Suipplementary Information for Experimental Design and Data Evaluation Considerations for HOVAMS

Electronic Supplementary Material
Accreditation and Quality Assurance, 2012, Springer, Heidelberg

Experimental design and data evaluation considerations
for comparisons of reference materials

David L. Duewer, Hugo Gasca Aragon, Katrice A. Lippa, Blaza Toman

National Institute of Standards and Technology (NIST)

Gaithersburg, MD 20899 USA

Disclaimer

Certain commercial software is identified in this report to specify the experimental procedure as completely as possible. In no case does such identification imply a recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the software is necessarily the best available for the purpose.

Contents

ANOVA-based tests for determining degrees of freedom...... S-

Software for estimation of expanded uncertainties...... S-

Excel “ANOVA: Single Factor”...... S-

SAS MIXED procedure...... S-

One-factor with replication...... S-

Two-factor nested...... S-

R lme4...... S-

One-factor with replication...... S-

Two-factor nested...... S-

OpenBUGS...... S-

One-Factor with replication...... S-

Two-factor nested...... S-

Exemplar GDR analysis...... S-

FREML...... S-

Data and commands...... S-

Tabular results...... S-

Graphical results...... S-

XLGENLINE...... S-

Data, commands, and tabular results...... S-

Graphical results...... S-

RegViz...... S-

Data and commands...... S-

Tabular results...... S-

Graphical results...... S-

References...... S-

RM Comparison Studies (Electronic Supplementary Material)

ANOVA-based tests for determining degrees of freedom

For a one-factor design with replication, the estimated number of degrees of freedom are

and for a two-factor nested design they are

For these designs, assuming random effects, there are two frequentist tests for the presence of a variance component based on the mean square errors: the classical significance test [[1]] and the less common test for a variance component estimate to be positive [[2]]. The 95 % level of confidence test for a variance component classical estimate uses the criteria:

where Mb is the between-group mean square, Mw is the within-group mean square, and Mr is the residual mean square. This test requires the variance component estimate being large enough for statistical significance. It will thus confirm the obvious outcome for small samples with large random effects but is insensitive to relatively small variance components.

The test for a variance component estimate to be positive uses the criteria:
.
This addresses whether the random effects are likely to be present regardless of the magnitude of the variance component estimate. This appears to be the test used in the SAS MIXED procedure. See below for an implementation of this test in R.

Software for estimation of expanded uncertainties

The following data and analyses procedures are intended as examples to guide use of the software systems and/or to benchmark analyses using other software systems. While many values in the different sets of results are useful for estimating variance components, the critical values for estimating expanded uncertainties on a mean result are in red font. The information provided by the Excel, SAS, and R implementations of the frequentist estimates discussed below all result in the same estimated values, although sometimes requiring additional calculations.

The same data are used in all examples. Data for two materials are provided for the one-factor with replication design, with and without appreciable between-campaign variance. Data for four exemplar materials are provided for the two-factor nested design, with and without appreciable between-aliquot and/or between-campaign variance.

Excel “ANOVA: Single Factor”

The Microsoft Office Excel 2003 “ANOVA: Single Factor” is an extremely easy-to-use tool that is adequate for evaluating one-factor designs with replication, but does require additional effort to interpret. The tool is provided in the “Analysis ToolPak” Excel add-in [[3]], an optional suite of data analysis tools freely available for most if not all versions of Excel. Most basic data analysis systems support one-factor ANOVA and it’s readily doable “by hand” [[4]].

Input data and exemplar command

Material A: No appreciable between-campaign variance

The “P-value” of 0.49 indicates that the “Between-Groups” (here, between-campaign) contribution to the variance is not significant. The standard uncertainty of the mean is therefore best calculated as the standard deviation/square root of the number of measurements. This tool provides neither the mean nor the SD, but they are easily calculated using Excel’s “AVERAGE(.)” and “STDEV(.)” functions. The mean is 50904.3, the standard deviation is 131.7, the standard uncertainty of the mean is , and the ±U95 is therefore t0.05,5 × 54 = 2.6 × 54 = 138.

