McCullagh, P. (1980). Regression models for ordinal data (with discussion). Journal of the Royal Statistical Society B 42, 109--142.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of (2.1) It is given by p sj = i z s Gamma1 ) j = 1; q (2.2) where fi j is a d dimensional regression parameter and z s Gamma1 is a vector of stochastic time dependent covariates of the same dimension. For the analysis of ordinal time series the cumulative odds model (see [22]) is given by P (Y s j j F s Gamma1 ) F ( j fl z s Gamma1 ) 2.3) where F denotes a cumulative distribution function, fl is a d dimensional regression parameter, z s Gamma1 stands for a vector of stochastic time dependent covariates of the same dimension while Gamma1 = 0 1 : ....

P. McCullagh. Regression models for ordinal data (with discussion). Journal of Royal Statistical Society, B42:109--142, 1980.

....prediction of human preferences, there has been rather little work in the machine learning and data mining area that speci cally targets this problem. The eld of statistics has developed several approaches to the problem of predicting ordinal variables, such as Ordinal Logistic Regression (e.g. [McCullagh, 1980, McCullagh Nelder, 1983] Some of these have also been studied in the eld of neural networks (e.g, Mathieson, 1996] Machine Learning, on the other hand, has started to look at the problem only recently. S. Kramer et al. Prediction of Ordinal Classes Using Regression Trees 3 [Potharst ....

McCullagh, P. (1980). Regression Models for Ordinal Data. Journal of the Royal Statistical Society Series B 42, 109-142.

....i j i j ij x F x F x x P x P y P y P j i P b a b a b a e b a a e b a a a = 5) where F denotes the cumulative density function of the logistic distribution. This model in (5) is called the ordered logit model, see for example McKelvey and Zavoina (1975) and McCullagh (1980) for some early applications. The parameters of the model can be estimated using the maximum likelihood technique. The likelihood function follows directly from equation (5) that is, n i y m i i j n i m y ij i j i ij ij ij x F x F y P y P L 1 1 1 1 1 class in ....

McCullagh, Peter (1980), Regression models for ordinal data (with discussion), Journal of the Royal Statistical Society B42, 109-142.

....we use a single n dimensional Markov random eld, so J = 1 in (3.1) and we drop the subscript j for notational convenience. Thus, x = x 1 ; x n ) is distributed according to (3. 2) Motivated by previous work on ordinal data based on grouped continuous models (Aitchison and Silvey, 1957; McCullagh, 1980), we de ne the weights for location i as w ij = 8 : 1 ( 1 x i ) if j = 1 1 ( j x i ) 1 ( j 1 x i ) if j = 2; k 1 1 1 ( k 1 x i ) if j = k (3.5) where ( is a speci ed continuous distribution function, and is a scale parameter. We ....

McCullagh, P. (1980) Regression models for ordinal data. Journal of the Royal Statistical Society B, 42, 109-142 (with discussion).

....by varying coefficients Goran Kauermann Ludwig Maximilians University Munich Institute for Statistics Akademiestrae 1 80799 Munich, Germany Abstract The paper presents a smooth regression model for ordinal data with longitudinal dependence structure. A marginal model with cumulative logit link (McCullagh 1980) is applied to cope for the ordinal scale and the main and covariate effects in the model are allowed to vary with time. Local fitting is pursued and asymptotic properties of the estimates are discussed. A data example demonstrates the exploratory flavor of the smooth model. In a second step, the ....

....of this paper is to discuss model (2) for longitudinal data with ordinal response variable. 2 We assume in the following that the response y ir takes values 1; q 1 which allow for an ordered interpretation. A widespread model for ordinal data is the cumulative model as introduced by McCullagh (1980). As varying coefficient model this is written as P (y ir kjx ir ; t ir ) Fffi 0k (t ir ) x ir fi x (t ir )g (3) for k = 1; q and F ( Delta) as known continuous distribution function. Frequently F ( Delta) is chosen as logistic distribution function. The q main effects fi 0k (t) are ....

