Quality Control Using Inferential Statistics in Weibull-based ...2013/01/22 · Quality Control Using Inferential Statistics in Weibull-based Reliability Analyses Reference Duffy,

Stephen F. Duffy1 and Ankurben Parikh2

Quality Control Using InferentialStatistics in Weibull-basedReliability Analyses

Reference

Duffy, Stephen F. and Parikh, Ankurben, “Quality Control Using Inferential Statistics in

Weibull-based Reliability Analyses,” Graphite Testing for Nuclear Applications: The

Significance of Test Specimen Volume and Geometry and the Statistical Significance of Test

Specimen Population, STP 1578, Nassia Tzelepi and Mark Carroll, pp. 1–18, doi:10.1520/

STP157820130122, ASTM International, West Conshohocken, PA 2014.3

ABSTRACT

Design codes and fitness-for-service protocols have recognized the need to

characterize the tensile strength of graphite as a random variable through the

use of probability density functions. Characterizing probability density functions

require more tests than typically needed to simply define an average value for

tensile strength. ASTM and the needs of nuclear design codes should dovetail on

this issue. The two-parameter Weibull distribution (an extreme-value

distribution) is adopted for the tensile strength of this material. The failure data

from bend tests or tensile tests are used to determine the Weibull modulus (m)

and Weibull characteristic strength (rh). To determine an estimate of the true

Weibull distribution parameters, maximum likelihood estimators are used. The

quality of the estimated parameters relative to the true distribution parameters

depends fundamentally on the number of samples taken to failure. The statistical

concepts of confidence intervals and hypothesis testing are presented pertaining

to their use in assessing the goodness of the estimated distribution parameters.

The inferential statistics tools enable the calculation of likelihood confidence

rings. The concept of how the true distribution parameters lie within a likelihood

ring with a specified confidence is presented. A material acceptance criterion is

Manuscript received August 15, 2013; accepted for publication March 10, 2014; published online July 18, 2014.1Ph.D., P.E., Cleveland State Univ., Cleveland, OH 44115, United States of America.2Cleveland State Univ., Cleveland, OH 44115, United States of America.3ASTM Symposium on Graphite Testing for Nuclear Applications: The Significance of Test Specimen Volume

and Geometry and the Statistical Significance of Test Specimen Population on Sept 19–20, 2013 in Seattle,

WA.

Copyright VC 2014 by ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959.

GRAPHITE TESTING FOR NUCLEAR APPLICATIONS: THE SIGNIFICANCE OF TEST SPECIMEN VOLUME

AND GEOMETRYAND THE STATISTICAL SIGNIFICANCE OF TEST SPECIMEN POPULATION 1

STP 1578, 2014 / available online at www.astm.org / doi: 10.1520/STP157820130122

defined here, and the criterion depends on establishing an acceptable

probability of failure of the component under design, as well as an acceptable

level of confidence associated with the estimated distribution parameter

determined using failure data from a specific type of strength test.

Keywords

graphite, Weibull, confidence bounds, likelihood ratio rings

IntroductionThis work presents the mathematical concepts behind statistical tools that, whencombined properly, lead to a simple quality control program for components fabri-cated from graphite. The data on mechanistic strength (which is treated as a randomvariable) should be used to accept or reject a graphite material for a given application.The two-parameter Weibull distribution is used to characterize the tensile strength ofgraphite. The Weibull distribution is an extreme-value distribution, and this facetmakes it the preferred distribution for characterizing a material’s minimum strength.

