Bayesian analysis of homogeneity studies in the production of ...citac.cc/adrian van der veen.pdfthe...

Bayesian analysis of homogeneity studies inthe production of reference materials

Adriaan M.H. van der Veen

Van Swinden Laboratorium, Delft, the Netherlands

21 April 2018CITAC Best Papers Award 2017, Sévres, France

A.M.H. van der Veen (VSL) Bayesian analysis of variance 21 April 2018 1 / 15

Rationale

Well known that the repeatability standard deviation (σ)sets a lower limit to the determination of thebetween-bottle standard deviation (τ)

Increase of the number of replicates not always possible

Need for a statistically sound approach for monitoring theperformance of a decanting method for synthetic naturalgas mixtures


Repeatability standard deviations of the naturalgas analyser

Table: Amount-of-substance fractions () expressed in % andrepeatability relative standard deviations (s) for natural gas

s

ethane 3.4978 0.012 %propane 4.7007 0.010 %iso-butane 0.1234 0.011 %n-butane 0.1225 0.012 %iso-pentane 0.0306 0.031 %n-pentane 0.0312 0.021 %n-hexane 0.0760 0.014 %carbon dioxide 5.0327 0.023 %nitrogen 0.4251 0.165 %methane 85.9601 0.011 %


Computation of the between-bottle standarddeviation

Compute the between-group variance

τ2 =MSbetween − MSwithin

n

Problems arise when MSbetween < MSwithin

Table: ANOVA results for nitrogen

Source of Variation SS ν MS F

Between Groups 2.92 × 10−10 9 3.25 × 10−11 0.663Within Groups 1.96 × 10−9 40 4.90 × 10−11

Total 2.25 × 10−9 49


Alternatives if MSbetween < MSwithin

set τ = 0 (e.g., ISO 13528:2015, ISO Guide 35:2017)

set τ =

√

√MSwithin

n(Ellison, 2015)

set τ =

√

√MSwithin

n4

√

√

√

2

(n − 1)(Van der Veen et al., 2001; ISO Guide 35:2017)


Bayesian modelling

“Solution” published in 2001 has never been viewed asentirely satisfactory

None of the three options permits a judgment on theperformance of the decanting method

Bayesian inference will provide a posterior probabilitydistribution for τ

Mean of the probability distribution of τ can be taken asthe estimate for the between-bottle standard deviation


Bayesian model (1)

Same underlying assumptions for the datanormally distributed datahomogeneity of the within-group standard deviationsno substantial prior information

Same statistical model as used in 2001 with classicalanalysis of variance

yj = μ + B + ϵj

where yj denotes the jth replicate on the ith mixture, μthe amount-of-substance fraction, B the bias of ithmixture, and ϵj the measurement error


Bayesian model (2)

Model parameters:Batch value for the amount-of-substance fraction μ

Within-group variance σ2 = vr(ϵj)

Between-group variance τ2 = vr(B)

For each of the parameters, a prior probability distributionis to be assigned


Prior distributions

Non-informative priors do not really reflect theinformation available

Weakly informative priorsallow a (weak) influence of our prior information (nominalamount-of-substance fraction, repeatability standarddeviation, between-bottle standard deviation)

improve performance of the Markov Chain Monte Carlomethod (MCMC)

do not require proving that the posterior probabilitydistribution is “proper”

should be assessed for actually behaving as “weaklyinformative”


Bayesian model (3)

μ ∼ N(μ0, σ20); σ0 = 0.05μ0

σ ∼ Cchy(0, Aσ); Aσ = 2mx(s)

τ ∼ Cchy(0, Aτ); Aτ = 5s1

Parameters for the folded Cauchy-distribution(t-distribution with ν = 1) set sufficiently wide to ensure“weakly informative” behaviour


Trace plots nitrogen

0.4245

0.4250

0.4255

5000 10000 15000 20000 25000

Iteration number

µ (c

mol

/mol

) chain

1

2

3

4

0.0009

0.0012

0.0015

0.0018

5000 10000 15000 20000 25000

Iteration number

σ/y

chain

1

2

3

4

0.000

0.001

0.002

0.003

5000 10000 15000 20000 25000

Iteration number

τ/y

chain

1

2

3

4


Posteriors for μ, σ and τ for nitrogen

0.4245 0.4250 0.4255

010

0030

00

µ (cmol/mol)

dens

ity

0.0000 0.0010 0.0020 0.0030

050

010

0015

00

τ/y

dens

ity

0.0008 0.0012 0.0016

050

015

0025

00

σ/y

dens

ity


Sensitivity analysis

1 E - 3 0 . 0 10 . 0 0

0 . 0 1

0 . 0 2

0 . 0 3

0 . 0 4

0 . 0 5 e t h a n e p r o p a n e i s o - b u t a n e n - b u t a n e n i t r o g e n c a r b o n d i o x i d e m e t h a n e

τ/µ

(%)

A / µ A.M.H. van der Veen (VSL) Bayesian analysis of variance 21 April 2018 13 / 15

Nitrogen: results and the alternatives

Table: Results of the MCMC calculation for nitrogen.

mean sd q0.025 q0.975

μ 0.4251443 0.0001680 0.4248122 0.4254757τ 0.0001720 0.0001439 0.0000061 0.0005320σ 0.0004808 0.0000557 0.0003752 0.0005935

Bayesian model: τ̂ = 1.72µmolmol−1, which lies inbetween

ISO Guide 35: ∗bb = 1.48µmolmol−1 and

Elisson:p

MSwithin/n = 3.13µmolmol−1.

Both ∗bb andp

MSwithin/n within the 95% coverageinterval of τ.


Conclusions

Bayesian analysis is to be preferred where classicalanalysis of variance and meta-analysis fail

Computations are involved (require Markov Chain MonteCarlo method)

Bayesian inference provides a wealth of information, wellbeyond GUM requirements

Classical methods compare well with the Bayesian modelproposed, as long as MSbetween > MSwithin


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