A Personal Tribute to Milton Sobel: Selecting the Best Treatment

Abstract

The paper begins with a sketch of Milton Sobel's life and work, which is followed by a summary of his pioneering collaborations in selecting the best among k(≥2) populations. It takes us back to Bechhofer et al. (1954), where a path-breaking two-stage selection methodology was developed for normal populations with unknown means and a common but unknown variance.

Every two-stage methodology begins with a pilot sample. However, no easy practical approach existed for determining an appropriate pilot size until recently. We have summarized two recent ideas to determine a pilot size.

Then, the methodology of Bechhofer et al. (1954) is extended for situations where population variances are with σ²(>0) being unknown but c _i (>0) being known, not all equal, i = 1,…, k(≥2). This new methodology does not need any new statistical table for its implementation beyond what was required by the original 1954 methodology.

Several illustrations are included from practical problems using real data. The paper ends with brief recollections of Milton Sobel and some personal thoughts.

Recommended by T. K. S. Solanky

Keywords:

• Clinical trial data
• Dermatology
• Framingham Heart Study
• Homeland security
• Horticulture data
• Implementation and data analysis
• Indifference zone approach
• Multiple comparisons
• Multiple of a common variance
• Pilot sample size
• Selection and ranking
• Two-stage methodologies

Subject Classifications:

• 62L10
• 62L05
• 62F07
• 60C05
• 62J15

ACKNOWLEDGMENTS

I am truly grateful to Pinyuen Chen and S. Panchapakesan for kindly going through an earlier version of this manuscript. They both gave me important suggestions and some key references that I had originally missed. Thanks to Pinyuen and Kesan for their help and encouragement. I also received valuable commentaries from two referees and an associate editor. Their thoughtful considerations helped me in preparing the final manuscript. I remain indebted to the referees and the associate editor. The sources of the photos used in Figure 4 (Hokwon Cho) and those in Figures 1 and 5 (Milton Sobel) are gratefully acknowledged.

Figure 1 Milton Sobel attending school, New York City.

Figure 2 Milton Sobel at the McMaster University conference celebrating Charlie Dunnett's 70th birthday, May 29–June 1, 1991.

Figure 3 Pinyuen Chen and Milton Sobel. University of Connecticut, Storrs, December 10, 1990.

Figure 4 Milton Sobel's birthday celebration, August 30, 1994, in their residence at Santa Barbara. Milton is seen with his wife, Florence (seated right), and a former Ph.D. student, Hokwon Cho (standing right) with his wife, Sookhyun (seated left).

Figure 5 Milton Sobel, a passionate chess player.

Figure 6 Milton Sobel and Herbert Robbins during the videotaping of a discussion. University of Connecticut–Storrs, December 10, 1990.

Notes

Recommended by T. K. S. Solanky