Medical Diagnostics NIAID Funding Opportunities Maria Y. Giovanni ...
Putting Into Practice What I Learned from FSU Statistics Professors Michael Proschan NIAID.
-
Upload
ross-prosper-wilkinson -
Category
Documents
-
view
214 -
download
0
Transcript of Putting Into Practice What I Learned from FSU Statistics Professors Michael 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!
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
( )P X A( ,1)X N
( ) 0 or 1P X A
The Great Basu!
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
( )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
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
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
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
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
The Indian Connection
(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
Influences on Teaching
• Can modify quincunx for non-iid rvs
• Permutation test in paired setting
T C Paired difference (T-C)
5 2 3
Influences on Teaching
• Can modify quincunx for non-iid rvs
• Permutation test in paired setting
C T Paired difference (T-C)
5 2 -3
Influences on Teaching
• 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
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
12
ii
d d
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,
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!