McCullagh, P. (1980). Regression model for ordinal data (with discussion). J.

....The method s shortcomings make further discussion unnecessary, and the reader is referred to the articles for further details. Many of the problems with ANOVA based techniques relate to the validity of the inference they make. A search for statistically valid methods leads naturally to McCullagh s (1980) regression models for ordered categorical data. These Generalized Linear Models (GLMs) are attractive since software is readily available for tting them. The method can be summarized as follows: instead of creating a pseudo measurement to be analyzed, the probability that an observation Y i ....

....are attractive since software is readily available for tting them. The method can be summarized as follows: instead of creating a pseudo measurement to be analyzed, the probability that an observation Y i from a sample of size n will fall in category j is modeled as a function of the predictors. McCullagh (1980) suggests a family of models of the form: link(Pr(Y i j) j X 0 i for j = 1; J 1 and i = 1; n; 1) where categories are labeled 1; J , link is a (known) monotone increasing function mapping the interval (0,1) onto the real line (1;1) j is a cutpoint , X i is ....

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McCullagh, P. (1980), \Regression Models for Ordinal Data," Journal of the Royal Statistical Society, Series B, 42, 109-142.

....Model, Sequential Model, Smoothing, Varying Coecient Model 1 1 Introduction We consider the ordinal response variable Y which takes values Y 2 f1; kg from a set of ordered categories. A widely used model for ordinal regression is the cumulative model which has been proposed by McCullagh (1980). This has the form P (Y rjx) Ff 0r x T g for r = 1; q = k 1 (1) where x is a set of explanatory quantities and the parameters 0r ful ll the restrictions 01 02 : 0q . In (1) the probability P (Y rjx) is linked to the linear predictor r = 0r x T via ....

McCullagh, P. (1980). Regression model for ordinal data (with discussion).

....the latent normal scores and the model parameters. The logistic #more commonly used in the Rasch model# and extreme value link functions are alternatives to the Gaussian link which also have natural interpretations, corresponding to proportional odds and proportional hazards models respectively #McCullagh 1980#. Weinvestigate the logistic link, rescaled by a factor of 4= p 2# to equate the slopes of the normal and logistic CDFs at zero. Equating the slopes yields parameter estimates for the # j and # k that are on a comparable scales for either link when the linear predictor is also near zero. ....

McCullagh, P. #1980#, Regression models for ordinal data,, Journal of the Royal Statistical Society - Series B,Vol. 42, 109-142.

....as the complexity, i.e. the size, of the derived decision tree. 3.2 Logistic Regression Since the data classes are ordered, an appropriate LR model to use here is the socalled Proportional Odds (PO) model. The PO model was introduced by Walker and Duncan [33] studied in detail by McCullagh [18], and compared to other LR models by [26] It is the most commonly used ordinal logistic regression model, and can be simply described as follows. Let Y denote the disease class variable, where the values 1, 2, 3, 4 correspond, respectively, to severe CTS, moderate CTS, mild CTS and NAD in our ....

....or loss functions, thereby taking into account the relative seriousness of the various misdiagnoses. In particular, the distance between the actual and computed diagnoses should play a part in constructing an appropriate loss function (see, for example, J. Anderson s comments on McCullagh s paper [18]) 24 APPENDIX This section contains the full details of the variables used in the analysis of the CTS dataset and the distributions of the non responses in the median nerve by diagnostic class. Table 5 presents the coding of the history symptom variables. A classical history of CTS is one of ....

P. MCCULLAGH. Regression models for ordinal data (with discussion). J. Royal Stat. Soc., Ser. B, 42:109--142, 1980.

....of latent derived variables linear combinations of the original predictor variables that give more explanatory power. They often can be thought of as a proxy for some underlying variable behind the mechanism of the process generating the data. For some models, such as the cumulative logit model (McCullagh, 1980), this argument is natural and well known. In elds such as plant ecology the idea is an important one. One can think of the role of C as choosing the best predictors from a linear combination of the original predictors, and A as regression coecients of these new predictors. Other motivations ....

