Simultaneous Confidence Intervals for the Population Cell Means, for Two-by-Two Factorial Data, that Utilize Uncertain Prior Information

Consider a two-by-two factorial experiment with more than one replicate. Suppose that we have uncertain prior information that the two-factor interaction is zero. We describe new simultaneous frequentist confidence intervals for the four 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.