In order to determine whether or not an effect is absent based on a statistical test, the recommended frequentist tool is the equivalence test. Typically, it is expected that an appropriate equivalence margin has been specified before any data are observed. Unfortunately, this can be a difficult task. If the margin is too small, then the test's power will be substantially reduced. If the margin is too large, any claims of equivalence will be meaningless. Moreover, it remains unclear how defining the margin afterwards will bias one's results. In this short article, we consider a series of hypothetical scenarios in which the margin is defined post-hoc or is otherwise considered controversial. We also review a number of relevant, potentially problematic actual studies from clinical trials research, with the aim of motivating a critical discussion as to what is acceptable and desirable in the reporting and interpretation of equivalence tests.
Background: trials to identify the minimal effective treatment duration are needed in different therapeutic areas, including bacterial infections, TB and Hepatitis--C. However, standard non-inferiority designs have several limitations, including arbitrariness of non-inferiority margins, choice of research arms and very large sample sizes. Methods: we recast the problem of finding an appropriate non-inferior treatment duration in terms of modelling the entire duration-response curve within a pre-specified range. We propose a multi-arm randomised trial design, allocating patients to different tr…
Background. Non-inferiority (NI) trials are increasingly used to evaluate new treatments expected to have secondary advantages over standard of care, but similar efficacy on the primary outcome. When designing a NI trial with a binary primary outcome, the choice of effect measure for the NI margin has an important effect on sample size calculations; furthermore, if the control event risk observed is markedly different from that assumed, the trial can quickly lose power or the results become difficult to interpret. Methods. We propose a new way of designing NI trials to overcome the issues rais…
Gini index is a widely used measure of economic inequality. This article develops a general theory for constructing a confidence interval for Gini index with a specified confidence coefficient and a specified width. Fixed sample size methods cannot simultaneously achieve both the specified confidence coefficient and specified width. We develop a purely sequential procedure for interval estimation of Gini index with a specified confidence coefficient and a fixed margin of error. Optimality properties of the proposed method, namely first order asymptotic efficiency and asymptotic consistency are…
For indications where only unstable reference treatments are available and use of placebo is ethically justified, three-arm `gold standard' designs with an experimental, reference and placebo arm are recommended for non-inferiority trials. In such designs, the demonstration of efficacy of the reference or experimental treatment is a requirement. They have the disadvantage that only little can be concluded from the trial if the reference fails to be efficacious. To overcome this, we investigate novel single-stage, adaptive test strategies where non-inferiority is tested only if the reference sh…
A Linear Relationship between Correlation and Cohen's Kappa for Binary Data and Simulating Multivariate Nominal and Ordinal Data with Specified Kappa Matrix
Under the null hypothesis, the marginal probability of the positive response is symmetric at any specified correlated coefficient, and the discordance probability is also symmetric to the positive response probability. The marginal distribution function of the discordant observation is monotonically decreasing with the increase of the discordance probability.And the minimum point of the distribution function is determined by the correlated coefficient.Based on the joint distribution of the two discordant observations, a confidence region of the possible values of two discordant variables is pr…
It will be recalled that the classical bivariate normal distributions have normal marginals and normal conditionals. It is natural to ask whether a similar phenomenon can be encountered involving Poisson marginals and conditionals. Reference to Arnold, Castillo and Sarabia's (1999) book on conditionally specified models will confirm that Poisson marginals will be encountered, together with both conditionals being of the Poisson form, only in the case in which the variables are independent. Instead, in the present article we will be focusing on bivariate distributions with one marginal and the …
The uniform distribution on matrices with specified row and column sums is often a natural choice of null model when testing for structure in two-way tables (binary or nonnegative integer). Due to the difficulty of sampling from this distribution, many approximate methods have been developed. We will show that by exploiting certain symmetries, exact sampling and counting is in fact possible in many nontrivial real-world cases. We illustrate with real datasets including ecological co-occurrence matrices and contingency tables.
We present a new approach for random sampling of contingency tables of any size and constraints based on a recently introduced $\textit{probabilistic divide-and-conquer}$ technique. A simple exact sampling algorithm is presented for $2\times n$ tables, as well as a generalization where each entry of the table has a specified marginal distribution.
This paper focuses on the influence of a misspecified covariance structure on false discovery rate for the large scale multiple testing problem. Specifically, we evaluate the influence on the marginal distribution of local fdr statistics, which are used in many multiple testing procedures and related to Bayesian posterior probabilities. Explicit forms of the marginal distributions under both correctly specified and incorrectly specified models are derived. The Kullback-Leibler divergence is used to quantify the influence caused by a misspecification. Several numerical examples are provided to …
