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

1 2 |

`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 |

`mInt` |
integer. Maximum order of the interactions in the models. |

`g` |
vector. Variance inflation factor for main 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 [ |

`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 |

`nStart` |
integer. Number of starting designs. |

`startDes` |
matrix. Matrix |

`top` |
integer. Highest MD follow-up designs recorded. |

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

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.

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 ( |

`Y` |
Response values from initial experiment ( |

`GAMMA` |
Variance inflation factor. |

`GAM2` |
If |

`BL` |
Number of blocks (>=1) accommodated in first columns of |

.

`COLS` |
Number of factors. |

`N` |
Number of candidate runs. |

`Xcand` |
Matrix of candidate runs. ( |

`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. ( |

`JFAC` |
Matrix with the factor numbers for each of the models. |

`CUT` |
Maximum interaction order considered. |

`MBEST` |
If |

`NTOP` |
Number of the top best designs. |

`TOPD` |
The D value for the best |

`TOPDES` |
Top |

`ESP` |
"Small number" provided to the ‘md’ FORTRAN subroutine. 1e-5 by default. |

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

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.

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

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.

`print.MD`

, `BsProb`

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 | ```
### 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)")
``` |

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