Regression Models with Ordinal Variables*
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Multivariate Ordinal Regression Models: an Analysis of Corporate Credit Ratings
Statistical Methods & Applications https://doi.org/10.1007/s10260-018-00437-7 ORIGINAL PAPER Multivariate ordinal regression models: an analysis of corporate credit ratings Rainer Hirk1 · Kurt Hornik1 · Laura Vana1 Accepted: 22 August 2018 © The Author(s) 2018 Abstract Correlated ordinal data typically arises from multiple measurements on a collection of subjects. Motivated by an application in credit risk, where multiple credit rating agencies assess the creditworthiness of a firm on an ordinal scale, we consider mul- tivariate ordinal regression models with a latent variable specification and correlated error terms. Two different link functions are employed, by assuming a multivariate normal and a multivariate logistic distribution for the latent variables underlying the ordinal outcomes. Composite likelihood methods, more specifically the pairwise and tripletwise likelihood approach, are applied for estimating the model parameters. Using simulated data sets with varying number of subjects, we investigate the performance of the pairwise likelihood estimates and find them to be robust for both link functions and reasonable sample size. The empirical application consists of an analysis of corporate credit ratings from the big three credit rating agencies (Standard & Poor’s, Moody’s and Fitch). Firm-level and stock price data for publicly traded US firms as well as an unbalanced panel of issuer credit ratings are collected and analyzed to illustrate the proposed framework. Keywords Composite likelihood · Credit ratings · Financial ratios · Latent variable models · Multivariate ordered probit · Multivariate ordered logit 1 Introduction The analysis of univariate or multivariate ordinal outcomes is an important task in various fields of research from social sciences to medical and clinical research. -
Analysis of Questionnaires and Qualitative Data – Non-Parametric Tests
Analysis of Questionnaires and Qualitative Data – Non-parametric Tests JERZY STEFANOWSKI Instytut Informatyki Politechnika Poznańska Lecture SE 2013, Poznań Recalling Basics Measurment Scales • Four scales of measurements commonly used in statistical analysis: nominal, ordinal, interval, and ratio scales •A nominal scale -> there is no relative ordering of the categories, e.g. sex of a person, colour, trademark, • Ordinal -> place object in a relative ordering, Many rating scales (e.g. never, rarely, sometimes, often, always) • Interval -> Places objects in order and equal differences in value denote equal differences in what we are measuring • Ratio -> similar interval measurement but also has a ‘true zero point’ and you can divide values. Coding values with numbers Numbers are used to code nominal or ordered values but they are not true numbers! Only for interval or ratio measurements they are proper number – e.g. you are allowed to perform algebraic operations (+, -, *, /) Most of our considerations as to statistical data analysis or prediction methods concern → numerical (interval, ratio) data. • In many domains we collect nominal or oridinal data • Use of Questionnaires or Survey Studies in SE! • Also collected in Web applications Types of Variables in Questionnaires • Yes/No Questions Any question that has yes or no as a possible response is nominal • Two or multi-values (options) • Gender (Femal vs. Male) • Activity nominal data type of 6 choices of activity in the park: • sport, •picnic, • reading, • walk (including with the dog), • meditation, •jog. Likert Scales • A special kind of survey question uses a set of responses that are ordered so that one response is greater (or preferred) than another. -
Ologit — Ordered Logistic Regression
