the analysis of rates using poisson regression models models are considered in which the underlying rate at which events occur can be represented by a regression function that describes the relation between the predictor variables and the unknown parameters estimates of the parameters can be obtained by means of iteratively reweighted least squares irls when the events of interest follow the poisson distribution the irls algorithm is equivalent to using the method of scoring to obtain maximum likelihood ml estimates the general poisson regression models include log linear quasilinear and intrinsically nonlinear models the approach considered enables one to concentrate on describing the relation between the dependent variable and the predictor variables through the regression model standard statistical packages that support irls can then be used to obtain ml estimates their asymptotic covariance matrix and diagnostic measures that can be used to aid the analyst in detecting outlying responses and extreme points in the model space applications of these methods to epidemiologic follow up studies with the data organized into a life table type of format are discussed the method is illustrated by using a nonlinear model derived from the multistage theory of carcinogenesis to analyze lung cancer death rates among british physicians who were regular cigarette smokers

congruence subgroups and generalized frobenius schur indicators we introduce generalized frobenius schur indicators for pivotal categories in a spherical fusion category mathcal c an equivariant indicator of an object in mathcal c is defined as a functional on the grothendieck algebra of the quantum double z mathcal c via generalized frobenius schur indicators the set of all equivariant indicators admits a natural action of the modular group using the properties of equivariant indicators we prove a congruence subgroup theorem for modular categories as a consequence all modular representations of a modular category have finite images and they satisfy a conjecture of eholzer in addition we obtain two formulae for the generalized indicators one of them a generalization of bantay s second indicator formula for a rational conformal field theory this formula implies a conjecture of pradisi sagnotti stanev as well as a conjecture of borisov halpern schweigert

a two state markov mixture model for a time series of epileptic seizure counts this paper discusses a model for a time series of epileptic seizure counts in which the mean of a poisson distribution changes according to an underlying two state markov chain the em algorithm dempster laird and rubin 1977 journal of the royal statistical society series b 39 1 38 is used to compute maximum likelihood estimators for the parameters of this two state mixture model and extensions are made allowing for nonstationarity the model is illustrated using daily seizure counts for patients with intractable epilepsy and results are compared with a simple poisson distribution and poisson regressions some simulation results are also presented to demonstrate the feasibility of this model

adjusted score tests of homogeneity for poisson processes abstract tests of homogeneity are being increasingly used for the analysis of event time data but relatively little attention has been paid to their distributional properties in settings with small to moderate sample sizes here we consider tests of homogeneity for recurrent event data in which the null model is a poisson process and the alternative is a mixed poisson process we examine score and adjusted score statistics in the context of parametric and semiparametric regression models where the adjustments correct for the bias induced by substituting the maximum likelihood estimates of the parameters into the test statistic we report the results of simulation studies suggesting that the adjusted score statistics are superior in terms of size and power we also find that the adjusted score test in the semiparametric regression model does not perform particularly well in small samples but the adjusted score test based on a piecewise exponential model has satisfactory performance the latter thus prov

variance component testing in generalised linear models with random effects summary there is considerable interest in testing for overdispersion correlation and heterogeneity across groups in biomedical studies in this paper we cast the problem in the framework of generalised linear models with random effects we propose a global score test for the null hypothesis that all the variance components are zero this test is a locally asymptotically most stringent test and is robust in the special sense that the test does not require specifying the joint distribution of the random effects we also propose individual score tests and their approximations for testing the variance components separately both tests can be easily implemented using existing statistical software we illustrate these tests with an application to the study of heterogeneity of mating success across males and females in an experiment on salamander matings and evaluate their performance through simulation

clustered mixed nonhomogeneous poisson process spline models for the analysis of recurrent event panel data summary a flexible semiparametric model for analyzing longitudinal panel count data arising from mixtures is presented panel count data refers here to count data on recurrent events collected as the number of events that have occurred within specific follow up periods the model assumes that the counts for each subject are generated by mixtures of nonhomogeneous poisson processes with smooth intensity functions modeled with penalized splines time dependent covariate effects are also incorporated into the process intensity using splines discrete mixtures of these nonhomogeneous poisson process spline models extract functional information from underlying clusters representing hidden subpopulations the motivating application is an experiment to test the effectiveness of pheromones in disrupting the mating pattern of the cherry bark tortrix moth mature moths arise from hidden but distinct subpopulations and monitoring the subpopulation responses was of interest within cluster random effects are used to account for correlation structures and heterogeneity common to this type of data an estimating equation approach to inference requiring only low moment assumptions is developed and the finite sample properties of the proposed estimating functions are investigated empirically by simulation

