Putting Into Practice What I Learned from FSU Statistics Professors

1. Putting Into Practice What I Learned from FSU Statistics Professors Michael Proschan NIAID

2. The Indian Connection • I recently tried to prove a theorem related to the monitoring of clinical trials • Last step: If and A is an event such that does not depend on μ, then • Pretty obvious, but how do you prove it? • Fred Flintstone called on The Great Gazoo!

3. The Indian Connection • I recently tried to prove a theorem related to the monitoring of clinical trials • Last step: If and A is an event such that does not depend on μ, then • Pretty obvious, but how do you prove it? • I call on The Great Basu!

4. The Indian Connection • is ancillary: its distribution does not depend on μ • X is a complete, sufficient statistic • Basu’s theorem: X is independent of

5. The Indian Connection • What I will remember most about Dr. Basu: • His ability to make the most complicated topics simple • “Let me ask you a question like this” • “Let me show you what he was trying to do” • His beautiful examples/counterexamples • 10 coin flips with P(heads)=p, test p=.5 against p>.5 at α=2-9; most powerful test throws out the last observation

6. The Indian Connection • The other half of the Indian connection was Dr. Sethuraman, who taught limit theory • I took that class at just the right time to solidify what I learned in Dr. McKeague’s probability • I learned so much from watching how Sethu thought • I also learned how to be careful about probability and asymptotic arguments

7. The Indian Connection • To work out asymptotics of monitoring clinical trials, we discuss a multivariate Slutsky theorem • To this day I worry it may be wrong because Sethu had me prove the following “theorem” • After I “proved” it on the board, Sethu pointed out the following counterexample

8. The Indian Connection

9. Not Very Probable! • I learned a lot from my probability professor, Dr. McKeague • Even though he hated it when I used Skorohod’s representation theorem! • Several years ago, my sister-in-law’s boyfriend, Pablo, said he was helping a doctor accused of overcharging Medicaid • He asked for my help to defend her

10. Not Very Probable! • State’s approach • Take random sample of the doctor’s Medicaid claims and compute sample mean overcharge • Construct 90% confidence interval for population mean overcharge, μ • Charge doctor nL, where • n is # of Medicare claims that year for the doctor • L is the lower limit of the confidence interval for μ

11. Not Very Probable! • I told Pablo I thought state’s approach was pretty reasonable • The only point of contention was whether the state really took a random sample • It appeared to be a convenience sample • Then I found out who the state’s expert witness was: • Dr. McKeague! • I’m not going against McKeague • They settled the case

12. Not Very Probable? • Recall the disputed election between Bush and Gore • Amazingly, almost an exact tie in popular vote • What is the probability of that? • From Dr. Leysieffer’s beautifully clear lecture notes on stochastic processes:

13. Not Very Probable? With 100 million voters, P(exact tie)≈1/18,000 Much more probable than you would think!

14. Linear Models • One area I have worked on is adaptive sample size calculation in clinical trials • Consider trial with paired differences X1,…,Xn, and want to test whether μ=0 • Sample size depends on σ2 • If we change sample size midstream based on updated within-trial variance, how different might the final variance be?

15. Linear Models

16. Linear Models

17. Linear Models • H called the Helmert transformation • By Helmert, if interim and final variance estimates are sk2 and sn2, • Makes it easy to derive the distribution of (n-1)sn2 given (k-1)sk2

18. Linear Models

19. Influences on Teaching • I learned different lessons about teaching from different professors • Clarity and organization • Dr. Leysieffer, Dr. Doss, Dr. Huffer • How to derive things yourself • My dad and the Indian connection (Drs. Basu and Sethuraman) • How to teach outside the box • Dr. Zahn

20. Influences on Teaching: • Quincunx is board with balls rolling down a triangular pattern of nails • Left or right bounce at row i is -1 or +1 independent of outcomes of previous rows • Each ball’s position at bottom represents sum of n iid displacements • Collection of balls in bins at bottom illustrates distribution of sum • Illustrates CLT if # rows large

22. Influences on Teaching • Can modify quincunx for non-iid rvs • Permutation test in paired setting T C Paired difference (T-C) 5 2 3

23. Influences on Teaching • Can modify quincunx for non-iid rvs • Permutation test in paired setting C T Paired difference (T-C) 5 2 -3

24. Influences on Teaching • Test statistic: • Sn is sum of independent, symmetric binary rvs • Is Sn asymptotically normal?

25. Influences on Teaching • Think about modified quincunx where horizontal distance between nails differs by row • When might normality not hold? • Suppose largest distance exceeds sum of all other distances • E.g., suppose

26. Very abnormal!

27. Influences on Teaching • The quincunx shows that some conditions are needed on the di to conclude asymptotic normality, but can noncomplying di arise as realizations of iid random variables? • Theorem:If the di are realizations from iid random variables with finite variance, then with probability 1,

28. Influences Beyond Statistics • Several professors helped me in ways that went beyond statistics • My dad • Dr. Hollander • Dr. Toler • Dr Zahn • He drove me to the edge, but brought me back!

29. Unforgettable Quotes • “This theorem is true only in general” • My dad • “Where is my duster” • Dr. Basu • “What belief, attitude, or position must have been present…” • Dr. Zahn • “Bob’s your uncle” • Dr. Meeter • “One upon n” (for 1/n) • Dr. Sethuraman