Title stata.com ologit — Ordered logistic regression Description Quick start Menu Syntax Options Remarks and examples Stored results Methods and formulas References Also see Description ologit fits ordered logit models of ordinal variable depvar on the independent variables indepvars. The actual values taken on by the dependent variable are irrelevant, except that larger values are assumed to correspond to “higher” outcomes. Quick start Ordinal logit model of y on x1 and categorical variables a and b ologit y x1 i.a i.b As above, and include interaction between a and b and report results as odds ratios ologit y x1 a##b, or With bootstrap standard errors ologit y x1 i.a i.b, vce(bootstrap) Analysis restricted to cases where catvar = 0 using svyset data with replicate weights svy bootstrap, subpop(if catvar==0): ologit y x1 i.a i.b Menu Statistics > Ordinal outcomes > Ordered logistic regression 1 2 ologit — Ordered logistic regression Syntax ologit depvar indepvars if in weight , options options Description Model offset(varname) include varname in model with coefficient constrained to 1 constraints(constraints) apply specified linear constraints SE/Robust vce(vcetype) vcetype may be oim, robust, cluster clustvar, bootstrap, or jackknife Reporting level(#) set confidence level; default is level(95) or report odds ratios nocnsreport do not display constraints display options control columns and column formats, row spacing, line width, display of omitted variables and base and empty cells, and factor-variable labeling Maximization maximize options control the maximization process; seldom used collinear keep collinear variables coeflegend display legend instead of statistics indepvars may contain factor variables; see [U] 11.4.3 Factor variables. -
Logit and Ordered Logit Regression (Ver
Getting Started in Logit and Ordered Logit Regression (ver. 3.1 beta) Oscar Torres-Reyna Data Consultant [email protected] http://dss.princeton.edu/training/ PU/DSS/OTR Logit model • Use logit models whenever your dependent variable is binary (also called dummy) which takes values 0 or 1. • Logit regression is a nonlinear regression model that forces the output (predicted values) to be either 0 or 1. • Logit models estimate the probability of your dependent variable to be 1 (Y=1). This is the probability that some event happens. PU/DSS/OTR Logit odelm From Stock & Watson, key concept 9.3. The logit model is: Pr(YXXXFXX 1 | 1= , 2 ,...=k β ) +0 β ( 1 +2 β 1 +βKKX 2 + ... ) 1 Pr(YXXX 1= | 1 , 2k = ,... ) 1−+(eβ0 + βXX 1 1 + β 2 2 + ...βKKX + ) 1 Pr(YXXX 1= | 1 , 2= ,... ) k ⎛ 1 ⎞ 1+ ⎜ ⎟ (⎝ eβ+0 βXX 1 1 + β 2 2 + ...βKK +X ⎠ ) Logit nd probita models are basically the same, the difference is in the distribution: • Logit – Cumulative standard logistic distribution (F) • Probit – Cumulative standard normal distribution (Φ) Both models provide similar results. PU/DSS/OTR It tests whether the combined effect, of all the variables in the model, is different from zero. If, for example, < 0.05 then the model have some relevant explanatory power, which does not mean it is well specified or at all correct. Logit: predicted probabilities After running the model: logit y_bin x1 x2 x3 x4 x5 x6 x7 Type predict y_bin_hat /*These are the predicted probabilities of Y=1 */ Here are the estimations for the first five cases, type: 1 x2 x3 x4 x5 x6 x7 y_bin_hatbrowse y_bin x Predicted probabilities To estimate the probability of Y=1 for the first row, replace the values of X into the logit regression equation. -
Estimating Heterogeneous Choice Models with Oglm 1 Introduction
Estimating heterogeneous choice models with oglm Richard Williams Department of Sociology, University of Notre Dame, Notre Dame, IN [email protected] Last revised October 17, 2010 – Forthcoming in The Stata Journal Abstract. When a binary or ordinal regression model incorrectly assumes that error variances are the same for all cases, the standard errors are wrong and (unlike OLS regression) the parameter estimates are biased. Heterogeneous choice (also known as location-scale or heteroskedastic ordered) models explicitly specify the determinants of heteroskedasticity in an attempt to correct for it. Such models are also useful when the variance itself is of substantive interest. This paper illustrates how the author’s Stata program oglm (Ordinal Generalized Linear Models) can be used to estimate heterogeneous choice and related models. It shows that two other models that have appeared in the literature (Allison’s model for group comparisons and Hauser and Andrew’s logistic response model with proportionality constraints) are special cases of a heterogeneous choice model and alternative parameterizations of it. The paper further argues that heterogeneous choice models may sometimes be an attractive alternative to other ordinal regression models, such as the generalized ordered logit model estimated by gologit2. Finally, the paper offers guidelines on how to interpret, test and modify heterogeneous choice models. Keywords. oglm, heterogeneous choice model, location-scale model, gologit2, ordinal regression, heteroskedasticity, generalized ordered logit model 1 Introduction When a binary or ordinal regression model incorrectly assumes that error variances are the same for all cases, the standard errors are wrong and (unlike OLS regression) the parameter estimates are biased (Yatchew & Griliches 1985). -
