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. Stevens 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.

In place of means and standard deviations, univariate statistics appropriate for ordinal data include the median, [8] : 59—61 other percentiles such as quartiles and deciles , [8] : 71 and the quartile deviation. 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 , [8] : — runs , [8] : — Smirnov , [8] : — and signed-ranks [8] : — tests. Test for two related or matched samples include the sign test [4] : 80—87 and the Wilcoxon signed ranks test. Tests for more than two related samples include the Friedman two-way analysis of variance by ranks [4] : — and the Page test for ordered alternatives.

Ordinal data can be considered as a quantitative variable. In logistic regression , the equation. 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:.

R is calculated by:. Classification methods have also been developed for ordinal data.

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The data are divided into different categories such that each observations are similar to each other. Dispersion is measured and minimized in each group to maximize classification results.

The dispersion function is used in information theory. There are several different models that can be used to describe the structure of ordinal data [11]. 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 model can be applied to nominal data. 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 proportional odds model has a very different structure to the other three models, and also a different underlying meaning. There are variants of all the models that use different link functions, such as the probit link or the complementary log-log link.

Erlbaum, Hillside, NJ, Individual Choice Behavior. Limited-Dependent and Qualitative Variables in Econometrics. Linear Structures. Charles Griffin, London, Marshall and D. Institutional performance. Leyland and H. Goldstein, editors, Multilevel Modelling of Health Statistics , pages — Regression models for ordinal data. McCullagh and J. Generalized Linear Models , 2nd edition. Conditional logit analysis of qualitative choice behavior.

Zarembka, editor, Frontiers in Econometrics. Academic Press, New York, Qualitative response models. Hildenbrand, editor, Advances in Econometrics , pages 1— Neuhaus, J. Kalbfleisch, and W. A comparison of cluster-specific and population-averaged approaches for analyzing correlated binary data. International Statistical Review , —35, Pendergast, S. Gange, M. Newton, M. Lindstrom, M. Palta, and M. A survey of methods for analyzing clustered binary response data.

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## The Assignment of Scores Procedure for Ordinal Categorical Data

Partial proportional odds models for ordinal response variables. Pinheiro and D. Approximations to the log-likelihood function in the nonlinear mixed-effects model. Journal of Computational and Graphical Statistics , —35, Prentice and L. Regression analysis of grouped survival data with application to breast cancer data.

Biometrics , —67, Rabe-Hesketh, A. Skrondal, and A. Reliable estimation of generalized linear mixed models using adaptive quadrature. The Stata Journal , —21, Working Paper , U. Maximum likelihood estimation of limited and discrete dependent variable models with nested random effects. Journal of Econometrics , —, Raman and D. A mixed-effects regression model for three-level ordinal response data.

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## Ordinal data

Raudenbush and A. Raudenbush, A. Bryk, Y. Cheong, and R. Scientific Software International, Chicago, Raudenbush, M. Yang, and M. Maximum likelihood for generalized linear models with nested random effects via high-order, multivariate Laplace approximation. Journal of Computational and Graphical Statistics , —, Revelt and K. Review of Economics and Statistics , —, Rijmen, F. Tuerlinckx, P. De Boeck, and P. A nonlinear mixed model framework for item response theory. Psychological Methods , —, An assessment of estimation procedures for multilevel models with binary responses.

Inference and missing data. Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph No. Scheike and T.

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A discrete survival model with random effects: An application to time to pregnancy. Singer and J. It's about time: Using discrete-time survival analysis to study duration and the timing of events. Journal of Educational and Behavioral Statistics , —, Skrondal and S. Multilevel logistic regression for polytomous data and rankings. Snijders and R. Spiegelhalter, A. Thomas, N. Best, and W. Stata Statistical Software: Release 9. Simulation-based estimation. Stiratelli, N.

Laird, and J. Random-effects models for serial observations with binary response. Stroud and D. Gaussian Quadrature Formulas. Ten Have. A mixed effects model for multivariate ordinal response data including correlated discrete failure times with ordinal responses. Ten Have, A. Kunselman, and L. A comparison of mixed effects logistic regression models for binary response data with two nested levels of clustering.

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Ten Have and D. Subject-specific and population-averaged continuation ratio logit models for multiple discrete time survival profiles. Ordinal probit: A generalization. Communications in Statistics: Theory and Methods , —11, Thissen and L. A taxonomy of item response models. Psychophysical analysis. American Journal of Psychology , —, Discrete Choice Methods with Simulation. Subjects Mathematics Nonfiction.

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Analysis of Ordinal Categorical Data, Second Edition provides an introduction to basic descriptive and inferential methods for categorical data, giving thorough coverage of new developments and recent methods. Special emphasis is placed on interpretation and application of methods including an integrated comparison of the available strategies for analyzing ordinal data. Practitioners of statistics in government, industry particularly pharmaceutical , and academia will want this new edition.

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