Continuation Ratio Model Research Articles

Is grand multiparity associated with offsprings' hospital-treated mental disorders? A 28-year follow-up of the North Finland 1966 birth cohort.

A child born to a grand multiparous (GMP) mother (i.e. a mother who has undergone six or more deliveries) is at increased risk of perinatal complications, but it is not known whether or not GMP status is associated with child's adulthood mental disorders. The data were obtained from the unselected, general population Northern Finland 1966 Birth Cohort (n = 11,017). The cohort members (children) were followed up prospectively to the age of 28 years. Using the National Hospital Discharge Register, a total of 89 DSM-III-R schizophrenia cases were identified, as well as 55 other psychoses, 87 personality disorders, 36 cases of alcoholism, 53 depressive disorders, and 67 anxiety and other non-psychotic disorders. The association between the mother's grand multiparity and the offspring's adult hospital-treated psychiatric morbidity was analysed using a continuation ratio model, which is a modification of logistic regression. Odds ratios were adjusted for social class, maternal antenatal depression, and wantedness of pregnancy. A total of 1320 mothers (12%) were GMPs. Maternal GMP status was not associated with offspring's schizophrenia, anxiety or other non-psychotic disorders. The risk of other psychoses (OR 2.3; 95% CI 1.2-4.7), alcoholism (OR 2.0; 95% CI 0.8-4.7) and depressive disorder (OR 2.2; 95% CI 1.0-4.5) was elevated among offspring of GMP mothers. It is possible that the mother's GMP status and the large family size associated with this are causal factors in the development of other psychoses than schizophrenia, alcoholism and depression among adult offspring.

A continuation ratio random effects model for repeated ordinal responses.

This paper presents methods of analysis for ordinal repeated measures. We use a generalization of the continuation ratio model with a random effect. We handle frailty by using a normal distribution and also a non-parametric distribution. The methodology is readily implemented in existing software and flexible enough to treat large data sets with uncommon time measurements as well as different numbers of repeated measures for each individual. The models are used to investigate how a group of explanatory variables influences the overall condition of patients treated for breast cancer.

Analysis of clustered data in receiver operating characteristic studies.

Clustered data is not simply correlated data, but has its own unique aspects. In this paper, various methods for correlated receiver operating characteristic (ROC) curve data that have been extended specifically to clustered data are reviewed. For those methods that have not yet been extended, suggestions for their application to clustered ROC studies are provided. Various methods with respect to their ability to meet either of two objectives of the analysis of clustered ROC data are compared to consider a variety of ROC indices and their accessibility to researchers. The available statistical methods for clustered data vary in the range of indices that can be considered and in their accessibility to researchers. Parametric models permit all indices to be considered but, owing to computational complexity, are the least accessible of available methods. Nonparametric methods are much more accessible, but only permit estimation and inference about ROC curve area. The jackknife method is the most accessible and permits any index to be considered. Future development of methods for clustered ROC studies should consider the continuation ratio model, which will permit the application of widely available software for the analysis of mixed generalized linear models. Another area of development should be in the adoption of bootstrapping methods to clustered ROC data.

Development of a clinical prediction model for an ordinal outcome: the World Health Organization Multicentre Study of Clinical Signs and Etiological agents of Pneumonia, Sepsis and Meningitis in Young Infants. WHO/ARI Young Infant Multicentre Study Group.

This paper describes the methodologies used to develop a prediction model to assist health workers in developing countries in facing one of the most difficult health problems in all parts of the world: the presentation of an acutely ill young infant. Statistical approaches for developing the clinical prediction model faced at least two major difficulties. First, the number of predictor variables, especially clinical signs and symptoms, is very large, necessitating the use of data reduction techniques that are blinded to the outcome. Second, there is no uniquely accepted continuous outcome measure or final binary diagnostic criterion. For example, the diagnosis of neonatal sepsis is ill-defined. Clinical decision makers must identify infants likely to have positive cultures as well as to grade the severity of illness. In the WHO/ARI Young Infant Multicentre Study we have found an ordinal outcome scale made up of a mixture of laboratory and diagnostic markers to have several clinical advantages as well as to increase the power of tests for risk factors. Such a mixed ordinal scale does present statistical challenges because it may violate constant slope assumptions of ordinal regression models. In this paper we develop and validate an ordinal predictive model after choosing a data reduction technique. We show how ordinality of the outcome is checked against each predictor. We describe new but simple techniques for graphically examining residuals from ordinal logistic models to detect problems with variable transformations as well as to detect non-proportional odds and other lack of fit. We examine an alternative type of ordinal logistic model, the continuation ratio model, to determine if it provides a better fit. We find that it does not but that this model is easily modified to allow the regression coefficients to vary with cut-offs of the response variable. Complex terms in this extended model are penalized to allow only as much complexity as the data will support. We approximate the extended continuation ratio model with a model with fewer terms to allow us to draw a nomogram for obtaining various predictions. The model is validated for calibration and discrimination using the bootstrap. We apply much of the modelling strategy described in Harrell, Lee and Mark (Statist. Med. 15, 361-387 (1998)) for survival analysis, adapting it to ordinal logistic regression and further emphasizing penalized maximum likelihood estimation and data reduction.

