Some Statistical Aspects of Predictive Medicine

Richard Simon, D.Sc.Chief, Biometric Research Branch

National Cancer Institutehttp://brb.nci.nih.gov

Biometric Research Branch Websitehttp://brb.nci.nih.gov

• Powerpoint presentations• Reprints• BRB-ArrayTools software• Web based tools for clinical trial design with

predictive biomarkers

Prediction for Informing Treatment Selection

• Most cancer treatments benefit only a minority of patients to whom they are administered

• Being able to predict which patients are (are not) likely to benefit from a treatment might – Save patients from unnecessary complications and

enhance their chance of receiving a more appropriate treatment

– Help control medical costs – Improve the success rate of clinical drug development

Prognostic vs Predictive Biomarkers

• Predictive biomarkers– Measured before treatment to identify who is

likely or unlikely to benefit from a particular treatment

• Prognostic biomarkers– Measured before treatment to indicate long-

term outcome for patients untreated or receiving standard treatment

In Oncology

• Recognition of the heterogeneity of tumors of the same primary site

• Availability of the tools of genomics for characterizing tumors

• Focus on molecularly targeted drugs

• Has resulted in – Increased interest in prediction problems– Need for new clinical trial designs

• p>n prediction problems in which number of variables is much greater than the number of cases– Many of the methods of statistics are based

on inference problems– Standard model building and evaluation

strategies are not effective for p>n prediction problems

Model Evaluation for p>n Prediction Problems

• Goodness of fit is not a proper measure of predictive accuracy

• Importance of Separating Training Data from Testing Data for p>n Prediction Problems

Simulation Training Validation

1

2

3

4

5

6

7

8

9

10

p=7.0e-05

p=0.70

p=4.2e-07

p=0.54

p=2.4e-13

p=0.60

p=1.3e-10

p=0.89

p=1.8e-13

p=0.36

p=5.5e-11

p=0.81

p=3.2e-09

p=0.46

p=1.8e-07

p=0.61

p=1.1e-07

p=0.49

p=4.3e-09

p=0.09

Separating Training Data from Testing Data

• Split-sample method

• Re-sampling methods– Leave one out cross validation– K-fold cross validation– Replicated split-sample– Bootstrap re-sampling

• “Prediction is very difficult; especially about the future.”

• SEARCH STRING: prediction future (name)

875,000 Einstein584,000 Twain364,000 Bohr113,000 Berra

• SEARCH STRING: prediction "especially * the future" (name)

31,200 Bohr18,500 Berra864 Einstein539 Twain

Prediction on Simulated Null DataSimon et al. J Nat Cancer Inst 95:14, 2003

Generation of Gene Expression Profiles

• 20 specimens (Pi is the expression profile for specimen i)

• Log-ratio measurements on 6000 genes

• Pi ~ MVN(0, I6000)

• Can we distinguish between the first 10 specimens (Class 1) and the last 10 (Class 2)?

Prediction Method

• Compound covariate predictor built from the log-ratios of the 10 most differentially expressed genes.

Number of misclassifications

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Pro

po

rtio

n o

f sim

ula

ted

da

ta s

ets

0.00

0.05

0.10

0.90

0.95

1.00

Cross-validation: none (resubstitution method)Cross-validation: after gene selectionCross-validation: prior to gene selection

Cross Validation

• With proper cross-validation, the model must be developed from scratch for each leave-one-out training set. This means that feature selection must be repeated for each leave-one-out training set.

• The cross-validated estimate of misclassification error is an estimate of the prediction error for the model developed by applying the specified algorithm to the full dataset

Permutation Distribution of Cross-validated Misclassification Rate of a Multivariate

Classifier Radmacher, McShane & Simon

J Comp Biol 9:505, 2002

• Randomly permute class labels and repeat the entire cross-validation

• Re-do for all (or 1000) random permutations of class labels

• Permutation p value is fraction of random permutations that gave as few misclassifications as e in the real data

Model Evaluation for p>n Prediction Problems

• Odds ratios and hazards ratios are not proper measures of prediction accuracy

• Statistical significance of regression coefficients are not proper measures of predictive accuracy

Evaluation of Prediction Accuracy

• For binary outcome– Cross-validated prediction error– Cross-validated sensitivity & specificity– Cross-validated ROC curve

