Bayesian framework for proficiency tests using auxiliary information on laboratories

Abstract

In this paper, we propose a Bayesian framework to analyse proficiency tests results that allows to combine prior information on laboratories and prior knowledge on the consensus value when no measurement uncertainties nor replicates are reported. For these proficiency tests, where the reported data is reduced to its minimum, we advocate that each piece of information related to the measurement process is valuable and can lead to a more reliable estimation of the consensus value and its associated uncertainty. The resulting marginal posterior distribution of the consensus value relies on the management of expert knowledge used to build prior distributions on the consensus value and the laboratory effects. The choices of priors are discussed to promote the method when the required auxiliary information is available. This new approach is applied on a simulated data set and on a real-life environmental proficiency test.

Appendices

Appendix 1: Posterior distributions under Jeffreys priors conditionally to the auxiliary information

Under the Jeffreys prior (see section "Conjugate priors"), the corresponding posterior distribution is given by the Bayes formula

$$\begin{aligned} \begin{aligned} \pi \left( \mu , \tau ^2 \vert {\varvec{x}} \right)&\propto l \left( {\varvec{x}} \vert \mu , \tau ^2 \right) \pi \left( \mu , \tau ^2 \right) \\&\propto \frac{1}{ \left( \tau ^2 \right) ^{p/2+1}} \exp \left\{ - \frac{1}{2 \tau ^2 } \left[ \sum _{i=1}^p \frac{\left( x_i -{\hat{\mu }}_\mathrm{ML} \right) ^2 }{\sigma _i^2} + \sum _{i=1}^p \frac{\left( \mu -{\hat{\mu }}_\mathrm{ML} \right) ^2 }{\sigma _i^2} \right] \right\} \\&\propto \frac{1}{ \left( \tau ^2 \right) ^{p/2+1}} \exp \left\{ - \frac{1}{2 \tau ^2 } \left[ p {\hat{\tau }}_\mathrm{ML}^2 +\frac{1}{\left( \sum _{i=1}^p \frac{1}{\sigma _i^2} \right) ^{-1} } \left( \mu - {\hat{\mu }}_\mathrm{ML} \right) ^2 \right] \right\} \\&\propto \frac{1}{ \left( \tau ^2 \right) ^{(p-1)/2+1}} \exp \left\{ - \frac{ p {\hat{\tau }}_\mathrm{ML}^2}{2 \tau ^2 } \right\} \frac{1}{ \left( \tau ^2 \right) ^{1/2}} \exp \left\{ -\frac{1}{2}\frac{\left( \mu - {\hat{\mu }}_\mathrm{ML} \right) ^2}{\tau ^2 \left( \sum _{i=1}^p \frac{1}{\sigma _i^2} \right) ^{-1}} \right\} \end{aligned} \end{aligned}$$

(55)

The posterior distributions are

$$\mu \vert \tau ^2, {\varvec{x}}\sim \mathrm {N} \left( {\hat{\mu }}_\mathrm{ML} , \tau ^2 \left( \sum _{i=1}^p \frac{1}{\sigma _i^2} \right) ^{-1} \right)$$

(56)

$$\tau ^2 \vert {\varvec{x}}\sim \mathrm {IG} \left( \frac{p-1}{2} , \frac{ p {\hat{\tau }}_\mathrm{ML}^2 }{2} \right)$$

(57)

The marginal posterior distribution of $\mu$ is obtained by integrating (56) out (57) and reads

$$\mu \vert {\varvec{x}} \sim \mathrm {T}_{p-1} \left( {\hat{\mu }}_\mathrm{ML}, u^2({\hat{\mu }}_\mathrm{prop})\right)$$

(58)

where $u^2({\hat{\mu }}_\mathrm{prop})=\left( u({\hat{\mu }}_\mathrm{prop}) \right) ^2 = {(p/(p-1)){\hat{\tau }}_\mathrm{ML}^2}\,/\,{\sum _{i=1}^p \frac{1}{\sigma _i^2}}$.

The posterior distribution (58) is centred at the maximum likelihood estimate ${\hat{\mu }}$. The uncertainty associated with the consensus value is then given by the standard deviation of the posterior distribution:

$$u({\hat{\mu }})=\sqrt{\frac{p-1}{p-3}}u({\hat{\mu }}_\mathrm{prop})$$

(59)

As the number of laboratories p increases (and so the degrees of freedom $p-1$), the student distribution may be approximated by the Gaussian distribution

$$\mu \vert {\varvec{x}} \sim \mathrm {N}\left( {\hat{\mu }}_\mathrm{ML}, u^2({\hat{\mu }}_\mathrm{prop}) \right)\,\text { for\,large} \, p$$

(60)

Appendix 2: Posterior distributions under conjugate prior and no auxiliary information

When no auxiliary information is available, the conjugate Gaussian/inverse gamma model to estimate the mean $\mu$ of a Gaussian sample $x_1,\ldots ,x_p$ with unknown variance $\tau ^2$ is

$$\begin{aligned} \begin{aligned} x_i&= \mu + \varepsilon _i \\ \varepsilon _i&\sim \mathrm {N}(0,\tau ^2) \\ \mu \vert \tau ^2&\sim \mathrm {N}(\mu _0,\tau ^2 \eta _0^2)\\ \tau ^2&\sim \mathrm {IG} \left( \frac{\nu _0}{2},\frac{\nu _0 s_0^2}{2} \right) \end{aligned} \end{aligned}$$

