The SURVEYPHREG Procedure
Contrasts and Hypothesis Tests
For a testable hypothesis 
, you can request different Wald tests by using the DF= option in the MODEL statement.
Let
where 
is a contrast vector or matrix that you specify, 
is the vector of regression parameters, 
is the estimated regression coefficients, 
is the estimated covariance matrix of , and 
is a matrix such that the following are true:
- has the same number of columns as .
- has full row rank.
The rank of
equals the rank of the matrix. All rows of
are estimable functions. The Wald F statistic that is computed by using the
matrix is equivalent to the Wald F statistic computed by using the matrix.
If is a full-rank matrix and all rows of are estimable functions, then is the same as . It is possible that such an matrix cannot be constructed for a given set of linear contrasts, in which case the contrasts are not testable. Let r be the rank of 
. Table 9 describes the Wald tests available in PROC SURVEYPHREG.
Table 9: Summary of Wald Tests
| Numerator | Denominator | |||
|---|---|---|---|---|
| Value of DF= | Test Request | Test Statistic | Degrees of Freedom | Degrees of Freedom |
| NONE | Chi-square | Q | r |  |
| v | Customized F | vQ/rd | r | v |
| DESIGN | Unadjusted F | Q/r | r | d |
| DESIGN (v) | Unadjusted F | Q/r | r | v |
| PARMADJ | Adjusted F | (d–r+1)Q/rd | r | d–r+1 |
| PARMADJ (v) | Adjusted F | (v–r+1)Q/rv | r | v–r+1 |
| DESIGNADJ | Adjusted F | Q/r | r | d |
The preceding development for Wald tests assumes that the estimated variance of 
, 
, is of the form 
for some estimate 
of the variance of 
. In this case, estimability, 
, ensures that this F statistic has a unique value no matter which kind of generalized inverse is used to compute it. However, when a design-based variance estimator is used to estimate the variability of , estimability does not ensure uniqueness. In this case, the F value is invariant to the choice of the generalized inverse if and only if 
is estimable and 
.
Although it is extremely rare, it is possible in practice that the preceding uniqueness condition is not satisfied. For example, if the number of clusters is less than the number of nonsingular parameters in the model, then the matrix of coefficients for testing the overall null does not satisfy the uniqueness condition. If this condition is not satisfied, then the F statistic for testing 
is not invariant to the choice of the 
-inverse of . In practical applications, the test statistic is compared with an F distribution as described in Table 9, but the value of the test statistic and therefore the inference might be different when a different -inverse is used. This F test is not recommended when the uniqueness condition is not satisfied. An alternative approach would be to increase the number of clusters or to find a parsimonious model so that the number of parameters is less than the number of clusters.