 Dataplot Vol 1 Auxiliary Chapter

RECIPE

Name:
RECIPE
Type:
Analysis Command
Purpose:
Compute REgression Confidence Intervals on PErcentiles (RECIPE).
Description:
RECIPE is used to compute approximate one-sided tolerance limits (or equivalently, confidence intervals on percentiles) for a wide range of situations where one is able to assume a normal probability model. Arbitrary regression models with or without a random effect can be analyzed. It is this ability to accomodate between batch variability that makes the RECIPE code perhaps unique.

The development of RECIPE was motivated for use in determining design allowables for composite materials in aircraft applications, particularly in the prescence of between-batch variability.

Note that an A-basis value is a (0.99,0.95) lower tolerance limit and a B-basis value is a (0.90,0.95) lower tolerance limit. Alternatively A- and B-basis values can be interpreted to be 95% lower confidence bounds on the 1st and 10th population percentiles respectively.

In Dataplot, the A BASIS and B BASIS commands can be used to compute A-basis and B-basis values for univariate data for normal, lognormal, and Weibull distributions.

RECIPE is used to calculate basis values for regression models with or without a random batch effect. This methodology was developed by NIST statistician Mark Vangel as part of his involvement with MIL-HDBK-17E (or MIL-17 Handbook). Dataplot adapted source code provided by Mark Vangel to implement this command within Dataplot.

More formally, let denote a mixed model with two components of variance. The fixed part of the model is , where X is an nxr known matrix having l <= n distinct rows, and is an unkown rXl vector. The random part is Z*b + e where Z is an nxm known matrix of zeros and ones, b follows a normal distribution with mean zero and standard deviation , and e follows a normal distribution with mean zero and standard deviation . Let w be an arbitrary rx1 vector, and assume , , and are estimable. RECIPE uses y to obtain approximate one-sided beta-content tolerance intervals for the population U where U follows a normal distribution with mean and standard deviation .

The above is only a brief outline of RECIPE. A complete discussion of this methodolgy is available at the following web site (including the original Fortran source code):

These web pages discuss both the mathematical model for RECIPE and its application to real problems. Additional discussion is contained in the papers listed in the Reference section. The example programs below give some typical applications of RECIPE.

Note that the input model for the original Fortran implementation of RECIPE is different than the Dataplot implementation.

RECIPE is typically preceeded by a regression or ANOVA analysis. RECIPE is then typically applied to the final regression or ANOVA model. RECIPE supports the following types of models:

1. 1-factor regression (including polynomial models)
2. multi-factor regression
3. multi-factor ANOVA

RECIPE is generally used in the context of the MIL-17 Handbook standards for obtaining A- and B-basis material property values.

The primary output from the RECIPE command is a set of tolerance values.

Syntax 1:
RECIPE FIT <y> <x> <batch> <xpred>
<SUBSET/EXCEPT/FOR qualification>
where <y> is the response (= dependent) variable;
<x> is the independent variable;
<batch> is a batch identifier variable;
<xpred> is a variable containing the points at which the tolerances are calculated;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax is used for linear regression and polynomial models.

The degree of the fit can be specified by entering the following command before the RECIPE FIT command:

RECIPE FIT DEGREE <N> where <N> is a positive integer that specifies the degree polynomial to fit (usually 1 or 2, but higher degrees are allowed).

If the RECIPE FIT DEGREE command is not entered, a linear fit is assumed.

Syntax 2:
RECIPE FIT <y> <x1 ... xk> <batch> <xpred1 ... xpredk>
<SUBSET/EXCEPT/FOR qualification>
where <y> is the response (= dependent) variable;
<x1 ... xk> is a list of one or more independent variables;
<batch> is a batch identifier variable;
<xpred1 ... xpredk> is a list of one or more variables containing the points at which the tolerances are calculated;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax is used for the multilinear regression case.

Syntax 3:
RECIPE ANOVA <y> <x1> ... <xk> <batch>
<SUBSET/EXCEPT/FOR qualification>
where <y> is the response (= dependent) variable;
<x1> ... <xk> are the k independent variables;
<batch> is a batch identifier variable;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax is used for multilinear and ANOVA models.

The <batch> variable is optional. To specify whether or not the last variable is a batch variable or one of the factor variables, enter the following command:

RECIPE FACTORS <N> where <N> is a positive integer that specifies the number of factor (indpendent) variables on the RECIPE ANOVA command.

Examples:
RECIPE ANOVA Y X BATCH
RECIPE ANOVA Y X BATCH SUBSET BATCH = 1 TO 3

RECIPE FIT Y X BATCH X2
RECIPE FIT Y X1 X2 BATCH X1P X2P

Note:
By default, Dataplot compute a 95% confidence interval for the 90% coverage interval (i.e., the 10th percentile).

