Estimating marginal and incremental effects on health outcomes using flexible link and variance function models

We propose an extension to the estimating equations in generalized linear models to estimate parameters in the link function and variance structure simultaneously with regression coefficients. Rather than focusing on the regression coefficients, the purpose of these models is inference about the mean of the outcome as a function of a set of covariates, and various functionals of the mean function used to measure the effects of the covariates. A commonly used functional in econometrics, referred to as the marginal effect, is the partial derivative of the mean function with respect to any covariate, averaged over the empirical distribution of covariates in the model. We define an analogous parameter for discrete covariates. The proposed estimation method not only helps to identify an appropriate link function and to suggest an underlying distribution for a specific application but also serves as a robust estimator when no specific distribution for the outcome measure can be identified. Using Monte Carlo simulations, we show that the resulting parameter estimators are consistent. The method is illustrated with an analysis of inpatient expenditure data from a study of hospitalists.     

Epidemiological Models with Non-Exponentially Distributed Disease Stages and Applications to Disease Control

SEIR epidemiological models with the inclusion of quarantine and isolation are used to study the control and intervention of infectious diseases. A simple ordinary differential equation (ODE) model that assumes exponential distribution for the latent and infectious stages is shown to be inadequate for assessing disease control strategies. By assuming arbitrarily distributed disease stages, a general integral equation model is developed, of which the simple ODE model is a special case. Analysis of the general model shows that the qualitative disease dynamics are determined by the reproductive number , which is a function of control measures. The integral equation model is shown to reduce to an ODE model when the disease stages are assumed to have a gamma distribution, which is more realistic than the exponential distribution. Outcomes of these models are compared regarding the effectiveness of various intervention policies. Numerical simulations suggest that models that assume exponential and non-exponential stage distribution assumptions can produce inconsistent predictions.     

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.     

Langevin machine: a neural network based on stochastically justifiable sigmoidal function

In neural networks the activation process controls the output as a nonlinear function of the input; and, this output remains bounded between limits as decided by a logistic function known as the sigmoid (S-shaped). Presently, by applying the considerations of Maxwell-Boltzmann statistics, the Langevin function is shown as the appropriate and justifiable sigmoid (instead of the conventional hyperbolic tangent function) to depict the bipolar nonlinear logic-operation enunciated by the collective stochastical response of artificial neurons under activation. That is, the graded response of a large network of neurons such as Hopfield's can be stochastically justified via the proposed model. The model is consistent with the established link between the Hopfield model and the statistical mechanics. The possible outcomes and implications of using the Langevin function (in lieu of conventional hyperbolic tangent and/or exponential sigmoids) in determining nonlinear decision boundaries, in characterizing the neural networks by the Langevin machine versus the Boltzmann machine, in sharpening and annealing schedules and in the optimization of nonlinear detector performance are discussed.     

A point mutation in domain 4-segment 6 of the skeletal muscle sodium channel produces an atypical inactivation state

We compared wild-type rat skeletal muscle NaChs (micro1) and a mutant NaCh (Y1586K) that has a single amino acid substitution, lysine (K) for tyrosine (Y), at position 1586 in the S6 transmembrane segment of domain 4. In Y1586K, macroscopic current decay is faster, the V(1/2) of the activation curve is shifted in the depolarized direction, and the fast-inactivation curve is less steep compared with mu1. After an 8-ms depolarization pulse, Y1586K recovers from inactivation much more slowly than mu1. The recovery is double exponential, suggesting recovery from two inactivation states. Varying the depolarization protocols isolates entry into an additional, "atypical" inactivation state in Y1586K that is distinct from typical fast or slow inactivation. Substitution of positively charged arginine (R) at Y1586 produces an inactivation phenotype similar to that of Y1586K. Substitution by negatively charged aspartic acid (D) or uncharged alanine (A) at Y1586 produces an inactivation phenotype similar to mu1. Our results suggest that the positive charge of lysine (K) produces the atypical inactivation state in Y1586K. We propose that a conformational change during depolarization alters the relative position of the 1586K residue in the D4-S6 segment and that atypical inactivation in Y1586K occurs via an electrostatic interaction in or near the inner pore region.     

