The SURVEYFREQ Procedure

Proportions

PROC SURVEYFREQ computes the estimate of the proportion in table cell (r, c) as the ratio of the estimated total for the table cell to the estimated overall total,

StartLayout 1st Row 1st Column ModifyingAbove upper P With caret Subscript r c 2nd Column equals 3rd Column ModifyingAbove upper N With caret Subscript r c Baseline slash ModifyingAbove upper N With caret 2nd Row 1st Column Blank 2nd Column equals 3rd Column left-parenthesis sigma-summation Underscript h equals 1 Overscript upper H Endscripts sigma-summation Underscript i equals 1 Overscript n Subscript h Baseline Endscripts sigma-summation Underscript j equals 1 Overscript m Subscript h i Baseline Endscripts delta Subscript h i j Baseline left-parenthesis r comma c right-parenthesis upper W Subscript h i j Baseline right-parenthesis slash left-parenthesis sigma-summation Underscript h equals 1 Overscript upper H Endscripts sigma-summation Underscript i equals 1 Overscript n Subscript h Baseline Endscripts sigma-summation Underscript j equals 1 Overscript m Subscript h i Baseline Endscripts upper W Subscript h i j Baseline right-parenthesis EndLayout

PROC SURVEYFREQ estimates the variances of the proportion estimates by using the variance estimation method that you request. If you request a replication method (bootstrap, BRR, jackknife, or replicate weights), the procedure estimates the variances as described in the section Replication Variance Estimation. By default, PROC SURVEYFREQ estimates variances by using the Taylor series method (which you can also request by specifying the VARMETHOD=TAYLOR option).

By using Taylor series linearization, the variance of a proportion estimate can be expressed as

ModifyingAbove normal upper V normal a normal r With caret left-parenthesis ModifyingAbove upper P With caret Subscript r c Baseline right-parenthesis equals sigma-summation Underscript h equals 1 Overscript upper H Endscripts ModifyingAbove normal upper V normal a normal r With caret Subscript h Baseline left-parenthesis ModifyingAbove upper P With caret Subscript r c Baseline right-parenthesis

where if n Subscript h Baseline greater-than 1,

StartLayout 1st Row 1st Column ModifyingAbove normal upper V normal a normal r With caret Subscript h Baseline left-parenthesis ModifyingAbove upper P With caret Subscript r c Baseline right-parenthesis 2nd Column equals 3rd Column StartFraction n Subscript h Baseline left-parenthesis 1 minus f Subscript h Baseline right-parenthesis Over n Subscript h Baseline minus 1 EndFraction sigma-summation Underscript i equals 1 Overscript n Subscript h Baseline Endscripts left-parenthesis e Subscript r c Superscript h i Baseline minus e overbar Subscript r c Superscript h Baseline right-parenthesis squared 2nd Row 1st Column e Subscript r c Superscript h i 2nd Column equals 3rd Column left-parenthesis sigma-summation Underscript j equals 1 Overscript m Subscript h i Baseline Endscripts left-parenthesis delta Subscript h i j Baseline left-parenthesis r comma c right-parenthesis minus ModifyingAbove upper P With caret Subscript r c Baseline right-parenthesis upper W Subscript h i j Baseline right-parenthesis slash ModifyingAbove upper N With caret 3rd Row 1st Column e overbar Subscript r c Superscript h 2nd Column equals 3rd Column left-parenthesis sigma-summation Underscript i equals 1 Overscript n Subscript h Baseline Endscripts e Subscript r c Superscript h i Baseline right-parenthesis slash n Subscript h Baseline EndLayout

and if n Subscript h Baseline equals 1,

ModifyingAbove normal upper V normal a normal r With caret Subscript h Baseline left-parenthesis ModifyingAbove upper P With caret Subscript r c Baseline right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column missing 2nd Column if n Subscript h Sub Superscript prime Subscript Baseline equals 1 for h prime equals 1 comma 2 comma ellipsis comma upper H 2nd Row 1st Column 0 2nd Column if n Subscript h Sub Superscript prime Subscript Baseline greater-than 1 for some 1 less-than-or-equal-to h prime less-than-or-equal-to upper H EndLayout

The standard error of the proportion is computed as

normal upper S normal t normal d normal upper E normal r normal r left-parenthesis ModifyingAbove upper P With caret Subscript r c Baseline right-parenthesis equals StartRoot ModifyingAbove normal upper V normal a normal r With caret left-parenthesis ModifyingAbove upper P With caret Subscript r c Baseline right-parenthesis EndRoot

Similarly, the estimate of the proportion in row r is

ModifyingAbove upper P With caret Subscript r dot Baseline equals ModifyingAbove upper N With caret Subscript r dot Baseline slash ModifyingAbove upper N With caret

And its variance estimate is

ModifyingAbove normal upper V normal a normal r With caret left-parenthesis ModifyingAbove upper P With caret Subscript r dot Baseline right-parenthesis equals sigma-summation Underscript h equals 1 Overscript upper H Endscripts ModifyingAbove normal upper V normal a normal r With caret Subscript h Baseline left-parenthesis ModifyingAbove upper P With caret Subscript r dot Baseline right-parenthesis

where if n Subscript h Baseline greater-than 1,

StartLayout 1st Row 1st Column ModifyingAbove normal upper V normal a normal r With caret Subscript h Baseline left-parenthesis ModifyingAbove upper P With caret Subscript r dot Baseline right-parenthesis 2nd Column equals 3rd Column StartFraction n Subscript h Baseline left-parenthesis 1 minus f Subscript h Baseline right-parenthesis Over n Subscript h Baseline minus 1 EndFraction sigma-summation Underscript i equals 1 Overscript n Subscript h Baseline Endscripts left-parenthesis e Subscript r dot Superscript h i Baseline minus e overbar Subscript r dot Superscript h Baseline right-parenthesis squared 2nd Row 1st Column e Subscript r dot Superscript h i 2nd Column equals 3rd Column left-parenthesis sigma-summation Underscript j equals 1 Overscript m Subscript h i Baseline Endscripts left-parenthesis delta Subscript h i j Baseline left-parenthesis r dot right-parenthesis minus ModifyingAbove upper P With caret Subscript r dot Baseline right-parenthesis upper W Subscript h i j Baseline right-parenthesis slash ModifyingAbove upper N With caret 3rd Row 1st Column e overbar Subscript r dot Superscript h 2nd Column equals 3rd Column left-parenthesis sigma-summation Underscript i equals 1 Overscript n Subscript h Baseline Endscripts e Subscript r dot Superscript h i Baseline right-parenthesis slash n Subscript h Baseline EndLayout

and if n Subscript h Baseline equals 1,

ModifyingAbove normal upper V normal a normal r With caret Subscript h Baseline left-parenthesis ModifyingAbove upper P With caret Subscript r dot Baseline right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column missing 2nd Column if n Subscript h Sub Superscript prime Subscript Baseline equals 1 for h prime equals 1 comma 2 comma ellipsis comma upper H 2nd Row 1st Column 0 2nd Column if n Subscript h Sub Superscript prime Subscript Baseline greater-than 1 for some 1 less-than-or-equal-to h prime less-than-or-equal-to upper H EndLayout

The standard error of the proportion in row r is computed as

normal upper S normal t normal d normal upper E normal r normal r left-parenthesis ModifyingAbove upper P With caret Subscript r dot Baseline right-parenthesis equals StartRoot ModifyingAbove normal upper V normal a normal r With caret left-parenthesis ModifyingAbove upper P With caret Subscript r dot Baseline right-parenthesis EndRoot

Computations for the proportion in column c are done in the same way.

Last updated: December 09, 2022