By the Neyman-Pearson lemma, the most powerful test of this hypothesis rejects the null hypothesis when the likelihood ratio exceeds some critical value. We wish to test the simple null hypothesis that \(\theta = 0\) versus the simple alternative hypothesis that \(\theta = 1\) at the \(\alpha = 0.05\) level. Assume the data are drawn from the normal distribution, \(N(\theta, 1)\) with unknown mean \(\theta\) and known variance \(\sigma^2 = 1\). In this post, we will analyze the following hypothesis test. It is interesting to speculate how differently statistical theory might have evolved if Fisher had been employed in medical or industrial research. Fisher, worked in agricultural research, where the outcome of a field trial is not available until long after the experiment has been designed and started. He classical theory of experimental design deals predominantly with experiments of predetermined size, presumably because the pioneers of the subject, particularly R. According to Armitage, one of the pioneers of sequential experiment design: The frequentist approach to hypothesis testing was pioneered just after the turn of the 20th century in England in order to analyze agricultural experiments. Below, we illustrate the perils of the naive approach to sequential testing (as this sort of procedure is known) and show how to perform a correct analysis of a fairly simple, yet illustrative introductory sequential experiment.Ī brief historical digression may be informative. While these reasons for continuous monitoring and early termination of certain experiments are quite compelling, if this method is applied naively, it can lead to wildly incorrect analyses. In medicine, it may be unethical to continue an experimental treatment which appears to have a detrimental effect, or to deny the obviously better experimental treatment to the control group until the predetermined sample size is reached. In business, experiments cost money, both in terms of the actual cost of data collection and in terms of the opportunity cost of waiting for an experiment to reach a set number of samples before acting on its outcome, which may have been apparent much earlier. There are many situations in which it is advantageous to monitor the status of an experiment and terminate it early if the conclusion seems apparent. In this post, we’ll explore the common scenario where we would like to monitor the status of an ongoing experiment and stop the experiment early if an effect becomes apparent. The dominant Neyman-Pearson hypothesis testing framework is subtle and easy to unknowingly misuse. This epansion has, however, come with a cost. Statistical powers of the proposed group sequential test are also presented.As the world becomes ever more data-driven, the basic theory of hypothesis testing is being used by more people and in more contexts than ever before. The results indicate that the type I error rate of the proposed test procedure is well preserved, while the type I error rate of the standard group sequential test is inflated as the population changes. A simulation was performed to evaluate the performance of the proposed method. A new group sequential test procedure that accounts for the effect of population changes is proposed. ![]() Under this model, we can make inference on the original target population based on additional data from the changed populations. In this paper, we consider changes in patient population related to some covariates of an on-going trial through a linear regression model. As a result, the original patient population may have changed to a similar but different patient population. In practice, however, this assumption is often not met because the trial may be modified after the review of the clinical data at interim. The standard group sequential test is statistically valid under the assumption that the patient population remains unchanged from one interim analysis to another. In clinical trials, a standard group sequential test with a fixed number of planned interim analyses is usually considered to assess the effect of a test treatment under study.
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