The VARCOMP Procedure

Confidence Limits

When no exact confidence limits exist, it is common practice to use approximate confidence limits. Two such approximations are the modified large-sample (MLS) method and the generalized confidence limit (GCL) method as discussed in Burdick, Borror, and Montgomery (2005). When analyzing a balanced one-way or two-way design, if you specify the CL= option with METHOD=TYPE1 or GRR, the VARCOMP procedure computes confidence limits by using either the MLS method (the default) or the GCL method. Generalized confidence limits are obtained by specifying the CL=GCL option in the MODEL statement.

MLS Confidence Limits

The method of MLS confidence limits was first introduced by Graybill and Wang (1980). It starts with approximate large-sample confidence limits; then it modifies the limits to be exact under certain parameter conditions.

For a balanced two-way crossed random model with interaction, formulas for the MLS method are given in Table 6. See Burdick, Borror, and Montgomery (2005) for the formulas for one-way or balanced two-way with no interaction models.

Confidence limits for parameters such as variances and their ratios might not contain the corresponding point estimates, because negative confidence bounds are increased to zero.

Table 6: 100(1 minus alpha)% MLS Confidence Limits

Parameter Lower Bound Upper Bound
mu Subscript y y overbar Subscript dot dot dot Baseline minus upper C StartRoot StartFraction upper K Over p o r EndFraction EndRoot y overbar Subscript dot dot dot Baseline plus upper C StartRoot StartFraction upper K Over p o r EndFraction EndRoot
gamma Subscript upper P ModifyingAbove gamma With caret Subscript upper P Baseline minus StartRoot upper V Subscript upper L upper P Baseline EndRoot slash left-parenthesis o r right-parenthesis ModifyingAbove gamma With caret Subscript upper P Baseline plus StartRoot upper V Subscript upper U upper P Baseline EndRoot slash left-parenthesis o r right-parenthesis
gamma Subscript upper M ModifyingAbove gamma With caret Subscript upper M Baseline minus StartRoot upper V Subscript upper L upper M Baseline EndRoot slash left-parenthesis p r right-parenthesis ModifyingAbove gamma With caret Subscript upper M Baseline plus StartRoot upper V Subscript upper U upper M Baseline EndRoot slash left-parenthesis p r right-parenthesis
gamma Subscript y ModifyingAbove gamma With caret Subscript y Baseline minus StartRoot upper V Subscript upper L upper T Baseline EndRoot slash left-parenthesis p o r right-parenthesis ModifyingAbove gamma With caret Subscript y Baseline plus StartRoot upper V Subscript upper U upper T Baseline EndRoot slash left-parenthesis p o r right-parenthesis
gamma Subscript upper R upper L Subscript upper R upper U Subscript upper R
rho Subscript upper P upper L Subscript upper R Baseline slash left-parenthesis 1 plus upper L Subscript upper R Baseline right-parenthesis upper U Subscript upper R Baseline slash left-parenthesis 1 plus upper U Subscript upper R Baseline right-parenthesis
rho Subscript upper M 1 slash left-parenthesis 1 plus upper U Subscript upper R Baseline right-parenthesis 1 slash left-parenthesis 1 plus upper L Subscript upper R Baseline right-parenthesis


The terms in Table 6 are defined as follows:

StartLayout 1st Row 1st Column upper V Subscript upper L upper P 2nd Column equals upper G 1 squared upper S Subscript upper P Superscript 4 Baseline plus upper H 3 squared upper S Subscript upper P upper O Superscript 4 Baseline plus upper G 13 upper S Subscript upper P Superscript 2 Baseline upper S Subscript upper P upper O Superscript 2 Baseline 2nd Row 1st Column upper V Subscript upper U upper P 2nd Column equals upper H 1 squared upper S Subscript upper P Superscript 4 Baseline plus upper G 3 squared upper S Subscript upper P upper O Superscript 4 Baseline plus upper H 13 upper S Subscript upper P Superscript 2 Baseline upper S Subscript upper P upper O Superscript 2 Baseline 3rd Row 1st Column upper V Subscript upper L upper M 2nd Column equals upper G 2 squared upper S Subscript upper O Superscript 4 Baseline plus upper G 3 squared left-parenthesis p minus 1 right-parenthesis squared upper S Subscript upper P upper O Superscript 4 Baseline plus upper G 4 squared p squared left-parenthesis r minus 1 right-parenthesis squared upper S Subscript upper E Superscript 4 Baseline 4th Row 1st Column upper V Subscript upper U upper M 2nd Column equals upper H 2 squared upper S Subscript upper O Superscript 4 Baseline plus upper H 3 squared left-parenthesis p minus 1 right-parenthesis squared upper S Subscript upper P upper O Superscript 4 Baseline plus upper H 4 squared p squared left-parenthesis r minus 1 right-parenthesis squared upper S Subscript upper E Superscript 4 Baseline 5th Row 1st Column upper V Subscript upper L upper T 2nd Column equals upper G 1 squared p squared upper S Subscript upper P Superscript 4 Baseline plus upper G 2 squared o squared upper S Subscript upper O Superscript 4 Baseline plus upper G 3 squared left-parenthesis p o minus p minus o right-parenthesis squared upper S Subscript upper P upper O Superscript 4 Baseline plus upper G 4 squared left-parenthesis p o right-parenthesis squared left-parenthesis r minus 1 right-parenthesis squared upper S Subscript upper E Superscript 4 Baseline 6th Row 1st Column upper V Subscript upper U upper T 2nd Column equals upper H 1 squared p squared upper S Subscript upper P Superscript 4 Baseline plus upper H 2 squared o squared upper S Subscript upper O Superscript 4 Baseline plus upper H 3 squared left-parenthesis p o minus p minus o right-parenthesis squared upper S Subscript upper P upper O Superscript 4 Baseline plus upper H 4 squared left-parenthesis p o right-parenthesis squared left-parenthesis r minus 1 right-parenthesis squared upper S Subscript upper E Superscript 4 Baseline 7th Row 1st Column upper L Subscript upper R 2nd Column equals StartFraction p left-parenthesis 1 minus upper G 1 right-parenthesis left-parenthesis upper S Subscript upper P Superscript 2 Baseline minus upper F 1 upper S Subscript upper P upper O Superscript 2 Baseline right-parenthesis Over p o left-parenthesis r minus 1 right-parenthesis upper S Subscript upper E Superscript 2 Baseline plus o left-parenthesis 1 minus upper G 1 right-parenthesis upper F 3 upper S Subscript upper O Superscript 2 Baseline plus o left-parenthesis p minus 1 right-parenthesis upper S Subscript upper P upper O Superscript 2 Baseline EndFraction 8th Row 1st Column upper U Subscript upper R 2nd Column equals StartFraction p left-parenthesis 1 plus upper H 1 right-parenthesis left-parenthesis upper S Subscript upper P Superscript 2 Baseline minus upper F 2 upper S Subscript upper P upper O Superscript 2 Baseline right-parenthesis Over p o left-parenthesis r minus 1 right-parenthesis upper S Subscript upper E Superscript 2 Baseline plus o left-parenthesis 1 plus upper H 1 right-parenthesis upper F 4 upper S Subscript upper O Superscript 2 Baseline plus o left-parenthesis p minus 1 right-parenthesis upper S Subscript upper P upper O Superscript 2 Baseline EndFraction 9th Row 1st Column upper G 1 2nd Column equals 1 minus upper F Subscript alpha slash 2 colon normal infinity comma p minus 1 Baseline 10th Row 1st Column upper G 2 2nd Column equals 1 minus upper F Subscript alpha slash 2 colon normal infinity comma o minus 1 Baseline 11th Row 1st Column upper G 3 2nd Column equals 1 minus upper F Subscript alpha slash 2 colon normal infinity comma left-parenthesis p minus 1 right-parenthesis left-parenthesis o minus 1 right-parenthesis Baseline 12th Row 1st Column upper G 4 2nd Column equals 1 minus upper F Subscript alpha slash 2 colon normal infinity comma p o left-parenthesis r minus 1 right-parenthesis Baseline 13th Row 1st Column upper H 1 2nd Column equals upper F Subscript 1 minus alpha slash 2 colon normal infinity comma p minus 1 Baseline minus 1 14th Row 1st Column upper H 2 2nd Column equals upper F Subscript 1 minus alpha slash 2 colon normal infinity comma o minus 1 Baseline minus 1 15th Row 1st Column upper H 3 2nd Column equals upper F Subscript 1 minus alpha slash 2 colon normal infinity comma left-parenthesis p minus 1 right-parenthesis left-parenthesis o minus 1 right-parenthesis Baseline minus 1 16th Row 1st Column upper H 4 2nd Column equals upper F Subscript 1 minus alpha slash 2 colon normal infinity comma p o left-parenthesis r minus 1 right-parenthesis Baseline minus 1 17th Row 1st Column upper F 1 2nd Column equals upper F Subscript 1 minus alpha slash 2 colon p minus 1 comma left-parenthesis p minus 1 right-parenthesis left-parenthesis o minus 1 right-parenthesis Baseline 18th Row 1st Column upper F 2 2nd Column equals upper F Subscript alpha slash 2 colon p minus 1 comma left-parenthesis p minus 1 right-parenthesis left-parenthesis o minus 1 right-parenthesis Baseline 19th Row 1st Column upper F 3 2nd Column equals upper F Subscript 1 minus alpha slash 2 colon p minus 1 comma o minus 1 Baseline 20th Row 1st Column upper F 4 2nd Column equals upper F Subscript alpha slash 2 colon p minus 1 comma o minus 1 Baseline 21st Row 1st Column upper G 13 2nd Column equals StartFraction left-parenthesis upper F 1 minus 1 right-parenthesis squared minus upper G 1 squared upper F 1 squared minus upper H 3 squared Over upper F 1 EndFraction 22nd Row 1st Column upper H 13 2nd Column equals StartFraction left-parenthesis 1 minus upper F 2 right-parenthesis squared minus upper H 1 squared upper F 2 squared minus upper G 3 squared Over upper F 2 EndFraction 23rd Row 1st Column upper K 2nd Column equals s Subscript upper P Superscript 2 Baseline plus s Subscript upper O Superscript 2 Baseline minus s Subscript upper P upper O Superscript 2 Baseline 24th Row 1st Column upper C 2nd Column equals StartFraction s Subscript upper P Superscript 2 Baseline StartRoot upper F Subscript 1 minus alpha colon 1 comma p minus 1 Baseline EndRoot plus s Subscript upper O Superscript 2 Baseline StartRoot upper F Subscript 1 minus alpha colon 1 comma o minus 1 Baseline EndRoot minus s Subscript upper P upper O Superscript 2 Baseline StartRoot upper F Subscript 1 minus alpha colon 1 comma left-parenthesis p minus 1 right-parenthesis left-parenthesis o minus 1 right-parenthesis Baseline EndRoot Over upper K EndFraction EndLayout

