Continuous Response Variables Research Articles

Cost effective prediction of the eutrophication status of lakes and reservoirs

Summary1. Eutrophication is a serious threat in many parts of the world, and identifying the environmental factors that determine the spatial distribution of eutrophicated waterbodies as well as the development of management tools is a challenge.2. In this study, data from the Ile‐de‐France region were analysed to determine if catchment scale environmental variables could predict concentrations of chlorophyll a (used as a proxy for eutrophication status) of artificial lakes and reservoirs.3. General additive models (GAM) and random forest models (RF) displayed greater predictive power than generalised linear models, indicating the importance of non‐monotonic relationships. Using RF modelling, very high predictive accuracy was achieved for both continuous and binomial (eutrophic or not) response variables (continuous: R2 = 0.715; binomial: kappa = 0.764, 89% of waterbodies were accurately predicted). The better predictive power and robustness of RF versus GAM was attributed to the formers ability to better handle complex interactions between predictors and to account for threshold effects.4. Our results confirmed the close link between the water quality of lakes and reservoirs and the characteristics of their catchments. Moreover, we also showed that (i) simple (e.g. linear and/or monotonic) relationships between catchment land use and water quality were only found for sub‐regional datasets, and (ii) land use needs to be considered in association with complementary environmental variables (hydromorphological variables) to best assess its impact on water quality.

The continuous cross-nested logit model: Formulation and application for departure time choice

Discrete choice models, like the multinomial logit (MNL), have long been recognized for their ability to capture a wide array of transport-related choice phenomena. However, a number of choices are continuous response variables (e.g., location, departure time, activity duration, and vehicle usage). This paper introduces the continuous cross-nested logit (CCNL) model. The CCNL model results from generalizing the discrete cross-nested logit (CNL) model for a continuous response variable, much like the continuous logit model emerges by generalizing the MNL. The model is formulated and shown to come from the generalized extreme value (GEV) class of models. In addition, the structure of utility correlations is presented. The model’s parameters are estimated for a work-tour departure time context using Bayesian estimation techniques and San Francisco Bay Area data. Empirical results suggest model predictions that are very similar to the continuous logit, but it out-performs the continuous logit in terms of out-of-sample prediction with these data. The CCNL also allows a more flexible choice behavior to emerge. Finally, a simple welfare example is illustrated and a number of model extensions are presented.

Childhood Malnutrition in Egypt using Geoadditive Gaussian and Latent Variable Models

Major progress has been made over the last 30 years in reducing the prevalence of malnutrition amongst children less than 5 years of age in developing countries. However, approximately 27% of children under the age of 5 in these countries are still malnourished. This work focuses on the childhood malnutrition in one of the biggest developing countries, Egypt. This study examined the association between bio-demographic and socioeconomic determinants and the malnutrition problem in children less than 5 years of age using the 2003 Demographic and Health survey data for Egypt. In the first step, we use separate geoadditive Gaussian models with the continuous response variables stunting (height-for-age), underweight (weight-for-age), and wasting (weight-for-height) as indicators of nutritional status in our case study. In a second step, based on the results of the first step, we apply the geoadditive Gaussian latent variable model for continuous indicators in which the 3 measurements of the malnutrition status of children are assumed as indicators for the latent variable "nutritional status".

Multiple Applications of Multivariate Adaptive Regression Splines Technique to Predict Rear-End Crashes at Unsignalized Intersections