.... Indeed, special cases of this have already been made, for example, the nonproportional odds model can be written logit Pr(Y jjx) j = j)0 T j x; j = 1; M: Under the parallelism assumption 1 = M ( say) this becomes the well known proportional odds model (McCullagh, 1980). The parallelism assumption corresponds to the reduced rank model (7) but with a known A = 1M and (unknown) C = hence r = 1. The parallelism assumption may be applied to other categorical data models such as the adjacentcategories and continuation ratio models. In the rest of this section we ....

McCullagh, P. (1980) Regression models for ordinal data. J. Roy. Statist. Soc. B, 42, 109-142.

....ROC curve can be obtained by fitting a parametric model to the ordinal categorical data and letting the threshold for test positivity vary over the entire range of the latent degree of suspicion variable. As discussed in the next section, the ordinal regression model with a cumulative link (McCullagh 1979, 1980) can be used to perform regression analysis of ROC curves. The focus of this paper is on the use of Bayesian hierarchical ordinal regression models in ROC analysis. We shall use the abbreviation HROC (Hierarchical ROC) for this type of analysis. The HROC models developed in this paper can include ....

....In addition, the models make it possible to estimate a correlation structure for the ordinal response as a method for studying the consistency of the response over multiple image interpretations. The HROC models discussed in this paper build on the cumulative link regression model pioneered by McCullagh (1979, 1980). In its simplest form, the cumulative link model relates the observed ordinal response to covariates through an unobserved continuous latent construct and a prespecified monotone link function. The model may contain both location and scale parameters, and was originally used to carry out fixed ....

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P. McCullagh (1980). Regression models for ordinal data (with discussion). Journal of the Royal Statistical Society Series B, 42, 109--142.

....in at least 15 of the 25 fields except for P. lapathifolium that was only observed in 12 fields, but was chosen for comparison with the other Polygonum species. logit(# ik )# # k ##log(# i ) 4 [1] Statistical model In this section we describe a model for ordinal count data due to McCullagh (1980), and show how this model can be used to predict the mean density of weeds in a field. The model is described for an extended Raunkir analysis with three rings, but can easily be extended to any number of rings. We consider 25 fields, each containing n plots, and since none of the fields are ....

McCullagh, P. 1980. Regression models for ordinal data. Journal of the Royal Statistical Society, 42, 109-142.

....a total of n = 1500 measurements. A pharmacokinetic model is fit, modeling an underlying continuous pain relief level j (t) with four continuous cutoff parameters j 0 ; j 3 estimated, so that y jt equals 0 if j (t) j 0 , 1 if j 0 j (t) j 1 , 4 if j (t) j 3 , as in McCullagh (1980). This setup is more complicated than a generalized linear model, but it still can be summarized as a vector y = y 1 ; yn ) of observed data (stringing together the data y jt from all subjects j and all time points t) a vector of parameters fi, and a model of independent outcomes: For ....

McCullagh, P. (1980). Regression models for ordinal data (with discussion). Journal of the Royal Statistical Society B 42, 109--142.

....Bayes factor; Discrete hazard function; Gibbs sampling; Marginal likelihood; Metropolis Hastings algorithm; Non nested models; Sequential probit; Training sample prior; Model comparison. 1 Introduction Ordinal response data is generally analyzed using the cumulative ordinal regression model (McCullagh, 1980). Underlying this model is a continuous latent variable that is observed in categorical form, depending on the value taken by the latent variable. In some circumstances, however, a different model may be more suitable in the development of an ordinal response model. For example, if one is studying ....

