department of biostatistics faculty research seminar series what am i doing? (besides teaching biost...

Department of BiostatisticsDepartment of Biostatistics Faculty Research Seminar SeriesFaculty Research Seminar Series

What am I doing?What am I doing?(Besides teaching BIOST 2083: Linear Models)(Besides teaching BIOST 2083: Linear Models)

Abdus S Wahed, Ph.D.Assistant Professor

Abdus S Wahed Faculty research seminar October 8, 2004

Survival Analysis Related to Multi-StageSurvival Analysis Related to Multi-Stage Randomization Designs in Clinical TrialsRandomization Designs in Clinical Trials

Skew-Symmetric DistributionsSkew-Symmetric Distributions

Statistical Modeling of Hepatitis C Viral DynamicsStatistical Modeling of Hepatitis C Viral Dynamics


Topics


Multi-stage Randomization Designs In Clinical TrialsMulti-stage Randomization Designs In Clinical Trials

Patients randomized to two or more treatments in the first Patients randomized to two or more treatments in the first stage (upon entry into the trial) stage (upon entry into the trial)

Those who Those who respondrespond to initial treatment are randomized to to initial treatment are randomized to two or more available treatments in the second stagetwo or more available treatments in the second stage

Those who Those who respondrespond to the second-stage treatment, they are to the second-stage treatment, they are randomized to two or more available treatments in the third randomized to two or more available treatments in the third stagestage

And so on…..And so on…..



All patients in CALGB clinical trial

InitialRandomizationStandard chemotherapy Chemotherapy + GMCSF

NoYes

Consent?

Respond?

YesNo

Respond?

Second Randomization

Maintenance I Maintenance II

Yes

Follow-up

NoNo



Question of Interest and Available AnswersQuestion of Interest and Available Answers

Which combination of therapies results in the Which combination of therapies results in the longest survival?longest survival?

Usual Analysis:Usual Analysis:– Separates out two stagesSeparates out two stages

Lunceford et al. (Lunceford et al. (Biometrics, 2002Biometrics, 2002):):– Defined treatment strategies such as:Defined treatment strategies such as:

““Treat with X followed by Y if respond to X and consents to Treat with X followed by Y if respond to X and consents to Y-randomization”Y-randomization”

– Consistent estimators for mean survival time under Consistent estimators for mean survival time under each strategyeach strategy



Question of Interest and Available AnswersQuestion of Interest and Available Answers

Wahed and Tsiatis (Wahed and Tsiatis (Biometrics, 2004Biometrics, 2004):):– Consistent and Consistent and efficientefficient estimators for mean survival estimators for mean survival

time (and survival probability) under each strategy time (and survival probability) under each strategy when there is no censoringwhen there is no censoring

Wahed and Tsiatis (Wahed and Tsiatis (Submitted, 2004Submitted, 2004):):– Consistent and Consistent and efficientefficient estimators for mean survival estimators for mean survival

time (and survival probability) under each strategy time (and survival probability) under each strategy for independent right censoringfor independent right censoring



Question of Interest and Current ResearchQuestion of Interest and Current Research

Recent work:Recent work:– How do you efficiently estimate quantiles of survival How do you efficiently estimate quantiles of survival

distribution for each treatment strategy? distribution for each treatment strategy?

– A clinical question of interest is what is the A clinical question of interest is what is the estimated mean survival for a population treated estimated mean survival for a population treated according to the policy according to the policy

““Treat with X followed by Y if respond to X and consents to Treat with X followed by Y if respond to X and consents to Y-randomization”Y-randomization”



Question of Interest and Current ResearchQuestion of Interest and Current Research

Work in progress Work in progress – Probability of randomization at any stage was Probability of randomization at any stage was

assumed to be independent of previous outcomeassumed to be independent of previous outcome but but can be generalized to depend on the data can be generalized to depend on the data collected prior to the randomization collected prior to the randomization

– Sample size determination (thanks to Dr. Majumder)Sample size determination (thanks to Dr. Majumder)

Other Issues Other Issues – Where censoring can depend on the observed dataWhere censoring can depend on the observed data– Log-rank-type tests for comparing treatment Log-rank-type tests for comparing treatment

strategiesstrategies



Statistical techniques I frequently employStatistical techniques I frequently employ

Martingles (related to censoring)Martingles (related to censoring)Semiparametric methods Semiparametric methods Inverse-probability-weightingInverse-probability-weightingCounterfactual random variables (even Counterfactual random variables (even when I am not interested in causal when I am not interested in causal inference)inference)Formal theory of monotone coarsening Formal theory of monotone coarsening (missingness)(missingness)




Main result Main result ((Derived distributions, Wahed, 2004Derived distributions, Wahed, 2004 ): ):

If If f(x)f(x) is a density with CDF is a density with CDF F(x),F(x), and and g(y)g(y) is is a density with support [0, 1], thena density with support [0, 1], then

h(z)=g[F(z)]f(z)h(z)=g[F(z)]f(z) (1)(1)

defines a probability density function.defines a probability density function.