Crash prediction models are used extensively in highway safety analysis. This paper discusses a recently developed data-mining technique to predict motor vehicle crashes: the multivariate adaptive regression splines (MARS) technique. MARS shows promising predictive power and does not suffer from a black-box limitation. Negative binomial (NB) and MARS models were fitted and compared with the use of extensive data collected on unsignalized intersections in Florida. Two models were estimated for rear-end crash frequency at three-and four-legged unsignalized intersections. Treatment of crash frequency as a continuous response variable to fit a MARS model was also examined by normalizing crash frequency with the natural logarithm of the annual average daily traffic. The combination of MARS with a machine learning technique (random forest) was explored and discussed. The significant factors that affected rear-end crashes were traffic volume on the major road, upstream and downstream distances to the nearest signalized intersection, median type on the major approach, land use at the intersection's influence area, and geographic location within the state. The study showed that MARS could predict crashes almost like the traditional NB models, and its goodness-of-fit performance was encouraging. The use of MARS to predict continuous response variables yielded more favorable results than its use to predict discrete response variables. The generated MARS models showed the most promising results after the covariates were screened by using random forest.

Performance of Selected Nonparametric Tests for Discrete Longitudinal Data Under Different Patterns of Missing Data

Comparison of changes over time of a continuous response variable between treatment groups is often of main interest in clinical trials. When the distributional properties of the continuous response variable are not regular enough, or when the response is discrete, nonparametric techniques have been used. The relative performances of selected repeated measures nonparametric two-sample tests proposed by Wei and Lachin, Koziol, Wei and Johnson, and the adapted Wilcoxon Rank-Sum test are compared through simulations based on quality of life data. The Wilcoxon Rank-Sum test is the most powerful and is not significantly affected by the different patterns of missing data.

Semiparametric Modeling and Estimation of Instrumental Variable Models

We apply Bayesian methods to a model involving a binary nonrandom treatment intake variable and an instrumental variable in which the functional forms of some of the covariates in both the treatment intake and outcome distributions are unknown. Continuous and binary response variables are considered. Under the assumption that the functional form is additive in the covariates, we develop efficient Markov chain Monte Carlo-based approaches for summarizing the posterior distribution and for comparing various alternative models via marginal likelihoods and Bayes factors. We show in a simulation experiment that the methods are capable of recovering the unknown functions and are sensitive neither to the sample size nor to the degree of confounding as measured by the correlation between the errors in the treatment and response equations. In the binary response case, however, estimation of the average treatment effect requires larger sample sizes, especially when the degree of confounding is high. The methods are applied to an example dealing with the effect on wages of more than 12 years of education.

Boosting ridge regression

There are several possible approaches to combining ridge regression with boosting techniques. In the simple or naive approach the ridge estimator is used to fit iteratively the current residuals yielding an alternative to the usual ridge estimator. In partial boosting only part of the regression parameters are reestimated within one step of the iterative procedure. The technique allows to distinguish between mandatory variables that are always included in the analysis and optional variables that are chosen only if relevant. The resulting procedure selects optional variables in a similar way as the Lasso, yielding a reduced set of influential variables, while allowing for regularized estimation of the mandatory parameters. The suggested procedures are investigated within the classical framework of continuous response variables as well as in the case of generalized linear models. The performance in terms of prediction and the identification of relevant variables is compared to several competitors as the Lasso and the more recently proposed elastic net. For the evaluation of the identification of relevant variables pseudo ROC curves are introduced.

An evaluation of differential susceptibility of barley varieties to lodging under varying environmental conditions

Information on the variation in susceptibility to lodging between barley varieties under different environments can be used for local prediction. It can also be used to identify varieties that are robust to environmental variation. Efficient methods to obtain analogous information on the local yielding capacity of crop varieties have recently been established. The present paper extends the use of these methods for the analysis of dichotomously measured performance characteristics, the susceptibility to lodging, in particular. The procedures examined were based on generalized linear models in which the expected conditional mean of the susceptibility to lodging was used as an explanatory variable to express the environmental effects of a given environment. Through the use of logistic transformation the approach extends previous methods developed for continuous response variables to binary response variables. Models were subsequently used to obtain measures of susceptibility to lodging of each barley variety in terms of environmental variation. While the emphasis is restricted to lodging, similar methods can also be applied to other performance characteristics.