Estimates of the true Weibull distribution parameters should be determinedusing maximum likelihood estimators (MLEs). The quality of the estimated param-eters relative to the true distribution parameters depends fundamentally on thenumber of samples taken to failure. The statistical concepts of confidence intervalsand hypothesis testing are employed to assess quality. Quality is defined by how

Nomenclature

H0 ¼null hypothesisH1 ¼ alternative hypothesisL ¼ likelihood functionL ¼natural log of the likelihood functionm ¼Weibull modulus~m ¼ estimated Weibull modulusn ¼ sample sizePf ¼probability of failureT ¼ test statistica ¼ significance levelb ¼probability of a Type II errorc ¼ confidence level

H0 ¼ vector of all the maximum likelihood estimator parameter estimatesHc

0 ¼ vector of all point estimates that are not maximum likelihood estimatorparameter estimates

rh ¼Weibull characteristic strength~rh ¼ estimated Weibull characteristic strength

2 STP 1578 On Graphite Testing for Nuclear Applications

close the estimated parameters are to the true distribution parameters. The qualityof the distribution parameters can have a direct effect on whether a certain grade ortype of graphite material is acceptable for a given application. Both inferential sta-tistics concepts (i.e., confidence intervals and hypothesis testing) enable the calcula-tion of likelihood confidence rings. Work showing how the true distributionparameters lie within a likelihood ring with a specified confidence is presented here.The size of the ring has direct bearing on the quality of the estimated parameters.

One must specify and associate an acceptable level of confidence with the esti-mated distribution parameters. This work shows how to construct likelihood ratioconfidence rings that establish an acceptance region based on a given level of confi-dence. Material performance curves are presented that are based on an acceptablecomponent probability of failure. Combining the two elements (i.e., the materialperformance curve and a likelihood ratio ring) allows the design engineer to deter-mine whether a material is suited for the component design at hand. The result is asimple approach to a quality assurance criterion.

Point Estimates and Confidence BoundsData related to the tensile strength of graphite can be generated through the use oftensile tests outlined in ASTM C565 [1], ASTM C749 [2], ASTM C781 [3], andASTM D7775 [4]. Bend tests are preferred for their simplicity, and flexural test pro-cedures are outlined in ASTM C651 [5]. Given data on the tensile strength ofgraphite, the first step is to ascertain values for the Weibull distribution parametersusing this information. MLEs, outlined in ASTM D7846 [6], are used to computepoint estimates. The next question is, have we estimated parameters to the best ofour ability? This is directly related to the fundamental question asked repeatedly,“How many samples must be tested?” The typical answer to this question seems tobe about 30. However, the appropriate question to ask is, “How many samples mustbe tested to establish a given confidence level for component reliability?” The workoutlined here answers this question quantitatively, utilizing interval estimates alongwith hypothesis testing. The methods outlined here are currently being imple-mented in the ASME Boiler and Pressure Vessel Code [7].

Confidence intervals are used to indicate the potential variability in estimatedparameters. Every time a sample is taken from the same population, a point esti-mate of the distribution parameters can be calculated. Successive samples producedifferent point estimate values, and thus the point estimates are treated as randomvariables. Thus interval estimates bracketing the true distribution parameters are asnecessary as point estimates. If the interval that brackets the true distribution pa-rameter contains the estimated parameter, then the estimate is consistent with thetrue value. Increasing the sample size will always narrow the interval bounds andprovide point estimates that approach the true distribution parameters. Intervalbounds on parameter estimates represent the range of values for the distributionparameters that are both reasonable and plausible.

DUFFY AND PARIKH, DOI 10.1520/STP157820130122 3

http://www.astm.org/Standards/C565



http://www.astm.org/Standards/D7775



Inferences and Hypothesis TestingAs noted above, there is a need to know whether a sample is large enough that thepoint estimates of the distribution parameters are in the same statistical neighbor-hood as the true population distribution parameters. The techniques for makingthis kind of assessment utilize inferential statistics. The type of inference focused onhere is the bounds on the true population parameters (which are never known)given a particular sample. Here a particular type of bound that is referred to as alikelihood ratio confidence bound is employed.

