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.     

Complicated relationship between autism with regression and epilepsy

We retrospectively evaluated a set of 205 children with autism and compared it to the partial sub-set of 71 (34.6%) children with a history of regression. From 71 children with regression, signs of epileptic processes were present in 43 (60.6%), 28 (65.12%) suffered clinical epileptic seizures, and 15 (34.9%) just had an epileptiform abnormality on the EEG. In our analysis, autistic regression is substantially more associated with epileptic process symptoms than in children with autism and no history of regression. More than 90% of children with a history of regression also show IQ < 70 and reduced functionality. Functionality and IQ further worsens with the occurrence of epileptic seizures (98% of children with regression and epilepsy have IQ < 70). We proved that low IQ and reduced functionality significantly correlate rather with epileptic seizures than just sub-clinical epileptiform abnormality on EEG. Clinical epileptic seizures associated with regression significantly influence the age of regression and its clinical type. The age of regression is higher compared to children with regression without epileptic seizures (in median: 35 months of age in patients with seizures while only 24 months in other patients). Patients with seizures revealed regression after 24th months of age in 68% of cases, while patients without seizures only in 27%. However, coincidence with epilepsy also increased the occurrence of regression before the 18th month of age (23% of patients), while only 4% of patients without epilepsy revealed regression before the 18th month. Epileptic seizures are significantly associated especially with behaviour regression rather than speech regression or regression in both behaviour and speech. Also epileptic seizures diagnosed before correct diagnosis of autism were significantly associated with delayed regression (both behavioural and speech regression).     

Fitting curves to data using nonlinear regression: a practical and nonmathematical review

Many types of data are best analyzed by fitting a curve using nonlinear regression, and computer programs that perform these calculations are readily available. Like every scientific technique, however, a nonlinear regression program can produce misleading results when used inappropriately. This article reviews the use of nonlinear regression in a practical and nonmathematical manner to answer the following questions: Why is nonlinear regression superior to linear regression of transformed data? How does nonlinear regression differ from polynomial regression and cubic spline? How do nonlinear regression programs work? What choices must an investigator make before performing nonlinear regression? What do the final results mean? How can two sets of data or two fits to one set of data be compared? What problems can cause the results to be wrong? This review is designed to demystify nonlinear regression so that both its power and its limitations will be appreciated.     

Embracing asymmetry in nature: How to account for skewness in ecological data

Asymmetric regression is an alternative to conventional linear regression that allows us to model the relationship between predictor variables and the response variable while accommodating skewness. Advantages of asymmetric regression include incorporating realistic ecological patterns observed in data, robustness to model misspecification and less sensitivity to outliers. Bayesian asymmetric regression relies on asymmetric distributions such as the asymmetric Laplace (ALD) or asymmetric normal (AND) in place of the normal distribution used in classic linear regression models. Asymmetric regression concepts can be used for process and parameter components of hierarchical Bayesian models and have a wide range of applications in data analyses. In particular, asymmetric regression allows us to fit more realistic statistical models to skewed data and pairs well with Bayesian inference. We first describe asymmetric regression using the ALD and AND. Second, we show how the ALD and AND can be used for Bayesian quantile and expectile regression for continuous response data. Third, we consider an extension to generalize Bayesian asymmetric regression to survey data consisting of counts of objects. Fourth, we describe a regression model using the ALD, and show that it can be applied to add needed flexibility, resulting in better predictive models compared to Poisson or negative binomial regression. We demonstrate concepts by analyzing a data set consisting of counts of Henslow’s sparrows following prescribed fire and provide annotated computer code to facilitate implementation. Our results suggest Bayesian asymmetric regression is an essential component of a scientist’s statistical toolbox.     

