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The TRANSREG Procedure |

For two reasons it is typically not appropriate to test hypotheses by using the output from PROC TRANSREG as input to other procedures such as the REG procedure. First, PROC REG has no way of determining how many degrees of freedom were used for each transformation. Second, the Type II sums of squares for the tests of the individual regression coefficients are not correct for the transformation regression model since PROC REG, as it evaluates the effect of each variable, cannot change the transformations of the other variables. PROC TRANSREG uses the correct degrees of freedom and sums of squares.

In an ordinary univariate linear model, there is one parameter for each
independent variable, including the intercept. In the transformation
regression model, many of the "variables" are used internally in
the bases for the transformations. Each basis column has one parameter
or *scoring* coefficient, and each linearly independent column has
one model degree of freedom associated with it. Coefficients applied to
transformed variables, *model coefficients*, do not enter into the
degrees of freedom calculations. They are by-products of the
standardizations and can be absorbed into the transformations by
specifying the ADDITIVE *a-option*. The word *parameter* is
reserved for model and scoring coefficients that have a degree of
freedom associated with them.

For expansions, there is one model parameter for each variable created
by the expansion (except for all missing CLASS columns and expansions
that have an implicit intercept). Each IDENTITY variable has one model
parameter. If there are *m* POINT variables, they expand to *m*+1
variables and, hence, have *m*+1 model parameters. For *m* EPOINT
variables, there are 2*m* model parameters. For *m* QPOINT variables,
there are *m*(*m*+3)/2 model parameters. If a variable with *m*
categories is designated CLASS, there are *m*-1 parameters. For BSPLINE
and PSPLINE variables of DEGREE=*n* with NKNOTS=*k*, there are *n*+*k*
parameters. Note that one of the *n*+*k*+1 BSPLINE columns and one of the
*m* CLASS(variable / ZERO=NONE) columns are not counted due to the
implicit intercept.

There are scoring parameters for missing values in nonexcluded observations. Each ordinary missing value (.) has one scoring parameter. Each different special missing value (._ and .A through .Z) within each variable has one scoring parameter. Missing values specified in the UNTIE= and MONOTONE= options follow the rules for UNTIE and MONOTONE transformations, which are described later in this chapter.

For all nonoptimal transformations (LOG, LOGIT, ARSIN, POWER, EXP, RANK), there is one parameter per variable in addition to any missing value scoring parameters.

For SPLINE, OPSCORE, and LINEAR transformations,
the number of scoring parameters is the
number of basis columns that are used internally to find the
transformations minus 1 for the intercept. The number of scoring
parameters for SPLINE variables is the same as the number of model
parameters for BSPLINE and PSPLINE variables. If DEGREE=*n* and
NKNOTS=*k*, there are *n*+*k* scoring parameters. The number of scoring
parameters for OPSCORE, SMOOTH, and SSPLINE variables is the same as the
number of model parameters for CLASS variables. If there are *m*
categories, there are *m*-1 scoring parameters. There is one parameter
for each LINEAR variable. For SPLINE, OPSCORE, LINEAR, MONOTONE, UNTIE,
and MSPLINE transformations, missing value scoring parameters are computed as described
previously with the nonoptimal transformations.

The number of scoring parameters for MONOTONE, UNTIE, and MSPLINE
transformations is less precise than for SPLINE, OPSCORE, and LINEAR
transformations. One way of handling a MONOTONE transformation is
to treat it as if it were
the same as an OPSCORE transformation. If there are *m* categories, there are *m*-1
potential scoring parameters. However, there are typically fewer
than *m*-1 unique parameter estimates since some of those *m*-1 scoring
parameter estimates may be tied during the optimal scaling to impose the
order constraints. Imposing ties on the scoring parameter estimates is
equivalent to fitting a model with fewer parameters. So there are two
available scoring parameter counts: *m*-1 and a smaller number that is
determined during the analysis. Using *m*-1 as the model degrees of
freedom for MONOTONE variables (treating OPSCORE and MONOTONE
transformations the same way) is *conservative*,
since the MONOTONE scoring parameter estimates are more restricted than
the OPSCORE scoring parameter estimates. Using the smaller count (the
number of scoring parameter estimates that are different minus 1 for the
intercept) in the model degrees of freedom is *liberal*,
since the data and the model together are being used to determine the
number of parameters. PROC TRANSREG reports tests using both liberal
and conservative degrees of freedom to provide lower and upper bounds
on the "true" *p*-values.

For the UNTIE transformation, the conservative scoring parameter count is the number of distinct observations, whereas the liberal scoring parameter count is the number of scoring parameter estimates that are different minus 1 for the intercept. Hence, when you specify UNTIE, conservative tests have zero error degrees of freedom unless there are replicated observations.

For MSPLINE variables of DEGREE=*n* and NKNOTS=*k*, the conservative
scoring parameter count is *n*+*k*, whereas the liberal parameter count is
the number of scoring parameter estimates that are different, minus 1
for the intercept. A liberal degrees of freedom of 1 does not
necessarily imply a linear transformation. It just implies that *n*
plus *k* minus the number of ties imposed equals 1. An example of a one
degree-of-freedom nonlinear transformation is a two-piece linear
transformation in which the slope of one piece is 0.

The number of scoring parameters is determined during each iteration.
After the last iteration, enough information is available for the TEST
*a-option* to produce an ANOVA table that reports the overall fit of the
model. If you specify the SS2 *a-option*, further iterations are necessary to test
the contribution of each transformation to the overall model.

The liberal tests do not compensate for over-parameterization. For
example, requesting a spline transformation with *k* knots when a
linear transformation will suffice results in "liberal"
tests that are actually conservative because too many degrees of
freedom are being used for the transformations. Use as few knots as
possible to avoid this problem.

In ordinary multiple regression, an *F* test of the null hypothesis
that the coefficient for variable *x*_{j} is zero can be constructed by
comparing two linear models. One model is the full model with all
parameters, and the other is a reduced model that has all parameters
except the parameter for variable *x*_{j}. The difference between the
model sum of squares for the full model and the model sum of squares for
the reduced model is the Type II sum of squares for the test of the null
hypothesis that the coefficient for variable *x*_{j} is 0. The numerator
of the *F* test has one degree of freedom. The mean square error
for the full model is the denominator of the *F* test of variable
*x*_{j}. Note that the estimates of the coefficients for the two
models are not usually the same. When variable *x*_{j} is removed,
the coefficients for the other variables change to compensate for the
removal of *x*_{j}. In a transformation regression model, the
transformations of the other variables must be allowed to change and the
numerator degrees of freedom are not always ones. It is not correct to
simply let the model coefficients for the transformed variables change
and apply the new model coefficients to the old transformations computed
with the old scoring parameter estimates. In a transformation
regression model, further iteration is needed to test each
transformation because all the scoring parameter estimates for other
variables must be allowed to change to test the effect of variable
*x*_{j}. This can be quite time consuming for a large model if the DUMMY
*a-option* cannot be used to solve directly for the transformations.

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