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) |

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 structure(psihat:psi,loadings:L,criterion:crit,eigenvals:vals,\ status:iconv,iter:n) 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 iteration 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