To change the confidence interval calculated, enter

RECIPE CONFIDENCE <val>

where <val> defines the desired confidence. Typical values are 0.95, 0.90, or 0.99.

To change the coverage interval, enter

RECIPE PROBABILITY CONTENT <val>

where <val> defines the desired coverage. Typical values are 0.90 and 0.99.

Note:
By default, the tolerance values outout from the RECIPE commands are automatically saved in the internal variable TOL. If you want to save these values in a variable with a different name, then enter the following command before the RECIPE FIT or RECIPE ANOVA command:

RECIPE OUTPUT <varname>

where <varname> is the desired variable name. Note that after the RECIPE FIT or RECIPE ANOVA command, this variable can be used like any other Dataplot variable.

Note:
If the between batch variance is zero, then tolerance limits produced by RECIPE are exact. However, when between batch variability exists, the actual confidence level will depend on the ratio of between batch variability and to within batch variability , or equivalently, to the intraclass correlation The intraclass correlation is the correlation between observations from the same batch.

The nuisance parameter is unknown and typically the number of batches is too small to even estimate it well. We would like to have a tolerance limit procedure for which the actual confidence interval equals the nominal level, whatever might be. Although this goal is probably unobtainable in general, one can come extremely close for certain simple regression models. RECIPE provides tolerance intervals for which the confidence levels do not depend strongly on , and for which the actual confidence is generally fairly close to the nominal level.

In order to determine how close the actual and nominal confidence levels are, it is necessary to simulate. The following commands

RECIPE SIMCOV FIT
RECIPE SIMCOV ANOVA

correspond to the RECIPE FIT and RECIPE ANOVA commands and can be used to simulate the actual confidence. The RECIPE SIMCOV commands have the same syntax as the RECIPE commands.

The output from the RECIPE SIMCOV commands is a column of values for with the corresponding actual confidence level. One can expect this command to show that the RECIPE intervals are conservative when is near zero, somewhat anticonservative for intermediate values of , and nearly exact for near 1. For highly unbalanced data, the confidence may vary substatially from the nominal level when =1.

If the actual confidence levels differ too substantially from the nominal levels, this indicates that a Satterthwaite approximation used by RECIPE is not adequate and can be improved upon by replacing the Satterthwaite approximation with the appropriate quantile of a simulated pivotal random variable. By doing this, one can obtain an actual confidence level that is nearly exact for =1 and typically better than Satterthwaite approximation for intermediate values of as well.

If the RECIPE SIMCOV command shows that the Satterthwaite approximation is not adequate, enter the following command to use simulated pivotal values instead:

RECIPE SATTERTHWAITE NO

To restore the use of Satterthwaite approximation, enter

RECIPE SATTERTHWAITE NO

To control the number of correlation values at which to compute SIMCOV probabilites, enter

RECIPE CORRELATION <n>

where <n> is the desired number of values. The default is 11 (that is, p = 0, 0.1, 0.2, 0.3, ... , 0.9 1.0).

To control the number of replicates used in computing the SIMCOV confidence levels, enter

RECIPE SIMCOV REPLICATES <n>

where <n> is the number of replicates. The default is 10,000.

If Satterthwaite approximation is turned off (and simulated pivotal values are used instead), the number of replicates used in computing the simulated pivotal values can be specified by entering the command

RECIPE SIMPVT REPLICATES <n>

where is the number of replicates. The default is 10,000.

Default:
None
Synonyms:
None
Related Commands:
 FIT = Perform a lest squares fit. ANOVA = Perform an ANOVA. BASIS TOLERANCE LIMITS = Generate A- and B-basis tolerance limits. TOLERANCE LIMITS = Generate tolerance limits for the mean. ANDERSON-DARLING K SAMPLE TEST = Perform an Anderson-Darling k sample test. ANDERSON-DARLING TEST = Perform an Anderson-Darling goodness of fit test. GRUBBS TEST = Perform Grubbs test for outliers.
References:
"MIL-HDBK-17 Volume 1: Guidelines for Characterization of Structural Materials", Depeartment of Defense, chapter 8. The URL for MIL-HDBK-17 is http://mil-17.udel.edu/.

"A User's Guide to RECIPE: A FORTRAN Program for Determining One-Sided Tolerance Limits for Mixed Models With Two Components of Variance Version 1.0", Mark Vangel, unpublished document available online at http://www.itl.nist.gov/div898/software/recipe/recidoc.pdf.

"New Methods for One-Sided Tolerance Limits for a One-Way Balanced Random Effects ANOVA Model", Mark Vangel, Technometrics, Vol. 34, 1992, pp. 176-185.

"One-Sided B-Content Tolerance Limits for Mixed Models with Two Components of Variance", Mark Vangel, Technometrics, submitted.