The basic issue is this: consider an infinitely large population with a known fre-quency distribution but unknown distribution parameters. Because of diminishedknowledge of the overall population available from a small sample taken from theinfinitely large population, that sample will generate a frequency distribution that isdifferent from that of the parent population—different in the sense that the distri-bution parameters estimated from the small sample (a subset) will not be the sameas the parameters of the parent population. The population and the sample can becharacterized by the same frequency distribution, but the two will have differentdistribution parameters. As the sample size increases, the frequency distributionassociated with the sample more closely resembles that of the parent population(i.e., the estimated parameters approach the true distribution parameters of the par-ent population).

Hypothesis testing is used to establish whether the true distribution parameterslie close to the point estimates. Two statistical hypotheses are proposed concerningthe estimated parameter values. The first, H1, is referred to as the alternative hy-pothesis. The latter hypothesis, H0, is referred to as the null hypothesis. Bothhypotheses are then tested with the samples taken from the parent population. Thegoal of the analyst is to decide whether there is enough evidence (data) to refute thenull hypothesis H0. That decision is made based on the value of a test statistic.

Here that test statistic is the ratio of two likelihood functions whose probabilityis known under the assumption that H0, the null hypothesis, is true. If the test sta-tistic takes on a value rarely encountered using the data collected, then the test sta-tistic indicates that the null hypothesis is unlikely and H0 is rejected. The value ofthe test statistic at which the rejection is made defines a rejection region. The proba-bility that the test statistic falls into the rejection region by chance is referred to asthe significance level, denoted by a. The significance level is defined as the probabil-ity of mistakenly rejecting a hypothesis when the hypothesis is valid.

Rejecting HypothesesMaking a decision regarding a hypothesis is associated with a statistical event withan attending probability, so an ability to assess the probability of making incorrectdecisions is required. Fisher [8] established a method for quantifying the amount ofevidence required in order for an event to be deemed unlikely to occur by chance.


He originally defined this quantity as the significance level. Significance levels aredifferent than confidence levels, but the two are related.

The significance level and the confidence level are functionally related throughthe following simple expression, with the confidence level denoted by c:

c ¼ 1� a(1)

The confidence level is associated with a range, or more specifically with bounds oran interval, within which a true population parameter resides. The confidence leveland, through the equation above, the significance level are chosen a priori based onthe design application at hand.

Given a significance level a defined by Eq 1, a rejection region can be estab-lished. This is known as the critical region for the test statistic selected. For our pur-poses, the observed tensile strength data for a graphite material are used todetermine whether the computed value of the test statistic associated with a hypoth-esis (not the parameter estimates) lies within or outside the rejection region. Theamount of data helps define the size of the critical region. If the test statistic iswithin the rejection region, then we say the hypothesis is rejected at the 100a % sig-nificance level. If a is quite small, then the probability of rejecting the null hypothe-sis when it is true can be made quite small.

Type I and Type II ErrorsConsider that for a true distribution parameter h there is a need to test the null hy-pothesis that h¼ h0 (in this context h0 is a stipulated value) against the alternativethat h= h0 at a significance level a. Under these two hypotheses, a confidence inter-val can be constructed that contains the true population parameters with a proba-bility of c¼ (1� a). In addition, this interval also contains the value h0.

Mistakes can be made in rejecting the null hypothesis given above. In hypothe-sis testing, two types of errors are possible.

1. Type I error: the rejection of the null hypothesis (H0) when it is true. Theprobability of committing a Type I error is denoted by a.

2. Type II error: the failure to reject the hypothesis (H0) when the alternativehypothesis (H1) is true. The probability of committing a Type II error is denotedby b.In either situation, judgment of the null hypothesis H0 is incorrect.

Now consider the situations in which correct decisions have been made. In thefirst case, the null hypothesis is not rejected and the null hypothesis is true. Theprobability of making this choice is c¼ (1� a). This is the same probability associ-ated with the confidence interval for the true population distribution parametersdiscussed. For the second situation, the probability of correctly rejecting the null hy-pothesis is the statistical complement of a Type II error, that is, (1�b). In statisticsthis is known as the power of the test of the null hypothesis. This quantity is used


to determine the sample size. Maximizing the probability of making a correct deci-sion requires high values of (1� a) and (1� b).

