Comparison of Fatigue Data Using the Maximum Likelihood Method

Comparison of fatigue data using the maximumlikelihood method

Luca Goglio, Massimo Rossetto *

Dipartimento di Meccanica, Politecnico di Torino, Corso Duca degli Abruzzi, 24, Torino 10129, Italy

Received 30 October 2002; accepted 31 October 2002

Abstract

The paper describes procedures to compare the fatigue behaviour of two populations corresponding to batches thatdi!er for a factor on the basis of fatigue data of two samples. Fatigue data are elaborated by means of the maximumlikelihood method to obtain the SNP curves and their confidence intervals. The likelihood ratio test is used to identifythe most adequate distribution and model. To analyse the influence of the factor, two integrated procedures areadopted: the first one is based on the LR test, the second is an empirical procedure, based on the comparison of theconfidence intervals of the curves.! 2003 Elsevier Ltd. All rights reserved.

Keywords: Fatigue data; Analysis; Comparison; Maximum likelihood; Likelihood ratio test

1. Introduction

In this paper we describe a statistical procedure to compare the SNP curves of two or more samples toassess the influence of a factor and to state if the samples are extracted from the same population (at leastwith respect to the fatigue behaviour).

In literature the comparison is usually performed visually, by plotting the mean curves (B50), with orwithout the data points, on the same graph (for example, [1–7]); usually the data are fitted by means of aleast square method that has some limitations because it is assumed that the scatter, as a function of thestress level, is uniform and runouts are not considered. In some cases also the scatter bands of the curves,that can help in decisions, are plotted [8,9], and data are elaborated using the maximum likelihood method[8].

The visual comparison of the curves can be su"cient only if there is a large di!erence in the obtainedcurves and original data, but in some case some doubts can arise. In these cases it is desirable to have atdisposal statistical tests allowing for a convincing comparison of the data.

*Corresponding author. Fax: +39-11-5646999.E-mail address: [email protected] (M. Rossetto).

0013-7944/$ - see front matter ! 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0013-7944(03)00009-2

Engineering Fracture Mechanics 71 (2004) 725–736www.elsevier.com/locate/engfracmech

A more sophisticated method consists in comparing the fatigue data for each stress level, or selectedones, using the classical statistical tests [10,11], but this method requires to have a large number of data foreach level. Moreover, all the samples must be obtained at the same levels, and the comparison is not madeon overall data.

In [12] the comparison is performed using also the analysis of variance and of covariance, but thisstatistical methods requires the homogeneity of variance (scatter constant as a function of stress level) andthe data cannot contain runouts.

In [13] the likelihood ratio test is proposed to compare K samples fitted with the same model by meansof the maximum likelihood method.

In this paper also a graphical method based on the comparison of the SNP curves with their confidenceinterval, more convincing, in our opinion, in respect to the usual graphical methods, is presented.

For a correct comparison of the data, and in any case to have reliable SNP curves, it is also important tostate the most adequate model to fit the data, considering the type of diagram (semilog or log–log), thetheoretical distribution that has to be considered and the type of model as a function of the stress level(linear or nonlinear scale parameters of distribution, constant or non-constant shape parameter, etc.).

Even if a lot of theoretical distribution have been proposed [14–18], the most used are two parameterslog-normal and Weibull distributions. In some cases three-parameter distributions are proposed [11,14], butusually there are some di"culties in the determination of the location parameter (corresponding to the B!0

curve) due to the low number of fatigue data in the samples.To state the theoretical distribution that best fit the data it is possible to apply the usual statistical test to

the data obtained at a single stress levels, based essentially on the linear regression of the (transformed) dataon probability plots [14]. Also in this case the method considers the single stress level and not on overalldata, and runouts cannot be considered.

In [19] Little describes a statistical test to discriminate between the log-normal and Weibull distributionconsidering all data simultaneously (all stress levels, failures and runouts); this test is based on the ratiobetween the maximum likelihood obtained using the two distributions. A similar test that uses the generallikelihood ratio test is proposed by Nelson in [13] on the basis of a theoretical work of Farewell andPrentice [20].

For the choice of the most adequate model in [13,21] a test, again based on likelihood ratio test, ispresented. Also the use of graphs for the raw and normalised residuals, which are evaluated as distance ofthe data from the mean value divided by the standard deviation, can help in the decision.

In this paper a procedure to fit and to compare the fatigue data of two samples that integrate thelikelihood ratio tests proposed by Nelson in [13] and graphical methods is presented. The procedure isdeveloped in two steps: determination of models that best fits the data of the two sample and comparison ofthe data. Some consideration about some possible di"culties are reported.

Furthermore an example concerning the influence of defects on the fatigue behaviour of M12 weldedstuds with mean stress of 400 MPa is reported.

2. Determination of the model for the SNP curves

To determine the SNP curves and their confidence limits the maximum likelihood method [13,21] hasbeen used. Unlike the methods based on least squares, this allows for considering also interrupted tests(runouts) and life data a!ected by scatter variable as a function of the stress level. For computationalconvenience usually the logarithm of the likelihood is used (£ ! maximum of the log likelihood).