A Bootstrap-Based Test for Establishing Noninferiority in Clinical Trials

A randomized, active-control clinical trial setting with the objective of testing noninferiority for a continuous response variable is considered. Noninferiority margin is based on the concept of preserving a certain fraction of the active control effect. Noninferiority is established if the ratio of the lower (upper) limit of the two-sided 95% confidence interval for the treatment difference to the estimated mean of the active control is greater (less) than a certain fraction. The nominal significance level is not maintained by the above confidence interval-based noninferiority test. We use bootstrapping to derive an accurate lower (upper) limit of the same confidence interval, which approximates the nominal significance level better and improves the power.

Inferences of variance function – a parametric robust way

Tsou (2003a) proposed a parametric procedure for making robust inference for mean regression parameters in the context of generalized linear models. This robust procedure is extended to model variance heterogeneity. The normal working model is adjusted to become asymptotically robust for inference about regression parameters of the variance function for practically all continuous response variables. The connection between the novel robust variance regression model and the estimating equations approach is also provided.

Generalized Linear Models in Family Studies

Generalized linear models (GLMs), as defined by J. A. Nelder and R. W. M. Wedderburn (1972), unify a class of regression models for categorical, discrete, and continuous response variables. As an extension of classical linear models, GLMs provide a common body of theory and methodology for some seemingly unrelated models and procedures, such as the logistic, Poisson, and probit models, that are increasingly used in family studies. This article provides an overview of the principle and the key components of GLMs, such as the exponential family of distributions, the linear predictor, and the link function. To illustrate the application of GLMs, this article uses Canadian national survey data to build an example focusing on the number of close friends among older adults. The article concludes with a discussion of the strengths and weaknesses of GLMs.

Comments about Joint Modeling of Cluster Size and Binary and Continuous Subunit‐Specific Outcomes

In longitudinal studies and in clustered situations often binary and continuous response variables are observed and need to be modeled together. In a recent publication Dunson, Chen, and Harry (2003, Biometrics 59, 521-530) (DCH) propose a Bayesian approach for joint modeling of cluster size and binary and continuous subunit-specific outcomes and illustrate this approach with a developmental toxicity data example. In this note we demonstrate how standard software (PROC NLMIXED in SAS) can be used to obtain maximum likelihood estimates in an alternative parameterization of the model with a single cluster-level factor considered by DCH for that example. We also suggest that a more general model with additional cluster-level random effects provides a better fit to the data set. An apparent discrepancy between the estimates obtained by DCH and the estimates obtained earlier by Catalano and Ryan (1992, Journal of the American Statistical Association 87, 651-658) is also resolved. The issue of bias in inferences concerning the dose effect when cluster size is ignored is discussed. The maximum-likelihood approach considered herein is applicable to general situations with multiple clustered or longitudinally measured outcomes of different type and does not require prior specification and extensive programming.

Discontinuities in indirect estimation: An application to EAR models

Among the simulation-based methods, indirect estimation techniques like Indirect Inference (I NDI NF) and Efficient Method of Moments (E MM) provide a simple solution to many computational problems associated with intractable Likelihood functions. Optimisation of the objective function can be critical in presence of not continuous response variables like, for instance, binary choice or discrete choice models, limited dependent variables, switching regime models. In particular, gradient-based optimisation algorithms can face difficulties when the not continuous response involves discontinuities in the objective function. A simple computational tool is suggested to “empirically” solve the problem. The case study is E MM applied to the autoregressive model with exponential marginal distribution (E AR). The proposed solution is also compared with the performance of the Conditional Least Squares estimation, suitable for this autoregressive model, by a set of Monte Carlo experiments.