Phylogeny, regression, and the allometry of physiological traits

Physiological and ecological allometries often pose linear regression problems characterized by (1) noncausal, phylogenetically autocorrelated independent (x) and dependent (y) variables (characters); (2) random variation in both variables; and (3) a focus on regression slopes (allometric exponents). Remedies for the phylogenetic autocorrelation of species values (phylogenetically independent contrasts) and variance structure of the data (reduced major axis [RMA] regression) have been developed, but most functional allometries are reported as ordinary least squares (OLS) regression without use of phylogenetically independent contrasts. We simulated Brownian diffusive evolution of functionally related characters and examined the importance of regression methodologies and phylogenetic contrasts in estimating regression slopes for phylogenetically constrained data. Simulations showed that both OLS and RMA regressions exhibit serious bias in estimated regression slopes under different circumstances but that a modified orthogonal (least squares variance-oriented residual [LSVOR]) regression was less biased than either OLS or RMA regressions. For strongly phylogenetically structured data, failure to use phylogenetic contrasts as regression data resulted in overestimation of the strength of the regression relationship and a significant increase in the variance of the slope estimate. Censoring of data sets by simulated extinction of taxa did not affect the importance of appropriate regression models or the use of phylogenetic contrasts.     

A new method for dealing with measurement error in explanatory variables of regression models

We introduce a new method, moment reconstruction, of correcting for measurement error in covariates in regression models. The central idea is similar to regression calibration in that the values of the covariates that are measured with error are replaced by "adjusted" values. In regression calibration the adjusted value is the expectation of the true value conditional on the measured value. In moment reconstruction the adjusted value is the variance-preserving empirical Bayes estimate of the true value conditional on the outcome variable. The adjusted values thereby have the same first two moments and the same covariance with the outcome variable as the unobserved "true" covariate values. We show that moment reconstruction is equivalent to regression calibration in the case of linear regression, but leads to different results for logistic regression. For case-control studies with logistic regression and covariates that are normally distributed within cases and controls, we show that the resulting estimates of the regression coefficients are consistent. In simulations we demonstrate that for logistic regression, moment reconstruction carries less bias than regression calibration, and for case-control studies is superior in mean-square error to the standard regression calibration approach. Finally, we give an example of the use of moment reconstruction in linear discriminant analysis and a nonstandard problem where we wish to adjust a classification tree for measurement error in the explanatory variables.     

Analytical methods matter too: Establishing a framework for estimating maximum metabolic rate for fishes

Advances in experimental design and equipment have simplified the collection of maximum metabolic rate (MMR) data for a more diverse array of water‐breathing animals. However, little attention has been given to the consequences of analytical choices in the estimation of MMR. Using different analytical methods can reduce the comparability of MMR estimates across species and studies and has consequences for the burgeoning number of macroecological meta‐analyses using metabolic rate data. Two key analytical choices that require standardization are the time interval, or regression window width, over which MMR is estimated, and the method used to locate that regression window within the raw oxygen depletion trace. Here, we consider the effect of both choices by estimating MMR for two shark and two salmonid species of different activity levels using multiple regression window widths and three analytical methods: rolling regression, sequential regression, and segmented regression. Shorter regression windows yielded higher metabolic rate estimates, with a risk that the shortest windows (<1‐min) reflect more system noise than MMR signal. Rolling regression was the best candidate model and produced the highest MMR estimates. Sequential regression models consistently produced lower relative estimates than rolling regression models, while the segmented regression model was unable to produce consistent MMR estimates across individuals. The time‐point of the MMR regression window along the oxygen consumption trace varied considerably across individuals but not across models. We show that choice of analytical method, in addition to more widely understood experimental choices, profoundly affect the resultant estimates of MMR. We recommend that researchers (1) employ a rolling regression model with a reliable regression window tailored to their experimental system and (2) explicitly report their analytical methods, including publishing raw data and code.     

Predictive Performance of the Methods of Restricted and Mixed Regression Estimators

This article considers the problem of simultaneous prediction of actual and average values of the study variable in a linear regression model when a set of linear restrictions binding the regression coefficients is available, and analyzes the performance properties of predictors arising from the methods of restricted regression and mixed regression besides least squares.     

