Ordinal data is a categorical, statistical data type where the variables have natural, ordered categories and the distances between the categories are not known.
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In multiple regression/correlation analysis, ordinal data can be accommodated using power polynomials and through normalization of scores and ranks.
R can be found by letting be the row scores and be the column scores. Let be the mean of the row scores while . Then is the marginal row probability and is the marginal column probability. R is calculated by:
Note that in the model definitions below, the values of and 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.
This model can be generalized by defining the model using instead of , 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.
This model does not impose an ordering on the categories and so can be applied to nominal data as well as ordinal data.
This is a more parsimonious, and more specialised, model than the baseline category logit model: can be thought of as similar to .
The non-ordered stereotype model has the same form as the ordered stereotype model, but without the ordering imposed on . This model can be applied to nominal data.
Note that the fitted scores, , indicate how easy it is to distinguish between the different levels of . If then that indicates that the current set of data for the covariates do not provide much information to distinguish between levels and , but that does not necessarily imply that the actual values and are far apart. And if the values of the covariates change, then for that new data the fitted scores and might then be far apart.
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 . The adjacent categories logit model can also be thought of as a special case of the ordered stereotype model, where , i.e. the distances between the are defined in advance, rather than being estimated based on the 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.
also use color or grayscale shading to display ordinal data.
Calculation of 'Effect Size' (Cliff's Delta d) using ordinal data has been recommended as a measure of statistical dominance.
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