Smoothed multiple endpoint procedures.

Classical multiple endpoint testing procedures such as Scheffe,

Tukey, Bonferroni etc. are oftentimes too conservative. This is

particularly true if the number of endpoints is large as in many recent

applications. In an effort to declare significance of endpoints more

rapidly than classical multiple comparison methods permit, procedures with

stepwise structure were introduced. Most recent procedures have a

stepwise structure, that is they are either single-step, step-up or

step-down. To evaluate and compare procedures the proposers have studied

properties of these stepwise procedures. One very surprising result of

considerable practical value is that the popular step-up procedure is

inadmissible, even for a vector risk function consisting of average size

as one component and average type II error as the other component. The

step-up and step-down procedures also have a disturbing practical

property; namely a small negative change in one variable accompanied with

reasonably sizeable positive changes in other variables can lead from many

rejections to many acceptances. The current proposal is to find 'smooth'

procedures that retain the desirable properties of stepwise procedures

while improving on the shortcomings. By representing the step-up

procedure as a linear combination of products of indicator functions a

smooth competitor to step-up is found and will be evaluated. Another idea

proposed is to find a parametric empirical Bayes procedure that will

control the average size component of the vector risk used to evaluate

procedures. A critical aspect of the proposal is to study the amount of

improvement the new methods have over step-up.

The investigators study the problem of how to decide which among

many possible individual entities are significant. For example, there are

thousands of a person's genes examined in a microarray. Which genes are

different or expressed differently when one examines cancer patients

compared to non-cancer patients. To be able to decide such could provide

a valuable diagnostic tool for early detection of cancer. This example,

is one of many, that illustrates the problem of multiple testing

procedures. Other areas of application include detection of covert

communications, detection of bioweapons, comparing several treatments with

a control, examination of mutual fund data in an effort to single out

successful funds, examining testing methodologies in education and

psychology settings. New and efficient statistical methodologies are

introduced and analyzed in this research undertaking.