Measurement error models and variance estimation in the presence of rounding error effects

Abstract

An approach to estimating measurement error variances for any instrument having round-off effects that might also have instrument bias is presented. Recently finite instrument resolution effects on error variances have been studied, but negligible instrument bias was assumed and the contexts were different than considered here. Our intent is to use repeated measurements on several standards to estimate the instrument’s random and systematic error variances. Recognizing that rounding impacts item bias and variance in a manner that depends on the true value, an approach is presented to estimate random error variance and instrument systematic error variance. The key finding is that item-specific bias can interfere with the estimation of overall instrument bias unless appropriate error modeling and associated inference steps are taken.

Appendix

Given repeated measurements of multiple standards, Equation 6 in the main text can be extended from one to N > 1 standards as

$$ \prod\limits_{i = 1}^{N} {\prod\limits_{j = 1}^{{n_{i} }} {P(M_{ij} = k_{ij} )} = } \prod\limits_{i = 1}^{N} {\prod\limits_{j = 1}^{{n_{i} }} {\{ \Upphi ((k_{ij} + 1/2 - x_{i} - T_{i} - B)/\sigma_{R} ) - \Upphi ((k_{ij} - 1/2 - x_{i} - T_{i} - B)/\sigma_{R} )} } \} $$

(8)

for true values x ₁, x ₂, …, x _N where Φ denotes the cumulative normal distribution. The data are the observed integer values k _ij and the unknowns are B and σ _R . Throughout this paper, the balance is assumed to round to the nearest integer, but accommodating any other balance resolution is straightforward. And the random balance errors are assumed to be normal but accommodating any other distribution is also straightforward.

Equation 8 can be used to find the maximum likelihood (ML) or Bayesian estimates of B and σ_R. Our strategy to find the Bayesian estimates is Markov Chain Monte Carlo (MCMC) as described in [22]. Briefly, MCMC is Monte Carlo integration using Markov chains, in a manner that provides observations from the posterior distribution of all parameters (B and σ _R in our case). As an aside, if we regard the true values as having a nominal (mean) value and a standard deviation, then uncertainty in the nominal standard values is easily accommodated, by modifying σ _R to include the standard deviation in the standard’s true value.

To implement MCMC, we used the open-source, widely distributed, and free statistical programming language R and the mcmc package. The metrop function in the mcmc package [21] implements the metropolis algorithm to propose new parameter values and accept them as observations from the posterior with appropriate probabilities. The main technical issue with MCMC is to check that the implementation produces samples from the correct stationary distribution. We used standard diagnostic plots and the guidance suggested with the mcmc package to assess convergence to the correct stationary distribution.

Here is the R code for the normally distributed balance errors case. First, we need to define the likelihood (Eq. 8). Let the vec object hold the values (B, σ _R ), yuse be a list (one component for each standard) of the observed data, and xuse be the vector of true (nominal) standard values. The log likelihood (named ludvn) is then:

The gamma function used as the prior for σ _R puts zero probability on negative values, so σ _R is constrained by its prior distribution to be nonnegative. The scaling parameter for the gamma is chosen so that the prior is noninformative, putting probability on a large range of nonnegative values. Similarly, the prior for B is noninformative, centered at 0 with a large standard deviation compared to the round-off interval.

The call to metrop in the mcmc package is:

• out1 = metrop(obj = ludvn,initial = init.temp, blen = 10,scale = 0.1,nbatch = 1000,

• x = xuse,y = yuse,round.interval = 1)

By trial and error, we found blen = 10 and scale = 0.1 led to good convergence results.

• The main summaries of the posterior in out1 are:

• mean(out1$batch [,1]) # estimate of B

• mean(out1$batch [,2]) # estimate of σ _R

• var(out1$batch [,1])^.5 # standard deviation of B

• var(out1$batch [,2])^.5 # standard deviation of σ _R

• quantile(out1$batch [,1],probs = c(.025,.975))# 95% posterior probability interval for B

• quantile(out1$batch [,2],probs = c(.025,.975)) # 95% posterior probability interval for σ _R

Example (Simulated Example 2 in section “Simulated example 2”):

• xuse = c(100.0,100.2,100.4,100.5)

• sigmaR = 0.05; n = 200; B = 0

• yuse = list(round(rnorm(n,xuse [1] +B,sigmaR),0),

• round(rnorm(n,xuse [2] +B,sigmaR),0),round(rnorm(n,xuse [3] +B,sigmaR),0),

• round(rnorm(n,xuse [4] +B,sigmaR),0))

• library(mcmc)

• # set initial values near true values or near ML estimates from nlminb

• init.temp = c(0+.01*rnorm(1),sigmaR * (1+.1* rnorm(1)))

• # set initial values near true values

We define effective random error variance as the sum of pure random error variance and the square of the item-specific bias.

If there is no overall instrument bias, then it is assumed that the item-specific biases will average to zero across a sufficiently large group of items.

Users who require estimates of B and σ_R but do not require the estimated standard deviations in these estimates can use nlminb or optim in R instead of MCMC. Using the same simulated example 2 from section “Simulated example 2”, nlminb produces:

# nlminb finds parameters that minimize the objective function, so use neg = T

temp = nlminb(start=c(B,sigmam),objective=ludvn,neg=T)