In our implementation of the method, developed in the MATLAB environment, we have consideredboth log–log and semilog diagrams, two distributions (two parameters log-normal and Weibull), and fourtype of the models of life as a function of the stress level or log of stress: linear and nonlinear mean,

726 L. Goglio, M. Rossetto / Engineering Fracture Mechanics 71 (2004) 725–736

constant and non-constant standard deviation in the case of log-normal distribution; linear and nonlinearlog of characteristic life, constant and non-constant shape factor in case of Weibull distribution.

In the case of log–log diagram a Weibull model [22] (also referred as Stromeyer model [23]) is considered.For the log-normal distribution the models are

adl l ! C1 " C2#log S $ log S% r ! exp#D1%bdl l ! C1 " C2#log S $ log S% r ! exp D1 " D2#log S $ log S%

! "

cdl l ! C1 " C2#log#S $ S0% $ log S% r ! exp#D1%ddl l ! C1 " C2#log#S $ S0% $ log S% r ! exp&D1 " D2#log#S $ S0% $ log S%'

#1%

where l is the mean of logN and r is the standard deviation of logN , Ci;Di; S0 are the coe"cients of themodels that maximise the log likelihood of the sample (£) and log S is the mean of log of stress; thisnormalisation of the stress allows a better numerical convergence and narrower confidence intervals forcoe"cient C1.

It is worth of note that using the maximum log likelihood method the coe"cient S0, sometimes referredto as fatigue limit [23,24], is only a mathematical coe"cient used to have a nonlinear model. In fact, forcomputational reasons, its value must be lower of the lowest stress level of the sample, and this valueusually di!ers from the 50% fatigue limit.

Considering a Weibull distribution the models are

adw ln h ! C1 " C2#log S $ log S% b ! exp#D1%bdw ln h ! C1 " C2#log S $ log S% b ! exp D1 " D2#log S $ log S%

! "

cdw ln h ! C1 " C2#log#S $ S0% $ log S% b ! exp#D1%ddw ln h ! C1 " C2#log#S $ S0% $ log S% b ! exp D1 " D2#log#S $ S0% $ log S%

! "

#2%

where h is characteristic life and b is the shape factor.In the case of semilog diagram the nonlinear behaviour of the mean or of the log of the characteristic life

is obtained using a hyperbolic model [25], thus the considered models for semilog diagram and log-normaldistribution are

asl l ! C1 " C2S r ! exp#D1%bsl l ! C1 " C2S r ! exp#D1 " D2S%csl l ! C1 " C2S " C3=S r ! exp#D1%dsl l ! C1 " C2S " C3=S r ! exp#D1 " D2S%

#3%

and considering a Weibull distribution:

asw ln h ! C1 " C2S b ! exp#D1%bsw ln h ! C1 " CsS b ! exp#D1 " D2S%csw ln h ! C1 " C2S " C3=S b ! exp#D1%dsw ln h ! C1 " C2S " C3=S b ! exp#D1 " D2S%

#4%

In this case it is not possible to normalise the stress because in the computational procedure some databecome infinite.

The analysis of the data requires some initial assumptions, namely the type of diagram, the theoreticaldistribution, the type of model for the distribution parameters. Therefore it is desirable to have at disposalstatistical tests and procedures allowing the best choices for the representation of the data.

2.1. Choice of the type of diagram

To the best of our knowledge, there is no statistical or empirical test to choose the type of diagram(semilog or log–log); therefore, this choice is due to experience and convenience in using the data.

L. Goglio, M. Rossetto / Engineering Fracture Mechanics 71 (2004) 725–736 727

2.2. Choice of the distribution

The choice between log-normal and Weibull distribution can be performed, as proposed by Nelsonin [13], using the likelihood ratio. The test requires to calculate the statistics:

Twl ! 2#£w " £l% #5%

where £w and £l are respectively the maximum log likelihood obtained using the Weibull and log-normaldistributions considering consistent types of model (with the same structure, for instance csl and csw). Twl isthen compared to the v2 distribution for the desired confidence level (1$ a) and for one degree of freedom;three cases are possible:

ii(i) Twl < 0: log-normal distribution fits better the data;i(ii) 0 < Twl < v2#1$ a; 1%: the Weibull distribution fits better but not significantly better;(iii) Twl > v2#1$ a; 1%: the Weibull distribution fits significantly better the data.

Therefore, to chose the distribution, the data must be elaborated using both Weibull and log-normaldistribution adopting all the type of models for the chosen type of diagram.

If cases (i) and (ii) occur for all type of models the log-normal distribution can be assumed, whereas cases(iii) and (ii) occur the Weibull distribution can be assumed. If cases (i) and (iii) occur for the di!erent typesof model the choice must be carried out contextually to the choice of the type of model.

In the following the likelihood ratio test used to state the most appropriate distribution will be referredto as ‘‘LR(D) test’’.

2.3. Choice of the model

The most adequate model for the available sample, after that the diagram and distribution have beenchosen, can be selected by applying both graphical and numerical methods.