So decision protocols must be designed so as to minimize the probability ofeither type of error in an optimal fashion. The probability of a Type I error is con-trolled by making a a suitable number, say, 1 in 10 or 1 in 20, or something smallerdepending on the consequence of making a Type I error. Minimizing the probabil-ity of making a Type II error is not straightforward. b is dependent on the alterna-tive hypothesis, on the sample size n, and on the true value of the distributionparameters tested. As discussed in the next section, the alternative hypothesis isgreatly influenced by the test statistic chosen to help quantify the decision.

Hence when hypothesis testing is applied to the distribution parameters, astatement of equality is made in the null hypothesis H0. Achieving statistical signifi-cance for this is akin to accepting that the observed results (the point estimates ofthe distribution parameters) are plausible values if the null hypothesis is notrejected. The alternative hypothesis does not in general specify particular values forthe true population parameters. However, as shown in a section that follows, the al-ternative hypothesis helps us establish bounds on the true distribution parameters.This is important when a confidence ring is formulated for a parameter distributionpair. The size of the ring can be enlarged or reduced based on two controllable pa-rameters, the significance level a and the sample size n.

The Likelihood Ratio Test StatisticThe test statistic used to influence decision making regarding the alternative hy-pothesis is a ratio of the natural log of two likelihood functions. In simple terms,one likelihood function is associated with a null hypothesis, and the other is associ-ated with an alternative hypothesis. For a probability density function with a singledistribution parameter, the general approaches to testing the null and alternativehypotheses are defined, respectively, as

H0 : h ¼ h0(2)

H1 : h ¼ h1(3)

Note that in the expression for the alternative hypothesis H1, the fact that h equalsh1 implies that h is not equal to h0, and the alternative hypothesis is consistent withthe discussion in the previous section. As they are used here, the hypothesesdescribe two complementary notions regarding the distribution parameters, andthese notions compete with each other. In this sense the hypotheses can be betterdescribed mathematically as

H0 : h 2 H0 ¼ ð1 h0; 2 h0; :::; r h0Þ(4)

H1 : h 2 Hc0 ¼ ð1 h1; 2 h1; :::; r h1Þ(5)

where r corresponds to the number of parameters in the probability density func-tion. Conceptually h0 and h1 are scalar values, whereas H0 and its complement Hc

0


are vectors of distribution parameters. A likelihood function associated with eachhypothesis can be formulated,

L0 ¼Yni¼1

f ðxijh 2 H0Þ(6)

for the null hypothesis and

L1 ¼Yni¼1

f ðxijh 2 Hc0Þ(7)

for the alternative function. The likelihood function L0 associated with the null hy-pothesis is evaluated using the maximum likelihood parameter estimates.

The sample population (i.e., graphite failure data) is assumed to be character-ized by a two-parameter Weibull distribution. There are methods to test the validityof this assumption. However, the material strength is characterized by a randomvariable, so it makes sense to use a minimum extreme-value distribution such as theWeibull distribution. Because this is a proof-of-concept effort focused on likelihoodratio rings, goodness-of-fit tests that can be used to discriminate between alternativeunderlying population distributions are left to others to pursue. A vector of distri-bution parameters whose components are the MLE parameter estimates is identi-fied as

ð1h0; 2h0Þ ¼ ð~h1; ~h2Þ¼ ð~m; ~rhÞ

(8)

where:~m¼maximum likelihood estimate of the Weibull modulus, and~rh¼maximum likelihood estimate of the characteristic strength.Now

H0 : h 2 H0 ¼ ð~m; ~rhÞ(9)

that is, H0 contains MLE parameter estimates, and

H1 : h 2 Hc0(10)

with Hc0 representing a vector of point estimates that are not MLE parameter esti-

mates. In essence, we are testing the null hypothesis that the true distribution pa-rameters are equal to the MLE parameter estimates, with an alternative hypothesisthat the true distribution parameters are not equal to the MLE parameter estimates.