unbiased testing in exponential family regression let x ij z i i 1 2 k j 1 2 n i be independent observations such that z i is a fixed r x 1 vector r can be 0 no z s observed or 1 2 k 1 and x ij is distributed according to a one parameter exponential family which is log concave with natural parameter i we test the hypothesis that z where 1 k z is the matrix whose ith row is z i and 1 r is a vector of parameters we focus on r 2 and z i 1 z i i 1 2 k z i z i 1 the null hypothesis on hand is thus of the form i 1 2 z i in such a case the model under the null hypothesis becomes logistic regression in the binomial case poisson regression in the poisson case and linear regression in the normal case we consider mostly the one sided alternative that the second order differences of the natural parameters are nonnegative such testing problems test goodness of fit vs alternatives in which the natural parameters behave in a convex way we find classes of tests that are unbiased and that lie in a complete class we also note that every admissible test of constant size is unbiased in some discrete situations we find the minimal complete class of unbiased admissible tests generalizations and examples are given

bootstrap tests for overdispersion in a zero inflated poisson regression model summaryridout hinde and demetrio 2001 biometrics57 219 223 derived a score test for testing a zero inflated poisson zip regression model against zero inflated negative binomial zinb alternatives they mentioned that the score test using the normal approximation might underestimate the nominal significance level possibly for small sample cases to remedy this problem a parametric bootstrap method is proposed it is shown that the bootstrap method keeps the significance level close to the nominal one and has greater power uniformly than the existing normal approximation for testing the hypothesis

information ratio test for model misspecification in quasi likelihood inference in this article we focus on the circumstances in quasi likelihood inference that the estimation accuracy of mean structure parameters is guaranteed by correct specification of the first moment but the estimation efficiency could be diminished due to misspecification of the second moment we propose an information ratio ir statistic to test for model misspecification of the variance covariance structure through a comparison between two forms of information matrix the negative sensitivity matrix and the variability matrix we establish asymptotic distributions of the proposed ir test statistics we also suggest an approximation to the asymptotic distribution of the ir statistic via a perturbation resampling method moreover we propose a selection criterion based on the ir test to select the best fitting variance covariance structure from a class of candidates through simulation studies it is shown that the ir statistic provides a powerful statistical tool to detect different scenarios of misspecific

on the single operation worst case time complexity of the disjoint set union problem we give an algorithm for the disjoint set union problem within the class of algorithms defined by tarjan which has o log n log log n single operation time complexity in the worst case also we define a class of algorithms for the disjoint set union problem which includes the class of algorithms defined by tarjan we prove that any algorithm from this class has at least omega log n log log n single operation time complexity in the worst case

predictive model assessment for count data we discuss tools for the evaluation of probabilistic forecasts and the critique of statistical models for count data our proposals include a nonrandomized version of the probability integral transform marginal calibration diagrams and proper scoring rules such as the predictive deviance in case studies we critique count regression models for patent data and assess the predictive performance of bayesian age period cohort models for larynx cancer counts in germany the toolbox applies in bayesian or classical and parametric or nonparametric settings and to any type of ordered discrete outcomes

analysis of two way layout of count data involving multiple counts in each cell abstract multiple counts may occur in each cell of an a b two way layout balanced or unbalanced of two fixed factors a and b standard log linear model analysis based on a poisson distribution assumption of the cell counts is not applicable here because of the unbalanced nature of the table or because the poisson distribution assumption is not valid we develop c tests for interaction and main effects assuming data to be poisson distributed and also assuming that data within the cells have extra over under dispersion beyond that explained by a poisson distribution for this we consider an extended negative binominal distribution and a semiparametric model using the quasi likelihood we show that in all situations the c tests for interaction are of very simple forms for c tests for the main effect in presence of no interaction such simplification is possible only under certain conditions a score test for detecting extra dispersion in presence of interaction is also obtained and is of sim

a score test for testing a zero inflated poisson regression model against zero inflated negative binomial alternatives count data often show a higher incidence of zero counts than would be expected if the data were poisson distributed zero inflated poisson regression models are a useful class of models for such data but parameter estimates may be seriously biased if the nonzero counts are overdispersed in relation to the poisson distribution we therefore provide a score test for testing zero inflated poisson regression models against zero inflated negative binomial alternatives