Complementary Log–Log Regression for the Estimation of Covariate‐Adjusted Prevalence Ratios in the Analysis of Data from Cross‐Sectional Studies

We assessed complementary log–log (CLL) regression as an alternative statistical model for estimating multivariable‐adjusted prevalence ratios (PR) and their confidence intervals. Using the delta method, we derived an expression for approximating the variance of the PR estimated using CLL regression. Then, using simulated data, we examined the performance of CLL regression in terms of the accuracy of the PR estimates, the width of the confidence intervals, and the empirical coverage probability, and compared it with results obtained from log–binomial regression and stratified Mantel–Haenszel analysis. Within the range of values of our simulated data, CLL regression performed well, with only slight bias of point estimates of the PR and good confidence interval coverage. In addition, and importantly, the computational algorithm did not have the convergence problems occasionally exhibited by log–binomial regression. The technique is easy to implement in SAS (SAS Institute, Cary, NC), and it does not have the theoretical and practical issues associated with competing approaches. CLL regression is an alternative method of binomial regression that warrants further assessment.     

Non-linear heteroscedastic regression model for determination of methotrexate in human plasma by high-performance liquid chromatography

Generalized least squares regression with variance function estimation was used to derive the calibration function for measurement of methotrexate plasma concentration and its results were compared with weighted least squares regression by usual weight factors and also with that of ordinary least squares method. In the calibration curve range of 0.05 to 100 microM, both heteroscedasticity and non-linearity were present therefore ordinary least squares linear regression methods could result in large errors in the calculation of methotrexate concentration. Generalized least squares regression with variance function estimation worked better than both the weighted regression with the usual weight factors and ordinary least squares regression and gave better estimates for methotrexate concentration.     

Photoperiodic control of the termination of breeding and the induction ofmoult in House Sparrows Passer domesticus

Two photoperiodic mechanisms controlling gonadal regression in birds have been identified: absolute photorefractoriness, typical of species with short breeding seasons, where gonadal regression occurs spontaneously during long days, and relative photorefractoriness, where a decrease in daylength is required to induce regression. An experiment was designed to test whether these simply represent extremes of one underlying mechanism. Three groups of male House Sparrows Passer domesticus were transferred from a short photoperiod, 8 h of light: 16 h of darkness per day (8L:16D) to long photoperiods of either 18L:6D, 16L:8D or 13L:11D. Gonadal maturation rates were similar in all three groups; gonadal regression and moult began latest in the 13L:11D group. Four additional groups of sparrows were transferred from 8L:16D to 18L:6D and then transferred to either 13L: 11D or 16L:8D prior to, or shortly after, the onset of gonadal regression. The decrease in daylength prior to regression had no effect on the timing of regression but did advance the onset of moult. Decrease in daylength after the onset of regression increased the rate of regression and the rate of moult. Because a decrease in daylength did not affect the timing of regression, the data do not support the hypothesis that absolute and relative photorefractoriness represent extremes of a single underlying photoperiodic control mechanism. The adaptive significance of the effects of decreasing daylength on the rate of regression and moult is discussed.     

Climate variations and salmonellosis transmission in Adelaide,South Australia: a comparison between regression models

This is the first study to identify appropriate regression models for the association between climate variation and salmonellosis transmission. A comparison between different regression models was conducted using surveillance data in Adelaide, South Australia. By using notified salmonellosis cases and climatic variables from the Adelaide metropolitan area over the period 1990–2003, four regression methods were examined: standard Poisson regression, autoregressive adjusted Poisson regression, multiple linear regression, and a seasonal autoregressive integrated moving average (SARIMA) model. Notified salmonellosis cases in 2004 were used to test the forecasting ability of the four models. Parameter estimation, goodness-of-fit and forecasting ability of the four regression models were compared. Temperatures occurring 2 weeks prior to cases were positively associated with cases of salmonellosis. Rainfall was also inversely related to the number of cases. The comparison of the goodness-of-fit and forecasting ability suggest that the SARIMA model is better than the other three regression models. Temperature and rainfall may be used as climatic predictors of salmonellosis cases in regions with climatic characteristics similar to those of Adelaide. The SARIMA model could, thus, be adopted to quantify the relationship between climate variations and salmonellosis transmission.     

