Comparison of Regression Methods for Modeling Intensive Care Length of Stay

Intensive care units (ICUs) are increasingly interested in assessing and improving their performance. ICU Length of Stay (LoS) could be seen as an indicator for efficiency of care. However, little consensus exists on which prognostic method should be used to adjust ICU LoS for case-mix factors. This study compared the performance of different regression models when predicting ICU LoS. We included data from 32,667 unplanned ICU admissions to ICUs participating in the Dutch National Intensive Care Evaluation (NICE) in the year 2011. We predicted ICU LoS using eight regression models: ordinary least squares regression on untransformed ICU LoS,LoS truncated at 30 days and log-transformed LoS; a generalized linear model with a Gaussian distribution and a logarithmic link function; Poisson regression; negative binomial regression; Gamma regression with a logarithmic link function; and the original and recalibrated APACHE IV model, for all patients together and for survivors and non-survivors separately. We assessed the predictive performance of the models using bootstrapping and the squared Pearson correlation coefficient (R²), root mean squared prediction error (RMSPE), mean absolute prediction error (MAPE) and bias. The distribution of ICU LoS was skewed to the right with a median of 1.7 days (interquartile range 0.8 to 4.0) and a mean of 4.2 days (standard deviation 7.9). The predictive performance of the models was between 0.09 and 0.20 for R², between 7.28 and 8.74 days for RMSPE, between 3.00 and 4.42 days for MAPE and between −2.99 and 1.64 days for bias. The predictive performance was slightly better for survivors than for non-survivors. We were disappointed in the predictive performance of the regression models and conclude that it is difficult to predict LoS of unplanned ICU admissions using patient characteristics at admission time only.     

Linking gene expression data with patient survival times using partial least squares

There is an increasing need to link the large amount of genotypic data, gathered using microarrays for example, with various phenotypic data from patients. The classification problem in which gene expression data serve as predictors and a class label phenotype as the binary outcome variable has been examined extensively, but there has been less emphasis in dealing with other types of phenotypic data. In particular, patient survival times with censoring are often not used directly as a response variable due to the complications that arise from censoring. We show that the issues involving censored data can be circumvented by reformulating the problem as a standard Poisson regression problem. The procedure for solving the transformed problem is a combination of two approaches: partial least squares, a regression technique that is especially effective when there is severe collinearity due to a large number of predictors, and generalized linear regression, which extends standard linear regression to deal with various types of response variables. The linear combinations of the original variables identified by the method are highly correlated with the patient survival times and at the same time account for the variability in the covariates. The algorithm is fast, as it does not involve any matrix decompositions in the iterations. We apply our method to data sets from lung carcinoma and diffuse large B-cell lymphoma studies to verify its effectiveness.     

Analysis of longitudinal count data with serial correlation

We propose a state space model for analyzing equally or unequally spaced longitudinal count data with serial correlation. With a log link function, the mean of the Poisson response variable is a nonlinear function of the fixed and random effects. The random effects are assumed to be generated from a Gaussian first order autoregression (AR(1)). In this case, the mean of the observations has a log normal distribution. We use a combination of linear and nonlinear methods to take advantage of the Gaussian process embedded in a nonlinear function. The state space model uses a modified Kalman filter recursion to estimate the mean and variance of the AR(1) random error given the previous observations. The marginal likelihood is approximated by numerically integrating out the AR(1) random error. Simulation studies with different sets of parameters show that the state space model performs well. The model is applied to Epileptic Seizure data and Primary Care Visits Data. Missing and unequally spaced observations are handled naturally with this model.     

Models and estimators linking individual-based and sample-based rarefaction,extrapolation and comparison of assemblages

