Evidence from small-subunit ribosomal RNA sequences for a fungal origin of Microsporidia

The phylum Microsporidia comprises a species-rich group of minute, single-celled, and intra-cellular parasites. Lacking normal mitochondria and with unique cytology, microsporidians have sometimes been thought to be a lineage of ancient eukaryotes. Although phylogenetic analyses using small-subunit ribosomal RNA (SSU-rRNA) genes almost invariably place the Microsporidia among the earliest branches on the eukaryotic tree, many other molecules suggest instead a relationship with fungi. Using maximum likelihood methods and a diverse SSU-rRNA data set, we have re-evaluated the phylogenetic affiliations of Microsporidia. We demonstrate that tree topologies used to estimate likelihood model parameters can materially affect phylogenetic searches. We present a procedure for reducing this bias: "tree-based site partitioning," in which a comprehensive set of alternative topologies is used to estimate sequence data partitions based on inferred evolutionary rates. This hypothesis-driven approach appears to be capable of utilizing phylogenetic information that is not available to standard likelihood implementations (e.g., approximation to a gamma distribution); we have employed it in maximum likelihood and Bayesian analysis. Applying our method to a phylogenetically diverse SSU-rRNA data set revealed that the early diverging ("deep") placement of Microsporidia typically found in SSU-rRNA trees is no better than a fungal placement, and that the likeliest placement of Microsporidia among non-long-branch eukaryotic taxa is actually within fungi. These results illustrate the importance of hypothesis testing in parameter estimation, provide a way to address certain problems in difficult data sets, and support a fungal origin for the Microsporidia.     

Unconditional Tests of Hypotheses Concerning Relative Risk

The problem of estimation of, and statistical tests for, the relative risk (=α) from samples in which there is matching on covariates have been discussed by various authors (Cox, 1970; MANTEL and HAENSZEL, 1959) using the approach of conditional variates. This paper discusses log LR (likelihood ratio) tests of hypotheses concerning α, as based on one or more sets of matched samples. Approximate X² tests are also developed for the hypotheses concerning α. An example is presented which illustrates the proposed tests of significance (Table 1).     

A Model for Censored Survival Data When There is an Intervening Event

A model is discussed for incorporating information from a time-dependent covariable (an intervening event) and covariables independent of time into the analysis of survival data. In the model, it is assumed that individuals are potentially subject to two paths to failure, one including the intervening event and the other not. Additional assumptions are that failure times associated with the two paths are independent and that the time to failure subsequent to the intervening event is dependent on the intervening event time. Allowing the underlying hazard rates for the model to follow a WEIBULL form, use of the model and methods for fitting and hypothesis testing are illustrated by application to a follow-up study involving industrial workers where disability retirement was the intervening event. Extensions of the model to accommodate grouped survival data are presented.     

The Wald Statistic in Proportional Hazards Hypothesis Testing

In survivorship modelling using the proportional hazards model of Cox (1972, Journal of the Royal Statistical Society, Series B, 34, 187–220), it is often desired to test a subset of the vector of unknown regression parameters β in the expression for the hazard rate at time t. The likelihood ratio test statistic is well behaved in most situations but may be expensive to calculate. The Wald (1943, Transactions of the American Mathematical Society 54, 426–482) test statistic is easier to calculate, but has some drawbacks. In testing a single parameter in a binomial logit model, Hauck and Donner (1977, Journal of the American Statistical Association 72, 851–853) show that the Wald statistic decreases to zero the further the parameter estimate is from the null and that the asymptotic power of the test decreases to the significance level. The Wald statistic is extensively used in statistical software packages for survivorship modelling and it is therefore important to understand its behavior. The present work examines empirically the behavior of the Wald statistic under various departures from the null hypothesis and under the presence of Type I censoring and covariates in the model. It is shown via examples that the Wald statistic's behavior is not as aberrant as found for the logistic model. For the single parameter case, the asymptotic non-null distribution of the Wald statistic is examined.