Score Tests for Exploring Complex Models: Application to HIV Dynamics Models

In biostatistics, more and more complex models are being developed. This is particularly the case in system biology. Fitting complex models can be very time‐consuming, since many models often have to be explored. Among the possibilities are the introduction of explanatory variables and the determination of random effects. The particularity of this use of the score test is that the null hypothesis is not itself very simple; typically, some random effects may be present under the null hypothesis. Moreover, the information matrix cannot be computed, but only an approximation based on the score. This article examines this situation with the specific example of HIV dynamics models. We examine the score test statistics for testing the effect of explanatory variables and the variance of random effect in this complex situation. We study type I errors and the statistical powers of this score test statistics and we apply the score test approach to a real data set of HIV‐infected patients.     

Small-scale plant species turnover in a limestone grassland: the carousel model and some comments on the niche concept

Abstract. This study reports on small-scale changes in the distribution of plant species in a 2.5 m² plot of grazed, species-rich Veronica spicata - Avenula pratensis grassland on shallow, dry, nutrient-poor soil in the Great Alvar area (Stora Alvaret) of southern Öland, southeastern Sweden. Multivari-ate analysis of 0.001 m² and 0.25 m² quadrats within the plot showed that there is little floristic variation without any trend in the plot. Average species richness varied little throughout the study period from 1986 to 1991 with 1986 averages of 7.0 on 0.001m², 16.3 on 0.01 m², and 26.1 on 0.25 m². On 0.001 m²the highest species number found was 12, on 0.01 m², 27. However, cumulative species richness, i.e. species number in the first year plus new species appearing in later years (averaged over 40 quadrats) increased over the same period, on 0.001 m² from 7.0 in 1986 to 14.9 in 1991, and on 0.01 m² from 16.3 to 24.1. Cumulative frequency, i.e. number of quadrats out of 40 where a species occurred in the first year or/and in later years, increased as well; the number of cumulatively highly frequent (> 80%) species increased from 5 in 1986 to 18 in 1991. Species mobility on the scale of the small quadrats used implies both appearance and disappearance from these quadrats. Using six examples, species mobility is shown to vary individualistically, both in rate and degree. We postulate that in homogeneous, grazed, nutrient- and water-deficient environments many species can reach virtually all microsites within the plot, which we express through the idea of the ‘carousel model’. We also question the usefulness of the niche concept and re-interpret it by stating that all species of this plant community have the same habitat niche, while most of them are short-lived and have the same regeneration niche. The essential variation amongst the species is their individual ability to establish or re-establish by making use of favourable conditions appearing in microsites in an unknown, complex spatio-temporal pattern.     

Symmetry and Asymmetry Models for Rectangular Tables

The models of complete, quasi and conditional symmetry as well as marginal homogeneity and diagonal asymmetry, applicable to square contingency tables are extended to non-square ones. Several properties of the models are investigated and estimation and test theory is presented. The utility of the new proposed models is discussed and illustrated by reanalyzing classical data sets.     

A Monte Carlo Test for Variance Homogeneity in Linear Models

A Monte Carlo procedure is proposed for testing homogeneity of variances in linear models. The method is applicable to a variety of common experimental designs. It is valid when errors are independently normally distributed. Under nonnormality the test is expected to behave robust in a similar fashion as Levene's test. Three examples are given to demonstrate the method.     

A simple test of the homogeneity of risk difference in sparse data: an application to a multicenter study

In this paper, we focus discussion on testing the homogeneity of risk difference for sparse data, in which we have few patients in each stratum, but a moderate or large number of strata. When the number of patients per treatment within strata is small (2 to 5 patients), none of test procedures proposed previously for testing the homogeneity of risk difference for sparse data can really perform well. On the basis of bootstrap methods, we develop a simple test procedure that can improve the power of the previous test procedures. Using Monte Carlo simulations, we demonstrate that the test procedure developed here can perform reasonable well with respect to Type I error even when the number of patients per stratum for each treatment is as small as two patients. We evaluate and study the power of the proposed test procedure in a variety of situations. We also include a comparison of the performance between the test statistics proposed elsewhere and the test procedure developed here. Finally, we briefly discuss the limitation of using the proposed test procedure. We use the data comparing two chemotherapy treatments in patients with multiple myeloma to illustrate the use of the proposed test procedure.     