Results for Material B: appreciable between-campaign variance

The “P-value” of 0.0004 indicates that there is a significant between-campaign contribution to the variance. The standard uncertainty on the mean is and v = 2. The ±U95 is thus estimated to be t0.05,2 × 70 = 4.3 × 70 = 301.

SAS MIXED Procedure

SAS [[5]] is an extensively documented and validated commercial data analysis system. The MIXED procedure [[6]] provides analysis of linear mixed-effects models.

It requires a good deal of experience and expertise to efficiently set up a SAS analysis and to select among the many options that are available for most of its component procedures, MIXED included. SAS is very definitely not freeware, but many institutions and universities do have licensed access.

One-factor with replication

Using SAS to analyze a one-factor completely balanced design is rather like using a sledgehammer to kill ants, but it does provide a “definitive” benchmark for less comprehensive (and less expensive) analysis systems.

Data and commands for the MIXED procedure

data CCQM_1F_Exemplar;

input material $char14. cmp rep y;

datalines;

MATERIAL A 1 1 50742

MATERIAL A 1 2 50760

MATERIAL A 1 3 51082

MATERIAL A 2 1 50912

MATERIAL A 2 2 50949

MATERIAL A 2 3 50981

MATERIAL B 1 1 23602

MATERIAL B 1 2 23576

MATERIAL B 1 3 23586

MATERIAL B 2 1 23431

MATERIAL B 2 2 23353

MATERIAL B 2 3 23459

MATERIAL B 3 1 23667

MATERIAL B 3 2 23645

MATERIAL B 3 3 23632

;

proc sort data= CCQM_1F_Exemplar;

by material;

run;

proc mixed data=CCQM_1F_Exemplar asycov noitprint;

class cmp rep;

model y = / solution ddfm=sat;

random cmp;

by material;

run;

Material A, no appreciable between-campaign variance

------material=MATERIAL A ------

The Mixed Procedure

Model Information

Data Set WORK.CCQM_1F_EXEMPLAR

Dependent Variable y

Covariance Structure Variance Components

Estimation Method REML

Residual Variance Method Profile

Fixed Effects SE Method Model-Based

Degrees of Freedom Method Satterthwaite

Class Level Information

Class Levels Values

cmp 2 1 2

rep 3 1 2 3

Dimensions

Covariance Parameters 2

Columns in X 1

Columns in Z 2

Subjects 1

Max Obs Per Subject 6

Number of Observations

Number of Observations Read 6

Number of Observations Used 6

Number of Observations Not Used 0

Covariance Parameter

Estimates

Cov Parm Estimate

cmp 0

Residual 17336

Asymptotic Covariance Matrix of Estimates

Row Cov Parm CovP1 CovP2

1 cmp

2 Residual 1.2022E8

Fit Statistics

-2 Res Log Likelihood 64.8

AIC (smaller is better) 66.8

AICC (smaller is better) 68.1

BIC (smaller is better) 65.5

Solution for Fixed Effects

Standard

Effect Estimate Error DF t Value Pr > |t|

Intercept 50904 53.7529 5 947.01 <.0001

The “Intercept” (mean) is 50904 with a “Standard Error” (standard uncertainty) of 54 and v = 5. The ±U95 estimate for the mean is thus t0.05,5 × 54 = 2.6 × 54 = 138.

