Some extensions of a linear model for categorical variables

The Grizzle-Starmer-Koch (GSK) model is extended to include the traditional log-linear model and a general class of Poisson and conditional Poisson distributions. Estimators of the model parameters are defined under general exact and stochastic linear constraints.     

Analysis of selection in populations observed over a sequence of consecutive generations

Summary A statistical model is presented for dealing with genotypic frequency data obtained from a single population observed over a run of consecutive generations. This model takes into account possible correlations that exist between generations by conditioning the marginal probability distribution of any one generation on the previously observed generation. Maximum likelihood estimates of the fitness parameters are derived and a hypothesis testing framework developed. The model is very general, and in this paper is applied to random-mating, selfing, parthenogenetic and mixed random-mating and selfing populations with respect to a single locus, g-allele model with constant genotypic fitness differences with all selection occurring either before or after sampling. The assumptions behind this model are contrasted with those of alternative techniques such as minimum chi-square or unconditional maximum likelihood estimation when the marginal likelihoods for any one generation are conditioned only on the initial conditions and not the previous generation. The conditional model is most appropriate when the sample size per generation is large either in an absolute sense or in relation to the total population size. Minimum chi-square and the unconditional likelihood are most appropriate when the population size is effectively infinite and the samples are small. Both models are appropriate when the samples are large and the population size is effectively infinite. Under these last conditions, the conditional model may be preferred because it has greater robustness with respect to small deviations from the underlying assumptions and has a greater simplicity of form. Furthermore, if any genetic drift occurs in the experiment, the minimum chi-square and unconditional likelihood approaches can create spurious evidence for selection while the conditional approach will not. Worked examples are presented.This study was supported in part by the U. S. Atomic Energy Commission, Contract AT (11-1) -1552 to the Department of Human Genetics (CFS), University of Michigan, and by National Science Foundation Grant BMS 74-17453 awarded to the author.     

The use of mixed logit models to reflect heterogeneity in capture-recapture studies

We examine issues in estimating population size N with capture-recapture models when there is variable catchability among subjects. We focus on a logistic-normal mixed model, for which the logit of the probability of capture is an additive function of a random subject and a fixed sampling occasion parameter. When the probability of capture is small or the degree of heterogeneity is large, the log-likelihood surface is relatively flat and it is difficult to obtain much information about N. We also discuss a latent class model and a log-linear model that account for heterogeneity and show that the log-linear model has greater scope. Models assuming homogeneity provide much narrower intervals for N but are usually highly overly optimistic, the actual coverage probability being much lower than the nominal level.     

Incorporating prior information into the analysis of contingency tables

Contingency tables are often analyzed using log-linear models and in some situations prior information on the value of parameters in the log-linear model is available. In this article we describe a prior-posterior procedure that incorporates prior information directly into the analysis through a multivariate normal prior for the log-linear parameters. The mode and curvature of the posterior density are proposed as summary statistics.     

Abundance estimation from multiple photo surveys: confidence distributions and reduced likelihoods for bowhead whales off Alaska

Maximum likelihood estimates of abundance are obtained from repeated photographic surveys of a closed stratified population with naturally marked and unmarked individuals. Capture intensities are assumed log-linear in stratum, year, and season. In the chosen model, an approximate confidence distribution for total abundance of bowhead whales, with an accompanying likelihood reduced of nuisance parameters, is found from a parametric bootstrap experiment. The confidence distribution depends on the assumed study protocol. A confidence distribution that is exact (except for the effect of discreteness) is found by conditioning in the unstratified case without unmarked individuals.     

Statistical methods for estimating doubling time in in vitro cell growth

Summary Doubling time has been widely used to represent the growth pattern of cells. A traditional method for finding the doubling time is to apply gray-scaled cells, where the logarithmic transformed scale is used. As an alternative statistical method, the log-linear model was recently proposed, for which actual cell numbers are used instead of the transformed gray-scaled cells. In this paper, I extend the log-linear model and propose the extended log-linear model. This model is designed for extra-Poisson variation, where the log-linear model produces the less appropriate estimate of the doubling time. Moreover, I compare statistical properties of the gray-scaled method, the log-linear model, and the extended log-linear model. For this purpose, I perform a Monte Carlo simulation study with three data-generating models: the additive error model, the multiplicative error model, and the overdispersed Poisson model. From the simulation study, I found that the gray-scaled method highly depends on the normality assumption of the gray-scaled cells; hence, this method is appropriate when the error model is multiplicative with the log-normally distributed errors. However, it is less efficient for other types of error distributions, especially when the error model is additive or the errors follow the Poisson distribution. The estimated standard error for the doubling time is not accurate in this case. The log-linear model was found to be efficient when the errors follow the Poisson distribution or nearly Poisson distribution. The efficiency of the log-linear model was decreased accordingly as the overdispersion increased, compared to the extended log-linear model. When the error model is additive or multiplicative with Gamma-distributed errors, the log-linear model is more efficient than the gray-scaled method. The extended log-linear model performs well overall for all three data-generating models. The loss of efficiency of the extended log-linear model is observed only when the error model is multiplicative with log-normally distributed errors, where the gray-scaled method is appropriate. However, the extended log-linear model is more efficient than the log-linear model in this case.     

