Estimation Criteria

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The CALIS Procedure

Estimation Criteria

The following five estimation methods are available in PROC CALIS:

unweighted least squares (ULS)
generalized least squares (GLS)
normal-theory maximum likelihood (ML)
weighted least squares (WLS, ADF)
diagonally weighted least squares (DWLS)

An INWGT= data set can be used to specify other than the default weight matrices W for GLS, WLS, and DWLS estimation.

In each case, the parameter vector is estimated iteratively by a nonlinear optimization algorithm that optimizes a goodness-of-fit function F. When n denotes the number of manifest variables, S denotes the given sample covariance or correlation matrix for a sample with size N, and C denotes the predicted moment matrix, then the fit function for unweighted least-squares estimation is

F_ULS = .5 Tr [ (S- C)² ]

For normal-theory generalized least-squares estimation, the function is

F_GLS = .5 Tr [ ( S^-1(S- C) )² ]

For normal-theory maximum likelihood estimation, the function is

F_ML = Tr(SC^-1) - n + ln(det(C)) - ln(det(S))

The first three functions can be expressed by the generally weighted least-squares criterion (Browne 1982):

F_GWLS = .5 Tr [ ( W^-1(S- C) )² ]

For unweighted least squares, the weight matrix W is chosen as the identity matrix I; for generalized least squares, the default weight matrix W is the sample covariance matrix S; and for normal-theory maximum likelihood, W is the iteratively updated predicted moment matrix C. The values of the normal-theory maximum likelihood function F_ML and the generally weighted least-squares criterion F_GWLS with W = C are asymptotically equivalent.

The goodness-of-fit function that is minimized in weighted least-squares estimation is

F_WLS = Vec(s_ij - c_ij)' W^-1 Vec(s_ij - c_ij)

where Vec(s_ij - c_ij) denotes the vector of the n(n+1)/2 elements of the lower triangle of the symmetric matrix S- C, and W = (w_ij,kl) is a positive definite symmetric matrix with n(n+1)/2 rows and columns.

If the moment matrix S is considered as a covariance rather than a correlation matrix, the default setting of W = (w_ij,kl) is the consistent but biased estimators of the asymptotic covariances $\sigma_{ij,kl}$ of the sample covariance s_ij with the sample covariance s_kl

w_ij,kl = s_ij,kl - s_ijs_kl

where

$s_{ij,kl} = {1 \over N} \sum_{r=1}^N{(z_{ri} - \overline{z_i}) (z_{rj} - \overline{z_j})(z_{rk} - \overline{z_k}) (z_{rl} - \overline{z_l})}$

The formula of the asymptotic covariances of uncorrected covariances (using the UCOV or NOINT option) is a straightforward generalization of this expression.

The resulting weight matrix W is at least positive semidefinite (except for rounding errors). Using the ASYCOV option, you can use Browne's (1984, formula (3.8)) unbiased estimators

$w_{ij,kl} &=& {N(N-1) \over (N-2)(N-3)} (s_{ij,kl} - s_{ij}s_{kl}) \ & & - {N \over (N-2)(N-3)} (s_{ik} s_{jl} + s_{il} s_{jk} - {2\over N-1} s_{ij} s_{kl})$

There is no guarantee that this weight matrix is positive semidefinite. However, the second part is of order O(N^-1) and does not destroy the positive semidefinite first part for sufficiently large N. For a large number of independent observations, default settings of the weight matrix W result in asymptotically distribution-free parameter estimates with unbiased standard errors and a correct $\chi^2$ test statistic (Browne 1982, 1984).

If the moment matrix S is a correlation (rather than a covariance) matrix, the default setting of W = (w_ij,kl) is the estimators of the asymptotic covariances $\sigma_{ij,kl}$ of the correlations S = (s_ij) (Browne and Shapiro 1986; DeLeeuw 1983)

$w_{ij,kl} & = & r_{ij,kl} - {1 \over 2} r_{ij}(r_{ii,kl} + r_{jj,kl}) - {1 \ove... ...\ & & + {1 \over 4} r_{ij}r_{kl} (r_{ii,kk} + r_{ii,ll} + r_{jj,kk} + r_{jj,ll})$

where

$r_{ij,kl} = {s_{ij,kl} \over \sqrt{s_{ii} s_{jj} s_{kk} s_{ll}}}$

The asymptotic variances of the diagonal elements of a correlation matrix are 0. Therefore, the weight matrix computed by Browne and Shapiro's formula is always singular. In this case the goodness-of-fit function for weighted least-squares estimation is modified to

$F_{WLS} = {\sum_{i=2}^n \sum_{j=1}^{i-1} \sum_{k=2}^n \sum_{l=1}^{k-1} w^{ij,kl} (s_{ij} - c_{ij})(s_{kl} - c_{kl}) } + r \sum_i^n (s_{ii} - c_{ii})^2$

where r is the penalty weight specified by the WPENALTY=r option and the w^ij,kl are the elements of the inverse of the reduced (n(n-1)/2) ×(n(n-1)/2) weight matrix that contains only the nonzero rows and columns of the full weight matrix W. The second term is a penalty term to fit the diagonal elements of the moment matrix S. The default value of r= 100 can be decreased or increased by the WPENALTY= option. The often used value of r=1 seems to be too small in many cases to fit the diagonal elements of a correlation matrix properly. If your model does not fit the diagonal of the moment matrix S, you can specify the NODIAG option to exclude the diagonal elements from the fit function.

Storing and inverting the huge weight matrix W in WLS estimation needs considerable computer resources. A compromise is found by implementing the DWLS method that uses only the diagonal of the weight matrix W from the WLS estimation in the minimization function

F_DWLS = Vec(s_ij - c_ij)' diag(W)^-1 Vec(s_ij - c_ij)

The statistical properties of DWLS estimates are still not known.

In generalized, weighted, or diagonally weighted least-squares estimation, you can change from the default settings of weight matrices W by using an INWGT= data set. Because the diagonal elements w_ii,kk of the weight matrix W are interpreted as asymptotic variances of the sample covariances or correlations, they cannot be negative. The CALIS procedure requires a positive definite weight matrix that has positive diagonal elements.

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