GÜNTER HEIMANN

Model-Based Tests forTQT Trials

Introduction

Summary of Methods

Type I Error

Summary andConclusions

version 0.2 1

Model-Based Tests for ThoroughQTc TrialsCSRC Thinktank MeetingWashington D.C., February 2, 2012

GÜNTER HEIMANNModeling & Simulation StatisticsNovartis AG, Basel

GÜNTER HEIMANN

Model-Based Tests forTQT Trials

Introduction

Summary of Methods

Type I Error

Summary andConclusions

version 0.2 2

Introduction

In this presentation I will discuss how a model-based analysisfor QTc data can be conducted in a statistically sound way, i.e.

I the type I error is controlledI the analysis is fully pre-specified

This approach can be applied to thorough QTc trials as well asto a combined analysis including SAD and MAD data.

Asymptotic results have been proven theoretically and aresupported by simulations for finite sample sizes.

GÜNTER HEIMANN

Model-Based Tests forTQT Trials

Introduction

Summary of Methods

Type I Error

Summary andConclusions

version 0.2 3

A First Look at Model-Based Analyses

Scatter plot with regression on FTY and Placebo data

conc (ng/ml)

QT

cI ch

an

ge

fro

m m

ea

n b

ase

line

(m

se

c)

0 5 10 15 20 25

-20

-10

01

02

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- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

10

placebo

low dose

high dose

Not all methods control the type I error!

Some methods are conservative power loss!

Where should we compare to the 10 msec bound

• at the estimated Cmax

• at the upper CI around

and at what -level (90%, 95%)?

GÜNTER HEIMANN

Model-Based Tests forTQT Trials

Introduction

Summary of Methods

Type I Error

Summary andConclusions

version 0.2 4

Model and Hypotheses

Model:

QTclkt = µ + pt + ϑPKlkt + εlkt

E[maxtPKlkt ] = Cmax

I adjusts for diurnal variation pt and PK concentrationsI any correlation structure within subject k allowedI additional fixed effects (such as a stratum effect for study

or cohort) can be addedI non-linear models also possible

Hypotheses: ϑCmax ≥ 10 versus ϑCmax < 10

I or corresponding adjustment for non-linear models

GÜNTER HEIMANN

Model-Based Tests forTQT Trials

Introduction

Summary of Methods

Type I Error

Summary andConclusions

version 0.2 5

Test Statistic and Test

Test statistic: ϑCmax

There are two approaches to obtain critical values for√n(ϑCmax − ϑCmax): the delta-method

I requires knowledge of joint distribution of√

n(ϑ− ϑ) and√n(Cmax − Cmax)

I can only be applied in certain linear models

the bootstrap

I is based on resampling of entire subjects (PK and QTc)I can be applied in linear and non-linear modelsI and is hence recommended

GÜNTER HEIMANN

Model-Based Tests forTQT Trials

Introduction

Summary of Methods

Type I Error

Summary andConclusions

version 0.2 6

Type I Error for Small Sample Sizes

1

All methods are slightly liberal for small sample sizes

GÜNTER HEIMANN

Model-Based Tests forTQT Trials

Introduction

Summary of Methods

Type I Error

Summary andConclusions

version 0.2 7

Type I Error for Moderate Sample Sizes

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Linear model, residual variance = 40, n = 100 per group, 1,000 simulations

popular rule

delta method

bootstrap

alpha = 0.05

H0

Delta method and bootstrap control type I error for moderate sample sizes (n>40)

GÜNTER HEIMANN

Model-Based Tests forTQT Trials

Introduction

Summary of Methods

Type I Error

Summary andConclusions

version 0.2 8

Type I Error is Controlled Asymptotically

1

Asymptotic power curves can be calculated explicitly! Delta method and bootstrap control the type I error asymptotically

GÜNTER HEIMANN

Model-Based Tests forTQT Trials

Introduction

Summary of Methods

Type I Error

Summary andConclusions

version 0.2 9

Type I Error Control is not Warranted

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Linear model, residual variance = 40, n = 300 per group, 10,000 simulations

popular rule

delta method

alpha = 0.05

H0

The simulated and the

asymptotic power curves

coincide for both methods!

The delta method controls

the type I error asymptotically!

The popular rule does not!

GÜNTER HEIMANN

Model-Based Tests forTQT Trials

Introduction

Summary of Methods

Type I Error

Summary andConclusions

version 0.2 10

Summary and Conclusions

I Statistical tests which control the type I error are availablefor linear and non-linear models.

I Simulations across wide range of scenarios (linear andnon-linear) demonstrate type I error control.

I Theoretical proofs available for linear models, confirmedby simulations.

I Asymptotic power curves are available for sample sizecalculations.

I Method can be adjusted to accomodate strata (studies &cohorts).

I However, these methods rely on additional assumptionsI the pre-specified model (linear, Emax, ...)I no histeresis

I Further research ongoing to overcome these issues