Vegetation—a Mosaic of Discrete Communities, or a Continuum

A simulation model, using the often observed log-linear relationship between the relative abundance and rank of species, shows that vegetation will tend to form a series of communities of constant composition along an environmental gradient even if the constituent species are initially responding independently to those environmental changes. The discreteness of these communities increases with increasing conformity to the log-linear relationship. An index quantifying this conformity (range 0-1.0) is defined and a method for estimating it from field vegetation data devised. In three vegetation data sets tested the conformity index was 0.0, 0.32 and 0.37.     

A model of physiological and allometric factors in the self-thinning curve

A generalized self-thinning curve for plants is derived from the modified Von Bertallanfy equation. When an asymptotic relation between photosynthesis per unit of leaf area and stocking density is assumed, the self-thinning curve thus derived is also asymptotic on a log-log scale but is fitted quite well by a log-linear approximation. The model predicts that the slope of the log-linear approximation is a function of (a) photosynthetic response to density and (b) the relation between leaf area and total aboveground biomass. Intercept of the log-linear approximation is a function of these plus maximum attainable biomass, site productivity, density at which maximum photosynthesis is attained, and the nature of carbon loss within the plant community. Linkages between various parameters within the model act to reduce differences in slope and intercept for species with different life history's and physiological requirements.     

MULTISITE MARK-RECAPTURE FOR CETACEANS: POPULATION ESTIMATES WITH BAYESIAN MODEL AVERAGING

Mark-recapture techniques are widely used to estimate the size of wildlife populations. However, in cetacean photo-identification studies, it is often impractical to sample across the entire range of the population. Consequently, negatively biased population estimates can result when large portions of a population are unavailable for photographic capture. To overcome this problem, we propose that individuals be sampled from a number of discrete sites located throughout the population's range. The recapture of individuals between sites can then be presented in a simple contingency table, where the cells refer to discrete categories formed by combinations of the study sites. We present a Bayesian framework for fitting a suite of log-linear models to these data, with each model representing a different hypothesis about dependence between sites. Modeling dependence facilitates the analysis of opportunistic photo-identification data from study sites located due to convenience rather than by design. Because inference about population size is sensitive to model choice, we use Bayesian Markov chain Monte Carlo approaches to estimate posterior model probabilities, and base inference on a model-averaged estimate of population size. We demonstrate this method in the analysis of photographic mark-recapture data for bottlenose dolphins from three coastal sites around NE Scotland.     

The effects of demography and linkage on the estimation of selection and mutation parameters

We explore the effects of demography and linkage on a maximum-likelihood (ML) method for estimating selection and mutation parameters in a reversible mutation model. This method assumes free recombination between sites and a randomly mating population of constant size and uses information from both polymorphic and monomorphic sites in the sample. Two likelihood-ratio test statistics were constructed under this ML framework: LRTγ for detecting selection and LRTκ for detecting mutational bias. By carrying out extensive simulations, we obtain the following results. When mutations are neutral and population size is constant, LRTγ and LRTκ follow a chi-square distribution with 1 d.f. regardless of the level of linkage, as long as the mutation rate is not very high. In addition, LRTγ and LRTκ are relatively insensitive to demographic effects and selection at linked sites. We find that the ML estimators of the selection and mutation parameters are usually approximately unbiased and that LRTκ usually has good power to detect mutational bias. Finally, with a recombination rate that is typical for Drosophila, LRTγ has good power to detect weak selection acting on synonymous sites. These results suggest that the method should be useful under many different circumstances.     

Hyperparameter estimation using stochastic approximation with application to population pharmacokinetics

A stochastic approximation algorithm is proposed for recursive estimation of the hyperparameters characterizing, in a population, the probability density function of the parameters of a statistical model. For a given population model defined by a parametric model of a biological process, an error model, and a class of densities on the set of the individual parameters, this algorithm provides a sequence of estimates from a sequence of individuals' observation vectors. Convergence conditions are verified for a class of population models including usual pharmacokinetic applications. This method is implemented for estimation of pharmacokinetic population parameters from drug multiple-dosing data. Its estimation capabilities are evaluated and compared to a classical method in population pharmacokinetics, the first-order method (NONMEM), on simulated data.     

