Sensitivity of test for overdispersion in Poisson regression

Overdispersion or extra-Poisson variation is very common for count data. This phenomenon arises when the variability of the counts greatly exceeds the mean under the Poisson assumption, resulting in substantial bias for the parameter estimates. To detect whether count data are overdispersed in the Poisson regression setting, various tests have been proposed and among them, the score tests derived by Dean (1992) are popular and easy to implement. However, such tests can be sensitive to anomalous or extreme observations. In this paper, diagnostic measures are proposed for assessing the sensitivity of Dean's score test for overdispersion in Poisson regression. Applications to the well-known fabric faults and Ames salmonella assay data sets illustrate the usefulness of the diagnostics in analyzing overdispersed count data.     

The evolution of fledging age in songbirds

In birds with altricial young an important stage in the life history is the age at fledging. In this paper we use an approach proven successful in the prediction of the optimal age at maturity in fish and reptiles to predict the optimal age of fledging in passerines. Integrating the effects of growth on future fecundity and survival leads to the prediction that the optimal age at fledging is given by a function that comprises survival to maturity, the exponent of the fecundity-body size relationship and nestling growth. Growth is described by the logistic equation with parameters, A, K and t(i). Assuming that the transitional mortality curve can be approximated by the nestling mortality, M(n), the optimal fledging age, t(f), is given by a simple formula involving the three growth parameters, nestling mortality (M(n)) and the exponent (d) of the fecundity-body size relationship. Predictions of this equation underestimate the true values by 11-16%, which is expected as a consequence of the transitional mortality function approximation. A transitional mortality function in which mortality is approximately 0.3-0.4 of nesting mortality (i.e. mortality declines rapidly after fledging) produces predictions which, on average, equal the observed values. Data are presented showing that mortality does indeed decline rapidly upon fledging.     

Perceptual interactions in odour mixtures: odour quality in binary mixtures of woody and fruity wine odorants

The qualitative perceptual interactions in three binary mixtures of wine odorants were studied: isoamyl acetate (fruity note)/whisky lactone (woody note), ethyl butyrate (fruity note)/whisky lactone (woody note) and ethyl butyrate (fruity note)/guaiacol (woody note). For each binary mixture, the perceived quality and intensity of 24 stimuli (four supra-threshold concentration levels of each of the two compounds and their 16 possible combinations) were evaluated in five replications by a trained panel of 13 subjects. The application of the Olsson predictive model for odour intensity and quality perception gave quite a good estimation of the evolution of single component identification in the mixture when the intensity proportion of unmixed components varied. However, this model was unable to account for the odour quality dominance in mixtures of iso-intense components. An alternative linear logistic model was proposed to study the qualitative dominance of the woody note in the three mixtures when the perceived intensities of each unmixed compound were equal.     

Relationship between communities and processes; new insights from a field study of a contaminated ecosystem

We used a 93‐year‐old mine waste contamination gradient in alluvial soil to explore the relationship between ecosystem level functioning and community structure in a chronically stressed ecosystem. The sensitivity of broad functional parameters (in situ soil respiration, microbial biomass, above and below ground plant biomass) and microbial diversity [phospholipid fatty acid (PLFA) abundance and richness] were compared. Functional responses were linear with respect to contaminants while thresholds were detected in the community structural response to contamination along the gradient. For example, in situ soil respiration was negatively and linearly correlated to contamination concentration (R = ?0.783, P < 0.01), but changes in microbial community structure only became evident where contaminant concentrations were greater than 28 times above background levels. Our results suggest that functional redundancy does not prevent depression of ecosystem function in the long‐term.     

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).     

Benchmark dose calculation from epidemiological data

A threshold for dose-dependent toxicity is crucial for standards setting but may not be possible to specify from empirical studies. Crump (1984) instead proposed calculating the lower statistical confidence bound of the benchmark dose, which he defined as the dose that causes a small excess risk. This concept has several advantages and has been adopted by regulatory agencies for establishing safe exposure limits for toxic substances such as mercury. We have examined the validity of this method as applied to an epidemiological study of continuous response data associated with mercury exposure. For models that are linear in the parameters, we derived an approximative expression for the lower confidence bound of the benchmark dose. We find that the benchmark calculations are highly dependent on the choice of the dose-effect function and the definition of the benchmark dose. We therefore recommend that several sets of biologically relevant default settings be used to illustrate the effect on the benchmark results and to stimulate research that will guide an a priori choice of proper default settings.     

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.