If we think of each experiment as a hypothesis test about the corresponding data stream, then what is needed is a combination of a multiple hypothesis test and a sequential hypothesis test. The preceding scenario occurs in a number of real applications including multiple endpoint (or multi-arm) clinical trials ( Jennison and Turnbull, 2000, Chapter 15), multi-channel changepoint detection ( Tartakovsky et al., 2003) and its applications to biosurveillance ( Mei, 2010), genetics and genomics ( Dudoit and van der Laan, 2008), acceptance sampling with multiple criteria ( Baillie, 1987), and financial trading strategies ( Romano and Wolf, 2005). The between-stream data may be very dissimilar in distribution and dimension, but at the same time may be highly correlated, or even duplicated exactly in some cases, since they all are related to some phenomenon.
The scientist would like to control the overall error rate of the battery of experiments in order to be able to draw statistically-valid conclusions for each experiment once all experimentation has ceased, but also needs to be as efficient as possible with the finite resources available by “dropping” certain experiments (i.e., stopping experimentation) when additional data is no longer needed from that stream to reach a conclusion.
The proposed procedure, which we call the sequential Holm procedure because of its inspiration from Holm’s (1979) seminal fixed-sample procedure, shows simultaneous savings in expected sample size and less conservative error control relative to fixed sample, sequential Bonferroni, and other recently proposed sequential procedures in a simulation study. Treating each experiment as a hypothesis test and adopting the familywise error rate (FWER) metric, we give a procedure that sequentially tests each hypothesis while controlling both the type I and II FWERs regardless of the between-stream correlation, and only requires arbitrary sequential test statistics that control the error rates for a given stream in isolation. The between-stream data may differ in distribution and dimension but also may be highly correlated, even duplicated exactly in some cases. The scientist would like to control the overall error rate in order to draw statistically-valid conclusions from each experiment, while being as efficient as possible. This paper addresses the following general scenario: A scientist wishes to perform a battery of experiments, each generating a sequential stream of data, to investigate some phenomenon. A confidence interval for the parameter θ, with confidence level or coefficient γ, is an interval ( u ( X ), v ( X ) ). Let X be a random sample from a probability distribution with statistical parameter θ, which is a quantity to be estimated, and φ, representing quantities that are not of immediate interest. 6.1 Confidence interval for specific distributions.4.1 Confidence procedure for uniform location.Likewise, greater variability in the sample produces a wider confidence interval, and a higher confidence level would demand a wider confidence interval. All else being the same, a larger sample would produce a narrower confidence interval. įactors affecting the width of the CI include the confidence level, the sample size, and the variability in the sample.
For example, out of all intervals computed at the 95% level, 95% of them should contain the parameter's true value. The confidence level represents the long-run proportion of corresponding CIs that contain the true value of the parameter. A confidence interval is computed at a designated confidence level the 95% confidence level is most common, but other levels, such as 90% or 99%, are sometimes used. In frequentist statistics, a confidence interval ( CI) is a range of estimates for an unknown parameter. The blue intervals contain the mean, and the red ones do not. At the center of each interval is the sample mean, marked with a diamond. The colored lines are 50% confidence intervals for the mean, μ. Each row of points is a sample from the same normal distribution.
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