David Allingham

Bayesian estimation of quantile distributions

Use of Bayesian modelling and analysis has become commonplace in many disciplines (finance, genetics and image analysis, for example). Many complex data sets are collected which do not readily admit standard distributions, and often comprise skew and kurtotic data. Such data is well-modelled by the very flexibly-shaped distributions of the quantile distribution family, whose members are defined by the inverse of their cumulative distribution functions and rarely have analytical likelihood functions defined. Without explicit likelihood functions, Bayesian methodologies such as Gibbs sampling cannot be applied to parameter estimation for this valuable class of distributions without resorting to numerical inversion. Approximate Bayesian computation provides an alternative approach requiring only a sampling scheme for the distribution of interest, enabling easier use of quantile distributions under the Bayesian framework. Parameter estimates for simulated and experimental data are presented.

Modelling the transmission dynamics of dengue in the presence of Wolbachia

Use of the bacterium Wolbachia is an innovative new strategy designed to break the cycle of dengue transmission. There are two main mechanisms by which Wolbachia could achieve this: by reducing the level of dengue virus in the mosquito and/or by shortening the host mosquitos lifespan. However, although Wolbachia shortens the lifespan, it also gives a breeding advantage which results in complex population dynamics. This study focuses on the development of a mathematical model to quantify the effect on human dengue cases of introducing Wolbachia into the mosquito population. The model consists of a compartment-based system of first-order differential equations; seasonal forcing in the mosquito population is introduced through the adult mosquito death rate. The analysis focuses on a single dengue outbreak typical of a region with a strong seasonally-varying mosquito population. We found that a significant reduction in human dengue cases can be obtained provided that Wolbachia-carrying mosquitoes persist when competing with mosquitoes without Wolbachia. Furthermore, using the Wolbachia strain WMel reduces the mosquito lifespan by at most 10% and allows them to persist in competition with non-Wolbachia-carrying mosquitoes. Mosquitoes carrying the WMelPop strain, however, are not likely to persist as it reduces the mosquito lifespan by up to 50%. When all other effects of Wolbachia on the mosquito physiology are ignored, cytoplasmic incompatibility alone results in a reduction in the number of human dengue cases. A sensitivity analysis of the parameters in the model shows that the transmission probability, the biting rate and the average adult mosquito death rate are the most important parameters for the outcome of the cumulative proportion of human individuals infected with dengue.

Parameter inference in small world network disease models with approximate Bayesian Computational methods

Abstract Small world network models have been effective in capturing the variable behaviour of reported case data of the SARS coronavirus outbreak in Hong Kong during 2003. Simulations of these models have previously been realized using informed “guesses” of the proposed model parameters and tested for consistency with the reported data by surrogate analysis. In this paper we attempt to provide statistically rigorous parameter distributions using Approximate Bayesian Computation sampling methods. We find that such sampling schemes are a useful framework for fitting parameters of stochastic small world network models where simulation of the system is straightforward but expressing a likelihood is cumbersome.

Bayesian hidden Markov model for DNA sequence segmentation: A prior sensitivity analysis

The sensitivity to the specification of the prior in a hidden Markov model describing homogeneous segments of DNA sequences is considered. An intron from the chimpanzee @a-fetoprotein gene, which plays an important role in embryonic development in mammals, is analysed. Three main aims are considered: (i) to assess the sensitivity to prior specification in Bayesian hidden Markov models for DNA sequence segmentation; (ii) to examine the impact of replacing the standard Dirichlet prior with a mixture Dirichlet prior; and (iii) to propose and illustrate a more comprehensive approach to sensitivity analysis, using importance sampling. It is obtained that (i) the posterior estimates obtained under a Bayesian hidden Markov model are indeed sensitive to the specification of the prior distributions; (ii) compared with the standard Dirichlet prior, the mixture Dirichlet prior is more flexible, less sensitive to the choice of hyperparameters and less constraining in the analysis, thus improving posterior estimates; and (iii) importance sampling was computationally feasible, fast and effective in allowing a richer sensitivity analysis.