Dr. Stein Replies

Back to table of contents Previous article Next article Letter to the EditorFull AccessDr. Stein RepliesMICHAEL D. STEIN, M.D., MICHAEL D. STEINSearch for more papers by this author, M.D., Providence, R.I.Published Online:1 May 2004https://doi.org/10.1176/appi.ajp.161.5.929-aAboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InEmail To the Editor: We welcome the opportunity to respond to Mr. Shi’s comments concerning our article describing the association between depression severity and risky injection drug use, measured as the frequency of sharing needles.One concern was our categorization of the intrinsically continuous response variable. As is commonly seen in studies of risk taking, the distribution of risky injection drug use was extremely skewed. The modal response (31.2%) was at the lower limit of zero and was not amenable to standard “normalizing” transformations, and the use of techniques like ordinal least-squares regression was untenable. Would we have observed the same association had we analyzed the continuous response? It depends on whether you use the product-moment (r=0.11, p=0.27) or rank-order (rs=0.23, p<0.02) correlation coefficient. Many would dichotomize the outcome (any risk versus no risk) and use logistic regression. We did and the results were completely consistent with those reported; an increase of one standard deviation in depression severity increased the odds (adjusted for all covariates described in the article) of any use of dirty needles (risky injection drug use) by a factor of 1.92 (z=2.42, p<0.02). We constructed a four-category ordered response to better reflect the relative risk and estimated the association by using the ordered logit model. In preparing this response, we estimated the full model by using both rank-transformed and log-transformed continuous response variables. Although neither was approximately normal, the effect of depression severity was significant with both transformations, and we are convinced of the robustness of our finding. Without providing a conceptual rationale, Mr. Shi posits that HIV status and contact with drug treatment services may moderate rather than mediate the association between depression severity and high-risk behavior involving injected drugs. We found no evidence that the association between depression severity and risky injection drug use was conditioned by either treatment contact or HIV status (relevant p values exceeded 0.50). Given our relatively small group size, the statistical power to detect moderator effects was low, and these hypotheses may merit future inquiry.We included total injection frequency as a covariate because we thought it a likely confound; its inclusion had virtually no substantive effect on the observed association between depression severity and risky injection drug use. Although it was not explicated in the published article, we examined the association between depression severity and total injection frequency by using a range of transformations, categorizations, and measures of association. There was no statistically reliable evidence that these variables were associated, and we discussed these issues.To satisfy Journal word limits, articles must often present reconstructed summaries of deeper research processes. Most of Mr. Shi’s questions and concerns were addressed in the process of inquiry but were left on “the cutting room floor.” Notwithstanding, the association of major depression and needle sharing has clear clinical implications and is worthy of continued study. FiguresReferencesCited byDetailsCited byNone Volume 161Issue 5 May 2004Pages 929-a-930 Metrics PDF download History Published online 1 May 2004 Published in print 1 May 2004

Latent semantic models for collaborative filtering

Collaborative filtering aims at learning predictive models of user preferences, interests or behavior from community data, that is, a database of available user preferences. In this article, we describe a new family of model-based algorithms designed for this task. These algorithms rely on a statistical modelling technique that introduces latent class variables in a mixture model setting to discover user communities and prototypical interest profiles. We investigate several variations to deal with discrete and continuous response variables as well as with different objective functions. The main advantages of this technique over standard memory-based methods are higher accuracy, constant time prediction, and an explicit and compact model representation. The latter can also be used to mine for user communitites. The experimental evaluation shows that substantial improvements in accucracy over existing methods and published results can be obtained.

Dynamic Latent Trait Models for Multidimensional Longitudinal Data

This article presents a new approach for analysis of multidimensional longitudinal data, motivated by studies using an item response battery to measure traits of an individual repeatedly over time. A general modeling framework is proposed that allows mixtures of count, categorical, and continuous response variables. Each response is related to age-specific latent traits through a generalized linear model that accommodates item-specific measurement errors. A transition model allows the latent traits at a given age to depend on observed predictors and on previous latent traits for that individual. Following a Bayesian approach to inference, a Markov chain Monte Carlo algorithm is proposed for posterior computation. The methods are applied to data from a neurotoxicity study of the pesticide methoxychlor, and evidence of a dose-dependent increase in motor activity is presented.

