Simultaneous confidence intervals for the population cell means, for two-by-two factorial data, that utilize uncertain prior information

TL;DR

This paper develops new simultaneous confidence intervals for two-by-two factorial experiment cell means that incorporate uncertain prior information about the interaction term, resulting in smaller expected volume when the prior is correct and reverting to standard intervals when data contradicts the prior.

Contribution

The paper introduces a novel method for constructing confidence intervals that utilize uncertain prior information, improving efficiency when the prior is accurate.

Findings

Intervals have smaller expected volume when the interaction is zero.

Intervals revert to standard Tukey intervals when data contradicts prior.

Application demonstrated on a real data set.

Abstract

Consider a two-by-two factorial experiment with more than 1 replicate. Suppose that we have uncertain prior information that the two-factor interaction is zero. We describe new simultaneous frequentist confidence intervals for the 4 population cell means, with simultaneous confidence coefficient 1-alpha, that utilize this prior information in the following sense. These simultaneous confidence intervals define a cube with expected volume that (a) is relatively small when the two-factor interaction is zero and (b) has maximum value that is not too large. Also, these intervals coincide with the standard simultaneous confidence intervals obtained by Tukey's method, with simultaneous confidence coefficient 1-alpha, when the data strongly contradict the prior information that the two-factor interaction is zero. We illustrate the application of these new simultaneous confidence intervals to a real data set.