Trans-Theta Logistics: A New Family of Population Growth Sigmoid Functions

Sigmoid functions have been applied in many areas to model self limited population growth. The most popular functions; General Logistic (GL), General von Bertalanffy (GV), and Gompertz (G), comprise a family of functions called Theta Logistic ([Formula: see text] L). Previously, we introduced a simple model of tumor cell population dynamics which provided a unifying foundation for these functions. In the model the total population (N) is divided into reproducing (P) and non-reproducing/quiescent (Q) sub-populations. The modes of the rate of change of ratio P/N was shown to produce GL, GV or G growth. We now generalize the population dynamics model and extend the possible modes of the P/N rate of change. We produce a new family of sigmoid growth functions, Trans-General Logistic (TGL), Trans-General von Bertalanffy (TGV) and Trans-Gompertz (TG)), which as a group we have named Trans-Theta Logistic (T [Formula: see text] L) since they exist when the [Formula: see text] L are translated from a two parameter into a three parameter phase space. Additionally, the model produces a new trigonometric based sigmoid (TS). The [Formula: see text] L sigmoids have an inflection point size fixed by a single parameter and an inflection age fixed by both of the defining parameters. T [Formula: see text] L and TS sigmoids have an inflection point size defined by two parameters in bounding relationships and inflection point age defined by three parameters (two bounded). While the Theta Logistic sigmoids provided flexibility in defining the inflection point size, the Trans-Theta Logistic sigmoids provide flexibility in defining the inflection point size and age. By matching the slopes at the inflection points we compare the range of values of inflection point age for T [Formula: see text] L versus [Formula: see text] L for model growth curves.     

Time-budget data: Log-linear and analysis of variance compared

Analysis of variance (ANOVA) and log-linear analyses of time-budget data from a study of sloth bear enclosure utilization are compared. Two sampling models that plausibly underlie such data are discussed. Either could lead to an analysis of variance, but only one to a log-linear analysis. Given an appropriate sampling model and appropriate data, there is much to recommend log-linear analysis, despite its unfamiliarity to most animal behaviorists. One need not worry whether distribution assumptions are violated. Moreover, the data analyzed are the data collected, not estimates derived from those data, and thus no power is lost through a data reduction step. No matter what analysis is used, effect size should be taken into consideration. Multiple R² can be used for ANOVA, but no directly comparable statistic exists for log-linear analyses. One possible candidate for a log-linear R² analog is discussed here, and appears to give sensible and interpretable results. © 1992 Wiley-Liss Inc.     

Testing for association between disease and linked marker loci: a log-linear-model analysis.

One approach frequently used for identifying genetic factors involved in the process of a complex disease is the comparison of patients and controls for a number of genetic markers near a candidate gene. The analysis of such association studies raises some specific problems because of the fact that genotypic and not gametic data are generally available. We present a log-linear-model analysis providing a valid method for analyzing such studies. When studying the association of disease with one marker locus, the log-linear model allows one to test for the difference between allelic frequencies among affected and unaffected individuals, Hardy-Weinberg (H-W) equilibrium in both groups, and interaction between the association of alleles at the marker locus and disease. This interaction provides information about the dominance of the disease susceptibility locus, with dominance defined using the epidemiological notion of odds ratio. The degree of dominance measured at the marker locus depends on the strength of linkage disequilibrium between the marker locus and the disease locus. When studying the association of disease with several linked markers, the model becomes rapidly complex and uninterpretable unless it is assumed that affected and unaffected populations are in H-W equilibrium at each locus. This hypothesis must be tested before going ahead in the analysis. If it is not rejected, the log-linear model offers a stepwise method of identification of the parameters causing the difference between populations. This model can be extended to any number of loci, alleles, or populations.     

A log-linear modeling framework for selective mixing.

Nonrandom mixing can significantly alter the diffusion path of an infectious disease such as AIDS that requires intimate contact. Recent attempts to model this effect have sought a general framework capable of representing both simple and arbitrarily complicated mixing structures, and of solving the balancing problem in a nonequilibrium multigroup population. Log-linear models are proposed here as a general framework for solving the first problem. This approach offers several additional benefits: The parameters used to govern the mixing have a simple, intuitive interpretation, the framework provides a statistically sound basis for the estimation of these parameters from mixing-matrix data, and the resulting estimates are easily integrated into compartmental models for diffusion. A modified selection model is proposed to solve the second problem of generalizing the selection process to nonequilibrium populations. The distribution of contacts under this model is derived and is found to satisfy the assumptions of statistical inference for log-linear models. Together these techniques provide an integrated and flexible framework for modeling the role of selective mixing in the spread of disease.     