The likelihood functions are now expressed as

~L0 ¼Yni¼1

f ðxij~m; ~rhÞ(11)


L1 ¼Yni¼1

f ðxijm;rhÞ(12)

A test statistic is introduced that is defined as the natural log of the ratio of the like-lihood functions,

T ¼ �2 ln L1

~L0

� �(13)

The Neyman–Pearson lemma [9] states that this likelihood ratio test is the mostpowerful test statistic available for testing the null hypothesis. We can rewrite thislast expression as

T ¼ 2 ~L� L� �

(14)

where

L ¼ ln ~L0� �

¼ lnYni¼1

f ðxij~m; ~rhÞ( )

(15)

~L ¼ ln ~L1� �

¼ lnYni¼1

f ðxijm; rhÞ( )

(16)

The natural log of the likelihood ratio of a null hypothesis to an alternative hy-pothesis is our test statistic, and its distribution can be determined in the limit asthe sample size approaches infinity. The test statistic is then used to form decisionregions where the null hypothesis can be accepted or rejected. A convenient resultattributable to Wilks [10] indicates that as the sample size n approaches infinity,the value �2 ln (T) will be asymptotically v2-distributed for a nested composite hy-pothesis. If one hypothesis can be derived as a limiting sequence of another, we saythat the two hypotheses are nested. In our case the sample (rX1, rX2,..., rXn) repre-senting the rth sample is drawn from a Weibull distribution under H0. These samesamples are used in the alternative hypothesis H1, and because their parent distribu-tion is assumed to be a Weibull distribution under both hypotheses, the twohypotheses are nested and conditions are satisfied for the application of Wilks’stheorem.

The test statistic is designed in such a way that the probability of a Type I errordoes not exceed a, a value that we control. Thus the probability of a Type I error isfixed, and we search for the test statistic that maximizes (1� b), where again b isthe probability of a Type II error. Where inferences are being made on parametersfrom a population characterized by a two-parameter Weibull distribution, thedegree of freedom for the v2 distribution is one, and the values of the v2 distributionare easily calculated. One can compute the likelihood ratio T and compare �2ln (T)to the v2 value corresponding to a desired significance level to define a rejectionregion. This is outlined in the next section. The value of the ratio of the two


likelihood functions defined above (L1/~L0) approaches 1 in the optimal criticalregion (i.e., the value of the test statistic T should be small). This is a result of mini-mizing a and maximizing (1� b). The ratio is high in the complementary region. Ahigh ratio corresponds to a high probability of a correct decision under H0. Thelikelihood ratio test implies that the null hypothesis should be rejected if the valueof the ratio is too small. How small is too small depends on the significance level ofthe test (i.e., on what probability of Type I error is considered tolerable).

Lower values of the likelihood ratio mean that the observed result is much lesslikely to occur under the null hypothesis than under the alternative hypothesis.Higher values of the likelihood ratio mean that the observed outcome is more orequally likely (or nearly as likely) to occur under the null hypothesis, and the nullhypothesis cannot be rejected.

The likelihood ratio test and its close relationship to the v2 test can be used todetermine what sample size will provide a reasonable approximation of the truepopulation parameters.

The Likelihood Ratio RingThe likelihood ratio confidence bounds are based on the inequality

T ¼ 2 ~L� L� �

¼ �2 ln L m;rhð ÞL ~m; ~rhð Þ

� �� v2

a;1

(17)

The equality in Eq 17 can be expressed as

L m;rhð Þ ¼L ~m; ~rhð Þ exp �v2a;12

� �(18)

Here, ~m and ~rh are maximum likelihood estimates of the distribution parametersbased on the data obtained from a sample. These parameter estimates are randomvariables (they vary from sample to sample), as are the test statistic T and v2. Equa-tion 17 stipulates a relationship between random variables. The true distributionparameters m and rh are fixed values, but they are unknown to us unless the popu-lation is completely sampled.