Regression models for ordinal responses: a review of methods and applications.

Epidemiologists are often interested in estimating the risk of several related diseases as well as adverse outcomes, which have a natural ordering of severity or certainty. While most investigators choose to model several dichotomous outcomes (such as very low birthweight versus normal and moderately low birthweight versus normal), this approach does not fully utilize the available information. Several statistical models for ordinal responses have been proposed, but have been underutilized. In this paper, we describe statistical methods for modelling ordinal response data, and illustrate the fit of these models to a large database from a perinatal health programme. Models considered here include (1) the cumulative logit model, (2) continuation-ratio model, (3) constrained and unconstrained partial proportional odds models, (4) adjacent-category logit model, (5) polytomous logistic model, and (6) stereotype logistic model. We illustrate and compare the fit of these models on a perinatal database, to study the impact of midline episiotomy procedure on perineal lacerations during labour and delivery. Finally, we provide a discussion on graphical methods for the assessment of model assumptions and model constraints, and conclude with a discussion on the choice of an ordinal model. The primary focus in this paper is the formulation of ordinal models, interpretation of model parameters, and their implications for epidemiological research. This paper presents a synthesized review of generalized linear regression models for analysing ordered responses. We recommend that the analyst performs (i) goodness-of-fit tests and an analysis of residuals, (ii) sensitivity analysis by fitting and comparing different models, and (iii) by graphically examining the model assumptions.

No-Observed-Adverse-Effect Levels in Severity Data

Toxicity data are often categorized by severity of response and dose level with the assumption that there is a tolerated dose below which there is no toxicity. For data from a controlled experiment, the largest observed dose at or below the tolerated dose is called the no-observed-adverse-effect level (NOAEL). The problem of identifying the NOAEL can be viewed statistically as estimating the maximal observed dose for which there is no increased severity or frequency of toxic response. We previously proposed a method based on the Akaike information criterion (AIC) for the case with only two response levels (presence or absence of a toxic endpoint). We show here that repeated applications of that method to suitably defined subsets of data provide the maximum penalized likelihood estimate of the NOAEL when there are multiple severity levels, under a slight modification of the continuation-ratio logit model. Three sets of data on controlled exposure of rodents are used to illustrate the method.

Logistic regression and continuation ratio models to estimate insect development under variable temperatures

Temperature strongly influences insect development. So a physiological scale, instead of a temporal scale, is classically used to introduce variable temperatures in phenological models. With a linear scale, we must estimate two parameters per stage, the base (temperature threshold) and the number of degree-days. Difficulties arise when neither of these parameters is known. Logistic regression and continuation ratio (CR) models may be applied, especially to stage-frequency data. In this paper, we test these two models by using them to estimate durations of development stages under variable temperatures. Simulations show that logistic regression applied to one stage gives relatively close estimates of development duration only when the base is known. Accuracy of parameter estimates depends only slightly on sample size. The CR model leads to unbiased estimates only in the case of a common base for all stages. When the base varies with stage, the CR model overestimates the durations and gives non-independent estimates. Thus, the continuation ratio model is inappropriate for field studies when the physiological scale varies with stage of development.

Modeling Insect Development Time of Two or More Larval Stages in the Field Under Variable Temperatures

A statistical method is proposed for modeling the relation between development and variable temperature with 2 or more stages. Classical models were built using laboratory data obtained by rearing insects at different constant temperatures; these models could not be simply extrapolated to natural conditions. More recent models incorporate a temperature-dependent development rate and variability between individuals. Generalized linear models applied to estimate rates of several stages require a physiological scale identical for all stages. The method we propose leads to separate stage duration estimates with a different physiological scale per stage. We apply the method to the 4 larval stages of a chestnut curculio, Curculio elephas Gyllenhal, from hatching to larval exit from the chestnut. Data came from experiments done in natural conditions in 1986 and 1992. The physiological scale is represented by degree-days, which leads to a statistically good fit for the 4 stages. Median duration and variability for each stage thus can be derived. Because the base estimate for larval stages of the chestnut weevil does not change with stage, our model can be compared with the continuation ratio model. The 2 approaches gave very similar median estimates, indicating robustness of our method. The advantage of our approach is its flexibility when stages differ according to their physiological base, which is observed, for example, for egg and larval weevil development and frequently with other species. However, the continuation ratio model generally leads to more accurate estimates in the case of a base common to each stage.