Material B, appreciable between-campaign variance

------material=MATERIAL B ------

The Mixed Procedure

Model Information

Data Set WORK.CCQM_1F_EXEMPLAR

Dependent Variable y

Covariance Structure Variance Components

Estimation Method REML

Residual Variance Method Profile

Fixed Effects SE Method Model-Based

Degrees of Freedom Method Satterthwaite

Class Level Information

Class Levels Values

cmp 3 1 2 3

rep 3 1 2 3

Dimensions

Covariance Parameters 2

Columns in X 1

Columns in Z 3

Subjects 1

Max Obs Per Subject 9

Number of Observations

Number of Observations Read 9

Number of Observations Used 9

Number of Observations Not Used 0

Covariance Parameter

Estimates

Cov Parm Estimate

cmp 14338

Residual 1167.44

Asymptotic Covariance Matrix of Estimates

Row Cov Parm CovP1 CovP2

1 cmp 2.1693E8 -151436

2 Residual -151436 454309

Fit Statistics

-2 Res Log Likelihood 88.7

AIC (smaller is better) 92.7

AICC (smaller is better) 95.1

BIC (smaller is better) 90.9

Solution for Fixed Effects

Standard

Effect Estimate Error DF t Value Pr > |t|

Intercept 23550 70.0636 2 336.12 <.0001

The mean is 23550 with a standard uncertainty of 70 and v = 2. The ±U95 estimate for the mean is thus t0.05,2 × 70 = 4.3 × 70 = 301.

Two-factor nested

Data and commands for the MIXED procedure

data CCQM_2F_Exemplar;

input material $char14. campaign aliquot rep y;

datalines;

MATERIAL C 1 1 1 37.451

MATERIAL C 1 1 2 38.416

MATERIAL C 1 2 1 37.720

MATERIAL C 1 2 2 38.268

MATERIAL C 2 1 1 38.054

MATERIAL C 2 1 2 37.386

MATERIAL C 2 2 1 37.934

MATERIAL C 2 2 2 38.484

MATERIAL D 1 1 1 34.045

MATERIAL D 1 1 2 34.562

MATERIAL D 1 2 1 34.031

MATERIAL D 1 2 2 33.249

MATERIAL D 2 1 1 34.421

MATERIAL D 2 1 2 34.320

MATERIAL D 2 2 1 33.852

MATERIAL D 2 2 2 34.512

MATERIAL E 1 1 1 27.370

MATERIAL E 1 1 2 27.436

MATERIAL E 1 2 1 27.170

MATERIAL E 1 2 2 27.467

MATERIAL E 2 1 1 27.508

MATERIAL E 2 1 2 27.429

MATERIAL E 2 2 1 27.431

MATERIAL E 2 2 2 27.651

MATERIAL F 1 1 1 7.105

MATERIAL F 1 1 2 6.983

MATERIAL F 1 2 1 7.067

MATERIAL F 1 2 2 7.042

MATERIAL F 2 1 1 7.132

MATERIAL F 2 1 2 7.096

MATERIAL F 2 2 1 7.172

MATERIAL F 2 2 2 7.422

;

proc sort data=CCQM_2F_Exemplar;

by material;

run;

proc mixed data=CCQM_2F_Exemplar asycov noitprint;

class campaign aliquot rep;

model y = / solution ddfm=sat;

random campaign aliquot(campaign);

by material;

run;

Material C, no appreciable between-aliquot or -campaign variance

------material=MATERIAL C ------

The Mixed Procedure

Model Information

Data Set WORK.CCQM_2F_EXEMPLAR

Dependent Variable y

Covariance Structure Variance Components

Estimation Method REML

Residual Variance Method Profile

Fixed Effects SE Method Model-Based

Degrees of Freedom Method Satterthwaite

Class Level Information

Class Levels Values

cmp 2 1 2

alq 2 1 2

rep 2 1 2

Dimensions

Covariance Parameters 3

Columns in X 1

Columns in Z 6

Subjects 1

Max Obs Per Subject 8

Number of Observations

Number of Observations Read 8

Number of Observations Used 8

Number of Observations Not Used 0

Covariance Parameter

Estimates

Cov Parm Estimate

cmp 0

alq(cmp) 0

Residual 0.1761

Asymptotic Covariance Matrix of Estimates

Row Cov Parm CovP1 CovP2 CovP3

1 cmp

2 alq(cmp)

