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The evaluation of the cumulative distribution function of a multivariate normal distribution is considered. The multivariate normal distribution can have any positive definite correlation matrix and any mean vector. The approach taken has two stages. In the first stage, it is shown how non-centred orthoscheme probabilities can be evaluated by using a recursive integration method. In the second stage, some ideas of Schläfli and Abrahamson are extended to show that any non-centred orthant probability can be expressed as differences between at most ("m" - 1)! non-centred orthoscheme probabilities. This approach allows an accurate evaluation of many multivariate normal probabilities which have important applications in statistical practice. Copyright 2003 Royal Statistical Society.

Non-inferiority (NI) trials are becoming increasingly popular. The main purpose of NI trials is to assert the efficacy of a new treatment compared with an active control by demonstrating that the new treatment maintains a substantial fraction of the treatment effect of the control. Most of the statistical testing procedures in this area have been developed for three-arm NI trials in which a new treatment is compared with an active control in the presence of a placebo. However, NI trials frequently involve comparisons of several new treatments with a control, such as in studies involving different doses of a new drug or different combinations of several new drugs. In seeking an adequate testing procedure for such cases, we use a new approach that modifies existing testing procedures to cover circumstances in which several new treatments are present. We also give methods and algorithms to produce the optimal sample size configuration. In addition, we also discuss the advantages of using different margins for the assay sensitivity test between the active control and the placebo and the NI test between the new treatments and the active control. We illustrate the new approach by using data from a clinical trial.