ICFTs: Function for calculating interaction contribution frequency...

In DoE.base: Full Factorials, Orthogonal Arrays and Base Utilities for DoE Packages

Function for calculating interaction contribution frequency tables

Description

Function ICFTs calculates interaction contribution frequency tables, function ICFT does the same for an entire (usually small) design with more detail.

Usage

ICFTs(design, digits = 3, resk.only = TRUE, kmin = NULL, kmax = ncol(design), 
   detail = FALSE, with.blocks = FALSE, conc = TRUE)
ICFT(design, digits = 3, with.blocks = FALSE, conc = TRUE, recode=TRUE)

Arguments

design	a factorial design. This can either be a matrix or a data frame in which all columns are experimental factors, or a special data frame of class design, which may also include response data. In any case, the design should be a factorial design; the functions are not useful for quantitative designs (like e.g. latin hypercube samples).
digits	integer; number of digits to round to
resk.only	logical; if all is TRUE, should only full resolution projections be considered? Choosing FALSE may cause very long run times.
kmin	integer; purpose is to continue an earlier run with additional larger projections
kmax	integer; limit on projection sizes to consider
detail	logical indicating whether calculation details are to be returned (see Groemping 2016). These will not be needed for normal use of the outcome, but may be interesting for special situations.
with.blocks	a logical, indicating whether or not an existing block factor is to be included into word counting. This option is ignored if design is not of class design. Per default, an existing block factor is ignored. For designs without a block factor, the option does not have an effect. If the design is blocked, and with.blocks is TRUE, the block factor is treated like any other factor.
conc	logical indicating whether ambiguities should be resolved concentrating the contribution on as few individual values as possible (default) or distributing it as evenly as possible (if FALSE)
recode	logical indicating whether or not to recode each column into normalized orthogonal coding with function contr.XuWu; if set to FALSE, it is the users responsibility to provide a design in a normalized orthogonal coding of choice

Details

The functions work for factors only and are not intended for quantitative variables.

Function ICFTs decomposes the projected $a_k$ values (most often: projected $a_3$ values) into single degree of freedom contributions from the respective $k$ factor interaction.

Function ICFT decomposes the all-factor interaction of the design given to it; it is intended for deep-dive investigations.

The ICFT itself is independent of the choice of normalized orthogonal coding, as are the singular values and the matrix of left singular vectors; in case of several identical singular values, the left singular vectors are not uniquely determined but are subject to arbitrary rotation. The right singular vectors depend on the choice of normalized orthogonal coding. They represent the directions of coefficient vectors for which the interaction contributions indicate the bias potential for the intercept (see Groemping 2016 for the maths behind this).

Value

Function ICFTs returns a list of lists with a component for each projection size considered. Each such component contains the following entries:

ICFT	interaction contribution frequency table for the projection size
ICs	individual interaction contributions (if requested by option detail)
sv2s	squared singular values (if requested by option detail)
mean.u2s	squared column means of left-singular vectors (if requested by option detail)

Function ICFT returns a list with the following components:

ICFT	interaction contribution frequency table for the projection size
ICs	Average R^2 frequency table for the projection size
sv2s	squared singular values of the model matrix
mean.u2s	squared column means of left-singular vectors in the rotated version (concentrated or even)
mm	model matrix of the interaction
u	(left singular vectors corresponding to the rotated version of ICFT (concentrated or even); these do not depend on the coding underlying the model matrix
v	(right singular vectors corresponding to the rotated version of ICFT (concentrated or even); these depend on the coding underlying the model matrix
c.worst	(v%*%c.worst is the worst case parameter vector for the model matrix mm in terms of bias of the average response for estimation of the intercept caused by the interaction under consideration

Warning

The functions have been checked on the types of designs for which they are intended (especially orthogonal arrays produced with oa.design). They may produce meaningless results for some other types of designs.

Author(s)

Ulrike Groemping

References

Groemping, U. (2017). An Interaction-Based Decomposition of Generalized Word Counts Suited to Assessing Combinatorial Equivalence of Factorial Designs. Reports in Mathematics, Physics and Chemistry, Report 1/2017. http://www1.bht-berlin.de/FB_II/reports/Report-2017-001.pdf, Department II, Berliner Hochschule fuer Technik (formerly Beuth University of Applied Sciences), Berlin.

Groemping, U. (2018). Coding Invariance in Factorial Linear Models and a New Tool for Assessing Combinatorial Equivalence of Factorial Designs. Journal of Statistical Planning and Inference 193, 1-14. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.jspi.2017.07.004")}.

See Also

See also GWLP and generalized.word.length

Examples

   oa24.bad <- oa.design(L24.2.13.3.1.4.1, columns=c(1,2,14,15))
   oa24.good <- oa.design(L24.2.13.3.1.4.1, columns=c(3,10,14,15))
   ## resolution is III in both cases, but the bad one has more words of length 3
   GWLP(oa24.bad)[4:5]
   ICFTs(oa24.bad)
   ICFTs(oa24.bad, conc=FALSE)
   GWLP(oa24.good)[4:5]
   ICFTs(oa24.good)
   ICFTs(oa24.good, conc=FALSE)
   ICFTs(oa24.good, resk.only=FALSE)
   
   ICFT(L18[,c(1,4,6)])
   ICFT(L18[,c(1,4,6)], conc=FALSE)

DoE.base documentation built on Nov. 15, 2023, 1:06 a.m.