Two Estimators of Mean Frequency in Products of Multinormal Probability Models

Two estimators, one additive the other multiplicative, are considered for mean frequencies in a complete three-way table. Using the mean square error criterion it is shown that preference for the additive estimator can be as high 7/8 in tables with row-column independence and in homogeneous tables. Extension to other estimators are discussed.     

Markov modelling of immunological and virological states in HIV-1 infected patients

The purpose of this study was to evaluate the evolution of HIV infected patients and to bring out some significant factors associated with this pathology. The main criteria revealing the State of illness is viral load measurement (VL). However the CD4 lymphocytes also represent an important marker as these reflect the State of the immune reservoir. Many studies have been carried out in this field and different models have been proposed with a view to a better understanding of this disease. Multi State Markov models defined in terms of CD4 counts, or in terms of viral load, have proved to be very useful tools for modelling HIV disease progression. The model we have developed in this study is based on both the CD4 lymphocytes counts and VL. Markov models are characterized by transition intensities. In this paper we explored several structures in succession. First, we used a homogeneous continuous time Markov process with four states defined by crossed values of CD4 and VL in a given patient at a given time. Then, the effect of certain covariates on the infection process was introduced into the model via the transition intensity functions, as with a Cox regression model. Since the hypothesis of homogeneity may be unrealistic in certain cases, we also considered piecewise homogeneous Markov models. Finally, the effects of covariates and time were combined in a piecewise homogeneous model with a covariate. We applied these methods to data from 1313 HIV-infected patients included in the NADIS cohort.     

Evidence for new Neanderthal teeth in Tabun Cave (Israel) by the application of self-organizing maps (SOMs)

Morphological and metrical study suggested that seven human teeth from Tabun Cave, Israel were part of the upper dentition of a single, probably Neanderthal, individual renumbered as Tabun BC7. An enamel fragment gave ESR age estimates of 82+/-14 ka (early U-uptake) and 92+/-18 ka (linear uptake) and an age estimate of 90(+30)(-16) ka using U-series disequilibrium. Although metrical analyses suggested Neanderthal affinities, definitive assessment was difficult as the values often fell into the ranges of both Neanderthal and Levantine early modern human samples. Therefore, two further classification analyses were conducted (neural networks using self-organizing maps and homogeneity analysis). Both identify Tabun BC7 as a Neanderthal. Neural networks are a promising tool for paleoanthropological studies as they can provide reliable classifications even with incomplete data.     

A heuristic test for homogeneity in species populations

Summary This paper elaborates on the notion of homogeneity in plant populations and its quantification. It describes a method for testing homogeneity without the need to make assumptions about generalized distributions. The implementation of the method is illustrated through actual numerical examples.This paper presents results from a study of plant populations for which a National Research Council of Canade grant has been received (L. Orlóci).     

Homogeneity test of rate ratios in stratified matched-pair studies

This paper investigates homogeneity test of rate ratios in stratified matched-pair studies on the basis of asymptotic and bootstrap-resampling methods. Based on the efficient score approach, we develop a simple and computationally tractable score test statistic. Several other homogeneity test statistics are also proposed on the basis of the weighted least-squares estimate and logarithmic transformation. Sample size formulae are derived to guarantee a pre-specified power for the proposed tests at the pre-given significance level. Empirical results confirm that (i) the modified score statistic based on the bootstrap-resampling method performs better in the sense that its empirical type I error rate is much closer to the pre-specified nominal level than those of other tests and its power is greater than those of other tests, and is hence recommended, whilst the statistics based on the weighted least-squares estimate and logarithmic transformation are slightly conservative under some of the considered settings; (ii) the derived sample size formulae are rather accurate in the sense that their empirical powers obtained from the estimated sample sizes are very close to the pre-specified nominal powers. A real example is used to illustrate the proposed methodologies.