study. Such designs are usually highly fractionated factorials and their aliases ... OBsMD package implements the objective Bayesian methodology proposed in ...
OBsMD: an R-package for Objective Bayesian Model Discrimination in Follow-up Designs Laura Deldossi and Marta Nai Ruscone Dipartimento di Scienze Statistiche - Universit`a Cattolica del Sacro Cuore L.go Gemelli 1, 20123 Milan, Italy Why Follow-up Design? Screening experiments are employed at the initial stages of investigation to discriminate, among many factors, those with potential effect over the response under study. Such designs are usually highly fractionated factorials and their aliases structure often leads to ambiguous conclusions regarding which combinations of factors are active. As a consequence it is necessary to augment the design with extra runs in order to find the best model.
OBsMD Package OBsMD package implements the objective Bayesian methodology proposed in Consonni and Deldossi (2013) in order to choose the optimal follow-up experiment, i.e. the experiment that better discriminates between competing models. OBsMD includes two functions which produce: I OBsProb: posterior probabilities of models and factors based on a multiplicity-correction prior on model space, and a robust prior on model specific parameters I OMD: optimal follow-up experiments, i.e. the identification of the extra-runs which maximize the model discrimination criterion represented by a weighted average of Kullback-Leibler divergences based on the predictive distribution between all pairs of models
OBsProb Function
OMD Function
OBsProb function computes the posterior probability of the models according to BFi0(y)Pi0 Pr(Mi|y) = P 1 + j6=0 BFj0(y)Pj0
OMD =
BFi0(y) =
n+1
−ti/2
ti + t0
×
Qi0(y)−(n−t0)/2 ti + 1
" 2F1
ti + 1 n − t0 ti + 3 (1 − Qi0(y)−1)(ti + t0) ; ; ; 2 2 2 n+1
#
OBsProb(X, y, blk , mFac, mInt, nTop) I X: matrix. Design matrix of dimension n × k, where n is the size of the screening experiment and k is the number of the (two-level) factors. I y: vector. Response vector of dimension n × 1. I blk: Number of blocking factors. I mFac: Maximum number of factors considered in the models. I mInt: Maximum interaction order considered in the models. I nTop: Number of models with the highest posterior probability to print in output.
Example Reactor experiment (Box, Hunter, Hunter, 1978, p.376) Run in the full design 2 7 12 13 19 22 25 32
P(Mi|y)P(Mj|y)KL(m(·|y, Mi), m(·|y, Mj)),
where: I m(·|y, Mi) is the predictive density for the vector of follow-up observations under model Mi using an objective prior for the parameters I KL(·, ·) is the Kullback-Leibler divergence OMD is computed for all the possible designs of nFoll runs chosen from 2k candidates of the full design. Designs with largest OMD are preferred: they allow to better discriminate between competing models.
The R function is:
The experimental plan consists in the analysis of eight runs 5 extracted from the original 2 factorial design, corresponding 5−2 to the 2 Resolution III experiment with generator I = ABD = ACE.
X i6=j
where
Fixed the dimension nFoll of the follow-up design, the function OMD computes the following criterion:
Run 1 2 3 4 5 6 7 8
A + + + +
OBsProb(X=X,y=y,blk=0,mFac=5,mInt=2,nMod=32)
B + + + +
C + + + +
D + + + +
E + + + +
Y 53 54 93 66 70 55 44 82
The R function is: OMD(X, y, nFac, blk, mInt, nMod, optop, osigtop, onftop, ojtop, nFoll, Xcand, nStart, Mbest, top) where most of the arguments derive from the ObsProb output and: I nMod: Number of competing models used to compute objective OMD I Xcand: matrix. Matrix of dimension 2k × (blk + k) of the candidate runs among which the extra runs are chosen. I nStart: Number of all the possible different designs of nFoll runs chosen from the 2k candidates of the full design. I Mbest: matrix. Matrix of dimension nStart × nFoll whose rows contain all the possible nStart designs.
Example - continue I Choice of the follow-up runs For fixed nFoll=4 extra runs, there are 52360 possible follow-up designs of 4 runs 5 chosen from 2 = 32 candidates. For each of them we compute the OMD criterion.
I Follow-up runs included After the addition of the extra runs associated to the highest OMD value, function OBsProb is recalled, and the posterior probabilities of the models recomputed
Future Extension Implementation of an exchange algorithm to search for the design with highest OMD, without need of computing OMD for every possible follow-up design. Main References CONSONNI, G., DELDOSSI, L. (2013): Objective Bayesian model discrimination in follow-up experimental designs. S.Co.2013, short paper. (Session F2, Wednesday, 9-11 am) MEYER, R.D. (1996): mdopt: FORTRAN programs to generate MD-optimal screening and follow-up designs, and analysis of data. Statlib, URL http://lib.stat.cmu.edu.
September 9 - 11, 2013
