Transcript INDIAN STATISTICAL INSTITUTE Platinum Jubilee

Acharya Nagarjuna University
Department of Statistics
Workshop on DoE & OR
Dec. 8 - 9, 2011
Pharmaceutical Laboratories :
Role of Designed Experiments
BIKAS K SINHA
Senior Professor of Statistics, ISI
&
Ex-Member
National Statistical Commission
Govt. of India
n
s factorial
Optimal
designs
when observations withinblocks are correlated
Computational Statistics & Data Analysis
Volume 50, Issue 10 , 2006, PP 2855-2862
Sethuraman / Raghavarao / Sinha
Venkat S. Sethuraman, Biomedical Data Sciences,
GlaxoSmithKline R&D, Philadelphia,USA
D Raghavarao : Department of Statistics,
Temple University, Philadelphia, USA
Laboratory Trials…
• In a pharmaceutical industry, the
concern was to develop once-daily
tablet formulation that would
improve patient compliance,
compared to a currently available
formulation administered twice daily.
Laboratory Expt….
• These laboratory experiments are
normally performed on healthy
human subjects (called volunteers),
who are administered several test
formulations separated by drugwashout days.
• Approved by Ethical Committee
Needed Signed Consent of V’s
Drug Formulations…
• Components : Each has at least two or three
options to be tried out.
• These are technically called levels and the
level combinations of all the components
constitute the drug formulations to be tested
out.
• In general, each volunteer is administered all
or selected drug formulations of the
constituent components under consideration.
Laboratory Results….
• The bio-availability of each formulation is
measured by AUC (Area Under time-plasma
concentration Curve) which is the response
variable of immediate interest. The drug
formulations involve several components such
as (i) type of polymers to prolong drug
release, (ii) amount of film coat on the tablet,
(iii) administration under fed or fasted state,
etc…..
• Use Factorial Expt. Concept
Concept of Blocking….
• The concept of BLOCKS relates to
different VOLUNTEERS.
• The availability of a volunteer for the
laboratory trial determines the block size
which is the number of different
formulations tested on the same volunteer.
• Different experimental conditions may
result in unequal block sizes as well.
• Some volunteers may opt for a few drug
formulations to be tested upon…..
Simplified Models….
• A simple statistical model assumes possible
existence of first-order [i.e., linear] effects
[called main effects] of the components only,
without any interaction effects.
• These linear effects are assumed to explain the
variation in the drug bioavailability.
• Different formulations of the drug are tested
on the volunteers in order to examine if there is
any differential effect of these formulations.
Designing Problems….
• In order that maximum information on the
first order effects may be extracted from the
experiment, we need to determine the
allocation of the drug formulations to
different volunteers forming the blocks.
• We assume : “s” levels of each of the “n”
components so that altogether sn drug
formulations are to be tested on “v”
volunteers in an optimal manner.
Basic Assumptions….
• All the sn drug formulations must be
tested collectively on these “v” volunteers
• Every volunteer must receive at least a
minimum number of drug formulations
• No two volunteers will receive the same
drug formulation
• Collectively….the volunteers must
exhaust all the sn drug formulations….
Model Descriptions….
• We assume a linear effects model [without any
quadratic/higher order terms] for ith
volunteer receiving fi drug formulations.
E [Yi] = Xi θ,
θ = [, 1, 2, …, n]

• D[Yi] = i of order fi ; i = 1, 2, …, v
• We assume an intraclass correlation structure
for
i’s of appropriate dimensions.
Unified Notations….
•
•
•
•
Levels of the factors [s odd OR even] :
0, +/- 1, +/- 2, ….OR +/- 1, +/- 3, +/- 5, …..
E[Y ir] =  + x ir11 + x ir2 2 +…+ x irn n
Volunteer #i; rth drug formulation in the
sequence; n = number of components
• i ’s…..per unit level effects of the components
• i = (1- )I +  J of dim. fi st ifi = sn
Optimal Design Problem…
• Most efficient estimation of the betaparameters….
• D-optimality Criterion involving ^’s for
Optimal Choice of ((x irh))’s…..
Results available: fi’s multiples of s….