Aims In ecology and conservation biology, the number of species counted in a biodiversity study is a key metric but is usually a biased underestimate of total species richness because many rare species are not detected. Moreover, comparing species richness among sites or samples is a statistical challenge because the observed number of species is sensitive to the number of individuals counted or the area sampled. For individual-based data, we treat a single, empirical sample of species abundances from an investigator-defined species assemblage or community as a reference point for two estimation objectives under two sampling models: estimating the expected number of species (and its unconditional variance) in a random sample of (i) a smaller number of individuals (multinomial model) or a smaller area sampled (Poisson model) and (ii) a larger number of individuals or a larger area sampled. For sample-based incidence (presence–absence) data, under a Bernoulli product model, we treat a single set of species incidence frequencies as the reference point to estimate richness for smaller and larger numbers of sampling units.Methods The first objective is a problem in interpolation that we address with classical rarefaction (multinomial model) and Coleman rarefaction (Poisson model) for individual-based data and with sample-based rarefaction (Bernoulli product model) for incidence frequencies. The second is a problem in extrapolation that we address with sampling-theoretic predictors for the number of species in a larger sample (multinomial model), a larger area (Poisson model) or a larger number of sampling units (Bernoulli product model), based on an estimate of asymptotic species richness. Although published methods exist for many of these objectives, we bring them together here with some new estimators under a unified statistical and notational framework. This novel integration of mathematically distinct approaches allowed us to link interpolated (rarefaction) curves and extrapolated curves to plot a unified species accumulation curve for empirical examples. We provide new, unconditional variance estimators for classical, individual-based rarefaction and for Coleman rarefaction, long missing from the toolkit of biodiversity measurement. We illustrate these methods with datasets for tropical beetles, tropical trees and tropical ants.Important findings Surprisingly, for all datasets we examined, the interpolation (rarefaction) curve and the extrapolation curve meet smoothly at the reference sample, yielding a single curve. Moreover, curves representing 95% confidence intervals for interpolated and extrapolated richness estimates also meet smoothly, allowing rigorous statistical comparison of samples not only for rarefaction but also for extrapolated richness values. The confidence intervals widen as the extrapolation moves further beyond the reference sample, but the method gives reasonable results for extrapolations up to about double or triple the original abundance or area of the reference sample. We found that the multinomial and Poisson models produced indistinguishable results, in units of estimated species, for all estimators and datasets. For sample-based abundance data, which allows the comparison of all three models, the Bernoulli product model generally yields lower richness estimates for rarefied data than either the multinomial or the Poisson models because of the ubiquity of non-random spatial distributions in nature.     

Generalized linear models with unspecified reference distribution

We propose a new class of semiparametric generalized linear models. As with existing models, these models are specified via a linear predictor and a link function for the mean of response Y as a function of predictors X. Here, however, the "baseline" distribution of Y at a given reference mean mu(0) is left unspecified and is estimated from the data. The response distribution when the mean differs from mu(0) is then generated via exponential tilting of the baseline distribution, yielding a response model that is a natural exponential family, with corresponding canonical link and variance functions. The resulting model has a level of flexibility similar to the popular proportional odds model. Maximum likelihood estimation is developed for response distributions with finite support, and the new model is studied and illustrated through simulations and example analyses from aging research.     

Analysis of Longitudinal Data of Epileptic Seizure Counts – A Two‐State Hidden Markov Regression Approach

This paper discusses a two‐state hidden Markov Poisson regression (MPR) model for analyzing longitudinal data of epileptic seizure counts, which allows for the rate of the Poisson process to depend on covariates through an exponential link function and to change according to the states of a two‐state Markov chain with its transition probabilities associated with covariates through a logit link function. This paper also considers a two‐state hidden Markov negative binomial regression (MNBR) model, as an alternative, by using the negative binomial instead of Poisson distribution in the proposed MPR model when there exists extra‐Poisson variation conditional on the states of the Markov chain. The two proposed models in this paper relax the stationary requirement of the Markov chain, allow for overdispersion relative to the usual Poisson regression model and for correlation between repeated observations. The proposed methodology provides a plausible analysis for the longitudinal data of epileptic seizure counts, and the MNBR model fits the data much better than the MPR model. Maximum likelihood estimation using the EM and quasi‐Newton algorithms is discussed. A Monte Carlo study for the proposed MPR model investigates the reliability of the estimation method, the choice of probabilities for the initial states of the Markov chain, and some finite sample behaviors of the maximum likelihood estimates, suggesting that (1) the estimation method is accurate and reliable as long as the total number of observations is reasonably large, and (2) the choice of probabilities for the initial states of the Markov process has little impact on the parameter estimates.     