Observation:Observation:– h(z)=f(z),h(z)=f(z), if if g(.)g(.) is uniform is uniform

– IfIf f f and and gg are symmetric, so isare symmetric, so is hh..

– If If gg is skewed and is skewed and ff is symmetric (or is symmetric (or asymmetric), then asymmetric), then hh is skewed. is skewed.



Innovation: Innovation: – BetaBetakk-normal distribution-normal distribution

TakeTake f f in (1) to be a standard normal distribution in (1) to be a standard normal distribution and and gg to be a beta distribution call the to be a beta distribution call the corresponding derived distribution from (1) corresponding derived distribution from (1) hh11

Take Take f f to be to be hh1 1 and and g g to be a beta distribution to be a beta distribution

and call the derived distributionand call the derived distribution h h22

Repeat Repeat kk-times.-times.




-4 -2 2 4

0.2

0.4

0.6

0.8

1

1.2

BetaN10,8,0,1BetaN10,3,0,1BetaN5,3,0,1BetaN5,1,0,1N0,1

Beta-normal DistributionsBeta-normal Distributions



N(0,1)

BetaN(5,1,0,1)

BetaN(5,3,0,1)

BetaN(10,3,0,1)

BetaN(10,8,0,1)

Innovation: Innovation:

– Triangular-normal distributionTriangular-normal distribution

– Beta-Gamma distributionBeta-Gamma distribution





Application:Application:– Distributions that are close to normal but Distributions that are close to normal but

have one tail extended (or squeezed ) can have one tail extended (or squeezed ) can be modeled by skew-normal distributionsbe modeled by skew-normal distributions

– Mixed effect modeling with non-normal error Mixed effect modeling with non-normal error distributionsdistributions






V(t ) = VV(t ) = V00 { A exp [- { A exp [-11(t – t(t – t00)]+)]+ (1- A) exp[-(1- A) exp[-2 2 (t – t(t – t00)]} t > t)]} t > t00 --- (4) --- (4)

where where 11 = ½ { ( c + = ½ { ( c + ) + [ ( c- ) + [ ( c- ) )22 + 4 ( 1 - + 4 ( 1 - ) c ) c ] ] ½½ } }22 = ½ { ( c + = ½ { ( c + ) - [ ( c- ) - [ ( c- ) )22 + 4 ( 1 - + 4 ( 1 - ) c ) c ] ] ½ ½ } } A = (A = ( c - c - 22 ) / ( ) / (1 1 - - 2 2 ))




1.1. Assumes Assumes being constant over time, which is not being constant over time, which is not the case with PEG-Interferon alpha-2a (Pegasysthe case with PEG-Interferon alpha-2a (Pegasys).).

2.2. Only works with the biphasic viral level declines. Only works with the biphasic viral level declines. (Herrmann et al., 2003 Hepatology)(Herrmann et al., 2003 Hepatology)

3.3. Ignores the possible correlations in viral levels over Ignores the possible correlations in viral levels over time.time.


0 5 10 15 20 25

050

0010

000

1500

0

days

pegI

FN



0 5 10 15 20 25

050

0010

000

1500

0

days

pegI

FN




= = ( ( (t) ) = (t) ) = maxmax * *(t) / ((t) / ( + + (t) )(t) )

(t)(t) = any function that describes the = any function that describes the pattern of drug concentration over timepattern of drug concentration over time




myoas

allE

0 200 400 600 800 1000

0.0

0.2

0.4

0.6

maxmax * *(t) (t)

( ( (t) ) = ___________(t) ) = ___________ + + (t)(t)

K

department of biostatistics faculty research seminar series what am i doing? (besides teaching biost...

Documents

current research work

survival probability

survival analysis

stage treatment

mean survival time

current research recent

clinical question

longest survival