• That means….each volunteer receives at
least ‘s’ formulations…..
• Recall : x irh refers to ith volunteer, rth drug
formulation and hth component
Theorem …..
• D-optimal design : r x irh = 0  i & h
• Interpretation : For every volunteer and every
component, algebraic sum of formulation
levels actually prescribed is 0 !
• Note : fi = Number of drug formulations
prescribed for Volunteer # i = multiple of s
• Goos (2002:Springer Lecture Notes) :fi = f
• Sethuraman et al (2006) : General case BUT
all f i’s are still multiples of s ……
Sketch of Proof…..
From ith Volunteer : Contribution to I(θ)
-1
Ii (θ) = [ X′i i Xi ]
where
i = (1- )I +  J is of order f
i
Summed over all i, this results into
I(θ) = A – B – C – D of order (n+1)
Details…..
A = (1- )-1 X′X
B =  (1- )-1 i ci UiUi′ where
ci = {1+  (fi -1)}-1 & Ui′ = (xi.1, xi.2,…, xi.n )
C = (1- )-1 ((c iJ)) where
c11 = sn ; c12 = 0; c22 = sn-1 (ar2) In
ar = ih xirh = sum of x-values for rth factor
D =  (1- )-1 ((d iJ)) where d11= cifi2 ;
d12= cifi Ui(1)′; d22 = ciUi(1)Ui(1)′; Ui = (1,Ui(1))
Information on -parameters…
Decompose I(θ)
and work out I() as
I()= I22.11 = I22 – I21I12 / I11
It follows that
I22=(1- )-1 [ sn-1( ar2)In - i ciUi(1)Ui(1)′)]
For D-optimality,
| I22 |  (1- )-n [ sn-1( ar2)]n with = iff
Ui(1) = 0 i i.e., r x irh = 0  i & h
Nature of D-Optimal Designs….
• Define Permutation Matrices P1,P2,…,Ps of
order s as follows :
P1=Is;Pi = right cyclic rotation of Pi-1;
i=2,3,…
For given s = 2k+1(odd), define
a = (-k, -(k-1), …, -1, 0, 1, …, k-1, k)′
Form sn-1 blocks each of size s by using
[a, Pi2a, Pi3a, …, Pina]
for i2, i3, …, in = {1, 2, …, s}.
D-Optimal Designs….
• Whenever fi’s are multiples of s, it is
enough to decompose the sn-1 blocks so
formed into subsets so that s. sn-1= sn = fi.
• Illustrative Examples follow for s = 2, 3 and
4.
D-Optimal Designs for s = 2
• Coded levels : -1 & 1
• Recall P1 = Is = I2 = [1 0; 0 1]
• P2 = right cyclic rotation of P1 = [0 1; 1 0]
• a = (-1 1)’
Case of n = 2 components : 22 = 4 formulations
•
Step I : [a a]; [a P2a]…2 blocks
•
[(-1 1) (-1 1)]; [(-1 1) (1 -1)]
• Volunteer -1 : First 2 col. (-1 -1) (1 1)
• Volunteer -2 : Last 2 col. (-1 1) (1 -1)
Case of Three Components….
n = 3: 23 = 8 formulations of 3 components
[a a a]; [a a P2a] [a P2a a] [a P2a P2a] : 4 blocks
[(-1 1) (-1 1) (-1 1)]; [(-1 1) (-1 1) (1 -1)]
[(-1 1) (1 -1) (-1 1)]; [(-1 1) (1 -1) (1 -1)]
Two Volunteers ----each with 4 formulations
V-1 : (-1 -1 -1) (1 1 1) (-1 -1 1) (1 1 -1)
V-2 : (-1 1 -1) (1 -1 1) (-1 1 1) (1 -1 -1)
• Likewise....we can handle ….
