Two-stage approaches to the analysis of occupancy data I: the homogeneous case (analysis of occupancy data)

Abstract

Occupancy models are used in statistical ecology to estimate species dispersion. The two components of an occupancy model are the detection and occupancy probabilities, with the main interest being in the occupancy probabilities. We show that for the homogeneous occupancy model there is an orthogonal transformation of the parameters that gives a natural two-stage inference procedure based on a conditional likelihood. We then extend this to a partial likelihood that gives explicit estimators of the model parameters. By allowing the separate modeling of the detection and occupancy probabilities, the extension of the two-stage approach to more general models has the potential to simplify the computational routines used there.     

SENSITIVITY ANALYSIS OF THE STOCHASTIC GROWTH RATE: THREE EXTENSIONS†

The perturbation analysis of population growth rate plays an important role in population biology. The sensitivity and/or elasticity (proportional sensitivity) of population growth rate to changes in the vital rates are regularly used (i) to predict the effects of environmental perturbations, (ii) to characterize selection gradients on life‐history traits, (iii) to evaluate management tactics, (iv) to analyse life table response experiments, and (v) to calculate the sampling variance in population growth rate. In a stochastic environment, population growth is described by the stochastic growth rate, which gives, with probability 1, the asymptotic time‐averaged growth rate of any realization. Tuljapurkar derived the sensitivity and elasticity of the stochastic growth rate to changes in the entries of the stochastic matrices. This paper extends his result to cover three cases, each of which has arisen recently in applications. The first gives the response of the stochastic growth rate to environment‐specific perturbations, applied only in a specified subset of the possible environments. The second gives the sensitivity and elasticity of the stochastic growth rate to changes in lower‐level parameters. The third applies to stochastic seasonal models, in which the projection matrix for each year is a periodic product of matrices describing seasonal transitions. In this case interest focuses on the sensitivity of the stochastic growth rate to changes in the entries of the seasonal matrices, not entries in the annual matrices. The paper describes examples of problems where each of these extensions is needed, and the algorithms for each of the new calculations.     

Dose-response models for time-to-response toxicity data

This paper concerns the modelling and analysis of toxicity data in which the time to some toxic response is recorded and of importance. The paper begins with a review of a variety of dose-response models, including log-linear, proportional hazards, multievent, and compartmental models, that have been suggested and used in this context. Models for competing risks are also reviewed and several possible approaches are considered and discussed in the context of dose response experiments. The formulation of a model for competing risks that utilizes cause-specific hazards gives a natural basis for the modelling and analysis of certain experimental situations in which only partial information is available on the time to response. Such situations arise, for example, when the presence of tumour is not clinically detectable in carcinogenicity trials. Likelihood construction is discussed, and appropriate likelihood contributions are derived. The paper concludes with a discussion of measures of risk which take account of the time dependence of the toxic response under study.     

An individual measure of relative survival

Summary.  Relative survival techniques are used to compare survival experience in a study cohort with that expected if background population rates apply. The techniques are especially useful when cause-specific death information is not accurate or not available as they provide a measure of excess mortality in a group of patients with a certain disease. Whereas these methods are based on group comparisons, we present here a transformation approach which instead gives for each individual an outcome measure relative to the appropriate background population. The new outcome measure is easily interpreted and can be analysed by using standard survival analysis techniques. It provides additional information on relative survival and gives new options in regression analysis. For example, one can estimate the proportion of patients who survived longer than a given percentile of the respective general population or compare survival experience of individuals while accounting for the population differences. The regression models for the new outcome measure are different from existing models, thus providing new possibilities in analysing relative survival data. One distinctive feature of our approach is that we adjust for expected survival before modelling. The paper is motivated by a study into the survival of patients after acute myocardial infarction.     

Equivariant prediction in finite populations

A theory of equivariant prediction is developed for predicting the population total in finite populations. Minimum risk equivariant predictors (MREP) are derived under the location, scale and locationscale superpopulation models. Under the general linear model, it is shown that the best(linear) unbiased predictor (B(L)UP) is an MREP.     

Marginal distribution of Markov-switching VAR processes

We make available simple and accurate closed-form approximations to the marginal distribution of Markov-switching vector auto-regressive (MS VAR) processes. The approximation is built upon the property of MS VAR processes of being Gaussian conditionally on any semi-infinite sequence of the latent state. Truncating the semi-infinite sequence and averaging over all possible sequences of that finite length yields a mixture of normals that converges to the unknown marginal distribution as the sequence length increases. Numerical experiments confirm the viability of the approach which extends to the closely related class of MS state space models. Several applications are discussed.     