Estimation of Population Regression Coefficient in Successive Sampling

An attempt has been made to derive the theory of successive sampling for estimation of regression coefficient. The situations considered are estimation of regression coefficient for current occasion, estimation of change of regression coefficients over two occasions and estimation of average of regression coefficients over two occasions. The expressions of optimum estimators along with their variances have been worked out. On comparing their efficiencies empirically, it has been observed that similar to the estimation of mean, successive sampling can also be used with advantage for estimation of regression coefficient.     

Regression of papillomas induced by cottontail rabbit papillomavirus is associated with infiltration of CD8+ cells and persistence of viral DNA after regression.

Cottontail rabbit papillomavirus (CRPV) is a highly oncogenic papillomavirus and has been successfully used as a model to develop protective vaccines against papillomaviruses. Papillomas induced by the virus may spontaneously regress, suggesting that CRPV can also serve as a model to develop therapeutic vaccines. As a first step toward this goal, we have analyzed immunologic and viral aspects associated with papilloma regression and have identified several features unique to regression. Immunohistochemical staining of biopsies from growing and regressing papillomas and from sites after complete regression showed infiltration of CD8+ cells into the basal and suprabasal layers of the epidermis only during active regression. In situ hybridizations with mRNA-specific probes were strongly positive for E6 and E7 mRNAs during regression, but no late mRNA was present. Viral DNA was detected by in situ hybridization during regression but not after regression. However, analysis by PCR revealed persistence of viral DNA for several months at the majority of regression sites. The results suggest that stimulation of a strong CD8+ response to virus-infected cells is important for an effective therapeutic vaccine and that special attention should be given to the suppression of latent infection.     

Regression analysis when covariates are regression parameters of a random effects model for observed longitudinal measurements

We consider regression analysis when covariate variables are the underlying regression coefficients of another linear mixed model. A naive approach is to use each subject's repeated measurements, which are assumed to follow a linear mixed model, and obtain subject-specific estimated coefficients to replace the covariate variables. However, directly replacing the unobserved covariates in the primary regression by these estimated coefficients may result in a significantly biased estimator. The aforementioned problem can be evaluated as a generalization of the classical additive error model where repeated measures are considered as replicates. To correct for these biases, we investigate a pseudo-expected estimating equation (EEE) estimator, a regression calibration (RC) estimator, and a refined version of the RC estimator. For linear regression, the first two estimators are identical under certain conditions. However, when the primary regression model is a nonlinear model, the RC estimator is usually biased. We thus consider a refined regression calibration estimator whose performance is close to that of the pseudo-EEE estimator but does not require numerical integration. The RC estimator is also extended to the proportional hazards regression model. In addition to the distribution theory, we evaluate the methods through simulation studies. The methods are applied to analyze a real dataset from a child growth study.     

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.     

Host factors associated with the kinetics of Epstein-Barr virus DNA load in patients with primary Epstein-Barr virus infection

The aims of this study were to elucidate the kinetics of Epstein-Barr virus (EBV) DNA load in serially collected peripheral blood mononuclear cells of patients with primary EBV infection, and to determine the correlated host factors. Blood samples were collected from 24 patients with primary EBV infection. EBV DNA copy numbers were measured using real-time polymerase chain reaction. Based on the kinetics of EBV DNA load, the 24 patients were divided into two groups: rapid regression and slow regression. Eighteen of the 24 patients (75%) were included in the slow regression and 6 (25%) in the rapid regression group. No statistically significant differences were observed between the two groups in clinical features and laboratory findings. However, acute phase (3 to 10 days after the onset of the illness) serum samples from six children in the slow regression and four in the rapid regression group revealed significantly higher serum interleukin (IL)-1β (P= 0.018), IL-12 (P= 0.009), tumor necrosis factor-α (P= 0.019), interferon-inducible protein 10, and monokine induced by interferon γ concentrations in the rapid regression than the slow regression group. On the other hand, sera from six children in the slow regression and four in the rapid regression group in the convalescent phase (14 to 21 days after the onset of the illness) showed no statistically significant differences between the two groups in these biomarker concentrations. Based on this, it was concluded that the kinetics of EBV DNA load can be divided to two different patterns after primary EBV infection, and immune response might be associated with viral clearance.     