The effect of Wolbachia on dengue outbreaks when dengue is repeatedly introduced

Use of the Wolbachia bacterium is a proposed new strategy to reduce dengue transmission, which results in around 390 million individuals infected annually. In places with strong variations in climatic conditions such as temperature and rainfall, dengue epidemics generally occur only at a certain time of the year. Where dengue is not endemic, the time of year in which imported cases enter the population plays a crucial role in determining the likelihood of outbreak occurrence. We use a mathematical model to study the effects of Wolbachia on dengue transmission dynamics and dengue seasonality. We focus in regions where dengue is not endemic but can spread due to the presence of a dengue vector and the arrival of people with dengue on a regular basis. Our results show that the time-window in which outbreaks can occur is reduced in the presence of Wolbachia-carrying Aedes aegypti mosquitoes by up to six weeks each year. We find that Wolbachia reduces overall case numbers by up to 80%. The strongest effect is obtained when the amplitude of the seasonal forcing is low (0.02-0.30). The benefits of Wolbachia also depend on the transmission rate, with the bacteria most effective at moderate transmission rates ranging between 0.08-0.12. Such rates are consistent with fitted estimates for Cairns, Australia.

The effect of Wolbachia on dengue dynamics in the presence of two serotypes of dengue: symmetric and asymmetric epidemiological characteristics

An innovative strategy to reduce dengue transmission uses the bacterium Wolbachia. We analysed the effects of Wolbachia on dengue transmission dynamics in the presence of two serotypes of dengue using a mathematical model, allowing for differences in the epidemiological characteristics of the serotypes. We found that Wolbachia has a greater effect on secondary infections than on primary infections across a range of epidemiological characteristics. If one serotype is more transmissible than the other, it will dominate primary infections and Wolbachia will be less effective at reducing secondary infections of either serotype. Differences in the antibody-dependent enhancement of the two serotypes have considerably less effect on the benefits of Wolbachia than differences in transmission probability. Even if the antibody-dependent enhancement rate is high, Wolbachia is still effective in reducing dengue. Our findings suggest that Wolbachia will be effective in the presence of more than one serotype of dengue; however, a better understanding of serotype-specific differences in transmission probability may be needed to optimize delivery of a Wolbachia intervention.

A Nonparametric Two-Sample Wald Test of Equality of Variances

We develop a test for equality of variances given two independent random samples of observations. The test can be expected to perform well when both sample sizes are at least moderate and the sample variances are asymptotically equivalent to the maximum likelihood estimators of the population variances. The test is motivated by and is here assessed for the case when both populations sampled are assumed to be normal. Popular choices of test would be the two-sample 𝐹 test if normality can be assumed and Levene’s test if this assumption is dubious. Another competitor is the Wald test for the difference in the population variances. We give a nonparametric analogue of this test and call it the 𝑅 test. In an indicative empirical study when both populations are normal, we find that when both sample sizes are at least 25 the 𝑅 test is nearly as robust as Levene’s test and nearly as powerful as the 𝐹 test.

Comparing Nonparametric Tests of Equality of Means for Randomized Block Designs

A number of nonparametric tests are compared empirically for a randomized block layout. We assess tests appropriate for when the data are not consistent with normality or when outliers invalidate traditional analysis of variance (ANOVA) tests. The objective is to assess, within this setting, tests that use ranks within blocks, the rank transform procedure that ranks the complete sample and continuous analogs of the Cochran–Mantel–Haenszel tests. The usual linear model is assumed, and our primary foci are tests of equality of means and component tests that assess linear and quadratic trends in the means. These tests include the traditional Page and Friedman tests. We conclude that the rank transform tests have competitive power and warrant greater use than is currently apparent.

Testing Equality of Variances for Multiple Univariate Normal Populations

To test for equality of variances in independent random samples from multiple univariate normal populations, the test of first choice would usually be the likelihood ratio test, the Bartlett test. This test is known to be powerful when normality can be assumed. Here two Wald tests of equality of variances are derived. The first test compares every variance with every other variance and was announced in Mather and Rayner (2002), but no proof was given there. The second test is derived from a quite different model using orthogonal contrasts, but is identical to the first. This second test statistic is similar to one given in Rippon and Rayner (2010), for which no empirical assessment has been given. These tests are compared with the Bartlett test in size and power. The Bartlett test is known to be nonrobust to the normality assumption, as is the orthogonal contrasts test. To deal with this difficulty an analogue of the new test is given. An indicative empirical assessment shows that it is more robust than the Bartlett test and competitive with the Levene test in its robustness to fat-tailed distributions. Moreover, it is a Wald test and has good power properties in large samples. Advice is given on how to implement the new test.