"Design Allowables From Regression Models Using Data From Several Batches", Mark Vangel, 12th Symposium on Composite Materials: Testing and Design, American Society of Testing and Materials, 1996.

Applications:
Materials Science
Implementation Date:
1997/9
Program 1:
. The complete analysis can be viewed by:
. LIST VANGEL31.DP
. RECIPE Tolerance Limits Analysis of Graphite/Epoxy Tape Strength
. Univariate Case
. ------------------------
. Step 1--Read in the data
.
skip 25
. ------------------------------------------------
. Step 6--Compute (recipe/normal) tolerance limits
.
recipe content 90
recipe confidence 95
recipe factors 1
let n = size y
let x = 1 for i = 1 1 n
let b = 1 for i = 1 1 n
recipe anova y x b
.
let tol1 = tol(1)
relative histogram y
drawdata tol1 0 tol1 +.01

**************************
**  recipe anova y x b  **
**************************

REGINI : WARNING: BETWEEN-BATCH VARIANCE CANNOT
BE ESTIMATED FROM THESE DATA. RESULTS
WILL BE BASED ON THE ASSUMPTION THAT THE
BETWEEN-BATCH VARIABILITY IS NEGLIGIBLE.

RECIPE ANALYSIS
(MARK VANGEL, NIST)

NUMBER OF OBSERVATIONS         =       38
NUMBER OF UNIQUE DESIGN POINTS =        1
NUMBER OF BATCHES              =        1

MODEL: ANOVA
RESSD FROM THE FITTED MODEL    =   0.3057080E+01
RESDF FROM THE FITTED MODEL    =        37.

PROBABILITY CONTENT            =   90.00000%
PROBABILITY CONFIDENCE         =   95.00000%
SATTERTHWAITE APPROXIMATION USED

TOLERANCE VALUES STORED IN VARIABLE TOL
RESIDUALS        STORED IN VARIABLE RES
PREDICTED VALUES STORED IN VARIABLE PRED

Program 2:
. The complete analysis can be viewed by:
. LIST VANGEL32.DP
. RECIPE Tolerance Limits Analysis of Graphite/Epoxy Tape Strength
. Regression Case
. Date--September 1997
.
. ------------------------
. Step 1--Read in the data
.
skip 25
.
. ------------------------------------------------
. Step 6--Compute (recipe/normal) tolerance limits
.
recipe content 90
recipe confidence 95
recipe degree 1
let x2 = sequence -75 25 200
recipe fit y x b x2
.
characters x blank box
lines blank solid dotted
plot y pred versus x and
plot tol versus x2

***************************
**  recipe fit y x b x2  **
***************************

RECIPE ANALYSIS
(MARK VANGEL, NIST)

NUMBER OF OBSERVATIONS         =       45
NUMBER OF UNIQUE DESIGN POINTS =        3
NUMBER OF BATCHES              =        3

MODEL: LINEAR FIT
RESSD FROM THE FITTED MODEL    =   0.1537953E+02
RESDF FROM THE FITTED MODEL    =        43.

PROBABILITY CONTENT            =   90.00000%
PROBABILITY CONFIDENCE         =   95.00000%
SATTERTHWAITE APPROXIMATION USED

TOLERANCE VALUES STORED IN VARIABLE TOL
RESIDUALS        STORED IN VARIABLE RES
PREDICTED VALUES STORED IN VARIABLE PRED

Program 3:
. The complete analysis can be viewed by:
. LIST VANGEL33.DP
. RECIPE Tolerance Limits Analysis of Graphite/Epoxy Fabric Strength
. ANOVA Case
. Date--September 1997
.
. ------------------------
. Step 1--Read in the data
.
skip 25
.
. ------------------------------------------------
. Step 6--Compute (recipe/normal) tolerance limits
.
recipe content 90
recipe confidence 95
recipe factors 1
recipe anova y x b
.
characters x blank box
lines blank blank blank
xtic offset 0.3 0.3
major xtic mark number 2
minor xtic mark number 0
plot y pred versus x and
plot tol versus x

**************************
**  recipe anova y x b  **
**************************

RECIPE ANALYSIS
(MARK VANGEL, NIST)

NUMBER OF OBSERVATIONS         =       30
NUMBER OF UNIQUE DESIGN POINTS =        2
NUMBER OF BATCHES              =        3

MODEL: ANOVA
RESSD FROM THE FITTED MODEL    =   0.8579769E+00
RESDF FROM THE FITTED MODEL    =        28.

PROBABILITY CONTENT            =   90.00000%
PROBABILITY CONFIDENCE         =   95.00000%
SATTERTHWAITE APPROXIMATION USED

TOLERANCE VALUES STORED IN VARIABLE TOL
RESIDUALS        STORED IN VARIABLE RES
PREDICTED VALUES STORED IN VARIABLE PRED

Date created: 6/5/2001
Last updated: 4/4/2003