(61)

Denoting ${\bar{x}}=\frac{1}{p} \sum _{i=1}^{p} x_i$, the likelihood can be factorized as

$$l({\varvec{x}}\vert \mu ,\tau ^2) \propto \frac{1}{\left( \tau ^2\right) ^{p/2} } \mathrm {exp} \left\{ -\frac{1}{2 \tau ^2} \left[ p (\mu - {\bar{x}})^2 + \sum _{i=1}^{p} (x_i -{\bar{x}})^2 \right] \right\}$$

(62)

The posterior distribution is obtained by applying the Bayes formula

$$\begin{aligned} \begin{aligned} \pi \left( \mu , \tau ^2 \vert {\varvec{x}} \right)&\propto l \left( {\varvec{x}} \vert \mu , \tau ^2 \right) \pi \left( \mu \vert \tau ^2 \right) \pi \left( \tau ^2 \right) \\ \end{aligned} \end{aligned}$$

(63)

where $l \left( {\varvec{x}} \vert \mu , \tau ^2 \right)$ is defined at Eq. 62.

After computation, the posterior distribution can be factorized as the product

$$\pi \left( \mu , \tau ^2 \vert {\varvec{x}} \right) \propto \pi \left( \tau ^2 \vert {\varvec{x}} \right) \pi \left( \mu \vert \tau ^2, {\varvec{x}} \right)$$

(64)

where $\mu \vert \tau ^2, {\varvec{x}} \sim \mathrm {N} \left( \mu _p, \tau ^2 \sigma _p^2 \right)$ and $\tau ^2 \vert {\varvec{x}} \sim \mathrm {IG} \left( \frac{\nu _n}{2}, \frac{\nu _n s_n^2}{2} \right)$ and

$$\begin{aligned} \begin{aligned} \mu _p&= \frac{p {\bar{x}} + \mu _0/\eta _0^2}{p + 1/\eta _0^2} \\ \sigma _p^2&= \frac{1}{\frac{1}{\eta _0^2} + p}\\ \nu _n&= \nu _0 + p \\ \nu _n s_n^2&= \nu _0 s_0^2 + \sum _{i=1}^p (x_i -{\bar{x}})^2 + \frac{({\bar{x}} - \mu _0)^2}{1/p + \eta _0^2} \end{aligned} \end{aligned}$$

(65)

The resulting marginal posterior distribution of the consensus value is

$$\mu \vert {\varvec{x}} \sim \mathrm {T}_{\nu _n}(\mu _p, \sigma _p^2 s_n^2)$$

(66)

with parameters $\nu _n, \mu _p, \sigma _p^2$ and $s_n^2$ defined at Eq. 65.

Note: Under Jeffreys prior, the marginal posterior distribution of the consensus value is

$$\mu \vert {\varvec{x}} \sim \mathrm {T}_{p-1}\left({\bar{x}},\frac{1}{p}\sum _{i=1}^{p}(x_i - {\bar{x}})^2\right)$$

(67)

Appendix 3: R code to generate samples from the posterior distribution of the consensus value under binary auxiliary information

Table 11 displays the R code used to sample from the marginal posterior distribution of the consensus value under Jeffreys prior and binary auxiliary information in the simulation study (plain grey curve Fig. 3).

The methodology is applied to $p=15$ laboratories for which a binary auxiliary information Ybin (line 4) is associated with measurement results x (line 3) so that the 4 highest results are (arbitrarily) associated with Ybin=1 and the others with Ybin=0, lines 4 to 7.

The transformation of the row data Ybin into the variance parameters stored in the matrix sigma2 requires to run a Gibbs sampling algorithm lines 11 to 35 to produce samples from latent continuous versions ytilde of Ybin. After an initial value has been given to the threshold c line 15, Gibbs sampling alternates between sampling in the posterior distribution of the latent variables given the current value of the threshold line 24 and sampling in the conditional posterior distribution of the threshold given the current sample of the latent variables line 27. The simulations of latent variables are stored in the matrix ytilde line 17 and simulations from the threshold are stored in the vector c. Note that the initial n_Gibbs number of simulations line 14 is reduced to the final L simulations line 34 to delete the burn in period of the chains and reduce autocorrelation of samples (thinning).

The link function line 39 transforms the output of the Gibbs sampler into the input of the algorithm given in Table 1 which performs the integration of the distribution of the consensus value over the distribution of the auxiliary information lines 41 to 50. The vector MU of length $L \times M$ stores the integrated posterior samples of the consensus value obtained under Jeffreys prior Eq. (58) lines 45 to 50, where mu_ML is the maximum likelihood estimate of $\mu$, tau_sq_ML is the maximum likelihood estimate of $\tau ^2$, tau_sq_ML*w_ML is the variance of the posterior distribution of $\mu$ obtained for a given realization sigma2[l,] of the vector ${\varvec{\sigma ^2}}$ with $l=1,\ldots ,$ L. For a given sigma2[l,], M (defined line 8) samples are drawn from the Student distribution Eq. 58 using the function rt_scaled from the R package metRology [5] and stored in the vector MU.

Lines 52 to 56 provide a plot of the resulting posterior density of the consensus value and of the threshold.

Table 11 R code to implement the algorithm given in Table 1 under Jeffreys prior distributions