1. Biostatistics 760
Random Thoughts

2. Upcoming Classes
Bios 761: Advanced Probability and Statistical Inference
Bios 767: Longitudinal Data Analysis
Bios 780: Theory and Methods for Survival Analysis
Bios 841: Statistical Consulting

3. Bios 761 Frequentist and Bayesian decision theory
Hypothesis testing: UMP tests, etc.
Bootstrap and other methods of inference
Stochastic processes:
Poisson processes
Markov chains
Martingales
Brownian motion

4. Bios 780 Time-to-event data Right censoring
Counting processes; martingales
Semiparametric approaches
Kaplan-Meier estimator
Log-rank statistic
Cox model
Data analysis

5. Bios 841 Consulting versus collaboration
Bringing it all together to solve problems
Communicating about statistics
Three real problems
Three journal style reports
One final oral presentation
Real time problem solving
What is the role of statistical theory?

6. A Few War Stories As a student: thesis on surrogates
As a postdoc: infectious diseases
As a new professor: cystic fibrosis (CF)*
Working on tenure: empirical processes
Empirical processes and cancer*
Chair of the DSMC for NICHD
Artificial intelligence and NSCLC

7. CF Neonatal Screening 1992: Joined Phil Farrell’s CF study team
1997: Farrell, Kosorok, Laxova, et al, published in NEJM
2004 (Oct. 15): CDC recommended CF newborn screening: the 1997 article was judged the only valid randomized trial
States offering CF newborn screening: 3 in 1997, 12 in 2004, 45 today

8. What Role Did “Theory” Play?
Used state-of-the-art statistical methods that were robust (GEE)
In other CF research we have used:
Current status methods (parametric, robust)
Constrained regression estimation
Semiparametric bootstrap inference
Martingale based survival analysis
New work using artificial intelligence

9. Empirical Processes and Cancer
Non-Hodgkin’s Lymphoma Prognostic Factors Project (1993, NEJM)
Cox proportional hazards model employed to ascertain risks of 5 prognostic factors: Age, performance Status, serum lactate dehydrogenase Level, number of extranodal disease Sites, tumor Stage
Diagnostics show the model fits poorly

10. What is the Problem? Poor survival function prediction
Possibly incorrect interpretation of risk factor effects
A model that adds a single parameter to the Cox model was developed and fit
This new model fits well (Kosorok,Lee and Fine, 2004)
Inference for the new model is complicated

11. What Does Theory Tell Us?
We can derive valid inferential tools for the new model: estimation and bootstrap
Robustness was also studied: we learn theoretically that the Cox model is robust to this kind of model misspecification:
The direction of the regression coefficients is preserved
Should use robust variance for Cox model

12. Theory Versus Applications
The title implies there is conflict between theory and applications
This isn’t true!
Theory provides a basis for correct thinking and problem solving for applications
Applications drive new theoretical development

13. Theory Can Be Impractical
Law of iterated logarithm: needs sample size of 108 (“asymptopia”).
Sometimes higher order approximations are needed before it becomes useful.
Sometimes computational properties of asymptotically optimal estimators are poor.
Some hard problems take years to solve.

14. Why Theory is Needed Often it does work for practical sample sizes.
Can reveal properties that are universally valid: simulation studies are limited to the scenarios investigated.
Theory can lead toward methodological solutions (Cook and Kosorok, 2004 JASA).
Theory can drive scientific discovery.
Some results are beautiful.

15. Data Mining Versus Inference
Data mining is summarizing and representing data no matter how complicated
Inference is determining valid measures of uncertainty
Patterns obtained from data mining can be misleading
Inference without data mining may miss important structure

16. The Core of Statistics Statistics is the science of science
How do we learn from our world and draw meaningful and valid conclusions from it?
Need both data mining and valid inference
Requires a unique kind of intuition
Needs many different intellectual perspectives
One of the most challenging of all fields

17. Everyone Needs Core Literacy
All statisticians need to know enough theory to have core literacy about statistics and to be able to problem solve
All statisticians need to know enough about applications to know what is important
All biostatisticians need to know enough statistical methods to be useful in practice
The purpose of a Ph.D. in Biostatistics is to enable the creation of new methodology

18. Semiparametric Inference
The study of statistical models with parametric and/or nonparametric parts
Can achieve trade-off between scientific meaning and model “robustness”
Estimation and inference are often hard
There exists an efficiency bound for parametric and some nonparametric parts
NPMLE, testing and estimating equations

19. Empirical Processes
Tools for complex model inference and high dimensional data
Can determine universal properties of semiparametric methods:
Consistency
Rate of convergence
Limiting distributions
Valid inference (empirical process bootstrap)
Empirical processes are everywhere

20. The Road Ahead
Whatever you choose to do, the core statistical theory classes will help you.
Be patient as your learn.
Be willing to work hard (struggle is good).
It takes many different kinds of thinkers with different learning styles.
There are important discoveries to be made in both applications and theory.