Non-orthogonal designs for measuring dispersion effects in sequential factor screening experiments using search linear models

“Dispersion” effects are considered in addition to “Location” effects of factors in the inferential procedure of sequential factor screening experiments with m factors each at two levels under search linear models. Search designs in measuring "Dispersion" and "Location" effects of factors are presented for both stage one and stage two of factor screening experiments with 4 ≤ m ≤ 10.     

The Performance of Gaussian and non Gaussian dynamic models in assessing market risk: The Implications for risk management

The developing markets are more volatile, unstable illiquid, and more prone to the external shocks. The non Gaussian VaR model gives more accurate risk models than Gaussian VaR models. Hence, the purpose of this study is to test if and how non Gaussian VaR models are comparatively better fit for risk modeling of developing markets than the Gaussian VaR models. The study measures the market risk for the daily closing price of Karachi Stock Exchange 100 index over the period of 2004–2016. The evaluation of VaR models suggests that non Gaussian dynamic model outperformed the Gaussian VaR models in developing markets.     

DYNAMICAL AND STATISTICAL MODELS FOR EXPLOITED POPULATIONS†

Human activities have indirectly modified the dynamics of many populations, accelerating considerably the natural rate of species extinction and raising strong concerns about biodiversity. In many such cases, the underlying ‘natural’ dynamics of the population has been modified by human‐induced increases in mortality, even if the populations are not exploited or harvested in the strict sense. Both dynamical and statistical models are needed to investigate the consequences of human‐induced mortality on the overall dynamics of a population. This paper reviews existing approaches and the potential of recent developments to help form a conceptual and practical framework for analysing the dynamics of exploited populations. It examines both the simple case of an extra source of mortality instantaneously in time, and the theory involved when both risks compete over a continuous time scale. This basic theory is expanded to structured populations, using matrix population models, with applications to the conservation biology of long‐lived vertebrates. The type and degree of compensation expected and approaches to detect it are reviewed, and ways of handling uncertainty are discussed.     

Asymptotic expansions and the reliability of tests in accelerated failure time models

This paper gives matrix formilae for the O(n^-1 ) cerrecti0n applicable to asymptotically efficient conditional moment tests. These formulae only require expectations of functions involving, at most, second order derivatives of the log-likelihood; unlike those previously providcd by Ferrari and Corddro(1994). The correction is used to assess the reliability of first order asymptotic theory for arbitrary residual-based diagnostics in a class of accelerated failure time models: this correction is always parameter free, depending only on the number of included covariates in the regression design. For all but one of the tests considered, first order theory is found to be extremely unreliable, even in quite large samples, although this may not be widely appreciated by applied workers.     

Double hierarchical generalized linear models (with discussion)

Summary.  We propose a class of double hierarchical generalized linear models in which random effects can be specified for both the mean and dispersion. Heteroscedasticity between clusters can be modelled by introducing random effects in the dispersion model, as is heterogeneity between clusters in the mean model. This class will, among other things, enable models with heavy-tailed distributions to be explored, providing robust estimation against outliers. The h -likelihood provides a unified framework for this new class of models and gives a single algorithm for fitting all members of the class. This algorithm does not require quadrature or prior probabilities.     

Selecting the Correct Number of Factors in Approximate Factor Models: The Large Panel Case With Group Bridge Estimators

This article proposes a group bridge estimator to select the correct number of factors in approximate factor models. It contributes to the literature on shrinkage estimation and factor models by extending the conventional bridge estimator from a single equation to a large panel context. The proposed estimator can consistently estimate the factor loadings of relevant factors and shrink the loadings of irrelevant factors to zero with a probability approaching one. Hence, it provides a consistent estimate for the number of factors. We also propose an algorithm for the new estimator; Monte Carlo experiments show that our algorithm converges reasonably fast and that our estimator has very good performance in small samples. An empirical example is also presented based on a commonly used U.S. macroeconomic dataset.     

Spatial measurement error in infectious disease models

Individual-level models (ILMs) for infectious disease can be used to model disease spread between individuals while taking into account important covariates. One important covariate in determining the risk of infection transfer can be spatial location. At the same time, measurement error is a concern in many areas of statistical analysis, and infectious disease modelling is no exception. In this paper, we are concerned with the issue of measurement error in the recorded location of individuals when using a simple spatial ILM to model the spread of disease within a population. An ILM that incorporates spatial location random effects is introduced within a hierarchical Bayesian framework. This model is tested upon both simulated data and data from the UK 2001 foot-and-mouth disease epidemic. The ability of the model to successfully identify both the spatial infection kernel and the basic reproduction number (R ₀) of the disease is tested.     

