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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
0
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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
1
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
0.28
0.7 0.75 0.8
pro
ba
bil
ity t
o r
eje
ct
slope
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
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
0.28
0.7 0.75 0.8
pro
ba
bil
ity t
o r
eje
ct
slope
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