3 Residual -0.00274 0.008863

Fit Statistics

-2 Res Log Likelihood 9.8

AIC (smaller is better) 11.8

AICC (smaller is better) 12.6

BIC (smaller is better) 10.5

Solution for Fixed Effects

Standard

Effect Estimate Error DF t Value Pr > |t|

Intercept 37.9641 0.1484 7 255.86 <.0001

The mean is 37.96 with a standard uncertainty of 0.15 and v = 7. The ±U95 is thus estimated to be t0.05,7 × 0.15 = 2.4 × 0.15 = 0.36. Note that the between-campaign and between-aliquot covariance estimates are both zero.

Material D, appreciable between-aliquot variance

------material=MATERIAL D ------

The Mixed Procedure

Model Information

Data Set WORK.CCQM_2F_EXEMPLAR

Dependent Variable y

Covariance Structure Variance Components

Estimation Method REML

Residual Variance Method Profile

Fixed Effects SE Method Model-Based

Degrees of Freedom Method Satterthwaite

Class Level Information

Class Levels Values

cmp 2 1 2

alq 2 1 2

rep 2 1 2

Dimensions

Covariance Parameters 3

Columns in X 1

Columns in Z 6

Subjects 1

Max Obs Per Subject 8

Number of Observations

Number of Observations Read 8

Number of Observations Used 8

Number of Observations Not Used 0

Covariance Parameter

Estimates

Cov Parm Estimate

cmp 0

alq(cmp) 0.02741

Residual 0.1656

Asymptotic Covariance Matrix of Estimates

Row Cov Parm CovP1 CovP2 CovP3

1 cmp

2 alq(cmp) 0.01152 -0.00685

3 Residual -0.00685 0.01371

Fit Statistics

-2 Res Log Likelihood 10.2

AIC (smaller is better) 14.2

AICC (smaller is better) 17.2

BIC (smaller is better) 11.6

Solution for Fixed Effects

Standard

Effect Estimate Error DF t Value Pr > |t|

Intercept 34.1240 0.1660 3 205.59 <.0001

The mean is 34.12 with a standard uncertainty of 0.17 and v = 3. The ±U95 is thus estimated to be t0.05,3 × 0.17 = 3.2 × 0.17 = 0.54. Note that the between-campaign covariance estimate is zero but that the between-aliquot estimate is greater than zero.

Material E, appreciable between-campaign variance

------material=MATERIAL E ------

The Mixed Procedure

Model Information

Data Set WORK.CCQM_2F_EXEMPLAR

Dependent Variable y

Covariance Structure Variance Components

Estimation Method REML

Residual Variance Method Profile

Fixed Effects SE Method Model-Based

Degrees of Freedom Method Satterthwaite

Class Level Information

Class Levels Values

cmp 2 1 2

alq 2 1 2

rep 2 1 2

Dimensions

Covariance Parameters 3

Columns in X 1

Columns in Z 6

Subjects 1

Max Obs Per Subject 8

Number of Observations

Number of Observations Read 8

Number of Observations Used 8

Number of Observations Not Used 0

Covariance Parameter

Estimates

Cov Parm Estimate

cmp 0.006784

alq(cmp) 0

Residual 0.01433

Asymptotic Covariance Matrix of Estimates

Row Cov Parm CovP1 CovP2 CovP3

1 cmp 0.000219 -0.00002

2 alq(cmp)

3 Residual -0.00002 0.000068

Fit Statistics

-2 Res Log Likelihood -6.7

AIC (smaller is better) -2.7

AICC (smaller is better) 0.3

BIC (smaller is better) -5.3

Solution for Fixed Effects

Standard

Effect Estimate Error DF t Value Pr > |t|

Intercept 27.4328 0.07200 1 381.03 0.0017

The mean is 27.433 with a standard uncertainty of 0.072 and v = 1. The ±U95 is thus estimated to be t0.05,1 × 0.072 = 12.7 × 0.072 = 0.91. Note that the between-aliquot covariance estimate is 0 but the between-campaign estimate is greater than 0.

