IBM® SPSS® Amos™ 28

Amos minimizes discrepancy functions (Browne, 1982; Browne, 1984) of the form,

(D1) .

Different discrepancy functions are obtained by changing the way f is defined. If means and intercepts are unconstrained and do not appear as explicit model parameters, and will be omitted and f will be written .

The discrepancy functions and are obtained by taking f to be

.

Except for an additive constant that depends only on the sample size, is –2 times the Kullback-Leibler information quantity (Kullback & Leibler, 1951). Strictly speaking, and do not qualify as discrepancy functions according to Browne's definition since .

For maximum likelihood estimation (Ml), are obtained by taking f  to be

(D2)

For generalized least squares estimation (Gls), are obtained by taking f to be

(D3) .

For asymptotically distribution-free estimation (Adf), are obtained by taking f to be

(D4) ,

where the elements of are given by Browne (1984) in Equations 3.1–3.4:

,

,

,

.

For 'scale free' least squares estimation (Sls), are obtained by taking f to be

(D5) ,

where .

For unweighted least squares estimation (Uls), are obtained by taking f  to be

(D6) .

The Emulisrel6 method can be used to replace (D1) with

(D1a) .

F is then calculated as .

When G = 1 and r = 1, (D1) and (D1a) are equivalent, giving

.

For maximum likelihood, asymptotically distribution-free, and generalized least squares estimation, both (D1) and (D1a) have a chi-square distribution for correctly specified models under appropriate distributional assumptions. Asymptotically, (D1) and (D1a) are equivalent. However, both formulas can exhibit some inconsistencies in finite samples. Suppose you have two independent samples and a model for each. Furthermore, suppose that you analyze the two samples simultaneously, but that, in doing so, you impose no constraints requiring any parameter in one model to equal any parameter in the other model. Then if you minimize (D1a), the parameter estimates obtained from the simultaneous analysis of both groups will be the same as from separate analyses of each group alone. Furthermore, the discrepancy function (D1a) obtained from the simultaneous analysis will be the sum of the discrepancy functions from the two separate analyses. Formula (D1) does not have this property when r is nonzero. Using formula (D1) to do a simultaneous analysis of the two groups will give the same parameter estimates as two separate analyses, but the discrepancy function from the simultaneous analysis will not be the sum of the individual discrepancy functions.

On the other hand, suppose you have a single sample to which you have fitted some model using Amos. Now suppose that you arbitrarily split the sample into two groups of unequal size and perform a simultaneous analysis of both groups, employing the original model for both groups, and constraining each parameter in the first group to be equal to the corresponding parameter in the second group. If you have minimized (D1) in both analyses, you will get the same results in both. However, if you use (D1a) in both analyses, the two analyses will produce different estimates and a different minimum value for F.

All of the inconsistencies just pointed out can be avoided by using (D1) with the choice r = 0, so that (D1) becomes

.