• Four Volunteers – each with 2 Formulations
D-Optimal Designs for s = 3
• Coded levels : -1 0 1
Recall P1 = Is = I3 = [1 0 0; 0 1 0; 0 0 1]
• P2 = right cyclic rotation of P1
•
= [0 0 1; 1 0 0; 0 1 0]
• P3 = right cyclic rotation of P2
•
= [0 1 0; 0 0 1; 1 0 0]
a = (-1 0 1)’
Case of Two Components
32 = 9 formulations divided into 3 blocks
Step I : [a a]; [a P2a] [a P3a] …3 blocks
• [(-1 0 1) (-1 0 1)];
• [(-1 0 1) (0 1 -1)];
• [(-1 0 1) ( 1 -1 0)]
V-1 : First 2 col. [ (-1 -1) (0 0) (1 1)]
V-2 : Next set of 2 col. [(-1 0) (0 1) (1 -1)]
V-3 : Last set of 2 col. [(-1 1) (0 -1) (1 0)]
Case of Three Components
33 = 27 formulations divided into 9 blocks
Step I : [a a a]; [a a P2a]; [a a P3a]
[a P2a a] [a P2a P2a] [a P2a P3a]
[a P3a a] [a P3a P2a] [a P3a P3a]
• [(-1 0 1) (-1 0 1) (-1 0 1)];
• [(-1 0 1) (-1 0 1) (0 1 -1)];
• [(-1 0 1) (-1 0 1) ( 1 -1 0)]
• etc….till the end
• [(-1 0 1) (1 -1 0) (1 -1 0)]
Key Reference
The end….
• Thanks for your attention !!!
• BKSinha [ISI, Kolkata]
• Guntur, December 8, 2011
INDIAN STATISTICAL
INSTITUTE
Platinum Jubilee Celebrations
[2006 – 2007]
• International Conference on
• Environmental and Ecological
Statistics with Applications
• Venue : Kolkata Campus
• Dates : March 21-23, 2007
Proposing Scientist….
• Bikas K Sinha
• Senior Professor of Statistics
[Stat-Math Division, ISI, Kolkata]
AND
• Member [2006 – 2009]
[National Statistical Commission]
Proposer’s Profile…
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Professor [ISI] since 1985
Mahalanobis Medal Recipient: 1980
UN Expert on Mission : 1990
US EPA Consultant [Env. & Ecology]: 1991
Int’l Stat Inst. Elected Member since 1985
Statistics Sectional President : Indian Science
Congress [2002]
• Member : National Stat. Commission [2006]
Organizing Committee
Int’l Conf. Env. & Eco. Stat. with Appls.
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Director, ISI…..Chairman
BKSinha….Vice-Chairman
Professor – in – Charge, Stat-Math Div
Head, Stat-Math Unit, Kolkata
Alok Goswami
GMSaha
Ratan Dasgupta
Aditya Bagchi
Debapriya Sengupta
Pulakesh Maiti [Convener]
Background Information….
• SURDAC Activities : 1990’s
• Collaboration of ISI Scientists with
Madhab Gadgil, Anil Gore, Paranjape
• 1993 : Int’l Conference in Statistical
Ecology
• Recent Collaboration with Anil Gore,
Paranjape, Abhik Gupta, Dilip Nath &
Others in North-East
Conference Thrust Areas
1. Toxic Release Inventory
(TRI)
• 2. Detection Limits
• 3. Combining Environmental
Indices
• 4. Cancer Growth Models
• 5. Pharmacokinetic Models
in Environmental Risk
Assessment
Thrust Areas …continued
• 6. Gene-Environment
Interaction Models and
Related Data Analysis
• 7. fMRI:Statistical Modeling
• 8. Health-Related Issues
[Arsenic Problem/Ozone Layer
/Environmental Health
Indices/Occupational Health
Hazards & Measures
Thrust Areas …continued
9.Environmental Awareness /
Health Issues in
Pharmaceutical Industries /
Women’s Health / Oceanography
& Marine Science / Forestry
Health Management]
• 10. Ecology : Ecological
Imbalances / Flora & Fauna /
Biodiversity Measures and
Related Issues
Confirmed Speakers….