• For survival outcome– Cross-validated Kaplan-Meier curves for predicted

high and low risk groups• Cross-validated K-M curves within levels of standard

prognostic staging system

– Cross-validated time-dependent ROC curves

i

-i

p dimensional vector of expression levels for case i

y =binary {0,1} class indicator for case i

p dimensional weights for linear classifier computed

for training set with case i omitted;

ix

-i

i

component j is zero

if variable j is not selected in feature selection step for

training set -i

score for case i computed from model developed

with case i omitted

y c

ix

-i

i1

i i1

i i1

predicted class

mis-classification error y (c)=y

sensitivity(c) y (c)=y

specificity(c) (1 ) y (c)=y

ROC curve sensitivity(c) vs 1-specificity(c)

i

n

i

n

ii

n

ii

I x c

I

y I

y I

LOOCV Error Estimates for Linear Classifiers

Cross-validated Kaplan-Meier Curves for Predicted High and Low Risk Groups

0

( ; )( ; ) ( ) exp( )

1 ( ; )

estimate of weights based on algorithm applied

to training set with case i omitted

Algorithm may involve feature selection,

i

f t xt x t x

F t x

penalized regression, ...

Classify case i as high risk if

c might be a percentile of { }

i i

i j

x c

x j i

Cross-Validated Time Dependent ROC Curve

-i

* landmark time of interest

sensitivity(c)=Pr[ x>c|t t*]

Pr[t<t*| x>c]Pr[ x>c] =

Pr[t t*]

Estimate 1st term in numerator using KM estimator

for cases with . Estimate 2nd tei

t

x c

-i

rm as proportion

of n cases with . Estimate denominator using KM

estimator for all n cases.

specificity(c)=Pr[ x c|t>t*]

Pr[t>t*| x c]Pr[ x c] =

Pr[t>t*]

Time-dependent

ix c

ROC is sensitivity(c) vs 1-specificity(c)

Is Accurate Prediction Possible For p>>n?

• Yes, in many cases, but standard statistical methods for model building and evaluation are often not effective

• Standard methods may over-fit the data and lead to poor predictions

• With p>n, unless data is inconsistent, a linear model can always be found that classifies the training data perfectly

Is Accurate Prediction Possible For p>>n?

• Some problems are easy; real problems are often difficult

• Simple methods like DLDA, nearest neighbor classifiers and shrunken centroid classifiers are as effective or more effective than more complex methods for many datasets

• Because of correlated variables, there are often many very distinct models that predict about equally well

• p>n prediction problems are not multiple testing problems

• The objective of prediction problems is accurate prediction, not controlling the false discovery rate– Parameters that control feature selection in prediction

problems are tuning parameters to be optimized for prediction accuracy

• Optimizaton by cross-validation nested within the cross-validation used for evaluating prediction accuracy

• Biological understanding is often a career objective; accurate prediction can sometimes be achieved in less time

Traditional Approach to Oncology Clinical Drug Development

• Phase III trials with broad eligibility to test the null hypothesis that a regimen containing the new drug is on average not better than the control treatment for all patients who might be treated by the new regimen

• Perform exploratory subset analyses but regard results as hypotheses to be tested on independent data

Traditional Clinical Trial Approaches

• Have protected us from false claims resulting from post-hoc data dredging not based on pre-defined biologically based hypotheses

• Have led to widespread over-treatment of patients with drugs to which many don’t need and from which many don’t benefit

• Are less suitable for evaluation of new molecularly targeted drugs which are expected to benefit only the patients whose tumors are driven by de-regulation of the target of the drug

Molecular Heterogeneity of Human Cancer

• Cancers of a primary site in many cases appear to represent a heterogeneous group of diverse molecular diseases which vary fundamentally with regard to – their oncogenecis and pathogenesis – their responsiveness to specific drugs

• The established molecular heterogeneity of human cancer requires the use new approaches to the development and evaluation of therapeutics

How Can We Develop New Drugs in a Manner More Consistent With Modern Tumor Biology and ObtainReliable Information About What Regimens Work for What Kinds of

Patients?

Using phase II data, develop predictor of response to new drugDevelop Predictor of Response to New Drug

Patient Predicted Responsive

New Drug Control

Patient Predicted Non-Responsive

Off Study

Evaluating the Efficiency of Enrichment and Stratification Clinical Trial Designs With

Predictive Biomarkers

• Simon R and Maitnourim A. Evaluating the efficiency of targeted designs for randomized clinical trials. Clinical Cancer Research 10:6759-63, 2004; Correction and supplement 12:3229, 2006

• Maitnourim A and Simon R. On the efficiency of targeted clinical trials. Statistics in Medicine 24:329-339, 2005.

