The log multinomial regression model for nominal outcomes with more than two attributes

An estimate of the risk or prevalence ratio, adjusted for confounders, can be obtained from a log binomial model (binomial errors, log link) fitted to binary outcome data. We propose a modification of the log binomial model to obtain relative risk estimates for nominal outcomes with more than two attributes (the "log multinomial model"). Extensive data simulations were undertaken to compare the performance of the log multinomial model with that of an expanded data multinomial logistic regression method based on the approach proposed by Schouten et al. (1993) for binary data, and with that of separate fits of a Poisson regression model based on the approach proposed by Zou (2004) and Carter, Lipsitz and Tilley (2005) for binary data. Log multinomial regression resulted in "inadmissable" solutions (out-of-bounds probabilities) exceeding 50% in some data settings. Coefficient estimates by the alternative methods produced out-of-bounds probabilities for the log multinomial model in up to 27% of samples to which a log multinomial model had been successfully fitted. The log multinomial coefficient estimates generally had lesser relative bias and mean squared error than the alternative methods. The practical utility of the log multinomial regression model was demonstrated with a real data example. The log multinomial model offers a practical solution to the problem of obtaining adjusted estimates of the risk ratio in the multinomial setting, but must be used with some care and attention to detail.     

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.     

Assessing the goodness of fit of logistic regression models in large samples: A modification of the Hosmer-Lemeshow test

Evaluating the goodness of fit of logistic regression models is crucial to ensure the accuracy of the estimated probabilities. Unfortunately, such evaluation is problematic in large samples. Because the power of traditional goodness of fit tests increases with the sample size, practically irrelevant discrepancies between estimated and true probabilities are increasingly likely to cause the rejection of the hypothesis of perfect fit in larger and larger samples. This phenomenon has been widely documented for popular goodness of fit tests, such as the Hosmer-Lemeshow test. To address this limitation, we propose a modification of the Hosmer-Lemeshow approach. By standardizing the noncentrality parameter that characterizes the alternative distribution of the Hosmer-Lemeshow statistic, we introduce a parameter that measures the goodness of fit of a model but does not depend on the sample size. We provide the methodology to estimate this parameter and construct confidence intervals for it. Finally, we propose a formal statistical test to rigorously assess whether the fit of a model, albeit not perfect, is acceptable for practical purposes. The proposed method is compared in a simulation study with a competing modification of the Hosmer-Lemeshow test, based on repeated subsampling. We provide a step-by-step illustration of our method using a model for postneonatal mortality developed in a large cohort of more than 300 000 observations.     

On a likelihood-based goodness-of-fit test of the beta-binomial model

When faced with proportion data that exhibit extra-binomial variation, data analysts often consider the beta-binomial distribution as an alternative model to the more common binomial distribution. A typical example occurs in toxicological experiments with laboratory animals, where binary observations on fetuses within a litter are often correlated with each other. In such instances, it may be of interest to test for the goodness of fit of the beta-binomial model; this effort is complicated, however, when there is large variability among the litter sizes. We investigate a recent goodness-of-fit test proposed by Brooks et al. (1997, Biometrics 53, 1097-1115) but find that it lacks the ability to distinguish between the beta-binomial model and some severely non-beta-binomial models. Other tests and models developed in their article are quite useful and interesting but are not examined herein.     

The use of a correlated binomial model for the analysis of certain toxicological experiments

In certain toxicological experiments with laboratory animals, the outcome of interest is the occurrence of dead or malformed fetuses in a litter. Previous investigations have shown that the simple one-parameter binomial and Poisson models generally provide poor fits to this type of binary data. In this paper, a type of correlated binomial model is proposed for use in this situation. First, the model is described in detail and is compared to a beta-binomial model proposed by Williams (1975). These two-parameter models are then contrasted for goodness of fit to some real-life data. Finally, numerical examples are given in which likelihood ratio tests based on these models are employed to assess the significance of treatment-control differences.     

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).     

Regression analysis of doubly censored failure time data using the additive hazards model

Doubly censored failure time data arise when the survival time of interest is the elapsed time between two related events and observations on occurrences of both events could be censored. Regression analysis of doubly censored data has recently attracted considerable attention and for this a few methods have been proposed (Kim et al., 1993, Biometrics 49, 13-22; Sun et al., 1999, Biometrics 55, 909-914; Pan, 2001, Biometrics 57, 1245-1250). However, all of the methods are based on the proportional hazards model and it is well known that the proportional hazards model may not fit failure time data well sometimes. This article investigates regression analysis of such data using the additive hazards model and an estimating equation approach is proposed for inference about regression parameters of interest. The proposed method can be easily implemented and the properties of the proposed estimates of regression parameters are established. The method is applied to a set of doubly censored data from an AIDS cohort study.     

