mapping_rule: Default Mapping Rule
In FAiR: Factor Analysis in R

Description Usage Arguments Details Value Author(s) References See Also Examples

It is not generally necessary to understand this help page if one merely wants to estimate a factor analysis model and this function should never be called directly. However, it is used as the default function to squash some coefficients to zero during semi-exploratory factor analysis and thus needs to be documented. Also, if one were to specify a different mapping rule function in the call to make_restrictions, then that function would need to have some of the same arguments as this one.

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mapping_rule(coefs, cormat, zeros = rep(ncol(coefs), ncol(coefs)),
row_complexity = NA_integer_, quasi_Yates = FALSE, weak_Thurstone = FALSE, 
Butler = FALSE,  viral = FALSE, mve = FALSE, communality = FALSE)

`coefs`	A primary pattern matrix with all cells filled in
`cormat`	A correlation matrix among primary factors with all cells filled in
`zeros`	An integer vector whose length is the same as the number of factors that indicates the requisite number of zeros in the corresponding column of the primary pattern matrix
`row\_complexity`	An integer vector, either of length one or the same length as the number of rows in `coefs`, which indicates the requisite number of non-zero coefficients in that row of the primary pattern matrix. If a scalar, this complexity is used for all rows of the primary pattern matrix. If `NA_integer_` (the default) this row-wise mapping rule is not enforced
`quasi\_Yates`	A logical indicating whether to enforce a mapping rule that is inspired by Yates (1987) to produce cohyperplanarity; see Details
`weak\_Thurstone`	A logical indicating whether to enforce a mapping rule that is based on a minimal version of Thurstone's (1947) second and third rules of thumb for identifying the structure; see Details
`Butler`	A logical indicating whether to enforce a mapping a mapping rule that is inspired by Butler (1969) to produce a unit complexity basis for the common factor space; see Details
`viral`	A logical indicating whether to enforce a mapping rule that puts some outcomes (namely double the number of factors) into two hyperplanes in common factor space; see Details
`mve`	A logical that must be `FALSE` because this mapping rule is (perhaps permanently) disabled
`communality`	A logical indicating whether to enforce a mapping rule that tries to put one zero per factor within a row of the primary pattern matrix that has a relatively large communality; see Details

The “mapping rule” is a deterministic function that takes a matrix with an insufficient number of exact zeros (often no zeros at all) and transforms it rather minimally to a matrix with the requisite number of exact zeros in the specified columns and / or rows. Hence, when the transformed matrix is returned, it should have enough zeros in the proper locations to potentially satisfy some theorem on model identification, typically Reiers<c3><b8>l's (1950). It is not possible to verify that all aspects of Reiers<c3><b8>l's (1950) theorem are satisfied, but another internal function verifies as much as it can. During the (genetic) optimization, the discrepancy function is called after the mapping rule is enforced and after the primary pattern matrix is checked for compliance with Reiers<c3><b8>l's (1950) theorem. Hence, optimization is intended to find the optimal estimates, subject to the restriction that the model is identified (with the help of the mapping rule).

By default, this function is copied into the mapping_rule slot in the call to make_restrictions and its formals are adjusted to correspond to the specified mapping rule. You may wish to do so manually in rare circumstances. If a different function is used to enforce a mapping rule, it should also have coefs, cormat, and zeros arguments as documented above.

The vignette has more details on Reiers<c3><b8>l's (1950) theorem and defines these mapping rules in symbols. To describe them briefly, the default “mapping rule” simply loops through the zeros argument and squashes the smallest zeros[p] cells (by magnitude) in the pth column of coefs to zero and returns the resulting matrix. This is the behavior depicted in the first example and can be brought about by leaving all the arguments that have default values at their defaults.

Only one mapping rule should be requested. The execption is that the default mapping rule is often called after a non-default mapping rule has been applied to make sure that that the pth column has zeros[p] zeros in it.

The weak_Thurstone mapping rule is similar but has an additional provision that no row of coefs may contain more than one (exact) zero. This mapping rule can be seen as minimally satisfying Thurstone's (1947) second and third rules for identification.

The row_complexity mapping rule is simply applied to the rows of the primary pattern matrix rather than the columns. Note that the “complexity” of an outcome is its number of non-zero coefficients. This mapping rule could loop over row_complexity (it actually utilizes apply) and squashes all but the largest (by magnitude) row_complexity[j] coefficients in the jth row of the reference structure matrix to zero and then rescales the result back to the primary pattern matrix. Then the default mapping rule is called, which may do nothing if there are now enough zeros in each column of coefs and the result is returned.

The Butler mapping rule concentrates the zeros within rows so that each factor correspons to an outcome of complexity one. Then, the default mapping rule is called to obtain more zeros for the pth factor.

The viral mapping rule is less drastic but concentrates two zeros into each of 2 * factors rows of coefs and is possibly useful if the number of factors is large. Then, the default mapping rule is called to obtain more zeros for the the pth factor.

The other mapping rules are more advanced and rely on the factor contribution matrix which is the element-by-element product of the the pattern and structure matrices. The rowSums of the factor contribution matrix is a vector of communalities. The communality mapping rule places one zero per factor in a row of coefs corresponding to a row of the factor contribution matrix with a high ratio of its arithmetic mean to its geometric mean. Then, the default mapping rule is called to obtain a sufficient number of zeros for the pth factor.

The quasi_Yates mapping rule places zeros in rows of coefs corresponding to rows of the factor contribution matrix with large differences between columns. The idea is to place zeros in rows where one factor is weak and another factor is strong in terms of explaining the variance in the corresponding outcome variable. This mapping rule is intended to achieve “cohyperplanarity” as discussed in Yates (1987), albeit not in reference to SEFA.

A matrix, specifically the primary pattern matrix with the requisite number of zeros in its columns and / or rows.

Ben Goodrich

Reiers<c3><b8>l, O. (1950) On the Identifiability of Parameters in Thurstone's Multiple Factor Analysis. Psychometrika, 15, 121–149.

make_restrictions, parameter.coef.SEFA-class

  ## This is just a demo; you should NOT call this function directly

  cormat <- diag(2)        # factor intercorrelation matrix
  coefs  <- matrix(rnorm(20), nrow = 10, ncol = 2) # primary pattern matrix
  zeros <- c(2, 2)         # we require two zeros per factor

  any(coefs == 0)          # FALSE

  ## Default mapping rule
  coefs_default <- mapping_rule(coefs, cormat, zeros)
  colSums(coefs_default == 0)      # c(2, 2)
  # Now the 2 smallest coefficients in each column are squashed to zero
  print(cbind(coefs, NA, coefs_default), digits = 3)

  ## row_complexity mapping rule
  coefs_row <- mapping_rule(coefs, cormat, zeros, row_complexity = 1)
  colSums(coefs_default == 0)      # at least two per factor
  # Now the smaller coefficient in each row is squashed to zero
  print(cbind(coefs, NA, coefs_row), digits = 3)

  ## The other mapping rules are sort of useless in the two factor case