All estimates and hypothesis tests assume that the model
is correctly specified and the errors are distributed
according to classical statistical assumptions.
The output displayed by PROC RSREG includes the following.
Estimation and Analysis of Variance
- The actual form of the coding operation
for each value of a variable is
where M is the average of the highest and
lowest values for the variable in the
design and S is half their difference.
The Subtracted off column contains the M values
for this formula for each factor variable,
and S is found in the Divided by column.
coded value = [1/S]( original value - M)
- The summary table for the response variable contains
the following information.
- Response Mean is the mean of
the response variable in the sample.
When a WEIGHT statement is used,
the mean is calculated by
- Root MSE estimates the standard deviation
of the response variable and is calculated as
the square root of the Total Error mean square.
- The R-Square value is R2, or
the coefficient of determination.
R2 measures the proportion of the
variation in the response that is attributed
to the model rather than to random error.
- The Coefficient of Variation is 100 times the
ratio of the Root MSE to the Response Mean.
- A table analyzing the significance of the terms of the regression
is displayed. Terms are brought into the regression in four steps: (1)
the Intercept and any covariates in the model,
(2) Linear terms like X1 and X2,
(3) pure Quadratic terms
like X1*X1 or X2*X2, and (4) Crossproduct terms like X1*X2.
- The Degrees of Freedom should be the same as
the number of corresponding parameters unless
one or more of the parameters are not estimable.
- Type I Sum of Squares, also called the sequential
sums of squares, measure the reduction in the
error sum of squares as sets of terms (Linear,
Quadratic, and so forth) are added to the model.
- R-Square measures the portion of total
R2 contributed as each set of terms (Linear,
Quadratic, and so forth) is added to the model.
- Each F Value tests the null hypothesis that
all parameters in the term are zero using the
Total Error mean square as the denominator.
This item is a test of a Type I hypothesis, containing
the usual F test numerator, conditional on the
effects of subsequent variables not being in the model.
- Pr > F is the significance value or probability
of obtaining at least as great an F ratio
given that the null hypothesis is true.
- The Total Error Sum of Squares can be
partitioned into Lack of Fit and Pure Error.
When Lack of Fit is significant, there is
variation around the model other than random error
(such as cubic effects of the factor variables).
- The Total Error Mean Square
estimates , the variance.
- F Value tests the null hypothesis that
the variation is adequately described by random error.
- A table containing the parameter estimates from the model is
- The Parameter Estimate column contains the
parameter estimates based
on the uncoded values of the factor variables.
If an effect is a linear combination of previous
effects, the parameter for the effect is not estimable.
When this happens, the degrees of freedom are zero,
the parameter estimate is set to zero, and the
estimates and tests on other parameters are
conditional on this parameter being zero.
- The Standard Error column contains the
estimated standard deviations of the parameter
estimates based on uncoded data.
- The t Value column contains
t values of a test of the null hypothesis that
the true parameter is zero when the uncoded
values of the factor variables are used.
- Pr > |T| gives the significance value
or probability of a greater absolute t
ratio given that the true parameter is zero.
- The Parameter Estimate from Coded Data
column contains the parameter estimates based on the
coded values of the factor variables.
These are the estimates used in the
subsequent canonical and ridge analyses.
- The sum of squares are partitioned by the Factors
in the model, and an analysis table is displayed.
The test on a factor, say X1, is a joint test
on all the parameters involving that factor.
For example, the test for X1 tests the null hypothesis that
the true parameters for X1, X1*X1, and X1*X2 are all zero.
- The Critical Value columns contains the values of the
factor variables that correspond to the
stationary point of the fitted response surface.
The critical values can be at a
minimum, maximum, or saddle point.
- The Eigenvalues and Eigenvectors are from
the matrix of
quadratic parameter estimates based on the coded data.
They characterize the shape of the response surface.
- Coded Radius is the distance from the
coded version of the associated point to the
coded version of the origin of the ridge.
The origin is given by the point at radius zero.
- Estimated Response is the estimated value of
the response variable at the associated point.
The Standard Error of this estimate is also given.
This quantity is useful for assessing the relative
credibility of the prediction at a given radius.
Typically, this standard error increases rapidly as
the ridge moves up to and beyond the design perimeter,
reflecting the inherent difficulty of making
predictions beyond the range of experimentation.
- Uncoded Factor Values are the values of the
uncoded factor variables that give the optimum
response at this radius from the ridge origin.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.