GDSARM {GDSARM}

Gauss-Dantzig Selector - Aggregation over Random Models (GDS-ARM)

Description

The GDS-ARM procedure consists of three steps. First, it runs the Gauss Dantzig Selector (GDS) nrep times, each time with a different set of nint randomly selected two-factor interactions. All m main effects are included in each GDS run. Second, the best ntop models are identified with the smallest BIC. Effects that appear in at least pkeep x ntop of the ntop models are then passed on to the third stage. In the third stage, stepwise regression is used. With n being the number of runs, the stepwise regression starts with at most n-3 selected effects from the previous step. The remaining effects from the previous step as well as all main effects are given a chance to enter into the model using the forward-backward stepwise regression. The function also has the option of using the modified GDS-ARM. The modified version incorporates effect heredity in two steps, first, for each model found by GDS, we ignore active interactions when at least one of the main effects is not active (for weak heredity) or when both main effects are not active (for strong heredity); and second, we do the same for the model found after the stepwise stage of GDS-ARM.

Usage

GDSARM(
  delta.n = 10,
  nint,
  nrep,
  ntop,
  pkeep,
  design,
  Y,
  cri.penter = 0.01,
  cri.premove = 0.05,
  opt.heredity = c("none"),
  seedvalue = 1234
)

Arguments

delta.n

a positive integer suggesting the number of delta values to be tried. delta.n equally spaced values of delta will be used strictly between 0 and max(|t(X)y|). The default value is set to 10.

nint

a positive integer representing the number of randomly chosen interactions. The suggested value to use is the ceiling of 20% of the total number of interactions, that is, for m factors, we have ceiling(0.2(m choose 2)).

nrep

a positive integer representing the number of times GDS should be run. The suggested value is (m choose 2).

ntop

a positive integer representing the number of top models to be selected among the nrep models. The suggested value is max(20, (nrep x nint)/(m(m-1)). The value of ntop should not exceed nrep.

pkeep

a number between 0 and 1 representing the proportion of ntop models in which an effect needs to appear in order to be selected for the stepwise regression stage.

design

a n \times m matrix of m two-level factors. The levels should be coded as +1 and -1.

Y

a vector of n responses.

cri.penter

the p-value cutoff for the most significant effect to enter into the stepwise regression model. The suggested value is 0.01.

cri.premove

the p-value cutoff for the least significant effect to exit from the stepwise regression model. The suggested value is 0.05.

opt.heredity

a string with either none, or weak, or strong. Denotes whether the effect-heredity (weak or strong) should be embedded in GDS-ARM. The default value is none as suggested in Singh and Stufken (2022).

seedvalue

a seed value that will fix the set of interactions being selected. The default value is seed to 1234.

Value

A list returning the selected effects as well as the corresponding important factors.

Source

Cand\'es, E. and Tao, T. (2007). The Dantzig selector: Statistical estimation when p is much larger than n. Annals of Statistics 35 (6), 2313–2351.

Dopico-Garc\' ia, M.S., Valentao, P., Guerra, L., Andrade, P. B., and Seabra, R. M. (2007). Experimental design for extraction and quantification of phenolic compounds and organic acids in white "Vinho Verde" grapes Analytica Chimica Acta, 583(1): 15–22.

Hamada, M. and Wu, C. F. J. (1992). Analysis of designed experiments with complex aliasing. Journal of Quality Technology 24 (3), 130–137.

Hunter, G. B., Hodi, F. S. and Eagar, T. W. (1982). High cycle fatigue of weld repaired cast Ti-6AI-4V. Metallurgical Transactions A 13 (9), 1589–1594.

Phoa, F. K., Pan, Y. H. and Xu, H. (2009). Analysis of supersaturated designs via the Dantzig selector. Journal of Statistical Planning and Inference 139 (7), 2362–2372.

Singh, R. and Stufken, J. (2022). Factor selection in screening experiments by aggregation over random models, 1–31. doi: 10.48550/arXiv.2205.13497

See Also

GDS_givencols, dantzig.delta

Examples

data(dataHamadaWu)
X = dataHamadaWu[,-8]
Y = dataHamadaWu[,8]
delta.n = 10
n = dim(X)[1]
m = dim(X)[2]
nint = ceiling(0.2*choose(m,2))
nrep = choose(m,2)
ntop = max(20, nint*nrep/(2*choose(m,2)))
pkeep = 0.25 
cri.penter = 0.01
cri.premove = 0.05
design = X
# GDS-ARM with default values
GDSARM(delta.n, nint, nrep, ntop, pkeep, X, Y, cri.penter, cri.premove)

# GDS-ARM with default values but with weak heredity
opt.heredity="weak" 
GDSARM(delta.n, nint, nrep, ntop, pkeep, X, Y, cri.penter, cri.premove, opt.heredity)


data(dataCompoundExt)
X = dataCompoundExt[,-9]
Y = dataCompoundExt[,9]
delta.n = 10
n = dim(X)[1]
m = dim(X)[2]
nint = ceiling(0.2*choose(m,2))
nrep = choose(m,2)
ntop = max(20, nint*nrep/(2*choose(m,2)))
pkeep = 0.25 
cri.penter = 0.01
cri.premove = 0.05
design = X
# GDS-ARM on compound extraction
GDSARM(delta.n, nint, nrep, ntop, pkeep, X, Y, cri.penter, cri.premove)

# GDS-ARM on compound extraction with strong heredity
opt.heredity = "strong"
GDSARM(delta.n, nint, nrep, ntop, pkeep, X, Y, cri.penter, cri.premove, opt.heredity)