Gamma generalized linear models for pharmacokinetic data

Summary .   This article considers the modeling of single-dose pharmacokinetic data. Traditionally, so-called compartmental models have been used to analyze such data. Unfortunately, the mean function of such models are sums of exponentials for which inference and computation may not be straightforward. We present an alternative to these models based on generalized linear models, for which desirable statistical properties exist, with a logarithmic link and gamma distribution. The latter has a constant coefficient of variation, which is often appropriate for pharmacokinetic data. Inference is convenient from either a likelihood or a Bayesian perspective. We consider models for both single and multiple individuals, the latter via generalized linear mixed models. For single individuals, Bayesian computation may be carried out with recourse to simulation. We describe a rejection algorithm that, unlike Markov chain Monte Carlo, produces independent samples from the posterior and allows straightforward calculation of Bayes factors for model comparison. We also illustrate how prior distributions may be specified in terms of model-free pharmacokinetic parameters of interest. The methods are applied to data from 12 individuals following administration of the antiasthmatic agent theophylline.     

The analysis of placement values for evaluating discriminatory measures

The idea of using measurements such as biomarkers, clinical data, or molecular biology assays for classification and prediction is popular in modern medicine. The scientific evaluation of such measures includes assessing the accuracy with which they predict the outcome of interest. Receiver operating characteristic curves are commonly used for evaluating the accuracy of diagnostic tests. They can be applied more broadly, indeed to any problem involving classification to two states or populations (D= 0 or 1). We show that the ROC curve can be interpreted as a cumulative distribution function for the discriminatory measure Y in the affected population (D= 1) after Y has been standardized to the distribution in the reference population (D= 0). The standardized values are called placement values. If the placement values have a uniform(0, 1) distribution, then Y is not discriminatory, because its distribution in the affected population is the same as that in the reference population. The degree to which the distribution of the standardized measure differs from uniform(0, 1) is a natural way to characterize the discriminatory capacity of Y and provides a nontraditional interpretation for the ROC curve. Statistical methods for making inference about distribution functions therefore motivate new approaches to making inference about ROC curves. We demonstrate this by considering the ROC-GLM regression model and observing that it is equivalent to a regression model for the distribution of placement values. The likelihood of the placement values provides a new approach to ROC parameter estimation that appears to be more efficient than previously proposed methods. The method is applied to evaluate a pulmonary function measure in cystic fibrosis patients as a predictor of future occurrence of severe acute pulmonary infection requiring hospitalization. Finally, we note the relationship between regression models for the mean placement value and recently proposed models for the area under the ROC curve which is the classic summary index of discrimination.     

Validation of an individual-based model of forest dynamics using self-thinning relationships

Self-thinning relationships link mean plant size and plant density for even-aged populations subjected to density-dependent mortality. Because the relationships are expressed at the population or the community level, they constitute a validation test for individual-based models of plant population dynamics. The model proposed here stems from forest gap models of the JABOWA/FORET-type and succeeds the validation test. This validates the growth and mortality individual-based local rules used in most gap models. The result arises in the model because some basic assumptions make density a negative exponential function of time and mean individual size a sigmoid function of time.     

Delayed-exponential approximation of a linear homogeneous diffusion model of neuron

The diffusion models of neuronal activity are general yet conceptually simple and flexible enough to be useful in a variety of modeling problems. Unfortunately, even simple diffusion models lead to tedious numerical calculations. Consequently, the existing neural net models use characteristics of a single neuron taken from the pre-diffusion era of neural modeling. Simplistic elements of neural nets forbid to incorporate a single learning neuron structure into the net model. The above drawback cannot be overcome without the use of the adequate structure of the single neuron as an element of a net. A linear (not necessarily homogeneous) diffusion model of a single neuron is a good candidate for such a structure, it must, however, be simplified. In the paper the structure of the diffusion model of neuron is discussed and a linear homogeneous model with reflection is analyzed. For this model an approximation is presented, which is based on the approximation of the first passage time distribution of the Ornstein-Uhlenbeck process by the delayed (shifted) exponential distribution. The resulting model has a simple structure and has a prospective application in neural modeling and in analysis of neural nets.Work supported by Polish Academy of Sciences grant # CPBP 04.01     