The Ordered Probit and Logit Models Have Come Into
ISSN 1444-8890 ECONOMIC THEORY, APPLICATIONS AND ISSUES Working Paper No. 42 Students’ Evaluation of Teaching Effectiveness: What Su rveys Tell and What They Do Not Tell by Mohammad Alauddin And Clem Tisdell November 2006 THE UNIVERSITY OF QUEENSLAND ISSN 1444-8890 ECONOMIC THEORY, APPLICATIONS AND ISSUES (Working Paper) Working Paper No. 42 Students’ Evaluation of Teaching Effectiveness: What Surveys Tell and What They Do Not Tell by Mohammad Alauddin1 and Clem Tisdell ∗ November 2006 © All rights reserved 1 School of Economics, The Univesity of Queensland, Brisbane 4072 Australia. Tel: +61 7 3365 6570 Fax: +61 7 3365 7299 Email: [email protected]. ∗ School of Economics, The University of Queensland, Brisbane 4072 Australia. Tel: +61 7 3365 6570 Fax: +61 7 3365 7299 Email: [email protected] Students’ Evaluations of Teaching Effectiveness: What Surveys Tell and What They Do Not Tell ABSTRACT Employing student evaluation of teaching (SET) data on a range of undergraduate and postgraduate economics courses, this paper uses ordered probit analysis to (i) investigate how student’s perceptions of ‘teaching quality’ (TEVAL) are influenced by their perceptions of their instructor’s attributes relating including presentation and explanation of lecture material, and organization of the instruction process; (ii) identify differences in the sensitivty of perceived teaching quality scores to variations in the independepent variables; (iii) investigate whether systematic differences in TEVAL scores occur for different levels of courses; and (iv) examine whether the SET data can provide a useful measure of teaching quality. It reveals that student’s perceptions of instructor’s improvement in organization, presentation and explanation, impact positively on students’ perceptions of teaching effectiveness. -
Lecture 6 Multiple Choice Models Part II – MN Probit, Ordered Choice
RS – Lecture 17 Lecture 6 Multiple Choice Models Part II – MN Probit, Ordered Choice 1 DCM: Different Models • Popular Models: 1. Probit Model 2. Binary Logit Model 3. Multinomial Logit Model 4. Nested Logit model 5. Ordered Logit Model • Relevant literature: - Train (2003): Discrete Choice Methods with Simulation - Franses and Paap (2001): Quantitative Models in Market Research - Hensher, Rose and Greene (2005): Applied Choice Analysis 1 RS – Lecture 17 Model – IIA: Alternative Models • In the MNL model we assumed independent nj with extreme value distributions. This essentially created the IIA property. • This is the main weakness of the MNL model. • The solution to the IIA problem is to relax the independence between the unobserved components of the latent utility, εi. • Solutions to IIA – Nested Logit Model, allowing correlation between some choices. – Models allowing correlation among the εi’s, such as MP Models. – Mixed or random coefficients models, where the marginal utilities associated with choice characteristics vary between individuals. Multinomial Probit Model • Changing the distribution of the error term in the RUM equation leads to alternative models. • A popular alternative: The εij’s follow an independent standard normal distributions for all i,j. • We retain independence across subjects but we allow dependence across alternatives, assuming that the vector εi = (εi1,εi2, , εiJ) follows a multivariate normal distribution, but with arbitrary covariance matrix Ω. 2 RS – Lecture 17 Multinomial Probit Model • The vector εi = (εi1,εi2, , εiJ) follows a multivariate normal distribution, but with arbitrary covariance matrix Ω. • The model is called the Multinomial probit model. It produces results similar results to the MNL model after standardization. -
