Counting the Degrees of Freedom
In a regression problem, the number of degrees of freedom
for the error estimate is the number of
observations in the data set minus the number of parameters.
The NOBS=, DFR= (RDF=), and DFE= (EDF=) options refer to degrees of
freedom in this sense. However, these values are not related to the
degrees of freedom of a test statistic used in a covariance or
correlation structure analysis. The NOBS=, DFR=, and DFE= options should
be used in PROC CALIS to specify only the effective number of observations
in the input DATA= data set.
In general, the number of degrees of freedom in a covariance or
correlation structure analysis is defined as the difference
between the number of nonredundant values q in the observed
n ×n correlation or covariance matrix S and the number
t of free parameters X used in the fit of the specified model,
df = q - t. Both values, q and t, are counted
differently in different situations by PROC CALIS.
The number of nonredundant
values q is generally equal to the number of lower triangular elements
in the n ×n moment matrix S including all diagonal elements,
minus a constant c dependent upon special circumstances,
-
q = n (n+1) / 2-c
The number c is evaluated by adding the following quantities:
- If you specify a linear structural equation model
containing exogenous manifest variables by using the
RAM or LINEQS statement, PROC CALIS adds to c the number
of variances and covariances among these manifest exogenous
variables, which are automatically set in the corresponding
locations of the central model matrices (see the section "Exogenous Manifest Variables").
- If you specify the DFREDUCE=i option, PROC CALIS adds
the specified number i to c. The number i can be
a negative integer.
- If you specify the NODIAG option to exclude the fit of the
diagonal elements of the data matrix S, PROC CALIS adds
the number n of diagonal elements to c.
- If all the following conditions hold,
then PROC CALIS adds to c the number of the diagonal locations:
- NODIAG and DFREDUC= options are not specified.
- A correlation structure is being fitted.
- The predicted correlation matrix contains constants on the diagonal.
In some complicated models, especially those using programming statements,
PROC CALIS may not be able to detect all the constant predicted
values. In such cases, you must specify the DFREDUCE= option
to get the correct degrees of freedom.
The number t is the number of different parameter names used
in constructing the model if you do not use programming statements
to impose constraints on the parameters. Using programming statements
in general introduces two kinds of parameters:
- independent parameters, which are
used only at the right-hand side of the expressions
- dependent parameters, which are used at least once
at the left-hand side of the expressions
The independent parameters
belong to the parameters involved in the estimation process,
whereas the dependent parameters are fully defined by the
programming statements and can be computed from the independent
parameters. In this case, the number t is the number of different
parameter names used in the model specification, but not used in
the programming statements, plus the number of independent parameters.
The independent parameters and their initial values can be defined
in a model specification statement or in a PARMS statement.
The degrees of freedom are automatically increased by the number of
active constraints in the solution. Similarly, the number of parameters
are decreased by the number of active constraints. This affects
the computation of many fit statistics and indices. Refer
to Dijkstra (1992) for a discussion of the validity of statistical
inferences with active boundary constraints. If the researcher believes
that the active constraints will have a small chance of occurrence in
repeated sampling, it may be more suitable to turn off the automatic
adjustment using the NOADJDF option.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.