Material F, appreciable between-campaign and -aliquot variance

------material=MATERIAL F ------

The Mixed Procedure

Model Information

Data Set WORK.CCQM_2F_EXEMPLAR

Dependent Variable y

Covariance Structure Variance Components

Estimation Method REML

Residual Variance Method Profile

Fixed Effects SE Method Model-Based

Degrees of Freedom Method Satterthwaite

Class Level Information

Class Levels Values

cmp 2 1 2

alq 2 1 2

rep 2 1 2

Dimensions

Covariance Parameters 3

Columns in X 1

Columns in Z 6

Subjects 1

Max Obs Per Subject 8

Number of Observations

Number of Observations Read 8

Number of Observations Used 8

Number of Observations Not Used 0

Covariance Parameter

Estimates

Cov Parm Estimate

cmp 0.008007

alq(cmp) 0.003443

Residual 0.009913

Asymptotic Covariance Matrix of Estimates

Row Cov Parm CovP1 CovP2 CovP3

1 cmp 0.000316 -0.00004

2 alq(cmp) -0.00004 0.000083 -0.00002

3 Residual -0.00002 0.000049

Fit Statistics

-2 Res Log Likelihood -7.7

AIC (smaller is better) -1.7

AICC (smaller is better) 6.3

BIC (smaller is better) -5.6

Solution for Fixed Effects

Standard

Effect Estimate Error DF t Value Pr > |t|

Intercept 7.1274 0.07812 1 91.23 0.0070

The mean is 7.127 with a standard uncertainty of 0.078 and v = 1. The ±U95 is thus estimated to be t0.05,1 × 0.078 = 12.7 × 0.078 = 0.99. Note that both the between-aliquot and between-campaign covariance estimates are greater than zero.

R lme4

R is an extremely powerful open-source environment for statistical computing and graphics [[7]]. The lme4 package [[8]] provides analysis of linear mixed-effects models.

Successfully getting R to run on your computer requires some “computer smarts” and successfully using R requires some programming and a bit of patience. However, R has become the tool of choice for many number-crunching tasks and learning to use it effectively is well worth the effort for anyone seriously interested in the analysis of data.

One-factor with replication

Input data and exemplar command

#======

# One Factor With Replication ANOVA for CCQM_1F_Exemplar Data #

#======

library(lme4)

# CCQM_1F_Exemplar Data

n<- 2

n.cmp<- 3

n.rep<- 3

camp<- as.factor(c(1,1,1,2,2,2,3,3,3))

d <- matrix(c(

50742, 50760, 51082, 50912, 50949, 50981, NA, NA, NA,

23602, 23576, 23586, 23431, 23353, 23459, 23667, 23645, 23632), n, n.cmp*n.rep, byrow=T)

material<- c("MATERIAL A","MATERIAL B")

value<- rep(0, n)

u.value<- rep(0, n)

u.camp<- rep(0, n)

u.replicate<- rep(0, n)

dof<- rep(0,n)

for (j in 1:n) {

y_ij <- d[j, c(1:(n.cmp*n.rep))]

value[j]<- mean(y_ij, na.rm=T)

dat <- data.frame(x=y_ij, camp=camp)

lmer.res<- lmer(x~1+(1|camp), data=dat, na.action=na.omit)

sigmas<- as.double(summary(lmer.res)@REmat[,4])

u.camp[j]<- sigmas[1]

u.replicate[j]<- sigmas[2]

u.value[j]<- sqrt(vcov(lmer.res)[1,1])

if (u.camp[j]==0) {

dof[j]<- length(y_ij[!is.na(y_ij)])-1

} else {

dof[j]<- length(levels(as.factor(camp)))-1

}

print( sprintf( "%14s %7.4f %7.4f %7.4f %7.4f %7.4f %2.0f",

material[j], value[j], u.value[j], qt(0.975,dof[j])*u.value[j], u.camp[j], u.replicate[j], dof[j] ) )

}

Results

[1] " MATERIAL A 50904.3333 53.7541 138.1792 0.0000 131.6700 5"

[1] " MATERIAL B 23550.1111 70.0638 301.4603 119.7400 34.1680 2"

The ±U95 estimates are 138 and 301.

