Best Model Discrimination (MD) Follow-Up Experiments

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Description

Best follow-up experiments based on the MD criterion are suggested to discriminate between competing models.

Usage

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MD(X, y, nFac, nBlk = 0, mInt = 3, g = 2,  nMod, p, s2, nf, facs, nFDes = 4,
Xcand, mIter = 20, nStart = 5, startDes = NULL, top = 20, eps = 1e-05)

Arguments

X

matrix. Design matrix of the initial experiment.

y

vector. Response vector of the initial experiment.

nFac

integer. Number of factors in the initial experiment.

nBlk

integer >=1. The number of blocking factors in the initial experiment. They are accommodated in the first columns of matrix X.

mInt

integer. Maximum order of the interactions in the models.

g

vector. Variance inflation factor for main effects (g[1]) and interactions effects (g[2]). If vector length is 1 the same inflation factor is used for main and interactions effects.

nMod

integer. Number of competing models.

p

vector. Posterior probabilities of the competing models.

s2

vector. Competing model variances.

nf

vector. Factors considered in each of the models.

facs

matrix. Matrix [nMod x max(nf)] of factor numbers in the design matrix.

nFDes

integer. Number of runs to consider in the follow-up experiment.

Xcand

matrix. Candidate runs to be chosen for the follow-up design.

mIter

integer. If 0, then user-entered designs startDes are evaluated, otherwise the maximum number of iterations for each Wynn search.

nStart

integer. Number of starting designs.

startDes

matrix. Matrix [nStart x nFDes]. Each row has the row numbers of the user-supplied starting design.

top

integer. Highest MD follow-up designs recorded.

eps

numeric. A small number (1e-5 by default) used for computations.

Details

The MD criterion, proposed by Meyer, Steinberg and Box is used to discriminate among competing models. Random starting runs chosen from Xcand are used for the Wynn search of best MD follow-up designs. nStart starting points are tried in the search limited to mIter iterations. If mIter=0 then startDes user-provided designs are used. Posterior probabilities and variances of the competing models are obtained from BsProb. The function calls the FORTRAN subroutine ‘md’ and captures summary results.

Value

A list with all input and output parameters of the FORTRAN subroutine MD. Most of the variable names kept to match FORTRAN code.

NSTART

Number of starting designs.

NRUNS

Number of runs used in follow-up designs.

ITMAX

Maximum number of iterations for each Wynn search.

INITDES

Number of starting points.

NO

Numbers of runs already completed before follow-up.

IND

Indicator; 0 indicates the user supplied starting designs.

X

Matrix for initial data (nrow(X)=N0; ncol(X)=COLS+BL).

Y

Response values from initial experiment (length(Y)=N0).

GAMMA

Variance inflation factor.

GAM2

If IND=1, GAM2 was used for interaction factors.

BL

Number of blocks (>=1) accommodated in first columns of X and Xcand

.

COLS

Number of factors.

N

Number of candidate runs.

Xcand

Matrix of candidate runs. (nrow(Xcand)=N, ncol(Xcand)=ncol(X)).

NM

Number of models considered.

P

Models posterior probability.

SIGMA2

Models variances.

NF

Number of factors per model.

MNF

Maximum number of factor in models. (MNF=max(NF)).

JFAC

Matrix with the factor numbers for each of the models.

CUT

Maximum interaction order considered.

MBEST

If INITDES=0, the first row of the MBEST[1,] matrix has the first user-supplied starting design. The last row the NSTART-th user-supplied starting design.

NTOP

Number of the top best designs.

TOPD

The D value for the best NTOP designs.

TOPDES

Top NTOP design factors.

ESP

"Small number" provided to the ‘mdFORTRAN subroutine. 1e-5 by default.

flag

Indicator = 1, if the ‘md’ subroutine finished properly, -1 otherwise.

Note

The function is a wrapper to call the FORTAN subroutine ‘md’, modification of Daniel Meyer's original program, ‘MD.f’, part of the mdopt bundle for Bayesian model discrimination of multifactor experiments.

