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Example 1
The Number of distinct sample moments referred to are sample means, variances, and
covariances. In most analyses, including the present one, Amos ignores means, so that
the sample moments are the sample variances of the four variables, recall1, recall2,
place1, and place2, and their sample covariances. There are four sample variances and
six sample covariances, for a total of 10 sample moments.
The Number of distinct parameters to be estimated are the corresponding
population variances and covariances. There are, of course, four population variances
and six population covariances, which makes 10 parameters to be estimated.
The Degrees of freedom is the amount by which the number of sample moments
exceeds the number of parameters to be estimated. In this example, there is a one-to-
one correspondence between the sample moments and the parameters to be estimated,
so it is no accident that there are zero degrees of freedom.
As we will see beginning with Example 2, any nontrivial null hypothesis about the
parameters reduces the number of parameters that have to be estimated. The result will
be positive degrees of freedom. For now, there is no null hypothesis being tested.
Without a null hypothesis to test, the following table is not very interesting:
If there had been a hypothesis under test in this example, the chi-square value would have
been a measure of the extent to which the data were incompatible with the hypothesis. A
chi-square value of 0 would ordinarily indicate no departure from the null hypothesis.
But in the present example, the 0 value for degrees of freedom and the 0 chi-square value
merely reflect the fact that there was no null hypothesis in the first place.
This line indicates that Amos successfully estimated the variances and covariances.
Sometimes structural modeling programs like Amos fail to find estimates. Usually,
when Amos fails, it is because you have posed a problem that has no solution, or no
unique solution. For example, if you attempt maximum likelihood estimation with
observed variables that are linearly dependent, Amos will fail because such an analysis
cannot be done in principle. Problems that have no unique solution are discussed
elsewhere in this user’s guide under the subject of identifiability. Less commonly,
Amos can fail because an estimation problem is just too difficult. The possibility of
such failures is generic to programs for analysis of moment structures. Although the
computational method used by Amos is highly effective, no computer program that
does the kind of analysis that Amos does can promise success in every case.
Chi-square = 0.00
Degrees of freedom = 0
Probability level cannot be computed
Minimum was achieved