Estimating numbers of homeless and homeless mentally ill people in north east Westminster by using capture-recapture analysis.

OBJECTIVES--To use routinely collected data to provide a reliable estimate of the size and psychiatric morbidity of the homeless population of a given geographical area by using capture-recapture analysis. DESIGN--A multiple sample, log-linear capture-recapture method was applied to a defined area of central London during 6 months. The method calculates the total homeless population from the sum of the population actually observed and an estimate of the unobserved population. Data were collected from local agencies used by homeless people. SUBJECTS--Homeless people in north east Westminster residing in bed and breakfast accommodation and hotels or sleeping rough who had contacted statutory or voluntary agencies in the area. RESULTS--2150 contacts by 1640 homeless people were recorded. The estimated unobserved population was 3293, giving a total homeless population for the period of around 5000 (SD 1250). Mental health problems were significantly less prominent in the unobserved compared with the observed population (23% (754) v 40% (627), P < 0.0001). For both groups the prevalence varied greatly with age and sex. CONCLUSIONS--Capture-recapture techniques can overcome problems of ascertainment in estimating populations of homeless and homeless mentally ill people. Prevalences of mental illness derived from surveys that do not correct for ascertainment are likely to be falsely inflated while at the same time underestimating the total size of the homeless mentally ill population. Population estimates derived from capture-recapture techniques may usefully provide a good basis for including homeless populations in capitation calculations for allocating funds within health services.     

Estimating population size for a continuous time frailty model with covariates in a capture-recapture study

A continuous time frailty capture-recapture model is proposed for estimating population size of a closed population with the use of observed covariates to explain individuals' heterogeneity in presence of a random effect. A conditional likelihood approach is used to derive the estimate of parameters, and the Horvitz-Thompson estimator is adopted to estimate the unknown population size. Asymptotic normality of the estimates is obtained. Simulation results and a real example show that the proposed method works satisfactorily.     

Population size estimation with interval censored counts and external information: Prevalence of multiple sclerosis in Rome

We discuss Bayesian log-linear models for incomplete contingency tables with both missing and interval censored cells, with the aim of obtaining reliable population size estimates. We also discuss use of external information on the censoring probability, which may substantially reduce uncertainty. We show in simulation that information on lower bounds and external information can each improve the mean squared error of population size estimates, even when the external information is not completely accurate. We conclude with an original example on estimation of prevalence of multiple sclerosis in the metropolitan area of Rome, where five out of six lists have interval censored counts. External information comes from mortality rates of multiple sclerosis patients.     

Testing for concordant equilibria between population samples

A substantial body of theory has been developed to assess the effect of evolutionary forces on the distribution of genotypes, both single and multilocus, within populations. One area where the potential for application of this theory has not been fully appreciated concerns the extent to which population samples differ. Within populations, the divergence of genotype or haplotype frequencies from that expected under Hardy-Weinberg (HW) or linkage equilibrium can be measured as disequilibria coefficients. To assess population samples for concordant equilibria, an analytical framework for comparing disequilibria coefficients between populations is necessary. Here we present log-linear models to evaluate such hypotheses. These models have broad utility ranging from conventional population genetics to genetic epidemiology. We demonstrate the use of these log-linear models (1) as a test for genetic association with disease and (2) as a test for different levels of linkage disequilibria between human populations.     

Measurement scale in maximum entropy models of species abundance

The consistency of the species abundance distribution across diverse communities has attracted widespread attention. In this paper, I argue that the consistency of pattern arises because diverse ecological mechanisms share a common symmetry with regard to measurement scale. By symmetry, I mean that different ecological processes preserve the same measure of information and lose all other information in the aggregation of various perturbations. I frame these explanations of symmetry, measurement, and aggregation in terms of a recently developed extension to the theory of maximum entropy. I show that the natural measurement scale for the species abundance distribution is log-linear: the information in observations at small population sizes scales logarithmically and, as population size increases, the scaling of information grades from logarithmic to linear. Such log-linear scaling leads naturally to a gamma distribution for species abundance, which matches well with the observed patterns. Much of the variation between samples can be explained by the magnitude at which the measurement scale grades from logarithmic to linear. This measurement approach can be applied to the similar problem of allelic diversity in population genetics and to a wide variety of other patterns in biology.