Residual analysis for spatial point processes (with discussion)

Summary.  We define residuals for point process models fitted to spatial point pattern data, and we propose diagnostic plots based on them. The residuals apply to any point process model that has a conditional intensity; the model may exhibit spatial heterogeneity, interpoint interaction and dependence on spatial covariates. Some existing ad hoc methods for model checking (quadrat counts, scan statistic, kernel smoothed intensity and Berman's diagnostic) are recovered as special cases. Diagnostic tools are developed systematically, by using an analogy between our spatial residuals and the usual residuals for (non-spatial) generalized linear models. The conditional intensity λ plays the role of the mean response. This makes it possible to adapt existing knowledge about model validation for generalized linear models to the spatial point process context, giving recommendations for diagnostic plots. A plot of smoothed residuals against spatial location, or against a spatial covariate, is effective in diagnosing spatial trend or co-variate effects. Q – Q -plots of the residuals are effective in diagnosing interpoint interaction.     

Semiparametric methods for response-selective and missing data problems in regression

Suppose that data are generated according to the model f ( y | x ; θ ) g ( x ), where y is a response and x are covariates. We derive and compare semiparametric likelihood and pseudolikelihood methods for estimating θ for situations in which units generated are not fully observed and in which it is impossible or undesirable to model the covariate distribution. The probability that a unit is fully observed may depend on y , and there may be a subset of covariates which is observed only for a subsample of individuals. Our key assumptions are that the probability that a unit has missing data depends only on which of a finite number of strata that ( y , x ) belongs to and that the stratum membership is observed for every unit. Applications include case–control studies in epidemiology, field reliability studies and broad classes of missing data and measurement error problems. Our results make fully efficient estimation of θ feasible, and they generalize and provide insight into a variety of methods that have been proposed for specific problems.     

Statistical issues in phase III human immunodeficiency virus vaccine trial design

Several scientific questions are of interest in phase III trials of prophylactic human immunodeficiency virus (HIV) vaccines. In this paper we focus on some issues related to evaluating the direct protective effects of a vaccine in reducing susceptibility, VE_S, and its effect on reducing infectiousness, VE_I. An estimation of VE_I generally requires information on contacts between the infective and susceptible individuals. By augmenting the primary participants of an HIV vaccine trial by their steady sexual partners, information can be collected that allows an estimation of VE_I as well as VE_S. Exposure to infection information, however, may be expensive and difficult to collect. A vaccine trial design can include a small validation set with good exposure to infection data to correct bias in a larger, simpler main study with only coarse exposure data. The large main study increases the efficiency of the small validation set. More research into the combination of different levels of information in vaccine trial design will yield more efficient and less biased estimates of the efficacy measures of interest.     

A Note on the Hodges–Lehmann Estimator and the Logistic Distribution

ABSTRACT

It is well known that the Hodges–Lehmann estimator is asymptotically efficient for the location parameter of the logistic distribution. In this article we give a simple and direct proof that this property also characterizes the logistic between all the symmetric location distributions under mild conditions. Using pseudolikelihood, we also show how to find from the Hodges–Lehmann estimator an asymptotically efficient estimator of the scale parameter of the logistic distribution.     

An Index of Local Sensitivity to Nonignorability for a Pseudolikelihood Method

For regression analysis of data with non response, sensitivity analysis is usually recommended. An index of local sensitivity to non ignorability (ISNI) (Troxel et al., 2004 Troxel , A. B. , Ma , G. , Heitjan , D. F. (2004). An index of local sensitivity to nonignorability. Statistica Sinica 14:1221–1237.[Web of Science ®] , [Google Scholar]) was derived to detect the sensitivity of maximum likelihood estimates to small departures from ignorability. However, ISNI requires specification of a parametric model for the missing-data mechanism. In this article, a local sensitivity index for a pseudolikelihood (PL) method that does not require specification of the mechanism is proposed. For bivariate data (x, y), when the non response mechanism is an arbitrary function of x + λy, this new index is defined as the first derivative of the PL estimate with respect to λ at λ = 0. The closed form was derived for normal regression data when the density function of the predictor x approximated by a kernel estimator in the PL method. The utility of this new local sensitivity index was illustrated through application on one dataset.     