....variables. 4 Fitting of related models In this section we consider the MCMC based fitting of two alternative classes of models for ordinal data. A method for comparing these models with the sequential model is discussed in Section 5. 4. 1 Cumulative ordinal model The cumulative model (McCullagh, 1980) is defined by the distribution Pr(Y jjfl; ffi) F (fl c j Gamma x 0 i ffi ) j = 1; J ; 8) where F ( Delta) is a specified cumulative distribution function, ffi is the regression parameter vector, and fl c = fl c 1 ; fl c J Gamma1 ) are the category specific cutpoints or ....

McCullagh, P. (1980), "Regression Models for Ordinal Data," Journal of the Royal Statistical Society, B 42, 109-127.

....of these coefficients. It should be noted that (3.9) does not impose any ordering on f1; 2; Kg. If this is intended this can be reached by some constraints for the weights w k (k = 1; K) The model (3. 9) then becomes the well known proportional odds model for ordered responses (McCullagh 1980, Anderson 1984) 3.3 Networks with one hidden layer A further extension of the logistic perceptron consists in introducing an additional layer of units, usually called the hidden layer , between input and output units. The units in this hidden layer are also called hidden units ; we first ....

McCullagh, P. (1980): Regression models for ordinal data (with discussion).

....Support Vector classification and Support Vector regression in the case of more than two ranks. 1 Introduction Problems of ordinal regression arise in many fields, e.g. in information retrieval (Herbrich et al. 1998) in econometric models (Tangian and Gruber 1995) and in classical statistics (McCullagh 1980; Anderson 1984) They can be related to the standard machine learning paradigm as follows: Given an i.i.d. sample S = f(x i ; y i )g i=1 P XY and a set H of mappings h from X to Y , a learning procedure selects one mapping h such that using a predefined loss l : Y Theta Y 7 R ....

....exists an ordering among the elements of Y . A variable of the above type exhibits an ordinal scale and can be thought of as the result of coarse measurement of a continuous variable (Anderson 1984) The ordinal scale leads to problems in defining an appropriate loss function for our task (see McCullagh 1980). In Section 2 we present a distribution independent model for ordinal regression, which is based on a loss function that acts on pairs of ranks. We give explicit uniform convergence bounds for the proposed loss function and show the relation between ordinal regression and preference learning. ....

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McCullagh, P. (1980). Regression models for ordinal data (with discussion). Journal of the Royal Statistical Society -- Series B 42, 109--142.

....under which M corresponds to a multinomial response model are easily expressed as follows: M is a multinomial response model if and only if M is a set of cosets of A. Equivalently, M = M A, in which it is understood that A # A# 1. Generally speaking, in models for ordinal responses (McCullagh, 1980), or models containing multiplicative e#ects (Anderson, 1984) M is not a subspace. For log linear models in which M is a subspace, the condition M = M A reduces to (1# A) #M, which is the familiar condition given by Palmgren (1981) see also McCullagh and Nelder (1989, p. 211) 4 ....

McCullagh, P. (1980). Regression models for ordinal data. J. Roy. Statist. Soc., Series B, 42, 109-142.

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McCullagh, P. (1980). Regression models for ordinal data (with discussion). Journal of the Royal Statistical Society B 42, 109--142.

No context found.

P. McCullagh. Regression models for ordinal data (with discussion) . Journal of the Royal Statistical Society, Series B, 42:109--142, 1980.

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McCullagh, P.: Regression Models for Ordinal Data. Journal of the Royal Statistical Society Series B, 42, 1980, 109--142.

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McCullagh, P., Regression models for ordinal data, Journal of the Royal Statistical Society, B42, (1980), 109-142.

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McCullagh,P. (1980). Regression models for ordinal data (with discussion).

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MCCULLAGH, P. Regression models for ordinal data (with discussion). J. Royal Stat. Soc., Ser. B, 42, 109-142, 1980.

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McCullagh,P. (1980). Regression models for ordinal data (with discussion). J. R. Statist.

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McCullagh,P. (1980). Regression models for ordinal data (with discussion). Journal of the Royal Statistical Society, Series B, 42, 109-142.

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