The natural variability of vital rates and associated statistics

The first concern of this work is the development of approximations to the distributions of crude mortality rates, age-specific mortality rates, age-standardized rates, standardized mortality ratios, and the like for the case of a closed population or period study. It is found that assuming Poisson birthtimes and independent lifetimes implies that the number of deaths and the corresponding midyear population have a bivariate Poisson distribution. The Lexis diagram is seen to make direct use of the result. It is suggested that in a variety of cases, it will be satisfactory to approximate the distribution of the number of deaths given the population size, by a Poisson with mean proportional to the population size. It is further suggested that situations in which explanatory variables are present may be modelled via a doubly stochastic Poisson distribution for the number of deaths, with mean proportional to the population size and an exponential function of a linear combination of the explanatories. Such a model is fit to mortality data for Canadian females classified by age and year. A dynamic variant of the model is further fit to the time series of total female deaths alone by year. The models with extra-Poisson variation are found to lead to substantially improved fits.     

Poisson vs normal-errors regression in Mac Nally (1996)

Abstract Mac Nally (1996), in describing the application of ‘hierarchical partitioning’ in regression modelling of species richness of breeding passerine birds with response variable the species count, rejects the use of Poisson regression in favour of normal-errors regression on an incorrect basis. Mac Nally uses a function of the residual sum of squares, the root-mean square prediction error (RMSPE), calculated from predictions from each regression and rejects the Poisson regression because its RMSPE was 20% larger. This note points out that the RMSPE will always be larger for the Poisson regression, given the same link function and linear predictor is used, even if the response is truly Poisson. References to appropriate methods of determining the most suitable response distribution and link function in the context of generalized linear models are given.     

Estimating equations for parameters in means and covariances of multivariate discrete and continuous responses.

Generalized estimating equations are introduced in an ad hoc fashion for the covariance matrix of a multivariate response. These equations are to be solved jointly with score equations from a generalized linear model for mean parameters. A class of quadratic exponential models is used to develop joint estimating equations for mean and covariance parameters in a more systematic fashion, and proposals for the use of such equations are developed. Comments on the relative merits of the ad hoc and model-based approaches to estimation are given and a regression illustration with a bivariate response is provided.     

Estimation and testing with overdispersed proportions using the beta-logistic regression model of Heckman and Willis

Methods are presented for modeling dose-related effects in proportion data when extra-binomial variability is a concern. Motivation is taken from experiments in developmental toxicology, where similarity among conceptuses within a litter leads to intralitter correlations and to overdispersion in the observed proportions. Appeal is made to the well-known beta-binomial distribution to represent the overdispersion. From this, an exponential function of the linear predictor is used to model the dose-response relationship. The specification was introduced previously for econometric applications by Heckman and Willis; it induces a form of logistic regression for the mean response, together with a reciprocal biexponential model for the intralitter correlation. Large-sample, likelihood-based methods for estimating and testing the joint proportion-correlation response are studied. A developmental toxicity data set illustrates the methods.     

On the impact of winter conditions on the dynamics of a population with non-overlapping generations: a model approach

The authors propose new type of models with non-overlapping generations. It is assumed that during winter period individuals are not active (as, for example, in insect populations in boreal forests) and some portion of population dyes. However the portion of population, that survives, Q, indirectly depends on feeding conditions in previous growing season. In the formal terms, Q = Q(u) is a decreasing function of the mean population size u (i.e., of the integral) over the growing period, and traditional discrete-time model therefore turns into a discrete-continuous one. Under any constant birth rate Y, the model is reduced to a discrete one in its general form, and a general result consists in global stability of the zero solution for any Y < 1, e.t., in population extinction from any initial state. In particular cases of dependence of Q(u) and different types of population self-limitation during growing season the general model results in a great variety of discrete models (including well known Moran-Ricker and Skellam models). For logistic growth of population during the growing season and exponential decrease in Q(u), the condition is obtained for a non-trivial steady state to exist, and the outcome is presented for bifurcation analysis with regard to parameter Y: cycles with typical period-doubling and chaotic dynamics.     

