The SURVEYPHREG Procedure

Testing the Global Null Hypothesis

The following statistics are available to test the global null hypothesis upper H 0 colon bold-italic beta equals bold 0. Let d be the usual degrees of freedom computed from the survey data by using the number of strata, clusters, or replicate weights; and let p be the number of estimable parameters in the null hypothesis upper H 0. For more information about computing d, see the section Degrees of Freedom.

The unadjusted likelihood ratio test statistic is expressed as

chi Subscript normal upper L normal upper R Superscript 2 Baseline equals 2 left-bracket log left-brace upper L left-parenthesis ModifyingAbove bold-italic beta With caret right-parenthesis right-brace minus log left-brace upper L left-parenthesis bold 0 right-parenthesis right-brace right-bracket

where L(dot) denotes the partial pseudo-likelihood that is described in the section Partial Likelihood Function for the Cox Model and ModifyingAbove bold-italic beta With caret denotes the estimated regression parameters. The p-value for the unadjusted test is computed by using a chi-square distribution with p degrees of freedom.

The unadjusted likelihood ratio statistic is sensitive to the scaling of the weights. PROC SURVEYPHREG computes an adjusted likelihood ratio test statistic that is invariant to the scaling of the weights. The adjusted test is similar to the second-order adjusted Rao-Scott chi-square tests. For more information, see Rao, Scott, and Skinner (1998), and Lumley and Scott (2013). The adjusted likelihood ratio test statistic is expressed as

chi Subscript normal upper R normal upper S Baseline 2 Superscript 2 Baseline equals StartFraction left-parenthesis n slash ModifyingAbove upper N With caret right-parenthesis chi Subscript normal upper L normal upper R Superscript 2 Baseline Over delta overbar Subscript period Baseline left-parenthesis 1 plus ModifyingAbove a With caret squared right-parenthesis EndFraction

where delta 1, delta 2, …, delta Subscript r are the positive eigenvalues from the generalized design effect matrix (Variance Ratios and Standard Error Ratios), delta overbar Subscript period Baseline equals 1 slash r sigma-summation Underscript i equals 1 Overscript r Endscripts delta Subscript i is the mean of the positive eigenvalues, ModifyingAbove a With caret squared equals left-parenthesis r minus 1 right-parenthesis Superscript negative 1 Baseline sigma-summation Underscript i equals 1 Overscript r Endscripts left-parenthesis delta Subscript i Baseline minus delta overbar Subscript period Baseline right-parenthesis squared slash delta overbar Subscript period Superscript 2 is the squared coefficient of variations of the positive eigenvalues, n is the number of observation units, and ModifyingAbove upper N With caret equals sigma-summation Underscript h i j Endscripts w Subscript h i j is the sum of the weights over all observation units. The p-value for the adjusted test is computed by using a chi-square distribution with r slash left-parenthesis 1 plus ModifyingAbove a With caret squared right-parenthesis degrees of freedom.

The usual assumptions that are required for a likelihood ratio test do not hold for the pseudo-likelihood that is used by PROC SURVEYPHREG (Rao, Scott, and Skinner 1998), leading to other methods of testing the global null hypothesis, such as the Wald test discussed in the following paragraph.

The Wald test uses the variance estimator that accounts for complex sampling such as stratification, clustering, and unequal weighting. Let upper Q equals ModifyingAbove bold-italic beta With caret prime left-bracket ModifyingAbove bold upper V With caret left-parenthesis ModifyingAbove bold-italic beta With caret right-parenthesis right-bracket Superscript negative 1 Baseline ModifyingAbove bold-italic beta With caret, where ModifyingAbove bold-italic beta With caret is the estimated regression parameters and ModifyingAbove bold upper V With caret left-parenthesis ModifyingAbove bold-italic beta With caret right-parenthesis is the estimated covariance matrix for ModifyingAbove bold-italic beta With caret. You can request the Wald tests that are described in the following table by using the DF= option in the MODEL statement.

Numerator Denominator
Value of DF= Test Request Test Statistic Degrees of Freedom Degrees of Freedom
NONE Chi-square Q p normal infinity
v Customized F vQ/pd p v
DESIGN Unadjusted F Q/p p d
DESIGN (v) Unadjusted F Q/p p v
PARMADJ Adjusted F (d–p+1)Q/pd p d–p+1
PARMADJ (v) Adjusted F (v-p+1)Q/pv p v-p+1
DESIGNADJ Adjusted F (d–p+1)Q/pd p d

Last updated: December 09, 2022