Zero-inflated Poisson and binomial regression with random effects: a case study

In a 1992 Technometrics paper, Lambert (1992, 34, 1-14) described zero-inflated Poisson (ZIP) regression, a class of models for count data with excess zeros. In a ZIP model, a count response variable is assumed to be distributed as a mixture of a Poisson(lambda) distribution and a distribution with point mass of one at zero, with mixing probability p. Both p and lambda are allowed to depend on covariates through canonical link generalized linear models. In this paper, we adapt Lambert's methodology to an upper bounded count situation, thereby obtaining a zero-inflated binomial (ZIB) model. In addition, we add to the flexibility of these fixed effects models by incorporating random effects so that, e.g., the within-subject correlation and between-subject heterogeneity typical of repeated measures data can be accommodated. We motivate, develop, and illustrate the methods described here with an example from horticulture, where both upper bounded count (binomial-type) and unbounded count (Poisson-type) data with excess zeros were collected in a repeated measures designed experiment.     

Parametric modal regression with varying precision

In this paper, we propose a simple parametric modal linear regression model where the response variable is gamma distributed using a new parameterization of this distribution that is indexed by mode and precision parameters, that is, in this new regression model, the modal and precision responses are related to a linear predictor through a link function and the linear predictor involves covariates and unknown regression parameters. The main advantage of our new parameterization is the straightforward interpretation of the regression coefficients in terms of the mode of the positive response variable, as is usual in the context of generalized linear models, and direct inference in parametric mode regression based on the likelihood paradigm. Furthermore, we discuss residuals and influence diagnostic tools. A Monte Carlo experiment is conducted to evaluate the performances of these estimators in finite samples with a discussion of the results. Finally, we illustrate the usefulness of the new model by two applications, to biology and demography.     

Estimating linear temporal trends from aggregated environmental monitoring data

Trend estimates are often used as part of environmental monitoring programs. These trends inform managers (e.g., are desired species increasing or undesired species decreasing?). Data collected from environmental monitoring programs is often aggregated (i.e., averaged), which confounds sampling and process variation. State-space models allow sampling variation and process variations to be separated. We used simulated time-series to compare linear trend estimations from three state-space models, a simple linear regression model, and an auto-regressive model. We also compared the performance of these five models to estimate trends from a long term monitoring program. We specifically estimated trends for two species of fish and four species of aquatic vegetation from the Upper Mississippi River system. We found that the simple linear regression had the best performance of all the given models because it was best able to recover parameters and had consistent numerical convergence. Conversely, the simple linear regression did the worst job estimating populations in a given year. The state-space models did not estimate trends well, but estimated population sizes best when the models converged. We found that a simple linear regression performed better than more complex autoregression and state-space models when used to analyze aggregated environmental monitoring data.     

Quasi Likelihood/Moment Method for Generalized and Restricted Generalized Poisson Regression Models and Its Application

This paper reviews the generalized Poisson regression model, the restricted generalized Poisson regression model and the mixed Poisson regression (negative binomial regression and Poisson inverse Gaussian regression) models which can be used for regression analysis of counts. The aim of this study is to demonstrate the quasi likelihood/moment method, which is used for estimation of the parameters of mixed Poisson regression models, also applicable to obtain the estimates of the parameters of the generalized Poisson regression and the restricted generalized Poisson regression models. Besides, at the end of this study an application related to this method for zoological data is given.     

Influences of size and sex on invasive species aggression and native species vulnerability: a case for modern regression techniques

Animal behaviour is of fundamental importance but is often overlooked in biological invasion research. A problem with such studies is that they may add pressure to already threatened species and subject vulnerable individuals to increased risk. One solution is to obtain the maximum possible information from the generated data using a variety of statistical techniques, instead of solely using simple versions of linear regression or generalized linear models as is customary. Here, we exemplify and compare the use of modern regression techniques which have very different conceptual backgrounds and aims (negative binomial models, zero-inflated regression, and expectile regression), and which have rarely been applied to behavioural data in biological invasion studies. We show that our data display overdispersion, which is frequent in ecological and behavioural data, and that conventional statistical methods such as Poisson generalized linear models are inadequate in this case. Expectile regression is similar to quantile regression and allows the estimation of functional relationships between variables for all portions of a probability distribution and is thus well suited for modelling boundaries in polygonal relationships or cases with heterogeneous variances which are frequent in behavioural data. We applied various statistical techniques to aggression in invasive mosquitofish, Gambusia holbrooki, and the concomitant vulnerability of native toothcarp, Aphanius iberus, in relation to individual size and sex. We found that medium sized male G. holbrooki carry out the majority of aggressive acts and that smaller and medium size A. iberus are most vulnerable. Of the regression techniques used, only negative binomial models and zero-inflated and expectile Poisson regressions revealed these relationships.     