Generalized Linear Models
CHAPTER 6 Generalized linear models 6.1 Introduction Generalized linear modeling is a framework for statistical analysis that includes linear and logistic regression as special cases. Linear regression directly predicts continuous data y from a linear predictor Xβ = β0 + X1β1 + + Xkβk.Logistic regression predicts Pr(y =1)forbinarydatafromalinearpredictorwithaninverse-··· logit transformation. A generalized linear model involves: 1. A data vector y =(y1,...,yn) 2. Predictors X and coefficients β,formingalinearpredictorXβ 1 3. A link function g,yieldingavectoroftransformeddataˆy = g− (Xβ)thatare used to model the data 4. A data distribution, p(y yˆ) | 5. Possibly other parameters, such as variances, overdispersions, and cutpoints, involved in the predictors, link function, and data distribution. The options in a generalized linear model are the transformation g and the data distribution p. In linear regression,thetransformationistheidentity(thatis,g(u) u)and • the data distribution is normal, with standard deviation σ estimated from≡ data. 1 1 In logistic regression,thetransformationistheinverse-logit,g− (u)=logit− (u) • (see Figure 5.2a on page 80) and the data distribution is defined by the proba- bility for binary data: Pr(y =1)=y ˆ. This chapter discusses several other classes of generalized linear model, which we list here for convenience: The Poisson model (Section 6.2) is used for count data; that is, where each • data point yi can equal 0, 1, 2, ....Theusualtransformationg used here is the logarithmic, so that g(u)=exp(u)transformsacontinuouslinearpredictorXiβ to a positivey ˆi.ThedatadistributionisPoisson. It is usually a good idea to add a parameter to this model to capture overdis- persion,thatis,variationinthedatabeyondwhatwouldbepredictedfromthe Poisson distribution alone. -
Ordered/Ordinal Logistic Regression with SAS and Stata1
Ordered/Ordinal Logistic Regression with SAS and Stata1 This document will describe the use of Ordered Logistic Regression (OLR), a statistical technique that can sometimes be used with an ordered (from low to high) dependent variable. The dependent variable used in this document will be the fear of crime, with values of: 1 = not at all fearful 2 = not very fearful 3 = somewhat fearful 4 = very fearful Ordered logit model has the form: This model is known as the proportional-odds model because the odds ratio of the event is independent of the category j. The odds ratio is assumed to be constant for all categories. Source: http://www.indiana.edu/~statmath/stat/all/cat/2b1.html Syntax and results using both SAS and Stata will be discussed. OLR models cumulative probability. It simultaneously estimates multiple equations. The number of equations it estimates will the number of categories in the dependent variable minus one. So, for our example, three equations will be estimated. The equations are: Pooled Pooled Categories compared to Categories Equation 1: 1 2 3 4 Equation 2: 1 2 3 4 Equation 3: 1 2 3 4 Each equation models the odds of being in the set of categories on the left versus the set of categories on the right. OLR provides only one set of coefficients for each independent variable. Therefore, there is an assumption of parallel regression. That is, the coefficients for the variables in the equations would not vary significantly if they were estimated separately. The intercepts would be different, but the slopes would be essentially the same. -
MATH 1070 Introductory Statistics Lecture Notes Descriptive Statistics and Graphical Representation
MATH 1070 Introductory Statistics Lecture notes Descriptive Statistics and Graphical Representation Objectives: 1. Learn the meaning of descriptive versus inferential statistics 2. Identify bar graphs, pie graphs, and histograms 3. Interpret a histogram 4. Compute individual items for a five number summary 5. Compute and use standard variance and deviation 6. Understand and use the Empirical Rule The Dichotomy in Statistics Statistics can be broadly divided into two separate categories : descriptive and inferential. Descriptive statistics are used to tell us about the data and convey information in a manner which has as little pain as possible. Descriptive statistics often take the form of graphs or charts, but can include such things as stem-and-leaf displays and