residuals in generalized linear models abstract generalized linear models are regression type models for data not normally distributed appropriately fitted by maximum likelihood rather than least squares typical examples are models for binomial or poisson data with a linear regression model for a given ordinarily nonlinear function of the expected values of the observations use of such models has become very common in recent years and there is a clear need to study the issue of appropriate residuals to be used for diagnostic purposes several definitions of residuals are possible for generalized linear models the statistical package glim baker and nelder 1978 routinely prints out residuals where v is the function relating the variance to the mean of y and is the maximum likelihood estimate of the ith mean as fitted to the regression model these residuals are the signed square roots of the contributions to the pearson goodness of fit statistic another choice of residual is the signed square root of the contribution to the devia

interval estimation of the over dispersion parameter in the analysis of one way layout of count data the over dispersion parameter is an important and versatile measure in the analysis of one way layout of count data in biological studies for example it is commonly used as an inverse measure of aggregation in biological count data its estimation from finite data sets is a recognized challenge many simulation studies have examined the bias and efficiency of different estimators of the over dispersion parameter for finite data sets see for example clark and perry biometrics 1989 45 309 316 and piegorsch biometrics 1990 46 863 867 but little attention has been paid to the accuracy of the confidence intervals cis of it in this paper we first derive asymptotic procedures for the construction of confidence limits for the over dispersion parameter using four estimators that are specified by only the first two moments of the counts we also obtain closed form asymptotic variance formulae for these four estimators in addition we consider the asymptotic ci based on the maximum likelihood ml estimator using the negative binomial model it appears from the simulation results that the asymptotic cis based on these five estimators have coverage below the nominal coverage probability to remedy this we also study the properties of the asymptotic cis based on the restricted estimates of ml extended quasi likelihood and double extended quasi likelihood by eliminating the nuisance parameter effect using their adjusted profile likelihood and quasi likelihoods it is shown that these cis outperform the competitors by providing coverage levels close to nominal over a wide range of parameter combinations two examples to biological count data are presented copyright 2010 john wiley sons ltd

testing goodness of fit for the poisson assumption when observations are not identically distributed abstract tests for detecting negative binomial departures from a poisson model are studied for the one way layout and regression through the origin cases approximations to the null and alternative distributions of these test statistics are presented locally optimal tests and tests suggested in the literature are compared in terms of asymptotic relative efficiency small sample comparisons are included

a specification test for univariate and multivariate proportional hazards models this paper applies white s 1982 econometrica 50 1 25 information matrix im test for correct model specification to proportional hazards models of univariate and multivariate censored survival data several alternative estimators of the test statistic are presented and their size performance examined white also suggested an estimator of the parameter covariance matrix that was robust to certain forms of model misspecification this has been subsequently proposed by others e g royall 1986 international statistical review 54 221 226 and applied by huster brookmeyer and self 1989 biometrics 45 145 156 as part of an independence working model iwm approach to multivariate censored survival data we illustrate how the im test can be used for both univariate data and as part of the iwm approach to multivariate data

econometric models for count data with an application to the patents r d relationship this paper focuses on developing and adapting statistical models of counts nonnegative integers in the context of panel data and using them to analyze the relationship between patents and r d expenditures since a variety of other economic data come in the form of repeated counts of some individual actions or events the methodology should have wide applications the statistical models we develop are applications and generalizations of the poisson distribution two important issues are i given the panel nature of our data how can we allow for separate persistent individual fixed or random effects ii how does one introduce the equivalent of disturbances in the equation into the analysis of poisson and other discrete probability functions the first problem is solved by conditioning on the total sum of outcomes over the observed years while the second problem is solved by introducing an additional source of randomness allowing the poisson parameter to be itself randomly distributed and compounding the two distributions lastly we develop a test statistic for the presence of serial correlation when fixed effects estimators are used in nonlinear conditional models

testing for overdispersion in poisson and binomial regression models abstract in this article a method for obtaining tests for overdispersion with respect to a natural exponential family is derived the tests are designed to be powerful against arbitrary alternative mixture models where only the first two moments of the mixed distribution are specified various tests for extra poisson and extra binomial variation are obtained as special cases the use of a particular test may be motivated by a consideration of the mechanism through which the overdispersion may arise the common occurrence of extra poisson and extra binomial variation has been noted by several authors however the poisson and binomial models remain valid in many instances and because of their simplicity and appeal it is of real interest to ascertain when they apply this paper develops a unifying theory for testing for overdispersion and generalizes tests previously derived including those by fisher 1950 collings and margolin 1985 and prentice 1986 it also shows the pearson statistic to be a sc

