Biostatistics 760

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Biostatistics 760

• Random Thoughts

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Upcoming Classes

• Bios 761 Advanced Probability and Statistical
  Inference
• Bios 763 Generalized Linear Model Theory and
  Applications
• Bios 767 Longitudinal Data Analysis
• Bios 780 Theory and Methods for Survival
  Analysis
• Bios 841 Statistical Consulting

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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

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Bios 780

• Time-to-event data
• Right censoring
• Counting processes martingales
• Semiparametric approaches
• Kaplan-Meier estimator
• Log-rank statistic
• Cox model
• Data analysis

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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?

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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

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CF Neonatal Screening

• 1992 Joined Phil Farrells 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

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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

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Empirical Processes and Cancer

• Non-Hodgkins 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

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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

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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

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Theory Versus Applications

• The title implies there is conflict between
  theory and applications
• This isnt true!
• Theory provides a basis for correct thinking and
  problem solving for applications
• Applications drive new theoretical development

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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.

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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.

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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

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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

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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

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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

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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

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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.