Use of nonlinear regression to analyze enzyme kinetic data: application to situations of substrate contamination and background subtraction

In a recent publication, A. Lundin, P. Arner, and J. Hellmér [Anal. Biochem. 177, 125-131 (1989)] describe a method whereby kinetic substrate assays can be performed when the assay mixture includes a significant contaminating levels of substrate. Their method requires various rearrangements of the data, and involves three separate linear regression calculations. We show how the same data may be analyzed directly, and far more simply, by nonlinear regression. Unlike the linear regression method, nonlinear regression allows direct calculation of the actual values for Km, Vmax, and the concentration of contaminating substrate (as well as estimates of their standard errors); the former method gives only apparent values. The nonlinear regression technique is also statistically a more valid means of analysis, as the rearrangements required to give linearized equations will considerably distort the error distribution and render simple unweighted linear regression inappropriate. The ease of incorporating extra parameters into standard equations when nonlinear regression is used is further illustrated by fitting enzyme reaction data which describe a first-order process when a significant nonspecific background is present. For this equation no simple rearranged linear plot is possible, but nonlinear regression is easily applied to determine the kinetic parameters.     

Regression trees for predicting mortality in patients with cardiovascular disease: What improvement is achieved by using ensemble‐based methods?

In biomedical research, the logistic regression model is the most commonly used method for predicting the probability of a binary outcome. While many clinical researchers have expressed an enthusiasm for regression trees, this method may have limited accuracy for predicting health outcomes. We aimed to evaluate the improvement that is achieved by using ensemble‐based methods, including bootstrap aggregation (bagging) of regression trees, random forests, and boosted regression trees. We analyzed 30‐day mortality in two large cohorts of patients hospitalized with either acute myocardial infarction (N = 16,230) or congestive heart failure (N = 15,848) in two distinct eras (1999–2001 and 2004–2005). We found that both the in‐sample and out‐of‐sample prediction of ensemble methods offered substantial improvement in predicting cardiovascular mortality compared to conventional regression trees. However, conventional logistic regression models that incorporated restricted cubic smoothing splines had even better performance. We conclude that ensemble methods from the data mining and machine learning literature increase the predictive performance of regression trees, but may not lead to clear advantages over conventional logistic regression models for predicting short‐term mortality in population‐based samples of subjects with cardiovascular disease.     

Quantile regression estimates of animal population trends

Ecologists often estimate population trends of animals in time series of counts using linear regression to estimate parameters in a linear transformation of multiplicative growth models, where logarithms of rates of change in counts in time intervals are used as response variables. We present quantile regression estimates for the median (0.50) and interquartile (0.25, 0.75) relationships as an alternative to mean regression estimates for common density-dependent and density-independent population growth models. We demonstrate that the quantile regression estimates are more robust to outliers and require fewer distributional assumptions than conventional mean regression estimates and can provide information on heterogeneous rates of change ignored by mean regression. We provide quantile regression trend estimates for 2 populations of greater sage-grouse (Centrocercus urophasianus) in Wyoming, USA, and for the Crawford population of Gunnison sage-grouse (Centrocercus minimus) in southwestern Colorado, USA. Our selected Gompertz models of density dependence for both populations of greater sage-grouse had smaller negative estimates of density-dependence terms and less variation in corresponding predicted growth rates (λ) for quantile than mean regression models. In contrast, our selected Gompertz models of density dependence with piecewise linear effects of years for the Crawford population of Gunnison sage-grouse had predicted changes in λ across years from quantile regressions that varied more than those from mean regression because of heterogeneity in estimated λs that were both less and greater than mean estimates. Our results add to literature establishing that quantile regression provides better behaved estimates than mean regression when there are outlying growth rates, including those induced by adjustments for zeros in the time series of counts. The 0.25 and 0.75 quantiles bracketing the median provide robust estimates of population changes (λ) for the central 50% of time series data and provide a 50% prediction interval for a single new prediction without making parametric distributional assumptions or assuming homogeneous λs. Compared to mean estimates, our quantile regression trend estimates for greater sage-grouse indicated less variation in density-dependent λs by minimizing sensitivity to outlying values, and for Gunnison sage-grouse indicated greater variation in density-dependent λs associated with heterogeneity among quantiles.