a note on permutation tests for variance components in multilevel generalized linear mixed models in many applications of generalized linear mixed models to multilevel data it is of interest to test whether a random effects variance component is zero it is well known that the usual asymptotic chi square distribution of the likelihood ratio and score statistics under the null does not necessarily hold in this note we propose a permutation test based on randomly permuting the indices associated with a given level of the model that has the correct type i error rate under the null results from a simulation study suggest that it is more powerful than tests based on mixtures of chi square distributions the proposed test is illustrated using data on the familial aggregation of sleep disturbance

level and index in the modular group it is shown that the index of a congruence subgroup of the modular group cannot be less than the level of the subgroup this allows a number of existence theorems about non congruence subgroups the level of a subgroup of the modular group can be defined in terms of the action on q we define a similar action to get information on congruence subgroups in fact we get a more powerful result but this appears to be the most useful version

congruence subgroups from representations of the three strand braid group abstract ng and schauenburg proved that the kernel of a 2 1 dimensional topological quantum field theory representation of sl 2 z is a congruence subgroup motivated by their result we explore when the kernel of an irreducible representation of the braid group b 3 with finite image enjoys a congruence subgroup property in particular we show that in dimensions two and three when the projective order of the image of the braid generator 1 is between 2 and 5 the kernel projects onto a congruence subgroup of psl 2 z and compute its level however we prove that for three dimensional representations the projective order is not enough to decide the congruence property for each integer of the form 2 l 6 with l odd we construct a pair of non congruence subgroups associated with three dimensional representations having finite image and 1 mapping to a matrix with projective order 2l our technique uses classification results of low dimensional braid group representations and the fricke wohlfahrt theorem in number theory

tests for detecting overdispersion in models with measurement error in covariates measurement error in covariates can affect the accuracy in count data modeling and analysis in overdispersion identification the true mean variance relationship can be obscured under the influence of measurement error in covariates in this paper we propose three tests for detecting overdispersion when covariates are measured with error a modified score test and two score tests based on the proposed approximate likelihood and quasi likelihood respectively the proposed approximate likelihood is derived under the classical measurement error model and the resulting approximate maximum likelihood estimator is shown to have superior efficiency simulation results also show that the score test based on approximate likelihood outperforms the test based on quasi likelihood and other alternatives in terms of empirical power by analyzing a real dataset containing the health related quality of life measurements of a particular group of patients we demonstrate the importance of the proposed methods by showing that the analyses with and without measurement error correction yield significantly different results copyright 2015 john wiley sons ltd

a variance components approach to categorical data models with heterogeneous cell populations analysis of spatial gradients in lung cancer mortality rates in north carolina counties a mixed categorical continuous variable model is proposed for the analysis of mortality rates this model differs from other available models such as weighted least squares and loglinear models in that the within cell populations are assumed to be heterogeneous in their levels of mortality risk heterogeneity implies that in addition to the sampling variance considered in other available models there will be a second component of variance due solely to within cell heterogeneity maximum likelihood procedures are presented for the estimation of the model parameters these procedures are based on the assumption that the distribution function for each cell death count is the negative binomial probability function this assumption is shown to be equivalent to assuming a mixture of poisson processes with the differential risk levels among individuals within each cell being governed by a two parameter gamma distribution the model is applied to data on lung cancer mortality for 1970 1975 for the 100 counties of north carolina the analysis shows that though a gradient in lung cancer mortality rates exist in space the gradient is restricted to specific demographic categories identified by race age and sex

