An essential textbook for any student or researcher in biology needing to design experiments, sample programs or analyse the resulting data The text begins with a revision of estimation and hypothesis testing methods, covering both classical and Bayesian philosophies, before advancing to the analysis of linear and generalized linear models Topics covered include linear and logistic regression, simple and complex ANOVA models (for factorial, nested, block, split-plot and repeated measures and covariance designs), and log-linear models Multivariate techniques, including classification and ordination, are then introduced Special emphasis is placed on checking assumptions, exploratory data analysis and presentation of results The main analyses are illustrated with many examples from published papers and there is an extensive reference list to both the statistical and biological literature The book is supported by a website that provides all data sets, questions for each chapter and links to software

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Experimental Design and Data Analysis for Biologists

Presents the likelihood methods for the analysis of cointegration in VAR models with Gaussian errors, seasonal dummies, and constant terms. Discusses likelihood ratio tests of cointegration rank and find the asymptotic distribution of the test statistics. Shows that the asymptotic distribution of the maximum likelihood estimator is mixed Gausssian.

Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models / Søren Johansen

The fine detail provided by sequencing-based transcriptome surveys suggests that RNA-seq is likely to become the platform of choice for interrogating steady state RNA. In order to discover biologically important changes in expression, we show that normalization continues to be an essential step in the analysis. We outline a simple and effective method for performing normalization and show dramatically improved results for inferring differential expression in simulated and publicly available data sets.

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A scaling normalization method for differential expression analysis of RNA-seq data

When more than one statistical test is performed in analysing the data from a clinical study, some statisticians and journal editors demand that a more stringent criterion be used for “statistical significance” than the conventional P<0051 Many well meaning researchers, eager for methodological rigour, comply without fully grasping what is at stake Recently, adjustments for multiple tests (or Bonferroni adjustments) have found their way into introductory texts on medical statistics, which has increased their apparent legitimacy This paper advances the view, widely held by epidemiologists, that Bonferroni adjustments are, at best, unnecessary and, at worst, deleterious to sound statistical inference

#### Summary points

Adjusting statistical significance for the number of tests that have been performed on study data—the Bonferroni method—creates more problems than it solves

The Bonferroni method is concerned with the general null hypothesis (that all null hypotheses are true simultaneously), which is rarely of interest or use to researchers

The main weakness is that the interpretation of a finding depends on the number of other tests performed

The likelihood of type II errors is also increased, so that truly important differences are deemed non-significant

Simply describing what tests of significance have been performed, and why, is generally the best way of dealing with multiple comparisons

Bonferroni adjustments are based on the following reasoning1-3 If a null hypothesis is true (for instance, two treatment groups in a randomised trial do not differ in terms of cure rates), a significant difference (P<005) will be observed by chance once in 20 trials This is the type I error, or α When 20 independent tests are performed (for example, study groups are compared with regard to 20 unrelated variables) and the null hypothesis holds for all 20 comparisons, the chance of at least one test being significant is no longer 005, but 064 …

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What's wrong with Bonferroni adjustments

This reference work and graduate level textbook considers a wide range of models and methods for analyzing and forecasting multiple time series. The models covered include vector autoregressive, cointegrated, vector autoregressive moving average, multivariate ARCH and periodic processes as well as dynamic simultaneous equations and state space models. Least squares, maximum likelihood, and Bayesian methods are considered for estimating these models. Different procedures for model selection and model specification are treated and a wide range of tests and criteria for model checking are introduced. Causality analysis, impulse response analysis and innovation accounting are presented as tools for structural analysis. The book is accessible to graduate students in business and economics. In addition, multiple time series courses in other fields such as statistics and engineering may be based on it. Applied researchers involved in analyzing multiple time series may benefit from the book as it provides the background and tools for their tasks. It bridges the gap to the difficult technical literature on the topic.

New Introduction to Multiple Time Series Analysis

Statistical Inference

The bioequivalence problem is of practical importance because the approval of most generic drugs in the United States and the European Community (EC) requires the establishment of bioequivalence between the brand-name drug and the proposed generic version. The problem is theoretically interesting because it has been recognized as one for which the desired inference, instead of the usual significant difference, is practical equivalence. The concept of intersection-union tests will be shown to clarify, simplify and unify bioequivalence testing. A test more powerful than the one currently specified by the FDA and EC guidelines will be derived. The claim that the bioequivalence problem defined in terms of the ratio of parameters is more difficult than the problem defined in terms of the difference of parameters will be refuted. The misconception that size-$\alpha$ bioequivalence tests generally correspond to $100(1 - 2 \alpha)%$ confidence sets will be shown to lead to incorrect statistical practices, and should be abandoned. Techniques for constructing $100(1 - \alpha)%$ confidence sets that correspond to size-$\alpha$ bioequivalence tests will be described. Finally, multiparameter bioequivalence problems will be discussed.

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Bioequivalence trials, intersection-union tests and equivalence confidence sets

For the one-sided hypothesis testing problem it is shown that it is possible to reconcile Bayesian evidence against H 0, expressed in terms of the posterior probability that H 0 is true, with frequentist evidence against H 0, expressed in terms of the p value. In fact, for many classes of prior distributions it is shown that the infimum of the Bayesian posterior probability of H 0 is equal to the p value; in other cases the infimum is less than the p value. The results are in contrast to recent work of Berger and Sellke (1987) in the two-sided (point null) case, where it was found that the p value is much smaller than the Bayesian infimum. Some comments on the point null problem are also given.

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Reconciling Bayesian and Frequentist Evidence in the One-Sided Testing Problem

The quality of a product might be determined by several parameters, each of which must meet certain standards before the product is acceptable. In this article, a method of determining whether all the parameters meet their respective standards is proposed. The method consists of testing each parameter individually and deciding that the product is acceptable only if each parameter passes its test. This simple method has some optimal properties including attaining exactly a prespecified consumer's risk and uniformly minimizing the producer's risk. These results are obtained from more general hypothesis-testing results concerning null hypotheses consisting of the unions of sets.

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Multiparameter Hypothesis Testing and Acceptance Sampling

For testing problems of the form H 0: v = v 0 with unknown nuisance parameter θ, various methods are used to deal with θ. The simplest approach is exemplified by the t test where the unknown variance is replaced by the sample variance and the t distribution accounts for estimation of the variance. In other problems, such as the 2 × 2 contingency table, one conditions on a sufficient statistic for 0 and proceeds as in Fisher's exact test. Because neither of these standard methods is appropriate for all situations, this article suggests a new method for handling the unknown θ. This new method is a simple modification of the formal definition of a p value that involves taking a maximum over the nuisance parameter space of a p value obtained for the case when θ is known. The suggested modification is to restrict the maximization to a confidence set for the nuisance parameter. After giving a brief justification, we give various examples to show how this new method gives improved results for 2 × 2 tabl...

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Roger L. Berger

Papers

Statistical Inference

Bioequivalence trials, intersection-union tests and equivalence confidence sets

Reconciling Bayesian and Frequentist Evidence in the One-Sided Testing Problem

Multiparameter Hypothesis Testing and Acceptance Sampling

P Values Maximized Over a Confidence Set for the Nuisance Parameter