However, if a is designated, then a value for v2 (i.e., a realization) is established.Once this realization is established for v2, a realization for the test statistic T can beestablished through Eq 17. For a given significance level, confidence bounds m0 and r0hcan be computed that satisfy Eq 18 (i.e., these bounds satisfy the following expression).

L m0; r0h� �

�L ~m; ~rhð Þ exp �v2a;12

� �¼ 0(19)

With a given value of m0, a pair of values can be found for r0h. This procedure isrepeated for a range of m0 values until there are enough values to produce a smooth


ring. These parameters map a contour ring in a plane perpendicular to the log like-lihood axis (see Fig. 1). A change in the significance level results in a different-sizedring. From the geometry in Fig. 1 we can see that the true distribution parametersthat are unknown to us will lie within the ring.

Aspects of Likelihood Confidence RingsIn order to present aspects of the likelihood confidence rings, Monte Carlo simula-tion is utilized to obtain test data. Using Monte Carlo simulation allows us the con-venience of knowing what the true distribution parameters are for a particulardataset. Here, it is arbitrarily assumed that the Weibull modulus is 17 and the Wei-bull characteristic strength is 400MPa.

Figure 2 shows the likelihood confidence ring for a 90 % confidence level and asample size of 10, along with the true distribution parameters and the estimated dis-tribution parameters. If the true distribution parameter pair were unknown, wewould be 90 % confident that the true distribution parameters were within the ring.If the Monte Carlo simulation process were continued nine more times (i.e., if wewere in possession of ten simulated datasets), then on average one of those datasetswould produce a likelihood confidence ring that did not contain the true distribu-tion parameter pair.

In Fig. 3 the effect of holding the sample size fixed and varying the confidencelevel is presented. The result is a series of nested likelihood confidence rings. Here

FIG. 1 Log likelihood frequency plot of L(m,rh) with likelihood confidence ring and

associated test statistic T.


we have one dataset and multiple rings associated with increments of the confi-dence level from 50 % to 95 %. Note that as the confidence level increases, the sizeof the likelihood confidence ring expands. For a given number of test specimens ina dataset, the area encompassed by the likelihood confidence ring expands as webecome more and more confident that the true distribution parameters are con-tained in the ring.

FIG. 2 Confidence ring contour for a sample size of 10 (m¼ 17, rh¼400).

FIG. 3 Dependence of likelihood confidence rings on c for a sample size of 30 (m¼ 17,

rh¼400).


The next figure, Fig. 4, depicts the effect of varying the sample size and holdingthe confidence level fixed at c¼ 90 %. The sample size was increased from n¼ 10 ton¼ 100. Note that all the likelihood confidence rings encompass the true distribu-tion parameters used to generate each sample. In addition, the area within the ringsgrows smaller as the sample size increases. As the sample size increases, we gain in-formation on the population and thereby reduce the region that could contain thetrue distribution parameters for a given level of confidence.

Figure 5 depicts a sampling procedure in which the size of the sample is heldfixed (i.e., n¼ 10) and the sampling process and ring generation have been repeated100 times. For a fixed confidence level of 90 %, one would expect that ten ringswould not encompass the true distribution parameters. Indeed that is the case. The90 likelihood confidence rings that encompassed the true distribution parametersare outlined in blue. The ten likelihood confidence rings that did not contain thedistribution parameters are outlined in dark orange.

Confidence Rings and Material AcceptanceThe material acceptance approach outlined here depends on several things. Firstone must have the ability to compute the probability of failure of the componentunder design. This probability is designated (Pf)component and is quantified using a

FIG. 4 Likelihood confidence rings for sample sizes ranging from 10 to 100 (m¼ 17,

rh¼400).


hazard rate format—that is, the probability of failure is expressed as a fraction witha numerator of 1. The method for computing this quantity is available in the ASMEBoiler and Pressure Vessel Code [7].