Regression analysis under non-standard situations: a pairwise pseudolikelihood approach

Regression analysis is one of the most used statistical methods for data analysis. There are, however, many situations in which one cannot base inference solely on f ( y ∣ x ; β), the conditional probability (density) function for the response variable Y , given x , the covariates. Examples include missing data where the missingness is non-ignorable, sampling surveys in which subjects are selected on the basis of the Y -values and meta-analysis where published studies are subject to 'selection bias'. The conventional approaches require the correct specification of the missingness mechanism, sampling probability and probability for being published respectively. In this paper, we propose an alternative estimating procedure for β based on an idea originated by Kalbfleisch. The novelty of this method is that no assumption on the missingness probability mechanisms etc. mentioned above is required to be specified. Asymptotic efficiency calculations and simulation studies were conducted to compare the method proposed with the two existing methods: the conditional likelihood and the weighted estimating function approaches.     

Approximate likelihood methods for estimating local recombination rates

Summary. There is currently great interest in understanding the way in which recombination rates vary, over short scales, across the human genome. Aside from inherent interest, an understanding of this local variation is essential for the sensible design and analysis of many studies aimed at elucidating the genetic basis of common diseases or of human population histories. Standard pedigree-based approaches do not have the fine scale resolution that is needed to address this issue. In contrast, samples of deoxyribonucleic acid sequences from unrelated chromosomes in the population carry relevant information, but inference from such data is extremely challenging. Although there has been much recent interest in the development of full likelihood inference methods for estimating local recombination rates from such data, they are not currently practicable for data sets of the size being generated by modern experimental techniques. We introduce and study two approximate likelihood methods. The first, a marginal likelihood, ignores some of the data. A careful choice of what to ignore results in substantial computational savings with virtually no loss of relevant information. For larger sequences, we introduce a 'composite' likelihood, which approximates the model of interest by ignoring certain long-range dependences. An informal asymptotic analysis and a simulation study suggest that inference based on the composite likelihood is practicable and performs well. We combine both methods to reanalyse data from the lipoprotein lipase gene, and the results seriously question conclusions from some earlier studies of these data.     

Efficient Bayesian inference for exponential random graph models by correcting the pseudo-posterior distribution

Exponential random graph models are an important tool in the statistical analysis of data. However, Bayesian parameter estimation for these models is extremely challenging, since evaluation of the posterior distribution typically involves the calculation of an intractable normalizing constant. This barrier motivates the consideration of tractable approximations to the likelihood function, such as the pseudolikelihood function, which offers an approach to constructing such an approximation. Naive implementation of what we term a pseudo-posterior resulting from replacing the likelihood function in the posterior distribution by the pseudolikelihood is likely to give misleading inferences. We provide practical guidelines to correct a sample from such a pseudo-posterior distribution so that it is approximately distributed from the target posterior distribution and discuss the computational and statistical efficiency that result from this approach. We illustrate our methodology through the analysis of real-world graphs. Comparisons against the approximate exchange algorithm of Caimo and Friel (2011) are provided, followed by concluding remarks.     

High dimensional multivariate mixed models for binary questionnaire data

Summary.  Questionnaires that are used to measure the effect of an intervention often consist of different sets of items, each set possibly measuring another concept. Mixed models with set-specific random effects are a flexible tool to model the different sets of items jointly. However, computational problems typically arise as the number of sets increases. This is especially true when the random-effects distribution cannot be integrated out analytically, as with mixed models for binary data. A pairwise modelling strategy, in which all possible bivariate mixed models are fitted and where inference follows from pseudolikelihood theory, has been proposed as a solution. This approach has been applied to assess the effect of physical activity on psychocognitive functioning, the latter measured by a battery of questionnaires.