Two-factor nested

Input data and exemplar command

#======

# Two-Factor Nested ANOVA for CCQM_2F_Exemplar Data #

======

library(lme4)

# CCQM_2F_Exemplar Data

n<- 4

n.cmp<- 2

n.ali<- 2

n.rep<- 2

camp<- as.factor(c(1,1,1,1,2,2,2,2))

aliquot<- as.factor(c(1,1,2,2,3,3,4,4))

d <- matrix(c(

37.451, 38.416, 37.720, 38.268, 38.054, 37.386, 37.934, 38.484,

34.045, 34.562, 34.031, 33.249, 34.421, 34.320, 33.852, 34.512,

27.370, 27.436, 27.170, 27.467, 27.508, 27.429, 27.431, 27.651,

7.105, 6.983, 7.067, 7.042, 7.132, 7.096, 7.172, 7.422

), n, n.cmp*n.ali*n.rep, byrow=T)

material<- c("MATERIAL C","MATERIAL D","MATERIAL E","MATERIAL F")

value<- rep(0, n)

u.value<- rep(0, n)

u.camp<- rep(0, n)

u.aliquot<- rep(0, n)

u.replicate<- rep(0, n)

dof<- rep(0,n)

for (i in 1:n) {

y_ijk <- d[i, 1:(n.cmp*n.ali*n.rep)]

value[i]<- mean(y_ijk)

dat <- data.frame(x=y_ijk, camp=camp, aliquot=aliquot)

# get the MS terms to test for the presence of variance components

tmp.aov<- summary(aov(x~camp*aliquot, data=dat))

MS<- tmp.aov[[1]][,3]

test.negative.var.camp<- pf(MS[2]/MS[1], n.cmp-1, n.cmp*(n.ali-1))>0.5

test.negative.var.aliquot<- pf(MS[3]/MS[2], n.cmp*(n.ali-1), n.cmp*n.ali*(n.rep-1))>0.5

lmer.res<- lmer(x~1+(1|camp)+(1|aliquot), data=dat)

sigmas<- as.double(summary(lmer.res)@REmat[,4])

u.camp[i]<- sigmas[2]

u.aliquot[i]<- sigmas[1]

u.replicate[i]<- sigmas[3]

u.value[i]<- sqrt(vcov(lmer.res)[1,1])

if (test.negative.var.camp) {

if (test.negative.var.aliquot) {

dof[i]<- n.cmp*n.ali*n.rep-1

} else {

dof[i]<- n.cmp*n.ali-1

}

} else {

dof[i]<- n.cmp-1

}

print( sprintf( "%14s %7.4f %7.4f %7.4f %2.0f %7.4f %7.4f %7.4f",

material[i], value[i], u.value[i], qt(0.975,dof[i])*u.value[i], dof[i], u.camp[i], u.aliquot[i],

u.replicate[i] ) )

}

Results

[1] " MATERIAL C 37.9641 0.1484 0.3509 7 0.0000 0.0000 0.4197"

[1] " MATERIAL D 34.1240 0.1660 0.5283 3 0.0000 0.1656 0.4069"

[1] " MATERIAL E 27.4327 0.0720 0.9148 1 0.0824 0.0000 0.1197"

[1] " MATERIAL F 7.1274 0.0781 0.9927 1 0.0895 0.0587 0.0996"

The ±U95 estimates are 0.35, 0.53, 0.91, and 0.99.