pictograms. Descriptive statistics, however, summarize the data and reduce it to a form which is not convertible back into the original data. We can only see what the presentor wants us to see. Inferential statistics make generalizations about data or make decisions when comparing datasets. The goal is to determine if the effects observed are random or can be attributed to the phenomenon under study. It is inferential statistics with which we will concern ourselves in this class. Basic Concepts for Statistics There are several terms which any student of statistics needs to be familiar with. These terms form the vocabulary of statistics, and are necessary to the understanding of the statistical techniques and the results. An individual number is a data point. If you are studying the temperature and you record today’s temperature at noon, then that is a data point. -
Ordered Logit Model
Title stata.com example 35g — Ordered probit and ordered logit Description Remarks and examples Reference Also see Description Below we demonstrate ordered probit and ordered logit in a measurement-model context. We are not going to illustrate every family/link combination. Ordered probit and logit, however, are unique in that a single equation is able to predict a set of ordered outcomes. The unordered alternative, mlogit, requires k − 1 equations to fit k (unordered) outcomes. To demonstrate ordered probit and ordered logit, we use the following data: . use http://www.stata-press.com/data/r13/gsem_issp93 (Selection from ISSP 1993) . describe Contains data from http://www.stata-press.com/data/r13/gsem_issp93.dta obs: 871 Selection for ISSP 1993 vars: 8 21 Mar 2013 16:03 size: 7,839 (_dta has notes) storage display value variable name type format label variable label id int %9.0g respondent identifier y1 byte %26.0g agree5 too much science, not enough feelings & faith y2 byte %26.0g agree5 science does more harm than good y3 byte %26.0g agree5 any change makes nature worse y4 byte %26.0g agree5 science will solve environmental problems sex byte %9.0g sex sex age byte %9.0g age age (6 categories) edu byte %20.0g edu education (6 categories) Sorted by: . notes _dta: 1. Data from Greenacre, M. and J Blasius, 2006, _Multiple Correspondence Analysis and Related Methods_, pp. 42-43, Boca Raton: Chapman & Hall. Data is a subset of the International Social Survey Program (ISSP) 1993. 2. Full text of y1: We believe too often in science, and not enough in feelings and faith. -
Using a General Ordered Logit Model to Explain the Influence of Hotel
sustainability Article Using a General Ordered Logit Model to Explain the Influence of Hotel Facilities, General and Sustainability-Related, on Customer Ratings Ioana-Nicoleta Abrudan 1,*, Ciprian-Marcel Pop 1 and Paul-Sorin Lazăr 2 1 Faculty of Economics and Business Administration, Babes, -Bolyai University, 58-60 T. Mihali St, 400591 Cluj-Napoca, Romania; [email protected] 2 Faculty of Business, Babes, -Bolyai University, 7 Horea St., 400174 Cluj-Napoca, Romania; [email protected] * Correspondence: [email protected]; Tel.: +40-752-028-500 Received: 7 October 2020; Accepted: 3 November 2020; Published: 9 November 2020 Abstract: The hotel market has become extremely competitive over the past years. Hotels try to differentiate themselves through their services and facilities. To make the best choice when searching for accommodation, guests increasingly use rating systems of booking sites. Using an ordered logit model (OLM), we identify, in our study, a sample that comprises of 635 hotels from Romania. These are the hotel facilities that significantly influence customer review scores (as an expression of customer satisfaction) on booking.com, the most widespread rating system. We also identify whether their impact on intervals of satisfaction levels vary. Some explanatory variables invalidate the Brant test for proportional odds assumption. Thus, for the final estimates, we use a generalized ordered logit model (GOLOGIT). The results show that food-related facilities, restaurants, and complimentary breakfasts, are very significant for customer ratings. Relevant hotel common facilities are the pool and parking spaces, while for the room—the flat-screen TV. It is interesting to note the negative influence of pets, which seem to disturb other tourists.