1. Department of Biostatistics Faculty Research Seminar Series What am I doing? (Besides teaching BIOST 2083: Linear Models) Abdus S Wahed, Ph.D. Assistant Professor Abdus S WahedFaculty research seminarOctober 8, 2004

2. Survival Analysis Related to Multi-Stage Randomization Designs in Clinical Trials Randomization Designs in Clinical Trials Skew-Symmetric Distributions Statistical Modeling of Hepatitis C Viral Dynamics Abdus S WahedFaculty research seminarOctober 8, 2004 Topics Department of Biostatistics Faculty Research Seminar Series

3. 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 WahedFaculty research seminarOctober 8, 2004 Department of Biostatistics Faculty Research Seminar Series

4. All patients in CALGB clinical trial Initial Randomization Standard chemotherapyChemotherapy + GMCSF No Yes Consent? Respond? Yes No Respond? Second Randomization Maintenance IMaintenance II Yes Follow-up No Abdus S WahedFaculty research seminarOctober 8, 2004 Department of Biostatistics Faculty Research Seminar Series

5. 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 Y- randomization” –Consistent estimators for mean survival time under each strategy Abdus S WahedFaculty research seminarOctober 8, 2004 Department of Biostatistics Faculty Research Seminar Series

6. 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 WahedFaculty research seminarOctober 8, 2004 Department of Biostatistics Faculty Research Seminar Series

7. 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 Y- randomization” Abdus S WahedFaculty research seminarOctober 8, 2004 Department of Biostatistics Faculty Research Seminar Series

8. 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 WahedFaculty research seminarOctober 8, 2004 Department of Biostatistics Faculty Research Seminar Series

9. 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 WahedFaculty research seminarOctober 8, 2004 Department of Biostatistics Faculty Research Seminar Series

10. 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) h(z)=g[F(z)]f(z) (1) defines a probability density function. Abdus S WahedFaculty research seminarOctober 8, 2004 Department of Biostatistics Faculty Research Seminar Series

11. 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 WahedFaculty research seminarOctober 8, 2004 Department of Biostatistics Faculty Research Seminar Series

12. Innovation: –Beta k -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) h 1 Take f to be h 1 and g to be a beta distribution and call the derived distribution h 2 Repeat k-times. Abdus S WahedFaculty research seminarOctober 8, 2004 Skew-Symmetric Distributions Department of Biostatistics Faculty Research Seminar Series

13. Beta-normal Distributions Abdus S WahedFaculty research seminarOctober 8, 2004 Department of Biostatistics Faculty Research Seminar Series N(0,1) BetaN(5,1,0,1) BetaN(5,3,0,1) BetaN(10,3,0,1) BetaN(10,8,0,1)

14. Innovation: –Triangular-normal distribution –Beta-Gamma distribution Abdus S WahedFaculty research seminarOctober 8, 2004 Skew-Symmetric Distributions Department of Biostatistics Faculty Research Seminar Series

15. 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 WahedFaculty research seminarOctober 8, 2004 Department of Biostatistics Faculty Research Seminar Series

16. Statistical Modeling of Hepatitis C Viral Dynamics Abdus S WahedFaculty research seminarOctober 8, 2004 Department of Biostatistics Faculty Research Seminar Series

17. Statistical Modeling of Hepatitis C Viral Dynamics Abdus S WahedFaculty research seminarOctober 8, 2004 Department of Biostatistics Faculty Research Seminar Series V(t ) = V 0 { A exp [- 1 (t – t 0 )]+ V(t ) = V 0 { A exp [- 1 (t – t 0 )]+ (1- A) exp[- 2 (t – t 0 )]} t > t 0 --- (4) (1- A) exp[- 2 (t – t 0 )]} t > t 0 --- (4) where where 1 = ½ { ( c +  ) + [ ( c-  ) 2 + 4 ( 1 -  ) c  ] ½ } 1 = ½ { ( c +  ) + [ ( c-  ) 2 + 4 ( 1 -  ) c  ] ½ } 2 = ½ { ( c +  ) - [ ( c-  ) 2 + 4 ( 1 -  ) c  ] ½ } 2 = ½ { ( c +  ) - [ ( c-  ) 2 + 4 ( 1 -  ) c  ] ½ } A = (  c - 2 ) / ( 1 - 2 ) A = (  c - 2 ) / ( 1 - 2 )

18. Statistical Modeling of Hepatitis C Viral Dynamics Abdus S WahedFaculty research seminarOctober 8, 2004 Department of Biostatistics Faculty Research Seminar Series 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.

19. Statistical Modeling of Hepatitis C Viral Dynamics Abdus S WahedFaculty research seminarOctober 8, 2004 Department of Biostatistics Faculty Research Seminar Series

20. Statistical Modeling of Hepatitis C Viral Dynamics Abdus S WahedFaculty research seminarOctober 8, 2004 Department of Biostatistics Faculty Research Seminar Series  =  (  (t) ) =  max *  (t) / (  +  (t) )   (t) = any function that describes the pattern of drug concentration over time

21. Statistical Modeling of Hepatitis C Viral Dynamics Abdus S WahedFaculty research seminarOctober 8, 2004 Department of Biostatistics Faculty Research Seminar Series  max *  (t)  max *  (t)  (  (t) ) = ___________  +  (t)  +  (t) K