OpenBUGS

OpenBUGS is an empirical Bayesian analysis freeware system that runs in several environments including R.. OpenBUGS is an actively developed and maintained lineal descendant of the pioneering (but now static) WinBUGS system [[9]]. While we have occasionally encountered problems running this code in WinBUGS, we have not observed problems with OpenBUGS.

One-Factor with replication

Model

### 2-level Bounded Nested ANOVA for CCQM_1F_Exemplar###

# ******** Specified parameters *******

# n0 number of materials

# n1 number campaigns per material

# n2 number repeat measurements per campaign

# RCRM[n0] vector of u(CRM)/crm

# Ufact....bounding factor for RCRMs

# y[n0,n1,n2] matrix of measurements

#

# ******** Interesting Estimated parameters *******

# m0[n0] vector of material-response means

# r1 scalar instrumental imprecision relative SD

# s0[n0] vector of between-campaign SDs

#

# ******** Intermediate estimated parameters *******

# m1[n0,n1] vector of campaign-response means

# sbnd[n0] vector of prior distribution on s0: RCRM*m0

# t0[n0] vector of inverse between-variances: 1/s0^2

# t1[n0] vector of inverse within-variances: 1/(r1*m0)^2

#

# ******** Priors *******

# m0 Normal, very broad centered at 0

# m1 Normal, data defined

# r1 Uniform, between 0 and 10% relative

# s0 Uniform, between 0 and RM’s assigned uncertainty

# y Normal, data defined

#

# The only tricky bits here are:

# 1) “sbnd” is used to limit the variability of the response to be

# no greater than that of the assigned uncertainty. They’re

# computed from the relative assigned uncertainties – see RCRM.

# 2) The relative instrumental imprecision is assumed to be the same

# for all materials

#

Model{

r1 ~ dunif(0.0,0.1)

for( i0 in 1 : n0) {

m0[i0] ~ dnorm(0,0.0000001)

sbnd[i0] <- Ufact*RCRM[i0]*abs(m0[i0])

s0[i0] ~ dunif(0.0,sbnd[i0])

t1[i0] <- 1/pow(r1*m0[i0],2)

t0[i0] <- 1/pow(s0[i0],2)

for( i1 in 1 : n1 ) {m1[i0,i1] ~ dnorm(m0[i0],t0[i0])

for( i2 in 1 : n2 ) {y[i0,i1,i2] ~ dnorm(m1[i0,i1],t1[i0])}}}}

Data and initialization parameters

# CCQM_1F_Exemplar data

list(n0=2,n1=3,n2=3,Ufact=1,

RCRM=c(0.00048,0.0058),

y=structure(.Data=c(50742,50760,51082,50912,50949,50981,NA,NA,NA,

23601,23576,23586,23431,23352,23459,23666,23644,23632),.Dim = c(2,3,3)))

# CCQM_1F_Exemplar Initialization values (not required but speeds things up)

list(m0=c(5.1E+04,2.4E+04))

Results

Mean sd MC_errorval2.5pcmedian val97.5pcstartsample

m0[1]50889.052.6291.160950780.050890.050991.010001100000

m0[2]23541.061.7550.2105223415.023542.023662.010001100000

The ±U95 for material “A” (here labeled m0[1]) is estimated to be (50991-50780)/2 = 106 and for material “B” (m0[2]) is (23662-23415)/2 = 124, both shorter than the 139 and 301 of the frequentist evaluations.

Two-factor nested

Model

### 2-Factor Nested ANOVA for CRM KCs###

# ******** Specified parameters *******

# n0 number of materials

# n1 number campaigns per material

# n2 number aliquots per campaign

# n3 number replicates per aliquot

# RCRM[n0] vector of u(CRM)/crm

# Ufact bounding factor for RCRMs

# s1hig highest allowable value for relative between-aliquot SD