The component probability of failure is modeled assuming the underlyingstrength is characterized by a two-parameter Weibull distribution. Thus a compo-nent probability of failure curve can be depicted in an m� rh graph as shown inFig. 6. Points along the curve represent parameter pairs equal to a specified proba-bility of failure. This curve is referred to as a material performance curve. We over-lay this graph with point estimates of the Weibull distribution parameters obtainedfrom tensile strength data that a typical material supplier might provide. Point esti-mates from these data that plot to the right of the material performance curve rep-resent a lower probability of failure. Conversely, point estimates to the left of thiscurve are associated with performance curves with a higher probability of failure.Thus the material performance curve defines two regions of the m� rh space, anacceptable performance region and a rejection region relative to a specified compo-nent probability of failure.

The material performance curve is easily married to a likelihood confidencering (discussed in previous sections). This allows the component fabricator todecide whether the material supplier is providing a material with high enough

FIG. 5 100 likelihood confidence rings. For all rings, n¼ 10 and c¼0.9 (m¼ 17, rh¼400).


quality predicated on the component design and the failure data. Keep in mind thatparameter estimates are estimates of the true distribution parameters of the popula-tion, values that are never known in real-life applications. However, through theuse of the likelihood confidence ring method we can define a region in some closeproximity of the estimated point parameter pair, knowing with some level of assur-ance that the true distribution parameters are contained within that region. If thatregion in its entirety falls to the right of the test specimen performance curve, thecomponent fabricator can accept the material with a known level of quality (i.e., thesignificance level). Not surprisingly, we define this procedure as the quality accep-tance criterion.

We have combined the two concepts, the likelihood confidence ring and thematerial performance curve, in one figure (Fig. 7). Here the material performancecurve given in Fig. 6 is overlain with the likelihood confidence ring from Fig. 2. Thisis a graphical representation of the quality assurance process. Rings that reside com-pletely to the right of the material performance curve would represent acceptablematerials. Those rings to the left would represent unacceptable materials and wouldbe rejected. In the specific case presented, the material performance curve cutsthrough the likelihood confidence ring. In this case there are certain regions of thelikelihood confidence ring that produce a safe design space, and there is a region ofthe likelihood confidence ring that produces an unsafe design space. In this situa-tion we know the distribution parameters, and they are purposely to the right of thematerial performance curve. But given the sample size, the ring did not resideentirely in the safe region. Moreover, in normal designs we never know the true

FIG. 6 Generic material performance curve.


distribution parameters, so we do not know where the true distribution parameterpair resides inside the likelihood confidence ring.

When the likelihood confidence ring resides totally to the left of the perform-ance curve, the choice to reject the material is quite clear. When the likelihoodconfidence ring lies completely to the right of the material performance curve,then once again, the choice is quite clear: accept the material. When the materialperformance curve slices through the likelihood confidence ring, we can shiftthe material performance curve to the left, as depicted in Fig. 8. This shift representsa reduction of component reliability or, alternatively, an increase in the componentprobability of failure. Alternatively, the confidence bound associated with likelihoodconfidence ring can be reduced so the ring shrinks enough such that the ring iscompletely to the right of the material performance curve. This is depicted in Fig. 9.

An interesting aspect of this approach is that it seems that the likelihood confi-dence rings give a good indication of which side of the material performance curvethe true distribution parameters lie on. If the material performance curve slicesthrough a likelihood confidence ring for a specified confidence level, then as thering size is diminished the ring becomes tangent to one side of the curve or theother. When this paper was written it was our experience that the side of the com-ponent reliability curve that the ring becomes tangent to matches with the side onwhich the true distribution parameters lie. It is emphasized that this is anecdotal.An example in which the true distribution parameters were chosen to the left of thematerial performance curve is depicted in Fig. 10. The true distribution parameters

FIG. 7 Material performance curve with likelihood confidence ring contour, n¼ 10

(m¼ 17, rh¼400).


FIG. 9 Material performance curves with likelihood confidence rings for changing

values of c (m¼ 17, rh¼400).