Dealing with the Phenomenon of Quasi-complete Separation and a Goodness of Fit Test in Logistic Regression Models in the Case of Long Data Sets

The phenomenon of quasi-complete separation that appears in the identification of the neuromuscular system called muscle spindle by a logistic regression model is considered. The system responds when it is affected by a number of stimuli. Both the response and the stimuli are very long binary sequences of events. In the logistic model, three functions are of special interest: the threshold, the recovery and the summation functions. The maximum likelihood estimates are obtained efficiently and very fast by using the penalized likelihood function. A validity test for the fitted model based on the randomized quantile residuals is proposed. The validity test is transformed to a goodness of fit test and the use of Q–Q plot is also discussed.

     

Spatial pattern of the Río Cuarto corn disease vector, Delphacodes kuscheli Fennah (Hom., Delphacidae), in oat fields in Argentina and design of sampling plans

The spatial pattern of the Río Cuarto Corn Disease vector, Delphacodes kuscheli (Hom., Delphacidae), was analysed in oat fields within the endemic area of the disease, during the growing seasons 1993 and 1994. The spatial pattern was analysed by fitting the probabilistic models Poisson and negative binomial and estimation of single-date and overall aggregation indices. The population of the different stage classes, sex, and wing forms showed a significant trend to aggregation as the negative binomial model fitted the observed frequency distributions in more than 78% of the cases (sampling dates) while the Poisson model fitted well in only 28% of cases or less. Single-date aggregation index, C _A, ranged from 0.3 to 0.84. Overall (whole season) aggregation index, C _A*, estimated through the Bliss and Owen's regression method, ranged from 0.18 (female adults) to 1.08 (nymphs I–II), indicating a moderate degree of aggregation compared with other planthopper species. There were no significant relationships between aggregation and population density. The minimum number of sampling units and critical lines for sequential sampling plans were calculated based on the estimation of C _A* for the precision levels ( D ) 0.1, 0.2 and 0.3. Even low degrees of aggregation, like that of adults, demand much more sampling effort than randomly distributed populations, particularly at high densities. General implications and limitations of the proposed sampling plans for monitoring the vector population abundance are discussed.     

Distribution of Polymorphus minutus among its intermediate hosts

Hirsch R. P. 1979. Distribution of Polymorphus minutus among its intermediate hosts. International journal for Parasitology10: 243–248. In 1971, Crofton investigated patterns of distribution of Polymorphus minutus in the intermediate host, Gammarus pulex. Among his conclusions were: (1) P. minutus populations occur in patterns similar to negative binomial distributions, and (2) parasite-induced host mortality results in patterns similar to truncated (high end) negative binomial distributions. Those conclusions, however, were not tested by statistical analyses. To test Crofton's observations, Chi-square goodness of fit tests were applied to data used by Crofton and an additional two stations sampled by Hynes & Nicholas in 1963. Analyses were expanded to include five theoretical distributions, four patterns of host mortality and various rates of host mortality. Truncated forms of negative binomial, positive binomial and Poisson distributions were also investigated where nontruncated distributions failed to fit observed distributions. It was found that negative binomial distributions most frequently describe patterns of P. minutus distribution with the exception of one population described by Poisson and another by positive binomial distributions. Crofton's assumption that truncated distributions result from parasite-induced host mortality seems unlikely in light of those analyses.     

Analysis of intercellular distributions of chromatid aberrations.

59 intercellular distributions of chemically induced and spontaneous chromatid aberrations were analyzed for goodness of fit in respect of the Poisson (PD), the geometrical (GD), and the negative binomial distributions (NBD). The data are excellently described by the NBD. This distribution can be deduced from a model out of the queueing theory and is based on the hypothesis of restitution. Estimators are obtained for the ratio of restitution and induction processes involved in the origin of chromosomal aberrations.     

Analysis of multi-type recurrent events in longitudinal studies; application to a skin cancer prevention trial

We consider the statistical modeling and analysis of replicated multi-type point process data with covariates. Such data arise when heterogeneous subjects experience repeated events or failures which may be of several distinct types. The underlying processes are modeled as nonhomogeneous mixed Poisson processes with random (subject) and fixed (covariate) effects. The method of maximum likelihood is used to obtain estimates and standard errors of the failure rate parameters and regression coefficients. Score tests and likelihood ratio statistics are used for covariate selection. A graphical test of goodness of fit of the selected model is based on generalized residuals. Measures for determining the influence of an individual observation on the estimated regression coefficients and on the score test statistic are developed. An application is described to a large ongoing randomized controlled clinical trial for the efficacy of nutritional supplements of selenium for the prevention of two types of skin cancer.     

