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Class Notes in Statistics and Econometrics Part 22

CHAPTER 43 Multiple Comparisons in the Linear Model. Due to the isomorphism of tests and confidence intervals, we will keep this whole discussion in terms of confidence intervals. 43.1. Rectangular Confidence Regions Assume you are interested in two linear combinations of β at the same time | CHAPTER 43 Multiple Comparisons in the Linear Model Due to the isomorphism of tests and confidence intervals we will keep this whole discussion in terms of confidence intervals. 43.1. Rectangular Confidence Regions Assume you are interested in two linear combinations of 3 at the same time i.e. you want separate confidence intervals for them. If you use the Cartesian product or the intersection depending on how you look at it of the individual confidence intervals the confidence level of this rectangular confidence region will of necessity be different that of the individual intervals used to form this region. If you want the 971 972 43. MULTIPLE COMPARISONS IN THE LINEAR MODEL joint confidence region to have confidence level 95 then the individual confidence intervals must have a confidence level higher than 95 i.e. they must be be wider. There are two main approaches for compute the confidence levels of the individual intervals one very simple one which is widely applicable but which is only approximate and one more specialized one which is precise in some situations and can be taken as an approximation in others. 43.1.1. Bonferroni Intervals. To derive the first method the Bonferroni intervals assume you have individual confidence intervals Ri for parameter . In order R1 to make simultaneous inferences about the whole parameter vector you 1 take the Cartesian product R1 xR2 x- -x Ri it is defined by G R1 x R2 x- -x Ri Ji. if and only if G Ri for all i. Usually it is difficult to compute the precise confidence level of such a rectangular set. If one cannot be precise it is safer to understate the confidence level. The following inequality from elementary probability theory called the Bonferroni 43.1. RECTANGULAR CONFIDENCE REGIONS 973 inequality gives a lower bound for the confidence level of this Cartesian product Given i events Ej with Pr Ej 1 aj then Pr p Ej 1 52ai- Proof Pr p Ej 1 Pr J Ej 1 Pr Ej . The so-called Bonferroni bounds therefore have the individual