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

Confidence sets that satisfy this first requirement can be classified into the following two groups. The first group consists of 1-α confidence sets with scaled expected volume that is less than or equal to 1 for all parameter values, so that these dominate the standard 1 -α confidence set. Examples of such confidence sets are the Stein-type confidence interval for the normal variance (see e.g. Maata and Casella, 1990 for a review) and Stein-type confidence sets for the multivariate normal mean (see e.g. Saleh, 2006 for a review). The second group consists of 1 -α confidence sets that satisfy this first requirement, when dominance of the usual 1-α confidence set is not possible (the scaled expected volume must exceed 1 for some parameter values). This second group includes confidence intervals described by Pratt (1961), Brown et al (1995) and Puza and O’Neill (2006ab). This second group also includes 1 -α confidence sets that satisfy the additional requirements that (a) the maximum (over the parameter space) of the scaled expected volume is not too much larger than 1 and (b) the confidence set reverts to the usual 1 -α confidence set when the data happen to strongly contradict the prior information. Confidence intervals that utilize uncertain prior information and satisfy these additional requirements have been proposed by Farchione and Kabaila (2008) and Kabaila and Giri (2009a).

Consider a 2 × 2 factorial experiment with c replicates, where c > 1. Label the factors A and B. Suppose that the parameters of interest are the four population cell means θ 00 , θ 10 , θ 01 , θ 11 where, for example, θ 10 denotes the expected response when factor A is high and factor B is low. Also suppose that, on the basis of previous experience with similar data sets and/or expert opinion and scientific background, we have uncertain prior information that the two-factor interaction is zero. Our aim is to find simultaneous frequentist confidence intervals for the population cell means, with simultaneous confidence coefficient 1 -α, that utilize this prior information.

Throughout this paper, we find that the simultaneous confidence intervals of interest define a cube. For convenience, we henceforth refer to this cube, rather than the corresponding simultaneous confidence intervals. Let θ = (θ 00 , θ 10 , θ 01 , θ 11 ).

The standard 1 -α confidence cube for θ is found using Tukey’s method (described e.g. on p.289 of Bickel and Doksum (1977)). We assess a 1 -α confidence cube for θ using the scaled expected volume of this confidence cube, defined to be the ratio (expected volume of this confidence cube)/(expected volume of standard 1 -α confidence cube). We say that this confidence cube utilizes the prior information if it has the following desirable properties. This confidence cube has scaled expected volume that (a) is significantly less than 1 when the two-factor interaction is zero and (b) has a maximum value that is not too much larger than 1. Also, this confidence cube coincides with the standard 1 -α confidence cube when the data strongly contradict the prior information that the two-factor interaction is zero.

The development of this new 1 -α confidence cube parallels the development by Kabaila and Giri (2009a) of a new 1 -α confidence interval for a specified simple effect, for 2 × 2 factorial data, that utilizes uncertain prior information that the two-factor interaction is zero. It is fortunate that the symmetries in the more complicated context considered in the present paper lead to simplifications that make the computation of the new 1 -α confidence cube feasible.

In Section 3 we provide a numerical illustration of the properties of this new confidence cube for 1 -α = 0.95 and c = 2. The two-factor interaction is described by the parameter β 12 in the regression model used for the experiment. The uncertain prior information is that β 12 = 0. Define the parameter γ = β 12 / var( β12 ), where β12 denotes the least squares estimator of β 12 . The scaled expected volume of the new confidence cube for θ is an even function of γ. The bottom panel of Figure 2 is a plot of the square root of the scaled expected volume of the new 0.95 confidence cube for

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