Developmental Strategy (II)

Develop Predictor of Response to New Rx

Predicted Non-responsive to New Rx

Predicted ResponsiveTo New Rx

ControlNew RX Control

New RX

Developmental Strategy (II)

• Do not use the diagnostic to restrict eligibility, but to structure a prospective analysis plan

• Having a prospective analysis plan is essential• “Stratifying” (balancing) the randomization is

useful to ensure that all randomized patients have tissue available but is not a substitute for a prospective analysis plan

• The purpose of the study is to evaluate the new treatment overall and for the pre-defined subsets; not to modify or refine the classifier

• R Simon. Using genomics in clinical trial design, Clinical Cancer Research 14:5984-93, 2008

• R Simon. Designs and adaptive analysis plans for pivotal clinical trials of therapeutics and companion diagnostics, Expert Opinion in Medical Diagnostics 2:721-29, 2008

• It can be difficult to identify a single completely defined classifier candidate prior to initiation of the phase III trial evaluating the new treatment

Cross-Validated Adaptive Signature Design

(In press)

Wenyu Jiang, Boris Freidlin, Richard Simon

Cross-Validated Adaptive Signature Design

End of Trial Analysis

• Compare T to C for all patients at significance level overall (e.g. 0.03)

– If overall H0 is rejected, then claim effectiveness of T for eligible patients

– Otherwise

Otherwise

• Partition the full data set into K parts P1 ,…,PK

• Form a training set by omitting one of the K parts, e.g. part k.– Trk={1,…,n}-Pk

• The omitted part Pk is the test set• Using the training set, develop a predictive

binary classifier B-k of the subset of patients who benefit preferentially from the new treatment compared to control

• Classify the patients i in the test set as sensitive B-k(xi)=1 or insensitive B-k(xi)=0– Let Sk={j in Pk : B-k(xi)=1}

• Repeat this procedure K times, leaving out a different part each time

• After this is completed, all patients in the full dataset are classified as sensitive or insensitive– Scv= Sk

• For patients classified as sensitive, compare outcomes for patients who received new treatment T to those who received control treatment C.– Outcomes for patients in Scv T vs outcomes for

patients in Scv C• Compute a test statistic Dsens

– e.g. the difference in response proportions or log-rank statistic for survival

• Generate the null distribution of Dsens by permuting the treatment labels and repeating the entire K-fold cross-validation procedure

• Perform test at significance level 0.05 - overall

• If H0 is rejected, claim superiority of new treatment T for future patients with expression vector x for which B(x)=1 where B is the classifier of sensitive patients developed using the full dataset

• The estimate of treatment effect for future sensitive patients is Dsens computed from the cross-validated sensitive subset Scv

• The stability of the sensitive subset {x:B(x)=1} can be evaluated based on applying the classifier development algorithm to non-parametric bootstrap samples of the full dataset {1,...,n}

70% Response to T in Sensitive Patients25% Response to T Otherwise

25% Response to C20% Patients Sensitive, n=400

ASD CV-ASD

Overall 0.05 Test 0.486 0.503

Overall 0.04 Test 0.452 0.471

Sensitive Subset 0.01 Test

0.207 0.588

Overall Power 0.525 0.731

Prediction Based Analysis of Clinical Trials

• Using cross-validation we can evaluate any classification algorithm for identifying the patients sensitive to the new treatment relative to the control using any set of covariates.

• The algorithm and covariates should be pre-specified. • The algorithm A, when applied to a dataset D should provide a

function B(x;A,D) that maps a covariate vector x to {0,1}, where 1 means that treatment T is prefered to treatment C for the patient.

• The algorithm can be simple or complex, frequentist or Bayesian

based. – Prediction effectiveness depends on the algorithm and the dataset– Complex algorithms may over-fit the data and provide poor results

• Including Bayesian models with many parameters and non-informative priors• Prediction effectiveness for the given clinical trial dataset can be

evaluated by cross-validation

cv

Estimated survival distribution for a set of n future patients

matched to clinical trial patients with regard to covariate vector x.

S set of clinical trial patients predicted sensitive to T in cross -validation

Patients in predicted sensitive subset are treated in future with T

and patients not in sensitive subset are treated with C

[ ( ; ) ( ) ( ; )] /

where || ||

Csens cv sens cv

sens cv

n KM t S T n n KM t S C n

n S

Standard Analysis Algorithm

• Test the overall H0 at 5% significance level

• If you reject H0 then treat all future patients with T– Expected survival KM(t;T)

• Otherwise treat all future patients with C– Expected survival KM(t;C)

Conclusions

• A more personalized oncology is rapidly developing based (so far) on information in the tumor genome

• Genomics has spawned new and interesting areas of biostatistics including methods for p>n prediction problems, systems biology and the design of predictive clinical trials

• There are important opportunities and great needs for young biostatisticians with rigorous training in biostatistics and high motivation for trans-disciplinary research in biology and biomedicine

Acknowledgements

• Kevin Dobbin• Boris Freidlin• Wenyu Jiang• Aboubakar Maitournam• Michael Radmacher• Jyothi Subramarian• Yingdong Zhao