putting into practice what i learned from fsu statistics professors michael proschan niaid

Putting Into Practice What I Learned from FSU Statistics

ProfessorsMichael Proschan

NIAID

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

( )P X A( ,1)X N

( ) 0 or 1P X A

The Great Gazoo!


• 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

( )P X A( ,1)X N

( ) 0 or 1P X A

The Great Basu!


• is ancillary: its distribution does not depend on μ

• X is a complete, sufficient statistic• Basu’s theorem: X is independent of

( )I X A

( )I X A

( ) ( ( ) 1)P X A P X A I X A

( ) [ ( ) 1]P X A P I X A 2[ ( )]P X A

( ) 0 or 1P X A


• 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


• 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


• 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

If in distribution and in distribution, and

and in probability, where and are constants,

then in distribution

n n

n n

n n n n

X X Y Y

a a b b a b

a X b Y aX bY


(0,1) ( "twittles like" (0,1))n nX N X N

(so also "twittles like" (0,1))n n nY X Y N

Let ( , ) iid (0,1)X Y N

Then in distribution and in distribution,

but 0 and (0,2)n n

n n

X X Y Y

X Y X Y N

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

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

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

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:

22 2 nn

n n

22 1(exact tie) (1/ 2) nnP

n n

With 100 million voters, P(exact tie)≈1/18,000

Not Very Probable?

Much more probable than you would think!

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?

1 -1 0 0 0

1 1 -2 0 0

1 1 1 -3 0

1 1 1 1 1

A

1 -1 0 0 0

2 21 1 -2

0 06 6 61 1 1 -3

, where 012 12 12 12

1 1 1 1 1

Y HX H

n n n n n

Linear Models

2 2 2|| || || || ' ' ' || ||Y H X X H H X X X X

2 2 i iY X

2

2 2 2- = i

i n i

XY Y X

n

12 2

1 1

= ( )n n

i ii i

Y X X

Linear Models

Linear Models

• H called the Helmert transformation• By Helmert, if interim and final variance

estimates are sk2 and sn

2,

• Makes it easy to derive the distribution of (n-1)sn

2 given (k-1)sk2

1 12 2 2 2

1 1

{( 1) , ( 1) } , k n

k n i ii i

k s n s Y Y

Linear Models

2 2 2 2 2 21 1 1 1{( 1) | ( 1) } ( ... | ... )n k n kP n s v k s u P Y Y v Y Y u

2 2 2 2 2 21 1 1 1 1{( ... ) ( ... ) | ... )k k n kP Y Y Y Y v Y Y u

2 2 2 21 1( ... ) ( ... )k n k nP u Y Y v P Y Y v u

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

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


• Can modify quincunx for non-iid rvs

• Permutation test in paired setting

T C Paired difference (T-C)

5 2 3


• Can modify quincunx for non-iid rvs

• Permutation test in paired setting

C T Paired difference (T-C)

5 2 -3


• Test statistic:

• Sn is sum of independent, symmetric binary rvs

• Is Sn asymptotically normal?

, w.p 1/ 2

+ w.p 1/ 2n i i i

i

S X X d

d


• 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

12

ii

d d

Very abnormal!


• 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,

2

1

(0,1) in distributionn

n

ii

SN

d

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!

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

putting into practice what i learned from fsu statistics professors michael proschan niaid

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