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Ordinal regression

About: Ordinal regression is a research topic. Over the lifetime, 1879 publications have been published within this topic receiving 65431 citations.


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TL;DR: In this paper, a mixture of normals prior replaces the usual single multivariate normal model for the latent variables, allowing for varying local dependence structure across the contingency table, and removing the problems related to the choice and resampling of cutoffs defined for these latent variables.
Abstract: This article proposes a probability model for k-dimensional ordinal outcomes, that is, it considers inference for data recorded in k-dimensional contingency tables with ordinal factors. The proposed approach is based on full posterior inference, assuming a flexible underlying prior probability model for the contingency table cell probabilities. We use a variation of the traditional multivariate probit model, with latent scores that determine the observed data. In our model, a mixture of normals prior replaces the usual single multivariate normal model for the latent variables. By augmenting the prior model to a mixture of normals we generalize inference in two important ways. First, we allow for varying local dependence structure across the contingency table. Second, inference in ordinal multivariate probit models is plagued by problems related to the choice and resampling of cutoffs defined for these latent variables. We show how the proposed mixture model approach entirely removes these problems. We ill...

114 citations

Book
01 Jan 2005
TL;DR: Data, models, and multidimensional scaling analysis nature of data analyzed in MDS measurement level of data shape of data conditionality of data missing data multivariate data classical MDS Euclidean model details of CMDS Replicated MDS Weighted MDS geometry of the weighted Euclideans model algebra of the weight-based model matrix algebra ofThe weighted Euclidan model Weirdness index flattened weights.
Abstract: 1. Model Selection Loglinear Analysis Loglinear Modeling Basics. A Two-Way Table. The Saturated Model. Main Effects. Interactions. Examining Parameters in a Saturated Model. Calculating the Missing Parameter Estimates. Testing Hypotheses about Parameters. Fitting an Independence Model. Specifying the Model. Checking Convergence. Chi-Square Goodness-of-Fit Tests. Hierarchical Models. Generating Classes. Selecting a Model. Evaluating Interactions. Testing Individual Terms in the Model. Model Selection Using Backward Elimination. 2. Logit Loglinear Analysis Dichotomous Logit Model. Loglinear Representation. Logit Model. Specifying the Model. Parameter Estimates for the Saturated Logit Model. Unsaturated Logit Model. Specifying the Analysis. Goodness-of-Fit Statistics. Observed and Expected Cell Counts. Parameter Estimates. Measures of Dispersion and Association. Polychotomous Logit Model. Specifying the Model. Goodness of Fit of the Model. Interpreting Parameter Estimates. Examining Residuals. Covariates. Other Logit Models. 3. Multinomial Logistic Regression The Logit Model. Baseline Logit Example. Specifying the Model. Parameter Estimates. Likelihood-Ratio Test for Individual Effects. Likelihood-Ratio Test for the Overall Model. Evaluating the Model. Calculating Predicted Probabilities and Expected Frequencies. Classification Table. Goodness-of-Fit Tests. Examining the Residuals. Pseudo-R-square Measures. Correcting for Overdispersion. Automated Variable Selection. Hierarchical Variable Entry. Specifying the Analysis. Step Output. Likelihood-Ratio Tests for Individual Effects. Matched Case-Control Studies. The Model. Creating the Difference Variables. The Data File. Specifying the Analysis. Examining the Results. 4. Ordinal Regression Fitting an Ordinal Logit Model. Modeling Cumulative Counts. Specifying the Analysis. Parameter Estimates. Testing Parallel Lines. Does the Model Fit? Comparing Observed and Expected Counts. Including Additional Predictor Variables. Overall Model Test. Measuring Strength of Association. Classifying Cases. Generalized Linear Models. Link Function. Fitting a Heteroscedastic Probit Model. Modeling Signal Detection. Fitting a Location-Only Model. Fitting a Scale Parameter. Parameter Estimates. Model-Fitting Information 5. Probit Regression Probit and Logit Response Models. Evaluating Insecticides. Confidence Intervals for Expected Dosages. Comparing Several Groups. Comparing Relative Potencies of the Agents. Estimates of Relative Median Potency Estimating the Natural Response Rate. More than One Stimulus Variable. 