Spatial modeling of data with excessive zeros applied to reindeer pellet‐group counts

We analyze a real data set pertaining to reindeer fecal pellet‐group counts obtained from a survey conducted in a forest area in northern Sweden. In the data set, over 70% of counts are zeros, and there is high spatial correlation. We use conditionally autoregressive random effects for modeling of spatial correlation in a Poisson generalized linear mixed model (GLMM), quasi‐Poisson hierarchical generalized linear model (HGLM), zero‐inflated Poisson (ZIP), and hurdle models. The quasi‐Poisson HGLM allows for both under‐ and overdispersion with excessive zeros, while the ZIP and hurdle models allow only for overdispersion. In analyzing the real data set, we see that the quasi‐Poisson HGLMs can perform better than the other commonly used models, for example, ordinary Poisson HGLMs, spatial ZIP, and spatial hurdle models, and that the underdispersed Poisson HGLMs with spatial correlation fit the reindeer data best. We develop R codes for fitting these models using a unified algorithm for the HGLMs. Spatial count response with an extremely high proportion of zeros, and underdispersion can be successfully modeled using the quasi‐Poisson HGLM with spatial random effects.     

Sensitivity analysis of common statistical models used to study the short-term effects of air pollution on health

The relationship between photochemical air pollutants (nitrogen dioxide and ozone) and emergency room admissions for asthma in Madrid (Spain) for the period 1995-1998 was analysed using the statistical models commonly used to studying the short-term effects of air pollution on health: linear and Cochrane-Orcutt regression, standard Poisson and Poisson corrected by overdispersion, Poisson autoregressive models, and generalised additive models. Linear regression models presented residual autocorrelation, Poisson regression models also showed overdispersion, and generalised additive models did not show residual autocorrelation and overdispersion was substantially reduced. Linear models provided biased estimates because our health outcome is non-normally distributed. Estimates from Poisson regression allowing for overdispersion and autocorrelation did not differ substantially from those reported by generalised additive models, which present the best model fit in terms of the absence of autocorrelation and reduction of overdispersion.     

Travel Time Estimation Using Freeway Point Detector Data Based on Evolving Fuzzy Neural Inference System

Travel time is an important measurement used to evaluate the extent of congestion within road networks. This paper presents a new method to estimate the travel time based on an evolving fuzzy neural inference system. The input variables in the system are traffic flow data (volume, occupancy, and speed) collected from loop detectors located at points both upstream and downstream of a given link, and the output variable is the link travel time. A first order Takagi-Sugeno fuzzy rule set is used to complete the inference. For training the evolving fuzzy neural network (EFNN), two learning processes are proposed: (1) a K-means method is employed to partition input samples into different clusters, and a Gaussian fuzzy membership function is designed for each cluster to measure the membership degree of samples to the cluster centers. As the number of input samples increases, the cluster centers are modified and membership functions are also updated; (2) a weighted recursive least squares estimator is used to optimize the parameters of the linear functions in the Takagi-Sugeno type fuzzy rules. Testing datasets consisting of actual and simulated data are used to test the proposed method. Three common criteria including mean absolute error (MAE), root mean square error (RMSE), and mean absolute relative error (MARE) are utilized to evaluate the estimation performance. Estimation results demonstrate the accuracy and effectiveness of the EFNN method through comparison with existing methods including: multiple linear regression (MLR), instantaneous model (IM), linear model (LM), neural network (NN), and cumulative plots (CP).     

Quasi-likelihood estimation for relative risk regression models

For a prospective randomized clinical trial with two groups, the relative risk can be used as a measure of treatment effect and is directly interpretable as the ratio of success probabilities in the new treatment group versus the placebo group. For a prospective study with many covariates and a binary outcome (success or failure), relative risk regression may be of interest. If we model the log of the success probability as a linear function of covariates, the regression coefficients are log-relative risks. However, using such a log-linear model with a Bernoulli likelihood can lead to convergence problems in the Newton-Raphson algorithm. This is likely to occur when the success probabilities are close to one. A constrained likelihood method proposed by Wacholder (1986, American Journal of Epidemiology 123, 174-184), also has convergence problems. We propose a quasi-likelihood method of moments technique in which we naively assume the Bernoulli outcome is Poisson, with the mean (success probability) following a log-linear model. We use the Poisson maximum likelihood equations to estimate the regression coefficients without constraints. Using method of moment ideas, one can show that the estimates using the Poisson likelihood will be consistent and asymptotically normal. We apply these methods to a double-blinded randomized trial in primary biliary cirrhosis of the liver (Markus et al., 1989, New England Journal of Medicine 320, 1709-1713).     