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 it is 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 excerpt

on performance of parametric and distribution free models for zero inflated and over dispersed count responses zero inflated poisson zip and negative binomial zinb models are widely used to model zero inflated count responses these models extend the poisson and negative binomial nb to address excessive zeros in the count response by adding a degenerate distribution centered at 0 and interpreting it as describing a non risk group in the population the zip zinb models a two component population mixture as in applications of poisson and nb the key difference between zip and zinb is the allowance for overdispersion by the zinb in its nb component in modeling the count response for the at risk group overdispersion arising in practice too often does not follow the nb and applications of zinb to such data yield invalid inference if sources of overdispersion are known other parametric models may be used to directly model the overdispersion such models too are subject to assumed distributions further this approach may not be applicable if information about the sources of overdispersion is unavailable in this paper we propose a distribution free alternative and compare its performance with these popular parametric models as well as a moment based approach proposed by yu et al statistics in medicine 2013 32 2390 2405 like the generalized estimating equations the proposed approach requires no elaborate distribution assumptions compared with the approach of yu et al it is more robust to overdispersed zero inflated responses we illustrate our approach with both simulated and real study data

on autocorrelation in a poisson regression model summary this paper develops a practical approach to diagnosing the existence of a latent stochastic process in the mean of a poisson regression model the asymptotic distribution of standard generalised linear model estimators is derived for the case where an autocorrelated latent process is present simple formulae for the effect of autocovariance on standard errors of the regression coefficients are also provided methods for adjusting for the severe bias in previously proposed estimators of autocovariance are derived and their behaviour is investigated applications of the methods to time series of monthly polio counts in the u s a and daily asthma presentations at a hospital in sydney are used to illustrate the results and methods

overdispersion tests for truncated poisson regression models abstract score tests of extra poisson variation in left or right truncated poisson regression models are derived for alternatives in the left or right truncated negative binomial family and for arbitrary truncation points using asymptotic expansions of the score function adjustment factors are derived for the tests which are expected to improve their small sample properties overdispersion tests for the untruncated poisson model in the econometric and statistical literature are special cases of the tests derived in this paper a monte carlo investigation of the new tests when data are truncated at zero show that the effect of truncation produces a significant size distortion and power reduction especially in small samples but also that the corrections suggested by asymptotic expansions improve the performance of the tests

accelerated intensity frailty model for recurrent events data in this article we propose an accelerated intensity frailty aif model for recurrent events data and derive a test for the variance of frailty in addition we develop a kernel smoothing based em algorithm for estimating regression coefficients and the baseline intensity function the variance of the resulting estimator for regression parameters is obtained by a numerical differentiation method simulation studies are conducted to evaluate the finite sample performance of the proposed estimator under practical settings and demonstrate the efficiency gain over the gehan rank estimator based on the aft model for counting process lin et al 1998 our method is further illustrated with an application to a bladder tumor recurrence data

universal deformations of the finite quotients of the braid group on 3 strands abstract we prove that the quotients of the group algebra of the braid group on 3 strands by a generic quartic and quintic relation respectively have finite rank this is a special case of a conjecture by broue malle and rouquier for the generic hecke algebra of an arbitrary complex reflection group exploring the consequences of this case we prove that we can determine completely the irreducible representations of this braid group of dimension at most 5 thus recovering a classification of tuba and wenzl in a more general framework

consistency and asymptotic normality of the maximum likelihood estimator in generalized linear models generalized linear models are used for regression analysis in a number of cases including categorical responses where the classical assumptions are violated the statistical analysis of such models is based on the asymptotic properties of the maximum likelihood estimator we present mild general conditions which respectively assure weak or strong consistency or asymptotic normality most of the previous work has been concerned with natural link functions in this case our normality condition though obtained by a different approach is closely related to a condition of haberman 1977a examples show how the general conditions reduce to weak requirements for special exponential families further for regressors with a compact range sufficient conditions are given which do not involve the unknown parameter and are therefore easy to check in practice responses with a bounded range e g categorical responses and stochastic regressors also are treated

regression based tests for overdispersion in the poisson model abstract a property of the poisson regression model is mean variance equality conditional on explanatory variables regression based tests for this property are proposed in a very general setting unlike classical statistical tests these tests require specification of only the mean variance relationship under the alternative rather than the complete distribution whose choice is usually arbitrary the optimal regression based test is easily computed as the t test from an auxiliary regression if a distribution under the alternative hypothesis is in fact specified and is in the katz system of distributions or is cox s local approximation to the poisson the score test for the poisson distribution is equivalent to the optimal regression based test

distribution free models for longitudinal count responses with overdispersion and structural zeros overdispersion and structural zeros are two major manifestations of departure from the poisson assumption when modeling count responses using poisson loglinear regression as noted in a large body of literature ignoring such departures could yield bias and lead to wrong conclusions different approaches have been developed to tackle these two major problems in this paper we review available methods for dealing with overdispersion and structural zeros within a longitudinal data setting and propose a distribution free modeling approach to address the limitations of these methods by utilizing a new class of functional response models frm we illustrate our approach with both simulated and real study data