Author(s)

R. Daniel Meyer. Adapted for R by Ernesto Barrios.

References

Meyer, R. D., Steinberg, D. M. and Box, G. E. P. (1996). "Follow-Up Designs to Resolve Confounding in Multifactor Experiments (with discussion)". Technometrics, Vol. 38, No. 4, pp. 303–332.

Box, G. E. P and R. D. Meyer (1993). "Finding the Active Factors in Fractionated Screening Experiments". Journal of Quality Technology. Vol. 25. No. 2. pp. 94–105.

See Also

print.MD, BsProb

Examples

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### Injection Molding Experiment. Meyer et al. 1996, example 2.
library(BsMD)
data(BM93.e3.data,package="BsMD")
X <- as.matrix(BM93.e3.data[1:16,c(1,2,4,6,9)])
y <- BM93.e3.data[1:16,10]
p <- c(0.2356,0.2356,0.2356,0.2356,0.0566)
s2 <- c(0.5815,0.5815,0.5815,0.5815,0.4412)
nf <- c(3,3,3,3,4)
facs <- matrix(c(2,1,1,1,1,3,3,2,2,2,4,4,3,4,3,0,0,0,0,4),nrow=5,
    dimnames=list(1:5,c("f1","f2","f3","f4")))
nFDes <- 4
Xcand <- matrix(c(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
                    -1,-1,-1,-1,1,1,1,1,-1,-1,-1,-1,1,1,1,1,
                    -1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,
                    -1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,
                    -1,1,1,-1,1,-1,-1,1,1,-1,-1,1,-1,1,1,-1),
                    nrow=16,dimnames=list(1:16,c("blk","f1","f2","f3","f4"))
                )
injectionMolding.MD <- MD(X = X, y = y, nFac = 4, nBlk = 1, mInt = 3,
            g = 2, nMod = 5, p = p, s2 = s2, nf = nf, facs = facs,
            nFDes = 4, Xcand = Xcand, mIter = 20, nStart = 25, top = 10)
summary(injectionMolding.MD)



### Reactor Experiment. Meyer et al. 1996, example 3.
par(mfrow=c(1,2),pty="s")
data(Reactor.data,package="BsMD")

# Posterior probabilities based on first 8 runs
X <- as.matrix(cbind(blk = rep(-1,8), Reactor.data[c(25,2,19,12,13,22,7,32), 1:5]))
y <- Reactor.data[c(25,2,19,12,13,22,7,32), 6]
reactor8.BsProb <- BsProb(X = X, y = y, blk = 1, mFac = 5, mInt = 3,
        p =0.25, g =0.40, ng = 1, nMod = 32)
plot(reactor8.BsProb,prt=TRUE,,main="(8 runs)")

# MD optimal 4-run design
p <- reactor8.BsProb$ptop
s2 <- reactor8.BsProb$sigtop
nf <- reactor8.BsProb$nftop
facs <- reactor8.BsProb$jtop
nFDes <- 4
Xcand <- as.matrix(cbind(blk = rep(+1,32), Reactor.data[,1:5]))
reactor.MD <- MD(X = X, y = y, nFac = 5, nBlk = 1, mInt = 3, g =0.40, nMod = 32,
        p = p,s2 = s2, nf = nf, facs = facs, nFDes = 4, Xcand = Xcand,
        mIter = 20, nStart = 25, top = 5)
summary(reactor.MD)

# Posterior probabilities based on all 12 runs
X <- rbind(X, Xcand[c(4,10,11,26), ])
y <- c(y, Reactor.data[c(4,10,11,26),6])
reactor12.BsProb <- BsProb(X = X, y = y, blk = 1, mFac = 5, mInt = 3,
        p = 0.25, g =1.20,ng = 1, nMod = 5)
plot(reactor12.BsProb,prt=TRUE,main="(12 runs)")