6. Kaplan-Meier Survival Analysis SPSS Procedures for Survival Data. Background. Calculating Length of Time. Estimating the Survival Function. Estimating the Conditional Probability. Estimating the Cumulative Probability of Survival. The SPSS Kaplan-Meier Table. Plotting Survival Functions. Comparing Survival Functions. Specifying the Analysis. Comparing Groups. Stratified Comparisons of Survival Functions. 7. Life Tables Background Studying Employment Longevity. The Body of a Life Table. Calculating Survival Probabilities. Assumptions Needed to Use the Life Table. Lost to Follow-up 1 Plotting Survival Functions. Comparing Survival Functions. 8. Cox Regression The Cox Regression Model. The Hazard Function. Proportional Hazards Assumption. Modeling Survival Times. Coding Categorical Variables. Specifying the Analysis. Testing Hypotheses about the Age Coefficient. Interpreting the Regression Coefficient. Baseline Hazard and Cumulative Survival Rates Including Multiple Covariates. The Model with Three Covariates. Global Tests of the Model. Plotting the Estimated Functions Checking the Proportional Hazards Assumption. Stratification. Log-Minus-Log Survival Plot. Identifying Influential Cases. Examining Residuals. Partial (Schoenfeld) Residuals. Martingale Residuals. Selecting Predictor Variables. Variable Selection Methods. An Example of Forward Selection. Omnibus Test of the Model At Each Step. Time-Dependent Covariates. Examining the Data. Specifying a Time-Dependent Covariate. Calculating Segmented Time-Dependent Covariates. Testing the Proportional Hazard Assumption with a Time-Dependent Covariate Fitting a Conditional Logistic Regression Model. The Data File Structure. Specifying the Analysis. Parameter Estimates. 9. Variance Components Examples Factors, Effects, and Models. Types of Factors. Types of Effects. Types of Models. Model for One-Way Classification. Estimation Methods. Negative Variance Estimates. Nested Design Model for Two-Way Classification. Univariate Repeated Measures Analysis Using a Mixed Model Approach Background Information. Model. Distribution Assumptions. Estimation Methods. 10. Linear Mixed Models The Linear Mixed Model. Background. 11. Nonlinear Regression Examples What Is a Nonlinear Model? Transforming Nonlinear Models. Intrinsically Nonlinear Models. Fitting a Logistic Population Growth Model. Estimating a Nonlinear Model. Finding Starting Values. Specifying the Analysis. Approximate Confidence Intervals for the Parameters. Bootstrap Estimates. Estimating Starting Values. Use Starting Values from Previous Analysis. Look for a Linear Approximation. Use Properties of the Nonlinear Model. Solve a System of Equations. Computational Issues. Additional Nonlinear Regression Options. Nonlinear Regression Common Models. Specifying a Segmented Model. 12. Two-Stage Least-Squares Regression Artichoke Data Demand-Price-Income Economic Model. Estimation with Ordinary Least Squares. Feedback and Correlated Errors. Two-Stage Least Squares. Strategy. Stage 1: Estimating Price. Stage 2: Estimating the Model. 2-Stage Least Squares Procedure. 13. Weighted Least-Squares Regression Diagnosing the Problem. Estimating the Weights. Estimating Weights as Powers. Specifying the Analysis. Examining the Log-Likelihood Functions. WLS Solutions. . Estimating Weights from Replicates. Diagnostics from the Linear Regression Procedure. 14. Multidimensional Scaling Data, Models, and Analysis of Multidimensional Scaling. Example: Flying Mileages. The Nature of Data Analyzed in MDS. The Measurement Level of Data. The Shape of Data. The Conditionality of Data. Missing Data. Multivariate Data. Classical MDS. Example: Flying Mileages Revisited. The Euclidean Model. Details of CMDS. Example: Ranked Flying Mileages. Repeated CMDS. Replicated MDS. Details of RMDS. Example: Perceived Body-Part Structure. Weighted MDS. Geometry of the Weighted Euclidean Model. Algebra of the Weighted Euclidean Model. Matrix Algebra of the Weighted Euclidean Model. Details of WMDS. Example: Perceived Body-Part Structure. The Weirdness Index. Flattened Weights.