Multivariate methods for clustered ordinal data with applications to survival analysis.

Clustered data are the rule in many clinical specialties such as ophthalmology. Methods have been developed for the treatment of clustered continuous or binary outcome data. Less attention has been given to ordinal outcomes which occur frequently in ophthalmology. For example, grading systems of cataract and diabetic retinopathy are commonly used where a photograph is graded by comparison with a series of reference photographs of increasing severity. Some commonly used methods for the analysis of ordered categorical data include the proportional odds and continuation ratio models. It is difficult, however, to incorporate clustering effects into these models. Instead, for clusters of size two, we propose a generalization of the adjacent category model given by log[Pr(i + 1,j)/Pr(i,j)] = ui + (j - 1) lambda + beta' x, where Pr(i,j) denotes the probability that the right (left) eye has grade i(j), x is a vector of (person or eye-specific) covariates for the right eye, u and beta are vectors of location and covariate parameters and lambda is a clustering parameter. Based on this model, and a similar model interchanging the role of i and j, we derived a closed-form expression for Pr(i,j) as a function of u, lambda and beta and use Newton-Raphson method to maximize the likelihood. An extension of the method allows for extra agreement along the diagonal and is then a generalization of the agreement plus linear-by-linear association model proposed by Agresti in the setting of no covariates. We apply these methods to a data set of 43 diabetic subjects from the Harvard Clinical Cataract Research Center, where cortical cataract grade was the outcome. We also extend this methodology to a survival setting, where both censored and uncensored outcomes are available for individual cluster members, and one wishes to take clustering into account. We apply the survival analysis model to a data set of 1807 children (two ears per child) in the greater Boston area, who were followed for the development of otitis media over the first year of life.

Statistical assessment of ordinal outcomes in comparative studies

Ordinal regression is a relatively new statistical method developed for analyzing ranked outcomes. In the past, ranked scales have often been analyzed without making full use of the ordinality of the data or, alternatively, by assigning arbitrary numerical scores to the ranks. While ordinal regression models are now available to make full use of ranked data, they are not used widely. This article, directed to clinical researchers and epidemiologists, provides a description of the properties of these methods. Using ordinal measures of back pain in a follow-up study of adolescent idiopathic scoliosis, we illustrate the advantages of these methods and describe how to interpret the estimated parameters. Comparisons with binary logistic regression are made to show why a single dichotomization of the ordinal scale may lead to incorrect inferences. Two ordinal models (the proportional odds and the continuation ratio models) are discussed, and the goodness-of-fit of these models is examined. We conclude that ordinal regression is a tool that is powerful, simple to use, and produces an interpretable parameter that summarizes the effect between groups over all levels of the outcome.

Modelling Tabular Data with an Ordered Outcome

A large amount of data that is considered within sociological studies consists of categorical variables that lend themselves to tabular analysis. In the sociological analysis of data regarding social class and educational attainment, for example, the variables of interest can often plausibly be considered as having a substantively interesting order. Standard log-linear models do not take ordinality into account, thereby potentially they may disregard useful information. Analyzing tables where the response variable has ordered categories through model building has been problematic in software packages such as GLIM (Aitken et al., 1989). Recent developments in statistical modelling have offered new possibilities and this paper explores one option, namely the continuation ratio model which was initially reported by Fienberg and Mason (1979). The fitting of this model to data in tabular form is possible in GLIM although not especially trivial and by and large this approach has not been employed in sociological research. In this paper I outline the continuation ratio model and comment upon how it can be fitted to data by sociologists using the GLIM software. In addition I present a short description of the relative merits of such an approach. Presenting this paper in an electronic format facilitates the possibility of replicating the analysis. The data is appended to the paper in the appropriate format along with a copy of the GLIM transcript. A dumped GLIM4 file is also attached.

Vector Generalized Additive Models

SUMMARY Vector smoothing is used to extend the class of generalized additive models in a very natural way to include a class of multivariate regression models. The resulting models are called ‘vector generalized additive models‘. The class of models for which the methodology gives generalized additive extensions includes the multiple logistic regression model for nominal responses, the continuation ratio model and the proportional and non-proportional odds models for ordinal responses, and the bivariate probit and bivariate logistic models for correlated binary responses. They may also be applied to generalized estimating equations.

