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Department of Biostatistics
Faculty Research Seminar Series
Abdus S Wahed, Ph.D.
Assistant Professor
What am I doing?
(Besides teaching BIOST 2083: Linear Models)
Abdus S Wahed
Faculty research seminar
October 8, 2004
Department of Biostatistics
Faculty Research Seminar Series
Topics
Survival Analysis Related to Multi-Stage
Randomization Designs in Clinical Trials
Skew-Symmetric Distributions
Statistical Modeling of Hepatitis C Viral Dynamics
Abdus S Wahed
Faculty research seminar
October 8, 2004
Department of Biostatistics
Faculty Research Seminar Series
Multi-stage Randomization Designs In Clinical Trials
Patients randomized to two or more treatments in the first
stage (upon entry into the trial)
Those who respond to initial treatment are randomized to two
or more available treatments in the second stage
Those who respond to the second-stage treatment, they are
randomized to two or more available treatments in the third
stage
And so on…..
Abdus S Wahed
Faculty research seminar
October 8, 2004
Department of Biostatistics
Faculty Research Seminar Series
All patients in CALGB clinical trial
Standard chemotherapy
No
Initial
Randomization
Chemotherapy + GMCSF
Respond?
Respond?
Yes
No
Maintenance I
No
Yes
Consent?
Yes
Second
Randomization
No
Maintenance II
Follow-up
Abdus S Wahed
Faculty research seminar
October 8, 2004
Department of Biostatistics
Faculty Research Seminar Series
Question of Interest and Available Answers
Which combination of therapies results in the
longest survival?
Usual Analysis:
– Separates out two stages
Lunceford et al. (Biometrics, 2002):
– Defined treatment strategies such as:
“Treat with X followed by Y if respond to X and consents to Yrandomization”
– Consistent estimators for mean survival time under
each strategy
Abdus S Wahed
Faculty research seminar
October 8, 2004
Department of Biostatistics
Faculty Research Seminar Series
Question of Interest and Available Answers
Wahed and Tsiatis (Biometrics, 2004):
– Consistent and efficient estimators for mean survival
time (and survival probability) under each strategy
when there is no censoring
Wahed and Tsiatis (Submitted, 2004):
– Consistent and efficient estimators for mean survival
time (and survival probability) under each strategy
for independent right censoring
Abdus S Wahed
Faculty research seminar
October 8, 2004
Department of Biostatistics
Faculty Research Seminar Series
Question of Interest and Current Research
Recent work:
– How do you efficiently estimate quantiles of survival
distribution for each treatment strategy?
– A clinical question of interest is what is the
estimated mean survival for a population treated
according to the policy
“Treat with X followed by Y if respond to X and consents to Yrandomization”
Abdus S Wahed
Faculty research seminar
October 8, 2004
Department of Biostatistics
Faculty Research Seminar Series
Question of Interest and Current Research
Work in progress
– Probability of randomization at any stage was
assumed to be independent of previous outcome
but can be generalized to depend on the data
collected prior to the randomization
– Sample size determination (thanks to Dr. Majumder)
Other Issues
– Where censoring can depend on the observed data
– Log-rank-type tests for comparing treatment
strategies
Abdus S Wahed
Faculty research seminar
October 8, 2004
Department of Biostatistics
Faculty Research Seminar Series
Statistical techniques I frequently employ
Martingles (related to censoring)
Semiparametric methods
Inverse-probability-weighting
Counterfactual random variables (even
when I am not interested in causal
inference)
Formal theory of monotone coarsening
(missingness)
Abdus S Wahed
Faculty research seminar
October 8, 2004
Department of Biostatistics
Faculty Research Seminar Series
Skew-Symmetric Distributions
Main result (Derived distributions, Wahed, 2004 ):
If f(x) is a density with CDF F(x), and g(y) is
a density with support [0, 1], then
h(z)=g[F(z)]f(z)
(1)
defines a probability density function.