A UNIFIED FRAMEWORK FOR MODELLING WILDLIFE POPULATION DYNAMICS†

This paper proposes a unified framework for defining and fitting stochastic, discrete‐time, discrete‐stage population dynamics models. The biological system is described by a state‐space model, where the true but unknown state of the population is modelled by a state process, and this is linked to survey data by an observation process. All sources of uncertainty in the inputs, including uncertainty about model specification, are readily incorporated. The paper shows how the state process can be represented as a generalization of the standard Leslie or Lefkovitch matrix. By dividing the state process into subprocesses, complex models can be constructed from manageable building blocks. The paper illustrates the approach with a model of the British grey seal metapopulation, using sequential importance sampling with kernel smoothing to fit the model.     

Arbitrage Pricing Theory as a Restricted Nonlinear Multivariate Regression Model Iterated Nonlinear Seemingly Unrelated Regression Estimates

By replacing the unknown random factors of factor analysis with observed macroeconomic variables, the arbitrage pricing theory (APT) is recast as a multivariate nonlinear regression model with across-equation restrictions. An explicit theoretical justification for the inclusion of an arbitrary, well-diversified market index is given. Using monthly returns on 70 stocks, iterated nonlinear seemingly unrelated regression techniques are employed to obtain joint estimates of asset sensitivities and their associated APT risk “prices.” Without the assumption of normally distributed errors, these estimators are strongly consistent and asymptotically normal. With the additional assumption of normal errors, they are also full-information maximum likelihood estimators. Classical asymptotic nonlinear nested hypothesis tests are supportive of the APT with measured macroeconomic factors.     

国际收入不平等变化的中国因素分析 ――基于控制经济因素的方法

 中国因素变化的贡献是国际收入不平等程度从1978年到2000年呈下降趋势的主要动力，在20世纪80和90年代的部分年份中国因素的贡献率超过100％。论文提出了国际收入不平等分析中的控制经济因素的方法，通过对中国因素的分析揭示，中国经济因素的变化是中国因素变化的主要原因，中国人口因素的变化是次要原因，但却是重要原因。所以，虽然经济增长是中国因素变化的主要原因，但人口增长的作用也非常重要。此结论同国际收入不平等分析中控制人口因素的方法的结论是相同的。     

Some algebra and geometry for hierarchical models, applied to diagnostics

Recent advances in computing make it practical to use complex hierarchical models. However, the complexity makes it difficult to see how features of the data determine the fitted model. This paper describes an approach to diagnostics for hierarchical models, specifically linear hierarchical models with additive normal or t -errors. The key is to express hierarchical models in the form of ordinary linear models by adding artificial `cases' to the data set corresponding to the higher levels of the hierarchy. The error term of this linear model is not homoscedastic, but its covariance structure is much simpler than that usually used in variance component or random effects models. The re-expression has several advantages. First, it is extremely general, covering dynamic linear models, random effect and mixed effect models, and pairwise difference models, among others. Second, it makes more explicit the geometry of hierarchical models, by analogy with the geometry of linear models. Third, the analogy with linear models provides a rich source of ideas for diagnostics for all the parts of hierarchical models. This paper gives diagnostics to examine candidate added variables, transformations, collinearity, case influence and residuals.     

SIMULATED MAXIMUM LIKELIHOOD APPLIED TO NON-GAUSSIAN AND NONLINEAR MIXED EFFECTS AND STATE–SPACE MODELS

The paper presents an overview of maximum likelihood estimation using simulated likelihood, including the use of antithetic variables and evaluation of the simulation error of the resulting estimates. It gives a general purpose implementation of simulated maximum likelihood and uses it to re‐visit four models that have previously appeared in the published literature: a state–space model for count data; a nested random effects model for binomial data; a nonlinear growth model with crossed random effects; and a crossed random effects model for binary salamander‐mating data. In the case of the last three examples, this appears to be the first time that maximum likelihood fits of these models have been presented.     

Bayesian Analysis of Two-Level Fractional Factorial Experiments with Non-Normal Responses

An intractable issue on screening experiments is to identify significant effects and to select the best model when the number of factors is large, especially for fractional factorial experiments with non-normal responses. In such cases, a three-stage Bayesian approach based on generalized linear models (GLMs) is proposed to identify which effects should be included in the target model where the principles of effect sparsity, hierarchy, and heredity are successfully considered. Three simulation experiments with non-normal responses which follow Poisson, binomial, and gamma distributions are presented to illustrate the performance of the proposed approach.     

PkStaMp Library for Constructing Optimal Population Designs for PK/PD Studies

We discuss a Matlab-based library for constructing optimal sampling schemes for pharmacokinetic (PK) and pharmacodynamic (PD) studies. The software relies on optimal design theory for nonlinear mixed effects models and, in particular, on the first-order optimization algorithm. The library includes a number of popular compartmental PK and combined PK/PD models and can be extended to include more models. An outline of inputs/outputs is provided, some algorithmic details and examples are presented, and future work is discussed.