The symbol upper F Subscript alpha colon d f Baseline italic 1 comma d f Baseline italic 2 represents the percentile of an F distribution with df1 and df2 degrees of freedom and area alpha to the left.

Generalized Confidence Limits

The method of generalized confidence limits was first introduced by Weerahandi (1993). The 100(1-alpha)% generalized confidence limits are determined as follows:

  1. Initialize the random number generator with the seed. The seed value is specified by the SEED= option.

  2. Sample N generalized pivot quantities (GPQ), defined to have a distribution that is independent of the parameters under study. The value N is specified by the NSAMPLE= option.

  3. Define the lower and upper limits as the alpha slash 2 and 1 minus alpha slash 2 quantiles of the sampled GPQ values.

Formulas for generalized confidence limits are given in Table 7, where Z denotes a standard normal random variable and upper W 1 comma upper W 2 comma upper W 3 comma and upper W 4 denote jointly independent chi-square random variables that are independent of Z with degrees of freedom p minus 1 comma o minus 1 comma left-parenthesis p minus 1 right-parenthesis left-parenthesis o minus 1 right-parenthesis and p o left-parenthesis r minus 1 right-parenthesis, respectively. The value of epsilon in Table 7 is specified by the EPSILON= option.

Table 7: 100(1 minus alpha)% Generalized Confidence Limits

Parameter GPQ
mu Subscript y y overbar Subscript dot dot dot Baseline minus upper Z StartRoot max left-bracket epsilon comma StartFraction left-parenthesis p minus 1 right-parenthesis s Subscript p Superscript 2 Baseline Over p o r upper W 1 EndFraction plus StartFraction left-parenthesis o minus 1 right-parenthesis s Subscript upper O Superscript 2 Baseline Over p o r upper W 2 EndFraction minus StartFraction left-parenthesis p minus 1 right-parenthesis left-parenthesis o minus 1 right-parenthesis s Subscript upper P upper O Superscript 2 Baseline Over p o r upper W 3 EndFraction right-bracket EndRoot
gamma Subscript upper P max left-bracket 0 comma StartFraction left-parenthesis p minus 1 right-parenthesis s Subscript p Superscript 2 Baseline Over o r upper W 1 EndFraction minus StartFraction left-parenthesis p minus 1 right-parenthesis left-parenthesis o minus 1 right-parenthesis s Subscript p o Superscript 2 Baseline Over p r upper W 3 EndFraction right-bracket
gamma Subscript upper M StartFraction left-parenthesis o minus 1 right-parenthesis s Subscript o Superscript 2 Baseline Over p r upper W 2 EndFraction plus StartFraction left-parenthesis p minus 1 right-parenthesis squared left-parenthesis 0 minus 1 right-parenthesis s Subscript p Superscript 2 Baseline o Over p r upper W 3 EndFraction plus StartFraction p o left-parenthesis r minus 1 right-parenthesis squared s Subscript upper E Superscript 2 Baseline Over r upper W 4 EndFraction
gamma Subscript y StartFraction left-parenthesis p minus 1 right-parenthesis s Subscript p Superscript 2 Baseline Over o r upper W 1 EndFraction plus StartFraction left-parenthesis o minus 1 right-parenthesis s Subscript upper O Superscript 2 Baseline Over p r upper W 2 EndFraction plus StartFraction left-parenthesis p o minus p minus o right-parenthesis left-parenthesis p minus 1 right-parenthesis left-parenthesis o minus 1 right-parenthesis s Subscript upper P upper O Superscript 2 Baseline Over p o r upper W 3 EndFraction plus StartFraction p o left-parenthesis r minus 1 right-parenthesis squared s Subscript upper E Superscript 2 Baseline Over r upper W 4 EndFraction
gamma Subscript upper R StartFraction GPQ left-parenthesis gamma Subscript upper P Baseline right-parenthesis Over GPQ left-parenthesis gamma Subscript upper M Baseline right-parenthesis EndFraction


In general, the GCL method provides a more accurate confidence interval with a shorter interval width than the MLS method. However, the greater accuracy comes at the cost of being somewhat nondeterministic, because of the reliance on simulation.

Last updated: December 09, 2022