The transducer model for contrast detection and discrimination: formal relations, implications, and an empirical test

The transducer function mu for contrast perception describes the nonlinear mapping of stimulus contrast onto an internal response. Under a signal detection theory approach, the transducer model of contrast perception states that the internal response elicited by a stimulus of contrast c is a random variable with mean mu(c). Using this approach, we derive the formal relations between the transducer function, the threshold-versus-contrast (TvC) function, and the psychometric functions for contrast detection and discrimination in 2AFC tasks. We show that the mathematical form of the TvC function is determined only by mu, and that the psychometric functions for detection and discrimination have a common mathematical form with common parameters emanating from, and only from, the transducer function mu and the form of the distribution of the internal responses. We discuss the theoretical and practical implications of these relations, which have bearings on the tenability of certain mathematical forms for the psychometric function and on the suitability of empirical approaches to model validation. We also present the results of a comprehensive test of these relations using two alternative forms of the transducer model: a three-parameter version that renders logistic psychometric functions and a five-parameter version using Foley's variant of the Naka-Rushton equation as transducer function. Our results support the validity of the formal relations implied by the general transducer model, and the two versions that were contrasted account for our data equally well.     

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.     

Kinetics of protein and DNA synthesis studied by mathematical modelling of flow cytometric protein and DNA histograms

Abstract. Mathematical models for histograms of cellular protein content as measured by flow cytometry were developed, based on theoretical protein distributions. These were derived from the age distribution of cells and the accumulation function for cellular protein content as a function of age within the cell cycle. A model assuming an exponential age distribution and an exponential protein. accumulation function was found to give the best representation of protein histograms of exponentially growing NHIK 3025 cells. This is in good agreement with the known kinetic behaviour of such cells. By the combined use of the protein histogram model and a similar model for DNA content, and assuming linear DNA accumulation during S, the fraction of cells in S, as a function of cellular protein content, was simulated. This function showed good agreement with values of the [³H]TdR labelling index scored in cells sorted by flow cytometry from 5-channel intervals of the protein histogram. The protein and DNA histogram models were combined into a two-dimensional model for correlated protein/DNA measurements. Comparison between simulated data and experimentally derived two-dimensional protein/DNA histograms gave further support to the cell kinetic assumptions underlying the models, but also identified some minor deviations which could not be recognized in the analysis of the one-dimensional histograms.     

Modes of Growth in Mammalian Cells

The increase of cell volume as a function of time was studied throughout the generation cycle in synchronous cultures of Chinese hamster cells using a Coulter aperture and a multichannel analyzer calibrated against known cell volumes. The experimental results were compared to a mathematical model of cell volume increase which considered the effect of the distribution of individual cell generation times on the progress of the population. Several modes of volume increase, including linear and exponential, were considered. The mean volume vs. time curve was rounded at the ends of the cycle even when linear growth was assumed. The experimental results show that cell volume increased in a smooth fashion as a function of time, with no discontinuities in rate detectable at periods when cells may have been undergoing metabolic shifts as, for example, through the phases associated with DNA synthesis, G₁, S, G₂. A statistical test on the comparison of the modal cell volume vs. time data to the predictions of linear and exponential growth models accepted both hypotheses within the resolution of these experiments. However, exponential growth was favored over linear growth in one cell line. Volume dispersion was almost constant with time in both sublines which is also consistent with exponential growth. Limitations of the electronic technique of volume measurement and indications for future experiments are discussed.     

Semiparametric models for missing covariate and response data in regression models

We consider a class of semiparametric models for the covariate distribution and missing data mechanism for missing covariate and/or response data for general classes of regression models including generalized linear models and generalized linear mixed models. Ignorable and nonignorable missing covariate and/or response data are considered. The proposed semiparametric model can be viewed as a sensitivity analysis for model misspecification of the missing covariate distribution and/or missing data mechanism. The semiparametric model consists of a generalized additive model (GAM) for the covariate distribution and/or missing data mechanism. Penalized regression splines are used to express the GAMs as a generalized linear mixed effects model, in which the variance of the corresponding random effects provides an intuitive index for choosing between the semiparametric and parametric model. Maximum likelihood estimates are then obtained via the EM algorithm. Simulations are given to demonstrate the methodology, and a real data set from a melanoma cancer clinical trial is analyzed using the proposed methods.     

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.