Statistical methods in regression and calibration analysis of chromosome aberration data

The method of iteratively reweighted least squares for the regression analysis of Poisson distributed chromosome aberration data is reviewed in the context of other fit procedures used in the cytogenetic literature. As an application of the resulting regression curves methods for calculating confidence intervals on dose from aberration yield are described and compared, and, for the linear quadratic model lambda = beta 0 + beta 1 chi + beta 2 chi 2 a confidence interval for the ratio beta 1/ beta 2 is given. Emphasis is placed on the rationale, interpretation and the limitations of various methods from a statistical point of view.     

Statistical analysis of functional response experiments

The aim of this paper is to highlight the benefits of a full data analysis of a functional response data set, compared to the usual one‐pass regression analysis. In a biological control setting where the choice of organism is often based on comparative studies of the functional responses, it is imperative to have both reliable estimates and a feeling of the degree of confidence one is willing to put on the figures. We analyzed a data set involving the freshwater predator Notonecta glauca (Hemiptera) preying on Asellus aquaticus during 24 h. The specific aim of the analysis was to test whether the functional response is of type II or type III. The different stages of a complete analysis are (1) a preliminary inspection of the data, (2) model building, (3) a model check and (4) a combination of the results with independent information. We argue that the analysis is best done with the predation rate as response and define a test for the location of its maximum. The existence of a maximum is typical for type III functional response. We explain why the binomial distribution is a natural error distribution, and how to implement the regression analysis within the family of generalized linear models using two competing link functions, the logit and the reciprocal. There is marked overdispersion which increases with increasing prey numbers. We use prior weights to take account of it. Using all available data, a type III functional response is warranted with the reciprocal link, but not with the logit link Model checks using Pearson residuals and regression diagnostics based on point deletions show that three points have a particularly strong influence on the parameter estimates. If these are deleted, the functional response type III is then warranted for both link functions. The complete analysis enables us to determine the various degrees of uncertainty and to draw biological conclusions with corresponding confidence. We are convinced that the data set shows a type III functional response, but we are less sure about which link function to choose. Furthermore, the marked overdispersion at high density, the regression diagnostics, as well as independent information on a change in the behaviour of the prey at high density, indicate that the experimental conditions may have changed as a function of the prey density.     

Effect of phosphatidylcholine bilayer thickness and molecular order on the binding of the antimicrobial peptide maculatin 1.1

Antimicrobial peptides (AMPs) interact directly with bacterial membrane lipids. Thus, changes in the lipid composition of bacterial membranes can have profound effects on the activity of AMPs. In order to understand the effect of bilayer thickness and molecular order on the activity of AMPs, the interaction of maculatin 1.1 (Mac1.1) with phosphatidylcholine (PC) model membranes composed of different monounsaturated acyl chain lengths between 14 and 22 carbons was characterised by dual polarisation interferometry (DPI) and 31P and 1H solid-state NMR techniques. The thickness and bilayer order of each PC bilayer showed a linear dependence on the acyl chain length. The binding of Mac1.1 exhibited a biphasic dependency between the amount of bound Mac1.1 and bilayer thickness, whereby the mass of bound peptide increased from C14 to C16 and then decreased from C16 to C22. Significant perturbation of 31P chemical shift anisotropy (CSA) values was only observed for DOPC (C18) and DEPC (C22), respectively. In the case of DEPC, the greater range in CSA indicated different headgroup conformations or environments in the presence of Mac1.1. Overall, the results indicated that there is a significant change in the bilayer order upon binding of Mac1.1 and this change occurred in a co-operative manner at higher concentrations of Mac1.1 with increasing bilayer thickness and order. Overall, an optimum bilayer thickness and lipid order may be required for effective membrane perturbation by Mac1.1 and increasing the bilayer thickness and order may counteract the activity of Mac1.1 and play a role in antimicrobial resistance to AMPs.