Consider a linear regression model with n-dimensional response vector, p-dimensional regression parameter β and independent normally distributed errors. Suppose that the parameter of interest is θ = a T β where a is a specified vector. Define the s-dimensional parameter vector τ = C T β − t where C and t are specified. Also suppose that we have uncertain prior information that τ = 0. Part of our evaluation of a frequentist confidence interval for θ is the ratio (expected length of this confidence interval)/(expected length of standard 1 − α confidence interval), which we call the scaled expected length of this interval. We say that a 1 −α confidence interval for θ utilizes this uncertain prior information if (a) the scaled expected length of this interval is significantly less than 1 when τ = 0, (b) the maximum value of the scaled expected length is not too large and (c) this confidence interval reverts to the standard 1 − α confidence interval when the data happen to strongly contradict the prior information. LetΘ = a Tβ andτ = C Tβ − t, whereβ is the least squares estimator of β. We consider the particular case that that E (τ − τ )(Θ − θ) = 0, so thatΘ andτ are independent. We present a new 1 − α confidence interval for θ that utilizes the uncertain prior information that τ = 0. The following problem is used to illustrate the application of this new confidence interval. Consider a 2 3 factorial experiment with 1 replicate. Suppose that the parameter of interest θ is a specified linear combination of the main effects. Assume that the three-factor interaction is zero. Also suppose that we have uncertain prior information that all of the two-factor interactions are zero. Our aim is to find a frequentist 0.95 confidence interval for θ that utilizes this uncertain prior information.
Consider a case-control study in which the aim is to assess a factor's effect on disease occurrence. We suppose that this factor is dichotomous. Also suppose that the data consists of two strata, each stratum summarized by a two-by-two table.A commonly-proposed two-stage analysis of this type of data is the following. We carry out a preliminary test of homogeneity of the stratum-specific odds ratios. If the null hypothesis of homogeneity is accepted then we find a confidence interval for the assumed common value (across strata) of the odds ratio. We examine the statistical properties of this two-stage analysis, based on the Woolf method, on confidence intervals constructed for the stratum-specific odds ratios, for large numbers of cases and controls for each stratum. We provide both a Monte Carlo simulation method and an elegant large-sample method for this examination. These methods are applied to obtain numerical results in the context of the large numbers of cases and controls for each stratum that arose in a real-life dataset. In this context, we find that the preliminary test of homogeneity of the stratum-specific odds ratios has a very harmful effect on the coverage probabilities of these confidence intervals.
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