fungible (version 2.2.2)

faAlign: Align the columns of two factor loading matrices

Description

Align factor loading matrices across solutions using the Hungarian algorithm to locate optimal matches. faAlign will match the factors of F2 (the input matrix) to those in F1 (the target matrix) to minimize a least squares discrepancy function or to maximize factor congruence coefficients (i.e., vector cosines).

Usage

faAlign(F1, F2, Phi2 = NULL, MatchMethod = "LS")

Value

F2

re-ordered and reflected loadings of F2.

Phi2

reordered and reflected factor correlations.

FactorMap

a 2 x k matrix (where k is the number of columns of F1) structured such that row 1: the original column order of F2; row 2: the sorted column order of F2.

UniqueMatch

(logical) indicates whether a unique match was found.

MatchMethod

"LS" (least squares) or "CC" (congruence coefficients, i.e., cosines).

CC

Congruence coefficients for the matched factors.

LS

Root-mean-squared-deviations (least squares criterion) for the matched factors.

Arguments

F1

target Factor Loadings Matrix.

F2

input Factor Loadings Matrix. F2 will be aligned with the target matrix, F1.

Phi2

optional factor correlation matrix for F2 (default = NULL).

MatchMethod

"LS" (Least Squares) or "CC" (congruence coefficients).

Author

Niels Waller

References

Kuhn, H. W. (1955). The Hungarian Method for the assignment problem. Naval Research Logistics Quarterly, 2, 83-97.

Kuhn, H. W. (1956). Variants of the Hungarian method for assignment problems. Naval Research Logistics Quarterly, 3, 253-258.

Papadimitriou, C. & Steiglitz, K. (1982). Combinatorial Optimization: Algorithms and Complexity. Englewood Cliffs: Prentice Hall.

See Also

Other Factor Analysis Routines: BiFAD(), Box26, GenerateBoxData(), Ledermann(), SLi(), SchmidLeiman(), faEKC(), faIB(), faLocalMin(), faMB(), faMain(), faScores(), faSort(), faStandardize(), faX(), fals(), fapa(), fareg(), fsIndeterminacy(), orderFactors(), print.faMB(), print.faMain(), promaxQ(), summary.faMB(), summary.faMain()

Examples

# This example demonstrates the computation of 
# non-parametric bootstrap confidence intervals
# for rotated factor loadings.


library(GPArotation)

data(HS9Var)

HS9 <- HS9Var[HS9Var$school == "Grant-White",7:15]

# Compute an R matrix for the HSVar9 Mental Abilities Data
R.HS9 <- cor(HS9)

varnames <- c( "vis.per", "cubes", 
            "lozenges", "paragraph.comp",
            "sentence.comp","word.mean",
            "speed.add", "speed.count.dots",
            "speed.discr")



# Extract and rotate a 3-factor solution
# via unweighted least squares factor extraction 
# and oblimin rotation. 

NFac <- 3
NVar <- 9
B <- 200      # Number of boostrap samples
NSubj <- nrow(HS9)

# Unrotated 3 factor uls solution 
 F3.uls <- fals(R = R.HS9, nfactors = NFac)
 
# Rotate via oblimin 
 F3.rot <- oblimin(F3.uls$loadings, 
                      gam = 0, 
                      normalize = FALSE)

 F3.loadings <- F3.rot$loadings
 F3.phi <- F3.rot$Phi
 
 # Reflect factors so that salient loadings are positive
 Dsgn <- diag(sign(colSums(F3.loadings^3)))
 F3.loadings <- F3.loadings %*% Dsgn
 F3.phi <- Dsgn %*% F3.phi %*% Dsgn
 
 rownames(F3.loadings) <- varnames
 colnames(F3.loadings) <- paste0("f", 1:3)
 colnames(F3.phi) <- rownames(F3.phi) <- paste0("f", 1:3)
 
 cat("\nOblimin rotated factor loadings for 9 Mental Abilities Variables")
 print( round(F3.loadings, 2))
 
 cat("\nFactor correlation matrix")
 print( round( F3.phi, 2))
 
  # Declare variables to hold bootstrap output
  Flist <- Philist <- as.list(rep(0, B))
  UniqueMatchVec <- rep(0, B)
  rows <- 1:NSubj
 
  # Analyze bootstrap samples and record results 
  for(i in 1:B){
    cat("\nWorking on sample ", i)
    set.seed(i)
    
    # Create bootstrap sanples
    bsRows <- sample(rows, NSubj, replace= TRUE)
    Fuls <- fals(R = cor(HS9[bsRows, ]), nfactors = NFac)
    # rotated loadings
    Fboot <- oblimin(Fuls$loadings,
                             gam = 0, 
                             normalize = FALSE)
    
    
    out <- faAlign(F1 = F3.loadings, 
                   F2 = Fboot$loadings, 
                   MatchMethod = "LS")
    
    Flist[[i]] <- out$F2 # aligned version of Fboot$loadings
    UniqueMatchVec[i] <- out$UniqueMatch
  }
  
  cat("\nNumber of Unique Matches: ", 
      100*round(mean(UniqueMatchVec),2),"%\n")

  
  #  Make a 3D array from list of matrices
  arr <- array( unlist(Flist) , c(NVar, NFac, B) )
  
  #  Get quantiles of factor elements over third dimension (samples)
  F95 <- apply( arr , 1:2 , quantile, .975 )
  F05 <- apply( arr , 1:2 , quantile, .025 )
  Fse <- apply( arr , 1:2, sd  )
  
  cat("\nUpper Bound 95% CI\n")
  print( round(F95,3))
  cat("\n\nLower Bound 95% CI\n")
  print( round(F05,3))
  
  # plot distribution of bootstrap estimates
  # for example element
  hist(arr[5,1,], xlim=c(.4,1),
       main = "Bootstrap Distribution for F[5,1]",
       xlab = "F[5,1]")
  
  print(round (F3.loadings, 2))
  cat("\nStandard Errors")
  print( round( Fse, 2))