114 citations

Journal ArticleDOI
TL;DR: For a large data set there seems to be no explicit preference for either a frequentist or Bayesian approach (if based on vague priors), and on relatively large data sets, the different software implementations of logistic random effects regression models produced similar results.
Abstract: Logistic random effects models are a popular tool to analyze multilevel also called hierarchical data with a binary or ordinal outcome. Here, we aim to compare different statistical software implementations of these models. We used individual patient data from 8509 patients in 231 centers with moderate and severe Traumatic Brain Injury (TBI) enrolled in eight Randomized Controlled Trials (RCTs) and three observational studies. We fitted logistic random effects regression models with the 5-point Glasgow Outcome Scale (GOS) as outcome, both dichotomized as well as ordinal, with center and/or trial as random effects, and as covariates age, motor score, pupil reactivity or trial. We then compared the implementations of frequentist and Bayesian methods to estimate the fixed and random effects. Frequentist approaches included R (lme4), Stata (GLLAMM), SAS (GLIMMIX and NLMIXED), MLwiN ([R]IGLS) and MIXOR, Bayesian approaches included WinBUGS, MLwiN (MCMC), R package MCMCglmm and SAS experimental procedure MCMC. Three data sets (the full data set and two sub-datasets) were analysed using basically two logistic random effects models with either one random effect for the center or two random effects for center and trial. For the ordinal outcome in the full data set also a proportional odds model with a random center effect was fitted. The packages gave similar parameter estimates for both the fixed and random effects and for the binary (and ordinal) models for the main study and when based on a relatively large number of level-1 (patient level) data compared to the number of level-2 (hospital level) data. However, when based on relatively sparse data set, i.e. when the numbers of level-1 and level-2 data units were about the same, the frequentist and Bayesian approaches showed somewhat different results. The software implementations differ considerably in flexibility, computation time, and usability. There are also differences in the availability of additional tools for model evaluation, such as diagnostic plots. The experimental SAS (version 9.2) procedure MCMC appeared to be inefficient. On relatively large data sets, the different software implementations of logistic random effects regression models produced similar results. Thus, for a large data set there seems to be no explicit preference (of course if there is no preference from a philosophical point of view) for either a frequentist or Bayesian approach (if based on vague priors). The choice for a particular implementation may largely depend on the desired flexibility, and the usability of the package. For small data sets the random effects variances are difficult to estimate. In the frequentist approaches the MLE of this variance was often estimated zero with a standard error that is either zero or could not be determined, while for Bayesian methods the estimates could depend on the chosen "non-informative" prior of the variance parameter. The starting value for the variance parameter may be also critical for the convergence of the Markov chain.

113 citations

Journal ArticleDOI
TL;DR: Threshold models for ordered categorical data as considered by McCullagh (1980) have become a standard tool in categorical regression and the class of sequential models is considered where response categories are reached successively step by step.

110 citations

Journal ArticleDOI
TL;DR: In this article, a simple agricultural plot experiment is considered with two sources of variation, namely between-plot variation and withinplot variation, and the maximum likelihood estimates can be obtained by iterative weighted least squares.
Abstract: Threshold models can be useful for analysing ordered categorical data, like ratings Such models provide a link between the ordinal scale of measurement and a linear scale on which treatments are supposed to act In this paper a simple agricultural plot experiment is considered with two sources of variation, namely between-plot variation and within-plot variation It is shown that for a threshold model with two sources of variation maximum likelihood estimates can be obtained by iterative weighted least squares

108 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023102
2022191
202188
202093
201979
201873