Fitting a random effects model to ordinal recurrent events using existing software

The continuation ratio model is a direct generalisation of the more familiar binary logistic model. In this paper, it is proposed to model ordinal recurrent events by generalising the logistic-normal model for binary recurrent events in a similar manner. This new model is implemented in the statistical software package SABRE.

A GENERALIZED LINEAR MODEL FOR ANALYSING RECEIVER OPERATING CHARACTERISTIC CURVES

We present a continuation ratio model for analysing ordinal categorical data and we apply the model to the problem of estimating receiver operating characteristic (ROC) curves. We apply the methods to post-prandial capillary blood glucose measurements as a criterion for a potential screening test for diabetes mellitus. One can obtain point estimates of sensitivity and specificity and their associated standard errors at any value along the observed range of post-prandial capillary blood glucose measurements. Also, in comparison to the models for ROCs described by Tosteson and Begg, ROC curves based on the continuation ratio model have the desired features that allow ROCs to be concave but not necessarily symmetric.

Standardized estimates from categorical regression models

We consider the problem of interpreting categorical regression models, such as the polytomous logistic model, the continuation-ratio model, the stereotype model, and the cumulative-odds model. We present a method to convert categorical regression coefficients into estimates of standardized fitted probabilities, probability differences and probability ratios. We use a delta-method approach to estimate standard errors. We then present a small simulation study to compare different transforms for setting confidence limits, and provide an illustration of our approach in an observational study of drug therapy of polymyositis.

Alternative models for ordinal logistic regression.

Armstrong and Sloan have reviewed two types of ordinal logistic models for epidemiologic data: the cumulative-odds model and the continuation-ratio model. I review here certain aspects of these models not emphasized previously, and describe a third type, the stereotype model, which in certain situations offers greater flexibility coupled with interpretational advantages. I illustrate the models in an analysis of pneumoconiosis among coal miners.

Analysis of failure time data with ordinal categories of response

When failure times are observed, additional information concerning the type of failure is often recorded. A method which simultaneously models the failure times and additional information in the form of ordinal categories is discussed. An application to clinical trial data, in which the failure times are times of onset of headache, and the headaches are classified into the ordinal categories mild, moderate and severe, illustrates how this method may be used and how the final model can be interpreted. The continuation ratio model, which is used in this method, is described in detail.

Ordinal regression models for epidemiologic data.

Health status is often measured in epidemiologic studies on an ordinal scale, but data of this type are generally reduced for analysis to a single dichotomy. Several statistical models have been developed to make full use of information in ordinal response data, but have not been much used in analyzing epidemiologic studies. The authors discuss two of these statistical models--the cumulative odds model and the continuation ratio model. They may be interpreted in terms of odds ratios, can account for confounding variables, have clear and testable assumptions, and have parameters that may be estimated and hypotheses that may be tested using available statistical packages. However, calculations of asymptotic relative efficiency and results of simulations showed that simple logistic regression applied to dichotomized responses can in some realistic situations have more than 75% of the efficiency of ordinal regression models, but only if the ordinal scale is collapsed into a dichotomy close to the optimal point. The application of the proposed models to data from a study of chest x-rays of workers exposed to mineral fibers confirmed that they are easy to use and interpret, but gave results quite similar to those obtained using simple logistic regression after dichotomizing outcome in the conventional way.

The Effects of Multi-Wave Mailings on the External Validity of Mail Surveys

Survey data, particularly mail questionnaires, are very useful in community development work. With relatively low cost, a practitioner can obtain valild information to determine community needs, support for programs and general attitudes and opinions of local citizens. Low response rates, however, can have serious effects on the validity of the data Previous research has shown that followup mailings are essential to obtaining a high response rate to mail surveys. This paper examines the potential for sample bias if the number of mailings is reduced. Differences between groups responding to three waves of mailings to a statewide Pennsylvania survey (N = 9,957) are examined via log-linear techniques, using continuation ratio models. The results indicate that initial respondents differ from laggard respondents on five demographic characteristics, but differences diminish between early laggard and late laggard respondents. The implication is that single mailings of questionnaires could cause serious threats t...

Multinomial regression models based on continuation ratios.

This paper concerns continuation ratio models for multinomial responses. These are conditional probabilities used in logit models to define the dependence of the multinomial proportions on explanatory variables and unknown parameters. A distinctive feature of these models is that if one models the various continuation ratios separately, then resulting estimates and test statistics are asymptotically independent. This allows the partitioning of likelihood ratio statistics and the search for effects in specific categories of an ordinal response variable. Models that use the same parameters for different continuation ratios are suitable for estimating more global differences. The fitting of these models to actual data is illustrated, including an example from a pharmaceutical study. The results show that different models are suitable for modelling complementary sorts of differences between multinomial response distributions.