Regression models for overdispersed jejunal surviving crypts data

The dose-response model concerns to establish a relationship between a dose and the magnitude of the response produced by the dose. A common complication in the dose-response model for jejunal crypts cell surviving data is overdispersion, where the observed variation exceeds that predicted from the binomial distribution. In this study, two different methods for analyzing jejunal crypts cell survival after regimens of several fractions are contrasted and compared. One method is the logistic regression approach, where the numbers of surviving crypts are predicted by the logistic function of a single dose of radiation. The other one is the transform-both-sides approach, where the arcsine transformation family is applied based on the first-order variance-stabilizing transformation. This family includes the square root, arcsine, and hyperbolic arcsine transformations, which have been used for Poisson, binomial, and negative binomial count data, as special cases. These approaches are applied to a data set from radiobiology. Simulation study indicates that the arcsine transformation family is more efficient than the logistic regression when there exists moderate overdispersion.     

Neutrality tests of highly polymorphic restriction-fragment-length polymorphisms.

The allele frequency data of Baird et al. were tested using Ewens-Watterson sampling theory for goodness of fit to the infinite-alleles model of neutral evolution. Although probes of both the HRAS-1 and D14S1 loci identify highly diverse restriction-fragment-length polymorphisms, the observed values of gene identity (F) and the common allele frequency (C) are not significantly different from the neutral expectation. Allele frequency distributions show a tendency toward a deficit in diversity for HRAS-1 and a slight excess diversity for D14S1. The direction of these departures is consistent with potential selective effects of the Harvey-ras oncogene and hitchhiking of the D14S1 locus to closely linked immunoglobulin genes. Direct chi 2-tests of goodness of fit of the observed and expected allele frequency distributions reveal significant departures in the caucasoid and Hispanic HRAS-1 distributions but not in any of the other tests.     

A simple method to fit geometric series and broken stick models in community ecology and island biogeography

Species abundance distributions are widely used in explaining natural communities, their natural evolution and the impacts of environmental disturbance. A commonly used approach is that of rank-abundance distributions. Favored, biologically founded models are the geometric series (GS) and the broken stick (BS) model. Comparing observed abundance distributions with those predicted by models is an extremely time-consuming task. Also, using goodness-of-fit tests for frequency distributions (like Chi-square or Kolmogorov–Smirnov tests) to compare observed with expected frequencies is problematic because the best way to calculate expected frequencies may be controversial. More important, the Chi-square test may prove if an observed distribution statistically differs from a model, but does not allow the investigator to choose among competing models from which the observed distribution does not differ. Both models can be easily tested by regression analysis. In GS, if a log scale is used for abundance, the species exactly fall along a straight line. The BS distribution shows up as nearly linear when a log scale is used for the rank axis. Regression analysis is proposed here as a simpler and more efficient method to fit the GS and BS models. Also, regression analysis (1) does not suffer from assumptions related to Chi-square tests; (2) obviates the need to establish expected frequencies, and (3) offers the possibility to choose the best fit among competing models. A possible extension of abundance-rank analysis to species richness on islands is also proposed as a method to discriminate between relict and equilibrial models. Examples of application to field data are also presented.     

Promotion time models with time-changing exposure and heterogeneity: application to infectious diseases

Promotion time models have been recently adapted to the context of infectious diseases to take into account discrete and multiple exposures. However, Poisson distribution of the number of pathogens transmitted at each exposure was a very strong assumption and did not allow for inter-individual heterogeneity. Bernoulli, the negative binomial, and the compound Poisson distributions were proposed as alternatives to Poisson distribution for the promotion time model with time-changing exposure. All were derived within the frailty model framework. All these distributions have a point mass at zero to take into account non-infected people. Bernoulli distribution, the two-component cure rate model, was extended to multiple exposures. Contrary to the negative binomial and the compound Poisson distributions, Bernoulli distribution did not enable to connect the number of pathogens transmitted to the delay between transmission and infection detection. Moreover, the two former distributions enable to account for inter-individual heterogeneity. The delay to surgical site infection was an example of single exposure. The probability of infection was very low; thus, estimation of the effect of selected risk factors on that probability obtained with Bernoulli and Poisson distributions were very close. The delay to nosocomial urinary tract infection was a multiple exposure example. The probabilities of pathogen transmission during catheter placement and catheter presence were estimated. Inter-individual heterogeneity was very high, and the fit was better with the compound Poisson and the negative binomial distributions. The proposed models proved to be also mechanistic. The negative binomial and the compound Poisson distributions were useful alternatives to account for inter-individual heterogeneity.     