• 1. Professor Bimal K Sinha : Univ. Maryland –
Baltimore County, USA
• 2. Professor Jerzy Filar : Univ. South Australia,
Australia
• 3. Professor Abhik Gupta : Department of
Environmental Science, Assam Univ., Silchar,
Assam
4. Professor Anil P Gore : Department of
Statistics, Univ. Pune
• 5. Dr. [Mrs.] S A Paranjape : Department of
Statistics, Univ. Pune
Confirmed Speakers….continued
• 6. Professor S N Dwivedi: AIIMS, New Delhi
• 7. Professor Tapio Nummi: University of
Tampere, Finland
• 8. Dr S. Asolekar : Centre for Env. Science &
Engg., IIT Mumbai
• 9. Dr D. N. Guha Mazumdar : Advisor, Pollution
Control, Govt. West Bengal
• 10. Dr D Chakrabarty, Director, School of
Environmental Studies, JU
Confirmed Speakers…continued
• 11. Dr.[Mrs.] Gitashree Das : North Eastern
Hill University, Assam
• 12. Dr. Tapan Chakrabarty : North Eastern
Hill University, Assam
• 13. Professor Dilip Nath : Gauhati
University
• 14. Dr. Kishore Das : Gauhati University
• 15. Professor Alok Goswami : Stat-Math
Unit, ISI, Kolkata
• 16. Dr. P Maiti : Economic Research Unit,
ISI, Kolkata
Confirmed Speakers…continued
• 17. Professor M Ghose : Agri. Science Unit,
ISI, Kolkata
• 18. Dr. Joydev Chattopdhyaya : Biological
Sciences Division, ISI, Kolkata
• 19. Professor Debapriya Sengupta : BIRU, ISI,
Kolkata
• 20. Professor Ratan DasGupta : Stat-Math
Unit, ISI, Kolkata
• 21. Professor Debasis Sengupta : Applied
Statistics Unit, ISI, Kolkata
22. Professor Bikas K Sinha : Stat-Math Unit,
ISI, Kolkata
Confirmation yet to be recd. from
• Dr. Olaf Berke, University of Guelph, ON,
Canada
• Prof. Dr. Leonard Held, Munich
• Prof. Sudip K Banerjee
Chairman, WB
Pollution Control Board
• Dr. Raman Sukumar, IISc, Centre for Ecological
Sciences, Bangalore
• Dr. Debashish Chatterjee, Kalyani University
TA/DA : Senior Scientists
[Rs. 1.50 Lakhs]
• 1. Professor Abhik Gupta :
Assam Univ., Silchar, Assam
2. Professor Anil P Gore : Pune
• 3. Dr. [Mrs.] S A Paranjape : Pune
• 4. Professor Dilip Nath : Assam
• 5.Professor S N Dwivedi:New Delhi
• 6. Dr Shyam R. Asolekar : Mumbai
• 7. Dr. Raman Sukumar : Bangalore
TA/DA: Young Scientists
[Rs. 1.20 Lakhs]
•
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1. Dr.[Mrs.] Gitashree Das : Assam
2.Dr. Tapan Chakrabarty : Assam
3. Dr. Kishore Das : Assam
Additional 10 from North-East
Regions and Mumbai-Pune Region
• Additional 30 from other regions
Proposed Publication : 2007
Env. & Ecology Conference
• Edited Volume :World Scientific Pub.
• Editors : BKSinha & AGoswami
• Editorial Board Members :
Bimal Sinha APGore DSengupta
Editorial Assistant : P Maiti
Updated Proposal :
Suggested at CC Meeting
• Combine Environment-Ecology,
Biodiversity & Sustainable Development
Topics together and publish a Single
Volume of approx. 500 pages
• Editorial Board : ?
• List of Contributors : ?
• Responsible Scientists : ?
• Deadlines : ?
Budget at a Glance…
• Total Budget : Rs. 5,70 Lakhs
• ISI [Sanctioned Budget] :
Rs. 3.50 Lakhs
• DST Funding [Requested] :
Rs. 2,20 Lakhs
DST Funding….
•
•
•
•
•
Specific Items
TA/DA : Young Scientists
TA/DA : Senior Scientists
Pre-Conference Printing
Publication of Proceedings
TOTAL
Rs. 1.00 Lakh
Rs. 0.60 Lakh
Rs. 0.10 Lakh
Rs. 0.50 Lakh
Rs. 2.20 Lakhs
•
•
Thank You !!!
BKSinha