When attempting to assess how well a model
fits a particular data set, one must realize at the outset that the classic
hypothesis-testing approach is inappropriate. Consider common factor analysis.
When maximum likelihood estimation became a practical reality, the *Chi-square*
"goodness-of-fit" statistic was originally employed in a sequential
testing strategy. According to this strategy, one first picked a small
number of factors, and tested the null hypothesis that this factor model
fit the population **S**
perfectly. If this hypothesis was rejected, the model was assumed to be
too simple (i.e., to have too few common factors) to fit the data. The
number of common factors was increased by one, and the preceding procedure
repeated. The sequence continued until the hypothesis test failed to reject
the hypothesis of perfect fit.

Steiger and Lind (1980) pointed out that this
logic was essentially flawed, because, for any population **S**
(other than one constructed as a numerical example directly from the common
factor model!) the *a priori* probability is essentially 1 that the
common factor model will not fit perfectly so long as degrees of freedom
for the *Chi-square* statistic were positive.

In essence, then, population fit for a covariance structure model with positive degrees of freedom is never really perfect. Testing whether it is perfect makes little sense. It is what statisticians sometimes call an "accept-support" hypothesis test, because accepting the null hypothesis supports what is generally the experimenter's point of view, i.e., that the model does fit.

Accept-support hypothesis tests are subject
to a host of problems. In particular, of course, the traditional priorities
between Type I and Type II error are reversed. If the proponent of a model
simply performs the *Chi-square* test with low enough power, the
model can be supported. As a natural consequence of this, hypothesis testing
approaches to the assessment of model fit *should* make some attempt
at power evaluation. Steiger and Lind (1980) demonstrated that performance
of statistical tests in common factor analysis could be predicted from
a noncentral *Chi-square* approximation. A number of papers dealing
with the theory and practice of power evaluation in covariance structure
analysis have been published (Matsueda & Bielby, 1986; Satorra and
Saris, 1985; Steiger, Shapiro, & Browne, 1985). Unfortunately, power
estimation in the analysis of a multivariate model is a difficult, somewhat
arbitrary procedure, and such power estimates have not, in general, been
reported in published studies.

The main reason for evaluating power is to
gain some understanding of precision of estimation in a particular situation,
to guard against the possibility that a model is "accepted"
simply because of insufficient power. An alternative (and actually more
direct) approach to the evaluation of precision is to *construct a confidence
interval on the population noncentrality parameter* (or some particularly
useful function of it). This approach, first suggested in the context
of covariance structure analysis by Steiger and Lind (1980) offers two
worthwhile pieces of information at the same time. It allows one, for
a particular model and data set to express (1) how bad a fit is in the
population, and (2) how precisely the *population *badness-of-fit
has been determined from the *sample *data.