the conditional distribution of goodness of fit statistics for discrete data abstract i consider the distribution of pearson s statistic and of the likelihood ratio goodness of fit statistic for discrete data in the important case where the data are extensive but sparse it is argued that the appropriate reference distribution is conditional on the sufficient statistic for the unknown regression parameters the first three conditional asymptotic cumulants are derived by edgeworth expansion and these are used for the computation of tail probabilities the principal advantage of the limit considered here as opposed to the more usual x 2 limit is that the cell counts need not be large

a hierarchical finite mixture model that accommodates zero inflated counts non independence and heterogeneity a number of mixture modeling approaches assume both normality and independent observations however these two assumptions are at odds with the reality of many data sets which are often characterized by an abundance of zero valued or highly skewed observations as well as observations from biologically related i e non independent subjects we present here a finite mixture model with a zero inflated poisson regression component that may be applied to both types of data this flexible approach allows the use of covariates to model both the poisson mean and rate of zero inflation and can incorporate random effects to accommodate non independent observations we demonstrate the utility of this approach by applying these models to a candidate endophenotype for schizophrenia but the same methods are applicable to other types of data characterized by zero inflation and non independence

mixed poisson regression models with covariate dependent rates this paper studies a class of poisson mixture models that includes covariates in rates this model contains poisson regression and independent poisson mixtures as special cases estimation methods based on the em and quasi newton algorithms properties of these estimates a model selection procedure residual analysis and goodness of fit test are discussed a monte carlo study investigates implementation and model choice issues this methodology is used to analyze seizure frequency and ames salmonella assay data

semiparametric estimation of count regression models this paper develops a semiparametric estimation approach for mixed count regression models based on series expansion for the unknown density of the unobserved heterogeneity we use the generalized laguerre series expansion around a gamma baseline density to model unobserved heterogeneity in a poisson mixture model we establish the consistency of the estimator and present a computational strategy to implement the proposed estimation techniques in the standard count model as well as in truncated censored and zero inflated count regression models monte carlo evidence shows that the finite sample behavior of the estimator is quite good the paper applies the method to a model of individual shopping behavior

tests for detecting overdispersion in poisson regression models abstract poisson regression models are widely used in analyzing count data this article develops tests for detecting extra poisson variation in such situations the tests can be obtained as score tests against arbitrary mixed poisson alternatives and are generalizations of tests of fisher 1950 and collings and margolin 1985 accurate approximations for computing significance levels are given and the power of the tests against negative binomial alternatives is compared with those of the pearson and deviance statistics one way to test for extra poisson variation is to fit models that parametrically incorporate and then test for the absence of such variation within the models for example negative binomial models can be used in this way cameron and trivedi 1986 lawless 1987a the tests in this article require only the poisson model to be fitted two test statistics are developed that are motivated partly by a desire to have good distributional approximations for computing significance levels simu

regression methods for poisson process data abstract this article is directed toward situations where individuals can experience repeated events and data on an individual consist of the number and occurrence times of events along with concomitant variables methods of regression analysis are presented based on poisson process and proportional intensity assumptions these include parametric and semi parametric approaches to model fitting model assessment and the treatment of random effects in addition insight is gained as to the central role of poisson and mixed poisson regression analysis of counts in these methods and as to the effects of unobserved heterogeneity on semi parametric analyses the methods in the article are based on the proportional intensity poisson process model for which an individual with given fixed covariate vector x has repeated events occur according to a nonhomogeneous poisson process with intensity function x t 0 t exp x estimation of and the baseline intensity 0 t are considered when 0 t is specifi

testing goodness of fit of a multinomial model against overdispersed alternatives the problem of testing the goodness of fit of the multinomial sampling model against an overdispersed dirichlet multinomial alternative is studied and an improved approximate sampling distribution is presented for the corresponding c ca test statistic the elements of this improvement include better control of test size and increased power against overdispersion the pearson chi square test is also considered for the detection of this form of overdispersion and its asymptotic efficiency relative to the c a test is obtained and shown to be always less than or equal to 1 monte carlo investigations suggest that the gain in power of the c a test relative to the pearson chi square test is small unless there is substantial inequality among the dirichlet multinomial sample sizes