotype model, where formula_39, i.e. the distances between the formula_25 are defined in advance, rather than being estimated based on the data. Comparisons between the models. The proportional odds model has a very different structure to the other three models, and also a different underlying meaning. Note that the size of the reference category in the proportional odds model varies with  formula_30, since formula_42 is compared to formula_43, whereas in the other models the size of the reference category remains fixed, as formula_44 is compared to formula_45 or formula_46. Different link functions. There are variants of all the models that use different link functions, such as the probit link or the complementary log-log link. Visualization and display. Ordinal data can be visualized in several different ways. Common visualizations are the bar chart or a pie chart. Tables can also be useful for displaying ordinal data and frequencies. Mosaic plots can be used to show the relationship between an ordinal variable  and a nominal or ordinal variable. A bump chart—a line chart that shows the relative ranking of items from one time point to the next—is also appropriate for ordinal data. Color or grayscale gradation can be used to represent the ordered nature of the data. A single-direction scale, such as income ranges, can be represented with a bar chart where increasing (or decreasing) saturation or lightness of a single color indicates higher (or lower) income. The ordinal distribution of a variable measured on a dual-direction scale, such as a Likert scale, could also be illustrated with  color in a stacked bar chart. A neutral color (white or gray) might be used for the middle (zero or neutral) point with contrasting colors used in the opposing directions from the midpoint, where increasing saturation or darkness of the colors could indicate categories at increasing distance from the midpoint. Choropleth maps also use color or grayscale shading to display ordinal data. Applications. The use of ordinal data can be found in most areas of research where categorical data are generated. Settings where ordinal data are often collected include the social and behavioral sciences and governmental and business settings where  measurements are collected from persons by observation, testing, or questionnaires. Some common contexts for the collection of ordinal data include survey research; and intelligence, aptitude, and personality testing.