On Estimating the Relationship between Longitudinal Measurements and Time‐to‐Event Data Using a Simple Two‐Stage Procedure

Summary Ye, Lin, and Taylor (2008, Biometrics 64 , 1238–1246) proposed a joint model for longitudinal measurements and time‐to‐event data in which the longitudinal measurements are modeled with a semiparametric mixed model to allow for the complex patterns in longitudinal biomarker data. They proposed a two‐stage regression calibration approach that is simpler to implement than a joint modeling approach. In the first stage of their approach, the mixed model is fit without regard to the time‐to‐event data. In the second stage, the posterior expectation of an individual's random effects from the mixed‐model are included as covariates in a Cox model. Although Ye et al. (2008) acknowledged that their regression calibration approach may cause a bias due to the problem of informative dropout and measurement error, they argued that the bias is small relative to alternative methods. In this article, we show that this bias may be substantial. We show how to alleviate much of this bias with an alternative regression calibration approach that can be applied for both discrete and continuous time‐to‐event data. Through simulations, the proposed approach is shown to have substantially less bias than the regression calibration approach proposed by Ye et al. (2008) . In agreement with the methodology proposed by Ye et al. (2008) , an advantage of our proposed approach over joint modeling is that it can be implemented with standard statistical software and does not require complex estimation techniques.     

Goodness-of-Fit Testing for the Logistic Regression Model when the Estimated Probabilities are Small

The distribution of the Hosmer-Lemeshow chi-square type goodness-of-fit tests (?_g, ?_g) for the logistic regression model are examined via simulations designed to examine their behavior when most of the estimated probabilities are small or are expected to fall in a few deciles. The results of the simulations show statistic ?_g should be used when the two outcome groups (y = 0, 1) are not well separated, Δ≤2, where Δ² is the Mahalanobis distance. Statistic ?_g should be used when Δ ≥ 8. Either statistic may be used when 2 ≦ Δ ≦ 8. All tests should be used with caution when the proportion in the sample with y = 1 is less than 0.1.     

A quantile count model of water depth constraints on Cape Sable seaside sparrows

1. A quantile regression model for counts of breeding Cape Sable seaside sparrows Ammodramus maritimus mirabilis (L.) as a function of water depth and previous year abundance was developed based on extensive surveys, 1992-2005, in the Florida Everglades. The quantile count model extends linear quantile regression methods to discrete response variables, providing a flexible alternative to discrete parametric distributional models, e.g. Poisson, negative binomial and their zero-inflated counterparts. 2. Estimates from our multiplicative model demonstrated that negative effects of increasing water depth in breeding habitat on sparrow numbers were dependent on recent occupation history. Upper 10th percentiles of counts (one to three sparrows) decreased with increasing water depth from 0 to 30 cm when sites were not occupied in previous years. However, upper 40th percentiles of counts (one to six sparrows) decreased with increasing water depth for sites occupied in previous years. 3. Greatest decreases (-50% to -83%) in upper quantiles of sparrow counts occurred as water depths increased from 0 to 15 cm when previous year counts were 1, but a small proportion of sites (5-10%) held at least one sparrow even as water depths increased to 20 or 30 cm. 4. A zero-inflated Poisson regression model provided estimates of conditional means that also decreased with increasing water depth but rates of change were lower and decreased with increasing previous year counts compared to the quantile count model. Quantiles computed for the zero-inflated Poisson model enhanced interpretation of this model but had greater lack-of-fit for water depths > 0 cm and previous year counts 1, conditions where the negative effect of water depths were readily apparent and fitted better with the quantile count model.     

A general framework for constraint approaches to adjusted risk differences

The risk difference is an intelligible measure for comparing disease incidence in two exposure or treatment groups. Despite its convenience in interpretation, it is less prevalent in epidemiological and clinical areas where regression models are required in order to adjust for confounding. One major barrier to its popularity is that standard linear binomial or Poisson regression models can provide estimated probabilities out of the range of (0,1), resulting in possible convergence issues. For estimating adjusted risk differences, we propose a general framework covering various constraint approaches based on binomial and Poisson regression models. The proposed methods span the areas of ordinary least squares, maximum likelihood estimation, and Bayesian inference. Compared to existing approaches, our methods prevent estimates and confidence intervals of predicted probabilities from falling out of the valid range. Through extensive simulation studies, we demonstrate that the proposed methods solve the issue of having estimates or confidence limits of predicted probabilities out of (0,1), while offering performance comparable to its alternative in terms of the bias, variability, and coverage rates in point and interval estimation of the risk difference. An application study is performed using data from the Prospective Registry Evaluating Myocardial Infarction: Event and Recovery (PREMIER) study.