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ulsfactor(s, m [, quiet:T,silent:T,start:psi0,uselogs:T], maxit:nmax,
  minit:nmin, print:T,crit:crvec]), REAL symmetric matrix s with no
  MISSING values, integers nmax > 0 and nmin > 0, crvec =
  vector(numsig, nsigsq, delta)

Keywords: factor analysis
You use ulsfactor() to find ULS (unweighted least squares) estimates of
the vector psi of factor analysis uniquenesses and the loading matrix L
by means of nonlinear least squares.  It uses macro ulsresids() to
compute residuals required by non-linear least squares macro levmar().

ulsfactor(r, m) estimates uniquenesses and loadings for a m-factor
analysis based on p by p symmetric correlation or variance-covariance
matrix r.  m > 0 must be an integer < (2*p + 1 - sqrt(8*p+1))/2.  The
default starting value for the uniquenesses is psi0 = 1/diag(solve(r)).

ulsfactor() prints the estimated uniquenesses psihat and loading matrix,
the minimized criterion value, and vals, a vector of numbers computed
from certain eigenvalues; see below.

The result is an "invisible" (assignable but not automatically printed)
variable of the form
The values of these components are as follows

   psi     length p vector of estimated uniquenesses
   L       p by m matrix of estimated loadings satisfying
           L' %*% L is diagonal
   crit    sum of squared "residuals" = vector(r - rhat)
   vals    v, where v is the vector of eigenvalues of r - dmat(psi)
   iconv   integer >= 0 indicating convergence status, with 0 and 4
           indicating non-convergence
   n       number of iterations

There are several keyword phrases that can be used as arguments to
control the behavior of ulsfactor().
  start:psi0     psi0 a REAL vector of length p with min(psi0) > 0
                 to be used as starting values for the iteration
  uselogs:T      log(psi) is used in the iteration instead of psi;
                 this ensures psi does not become negative
  maxit:nmax     No more than nmax iterations are to be used; the
                 default is 30
  minit:nmin     At least nmin iterations will be performed; the
                 default is 1
  crit:crvec     crvec = vector(numsig, nsigsq, delta), 3 criteria
                 for convergence; a negative criterion is ignored;
                 see macro levmar() for details
  quiet:T        uniquenesses, loadings and vals are not printed;
                 only certain warning messages are printed
  silent:T       nothing is printed except error messages
  print:T        macro levmar() will print a status report on every

Values of iconv indicate the following situations
 iconv  Meaning
   0    Not converged
   1    Converged with relative change in psi or log(psi) < 10^-numsig
   2    Converged with relative change in rss <= 10^-nsigsq
   3    Converged with ||gradient|| < delta
   4    Interation stopped; could not reduce rss.

With or without uselogs:T, there is no guarantee that the solution
found is admissible in the sense that r - dmat(psihat) is positive
semi-definite.  A warning is printed when is not.  With uselogs:T this
situation is sometimes indicated by an 'argument to solve() singular'
error message.

Without uselogs:T, there is no guarantee that min(psihat) > = 0.  A
warning is printed if it is not.

Every time macro ulsresids() is called by levmar(), it saves a copy of
the current value of psi in invisible variable _PSIGLS.  Type
print(_PSIGLS) to see this value.

ulsfactor() is very much a work in progress that may in fact be
abandoned in favor of other methods using optimization macros in macro
file Math.mac distributed with MacAnova.

Gary Oehlert 2003-01-15