Sample Size Calculation for Dichotomous Outcomes in Cluster Randomization Trials with Varying Cluster Size

It is common to assume a constant cluster size in the estimation of sample size for cluster randomized trials. Manatunga et al. (1) derived sample size estimation for continuous response variables in cases of varying cluster size. This paper discusses a parallel study for binary response variables. We compare the maximum cluster size method, the average cluster size method, and the proposed method.

Comparing five modelling techniques for predicting forest characteristics

Broad-scale maps of forest characteristics are needed throughout the United States for a wide variety of forest land management applications. Inexpensive maps can be produced by modelling forest class and structure variables collected in nationwide forest inventories as functions of satellite-based information. But little work has been directed at comparing modelling techniques to determine which tools are best suited to mapping tasks given multiple objectives and logistical constraints. Consequently, five modelling techniques were compared for mapping forest characteristics in the Interior Western United States. The modelling techniques included linear models (LMs), generalized additive models (GAMs), classification and regression trees (CARTs), multivariate adaptive regression splines (MARS), and artificial neural networks (ANNs). Models were built for two discrete and four continuous forest response variables using a variety of satellite-based predictor variables within each of five ecologically different regions. All techniques proved themselves workable in an automated environment. When their potential mapping ability was explored through simulations, tremendous advantages were seen in use of MARS and ANN for prediction over LMs, GAMs, and CART. However, much smaller differences were seen when using real data. In some instances, a simple linear approach worked virtually as well as the more complex models, while small gains were seen using more complex models in other instances. In real data runs, MARS and GAMS performed (marginally) best for prediction of forest characteristics.

The prevalence of negative studies with inadequate statistical power: an analysis of the plastic surgery literature.

Studies published in the medical literature often neglect to consider the statistical power needed to detect a meaningful difference between study groups. Small sample sizes tend to produce negative results because of low statistical power. Studies that cannot make conclusive statements about their hypotheses can waste resources, deter further research, and impede advances in clinical treatment. The current study reviewed three of the most frequently read plastic surgery journals from 1976 to 1996 to determine the prevalence of inadequately (<80 percent) powered clinical trials and experimental studies that found no difference (negative studies) in the response variable of interest between comparison groups. The statistical power of 54 negative studies using continuous response variables was calculated to detect a difference of 1 SD (+/-1 SD) in means between the comparative groups. The power of another 57 negative studies with dichotomous response (yes/no) variables was calculated to detect a relative change in proportions of 25 percent and 50 percent from the experimental to the control group. It was found that 85 percent of the studies with continuous response variables had inadequate power to detect the desired mean difference of +/-1 SD. In studies with dichotomous response variables, 98 percent had inadequate power to detect a desired 25 percent relative change in proportions, and 74 percent had inadequate power to detect a desired 50 percent relative change in proportions. These results indicate that many of the studies in the plastic surgery literature lack adequate power to detect a moderate-to-large difference between groups. The lack of power makes the interpretation of the studies with negative findings inconclusive. Proper study design dictates that investigators consider a priori the difference between groups that is of clinical interest, and the sample size per group that is needed to provide adequate statistical power to detect the desired difference.

A Correlated Probit Model for Joint Modeling of Clustered Binary and Continuous Responses

A difficulty in joint modeling of continuous and discrete response variables is the lack of a natural multivariate distribution. For joint modeling of clustered observations on binary and continuous responses, we study a correlated probit model that has an underlying normal latent variable for the binary responses. Catalano and Ryan have factored the model into a marginal and a conditional component and used generalized estimating equations methodology to estimate the effects. We propose a Monte Carlo expectation–conditional maximization algorithm for finding maximum likelihood estimates of the mixed model itself, extending and accelerating an algorithm for models with binary responses. We demonstrate the methodology with a developmental toxicity study measuring fetal weight and a binary malformation status for several litters of mice. A simulation study suggests that efficiency gains of joint fittings over separate fittings of the response variables occur mainly for small datasets with strong correlations between the responses within cluster.