Measurement Error and Conformity Control

Figure2.1shows the result of the systematic literature search I conducted related to measurement uncertainty and conformity control.

A: Uncertainty evaluation with symmetric error distribution D: Conformity area with

asymmetric error distribution

with asymmetric error distribution

C: Conformity area with symmetric error distribution

Legend:

Articles

Articles (conformity with asymmetriy) Structure nodes

5 30 180 800 5000

FIGURE 2.1: Result of the literature research (measurement uncer-tainty area)

Since my first research question (Q1) is related to the distribution properties of measurement error, the main goal of this review was to identify the most relevant studies that deal with asymmetric measurement error distributions in the field of conformity control. After the search I classified the papers using the survey de-scribed by Table2.1.

Nodes represent the reviewed papers and edges represent the citing relationship between them. Blue nodes illustrates the responses related to each question from Ta-ble2.1so they illustrate the result of the classification (in other words, the blue nodes represent the structure of the aforementioned survey). The reviewed and classified papers were colored with red, however there is a group of papers highlighted with green. I highlighted those nodes, because that group includes papers considering asymmetric measurement error with the aspect of conformity control (group D). In addition the size of the nodes represents the citation numbers (How many times they were cited by others.) in logarithmic scale.

86 studies were selected and categorized and 6 papers out of the 86 were classi-fied into group "D" (colored with green). Although many researches focused on the evaluation of measurement uncertainty even assuming asymmetric measurement error distributions, only a few considered the effect of asymmetric measurement un-certainty and its consequences in conformity control. I summarize the most relevant contributions in two steps, starting with groups "A", "B" and "C".

Groups "A", "B" and "C":

As part of the Six Sigma approach, measurement system analysis (MSA) and

R&R tests (repeatability and reproducibility) are often used to evaluate measure-ment uncertainty of a measuring device. These methods are useful to get knowledge about the performance of measurement device or system, however their purpose is to support the decision making about the validation of the device/system and do not consider the further consequences of the measurement uncertainty (AIAG,2010).

In 1995, the attitudes were changed related to the measurement uncertainty with the construction of the Guide to the Expression of Uncertainty in Measurement (GUM)(BIPM et al.,1995). GUM proposes the expression of the measurement un-certainty in two ways. On one hand, the measurement unun-certainty is expressed as a probability distribution derived from the measurement. On the other hand, this uncertainty can be described as an interval. In the first case, the standard devia-tion is used for the characterizadevia-tion of the distribudevia-tion (standard uncertainty). If the result of the measurement is obtained by combining the standard deviation of sev-eral input estimates, the standard deviation is called combined standard uncertainty.

In the second case, the length of the interval can be determined by the multiplica-tion of the combined standard uncertainty and a coverage factor k and called as expanded uncertainty. There are several guidelines that proposes 2 as a value of the coverage factork(BIPM et al.,1995, Eurachem,2007a, Heping and Xiangqian,2009, Rabinovich,2006, Jones and Schoonover,2002), producing 95.45 % confidence level, however this statement is only true if the combined uncertainty follows normal dis-tribution, otherwise the estimation of the confidence level is not correct (Vilbaste et al.,2010, Synek,2006).

Asymmetry of the distribution can also lead to incorrect estimation and incorrect decisions as well. The JCGM Guide 101 (BIPM et al.,1995) introduces that exponen-tial and gamma distributions are observable as asymmetric examples, furthermore, researches have shown that asymmetry can appear in combined standard uncer-tainty as well (Herrador and Gonzalez,2004, D’Agostini,2004, Pendrill,2014). Not only skewness can be the root cause of the over- or underestimation of confidence level. Kurtosis can also vary by different measurement devices or systems. Lep-tocurtic and platykurtic distributions are also observable by several measurement systems (Martens,2002, Pavlovcic et al.,2009).

If the measurement error distribution follows non-normal distribution, the confi-dence level will be estimated incorrectly and using thek=2 proposal, decision errors can be made, since the principal assumption of the proposal is not valid. Further-more, the rules based on the assumption of normal distribution do not consider the consequences of the decision errors, however they can lead to considerable prob-lems.

Group "D":

Figure2.2shows the structure of the papers classified in group "D" in details.

Uncertainty in conformity

FIGURE 2.2: Result of the literature research (measurement uncer-tainty area-Sub-graph)

There were also researches taking measurement uncertainty and the decision consequences into account. Rossi and Crenna showed that measurement uncertainty should not be treated as an interval or a simple standard deviation, but needs to be considered as a probability distribution in order to avoid incorrect decisions (Rossi and Crenna,2006). Williams (2008) pointed out that decision rules must be carefully defined when skewed measurement error distribution is assumed. Although, Forbes has proposed a method treating the conformance assessment as a Bayesian decision, he only considered the cost and revenue of the incorrect decisions (Forbes, 2006).

Later, Pendrill has developed a more comprehensive model considering measure-ment uncertainty in conformity sampling. The model included all the four decision outcomes (correct acceptance, false rejection, false acceptance and correct rejection) however, only correct decision-, and testing costs were considered during the calcu-lations (Pendrill,2008, Pendrill,2014).

The referenced papers made steps towards the risk-based aspect of the confor-mity control, they did not consider all the four decision outcomes in the calcula-tions and however, they also considered even asymmetric measurement error dis-tributions, the strength of the characteristics of the measurement error distribution (skewness, kurtosis) were not analyzed. Research Question Q1 is still valid after the literature review, since I did not find any paper that answered the question and in-vestigated how 3^rd and 4^th moments of measurement error distribution affects the decision outcomes during conformity control. In my thesis, I develop a risk-based model in the statistical process control including all the four decision outcomes and examine the impact of 3^rdand 4^thmoments of the measurement error distribution.