FIG. 8 Two parallel material performance curves with likelihood confidence ring (m¼ 17,

rh¼400).


are known because we are conducting a Monte Carlo simulation exercise to pro-duce failure data. As the confidence level decreases in Fig. 10, the rings become tan-gent to the curve on the rejection side.

SummaryThis effort focused on graphite materials and the details associated with calculatingpoint estimates for the Weibull distribution parameters associated with the tensilestrength. One can easily generate point estimates from failure data using maximumlikelihood estimators. More information regarding the population (i.e., more failuredata) always improves the quality of point estimates; the question becomes howmuch data is sufficient given the application. The work outlined here speaks directlyto this issue.

Hypothesis testing and the relationship it maintains with parameter estimationwere outlined. A test statistic was adopted that allows one to map out an acceptanceregion in the m� rh parameter distribution space. The theoretical support for theequations used to generate the likelihood rings was outlined. Inferential statisticsallowed us to generate confidence bounds on the true distribution parameters utiliz-ing the test data at hand. These bounds are dependent on the size of the sampleused to calculate point estimates. The effort focused on a particular type of confi-dence bound known as likelihood confidence rings.

Component reliability curves were discussed. The concepts of the likelihoodconfidence rings and the component probability of failure curves were combinedgraphically. This combination gives rise to a material qualification process. This

FIG. 10 Material performance curves with likelihood confidence rings for changing

values of c (m¼6, rh¼350).


process combines information regarding the reliability of the component and theparameter estimates to assess the quality of the material.

References

[1] ASTM C565-93(2010)e1: Test Methods for Tension Testing of Carbon and Graphite

Mechanical Materials, Annual Book of ASTM Standards, ASTM International, West Con-

shohocken, PA, 2010.

[2] ASTM C749-08(2010)e1: Test Method for Tensile Stress-Strain of Carbon and Graphite,

Annual Book of ASTM Standards, ASTM International, West Conshohocken, PA, 2010.

[3] ASTM C781-08: Practice for Testing Graphite and Boronated Graphite Materials for High-

temperature Gas-cooled Nuclear Reactor Components, Annual Book of ASTM Standards,

ASTM International, West Conshohocken, PA, 2008.

[4] ASTM D7775-11e1: Guide for Measurements on Small Graphite Specimens, Annual Book

of ASTM Standards, ASTM International, West Conshohocken, PA, 2011.

[5] ASTM C651-11: Test Method for Flexural Strength of Manufactured Carbon and Graphite

Articles Using Four-point Loading at Room Temperature, Annual Book of ASTM Stand-

ards, ASTM International, West Conshohocken, PA, 2011.

[6] ASTM D7846-13: Reporting Uniaxial Strength Data and Estimating Weibull Distribution

Parameters for Advanced Graphites, Annual Book of ASTM Standards, ASTM Interna-

tional, West Conshohocken, PA, 2013.

[7] ASME, “Article HHA-II-3000, Section III, Division 5, High Temperature Reactors, Rules

for Construction of Nuclear Facility Components,” ASME Boiler and Pressure Vessel

Code, ASME, New York, 2013.

[8] Fisher, R. A., “Theory of Statistical Estimation,” Proc. Cambridge Philos. Soc., Vol. 22,

1925, pp. 700–725.

[9] Neyman, J. and Pearson, E., “On the Problem of the Most Efficient Tests of Statistical

Hypotheses,” Philos. Trans. R. Soc. London Series A, Vol. 231, 1933, pp. 289–337.

[10] Wilks, S. S., “The Large Sample Distribution of the Likelihood Ratio for Testing Composite

Hypotheses,” Ann. Math. Stat., Vol. 9, No. 1, 1938, pp. 60–62.








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Quality Control Using Inferential Statistics in Weibull-based ...2013/01/22 · Quality Control Using Inferential Statistics in Weibull-based Reliability Analyses Reference Duffy,