Abdus S Wahed
Faculty research seminar
October 8, 2004
Department of Biostatistics
Faculty Research Seminar Series
Skew-Symmetric Distributions
Observation:
– h(z)=f(z), if g(.) is uniform
– If f and g are symmetric, so is h.
– If g is skewed and f is symmetric (or
asymmetric), then h is skewed.
Abdus S Wahed
Faculty research seminar
October 8, 2004
Department of Biostatistics
Faculty Research Seminar Series
Skew-Symmetric Distributions
Innovation:
– Betak-normal distribution
Take f in (1) to be a standard normal distribution
and g to be a beta distribution call the
corresponding derived distribution from (1) h1
Take f to be h1 and g to be a beta distribution
and call the derived distribution h2
Repeat k-times.
Abdus S Wahed
Faculty research seminar
October 8, 2004
Department of Biostatistics
Faculty Research Seminar Series
Beta-normal Distributions
BetaN(10,8,0,1)
N 0,1
1.2
BetaN
5, 1,0,1
BetaN
5, 3,0,1
BetaN
10 ,3,0,1
0.8
BetaN
10 ,8,0,1
0.6
1
BetaN(10,3,0,1)
BetaN(5,1,0,1)
BetaN(5,3,0,1)
0.4
N(0,1)
0.2
-4
Abdus S Wahed
-2
2
Faculty research seminar
4
October 8, 2004
Department of Biostatistics
Faculty Research Seminar Series
Skew-Symmetric Distributions
Innovation:
– Triangular-normal distribution
– Beta-Gamma distribution
Abdus S Wahed
Faculty research seminar
October 8, 2004
Department of Biostatistics
Faculty Research Seminar Series
Skew-Symmetric Distributions
Application:
– Distributions that are close to normal but
have one tail extended (or squeezed ) can
be modeled by skew-normal distributions
– Mixed effect modeling with non-normal error
distributions
Abdus S Wahed
Faculty research seminar
October 8, 2004
Department of Biostatistics
Faculty Research Seminar Series
Statistical Modeling of Hepatitis C Viral Dynamics
Abdus S Wahed
Faculty research seminar
October 8, 2004
Department of Biostatistics
Faculty Research Seminar Series
Statistical Modeling of Hepatitis C Viral Dynamics
V(t ) = V0 { A exp [-1(t – t0)]+
(1- A) exp[-2 (t – t0)]} t > t0
--- (4)
where
1 = ½ { ( c +  ) + [ ( c-  )2 + 4 ( 1 - ) c  ] ½ }
2 = ½ { ( c +  ) - [ ( c-  )2 + 4 ( 1 - ) c  ] ½ }
A = ( c - 2 ) / (1 - 2 )
Abdus S Wahed
Faculty research seminar
October 8, 2004
Department of Biostatistics
Faculty Research Seminar Series
Statistical Modeling of Hepatitis C Viral Dynamics
1.
Assumes  being constant over time, which is not
the case with PEG-Interferon alpha-2a (Pegasys).
2.
Only works with the biphasic viral level declines.
(Herrmann et al., 2003 Hepatology)
3.
Ignores the possible correlations in viral levels over
time.
Abdus S Wahed
Faculty research seminar
October 8, 2004
Department of Biostatistics
Faculty Research Seminar Series
15000
10000
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pegIFN
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pegIFN
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Statistical Modeling of Hepatitis C Viral Dynamics
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days
Abdus S Wahed
10
Faculty research seminar
October 8, 2004
Department of Biostatistics
Faculty Research Seminar Series
Statistical Modeling of Hepatitis C Viral Dynamics

= ( (t) ) =  max *(t) / ( + (t) )

(t) = any function that describes the
pattern of drug concentration over time
Abdus S Wahed
Faculty research seminar
October 8, 2004
Department of Biostatistics
Faculty Research Seminar Series
0.4
 max *(t)
( (t) ) = ___________
 + (t)
0.0
0.2
allE
0.6
Statistical Modeling of Hepatitis C Viral Dynamics
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Abdus S Wahed
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myoas
Faculty research seminar
October 8, 2004