The graphical method requires the construction of graphs for the raw and normalised residuals, whichare evaluated as distance of the data from the mean value divided by the standard deviation [13]. Nodi"culties arise in constructing such diagrams, but their interpretation requires experience, the larger thenumber of considered alternatives the higher the di"culty.

The numerical method uses again the likelihood ratio test [21,13]; in this case the test can be used whenthe models belong to the same set, i.e. when a model can be derived from a more general one by setting tozero the values of some coe"cients.

For instance in our case models asl can be derived from model dsl by setting to zero coe"cients C3 andD2, the same applies to other models.

The likelihood ratio test allows for checking whether the coe"cients added to obtain a higher degreemodel are statistically significant, i.e. if the addition significantly improves data representation [21].

The test requires to calculate the statistics:

Tpq ! 2#£p $ £q% #6%

where £p is the logarithm of the maximum likelihood obtained using the higher order model and £q is thelogarithm of the maximum likelihood obtained using the lower order model. The value Tpq is then comparedto the v2 distribution for the desired confidence level (1$ a) and a number of degrees of freedom equal tothe number of coe"cients being added (G). If Tpq > v2#1$ a;G% the higher order model fits significantlybetter the data.


If di!erent response of the test for the distribution for di!erent types of model occurred the data mustbe elaborated using both the distributions and the best combination of type of model and distribution mustbe chosen.

In the following the likelihood ratio test used to state the most appropriate type of model will be referredto as ‘‘LR(M) test’’.

Another possible procedure to assess the significance of a higher order model, consists in checking thatthe confidence interval of the added coe"cient does not enclose zero [21]. The (approximate) confidenceintervals are easy to obtain from the covariance matrix that can be evaluated considering the Fisher matrix,i.e. the matrix of negative second partial derivatives of £ with respect to coe"cients Ci and Di and S0.However, this procedure may lead to incorrect conclusions. In fact, whether the LR(M) test states that acoe"cient is statistically significant, its confidence interval does not enclose zero; but, whether the LR(M)test states that a coe"cient is not statistically significant, not necessarily the confidence interval encloseszero.

3. Comparison of two or more samples

To compare two, or more, samples two methods are here reported: the first one is an empirical graphicalmethod, valid in the case of two samples, the second one is a numerical method based on likelihood ratiotest.

3.1. Graphical method

The data obtained from the two samples are treated separately by means of the maximum likelihoodmethod, using the same model chosen by the method described previously.

Then the characteristic life h#S% for the Weibull distribution, or the mean life l#S% for the log-normaldistribution, with their related confidence interval at the chosen level (e.g. 95%) are plotted and superposedon the same graph. If the curve h#S% (l#S%) of one batch is included in the confidence interval of the curve ofthe other batch, and vice versa, from the statistical viewpoint the two curves may not be distinguished, thusthe characteristic life (mean life) of the two batches is statistically the same. If this is not the case, it may beconcluded that the behaviour of the batches is di!erent.

If the characteristic lives (mean lives) are the same, the curves corresponding to the highest interestingsurvival percentile (for instance 90%) are plotted together with their confidence intervals. If the curve of thesurvival percentile of one batch is included in the confidence interval of the curve of the other batch, andvice versa, from the statistical viewpoint the two curves may not be distinguished. In this case it may beassumed that the shape parameters b#S% (r#S%) are statistically equal. Thus the data from the two batchesrepresent two samples extracted from the same population (at least with respect to the fatigue behaviour).

The latter step is the most critical, because the functions of the percentile and the related confidenceinterval also depend on h#S% (or l#S%), that are numerically di!erent for the two samples. This makes theprocedure weak from the theoretical viewpoint, but we think that as far as application is concerned havingcurves not distinguishable at the highest percentile of interest justifies the assumption of unique population.

3.2. Likelihood ratio test method

This method allows to compare K samples elaborated using the same model with Q coe"cients and it isbased on the simultaneous comparison of the Q coe"cients [13].

The method requires to compute the maximum likelihood, for each sample (£i) and for the pooled data(£0). The test requires to calculate the statistics:


T ! 2X

£i#

$ £0$

#7%

and its value is compared to the v2 distribution for the desired confidence level (1$ a) and a number ofdegrees of freedom equal to Q.

If T 6 v2#1$ a;Q% the Q coe"cients of the samples not di!er significantly (at the chosen confidence level)and it is possible to state that the samples are extracted from the same population (at least with respect tothe fatigue behaviour).

If T > v2#1$ a;Q% at least one pairs of coe"cients di!er convincingly and it is possible to state that thesamples are extracted from di!erent population (with respect to the fatigue behaviour).

In the latter case, it is possible to compare the confidence intervals of the coe"cients to state why thesamples di!er. In fact it is possible, for instance, that the two samples di!er only for the dispersionparameter, but have the same mean; also the superposed plots can help in this comparison.

Obviously, after that it has been concluded that the data are all from the same population the fatiguecurves may be obtained considering the pooled data.

In the following the likelihood ratio test used to compare samples will be referred to as ‘‘LR(C) test’’.

4. Example: defective or non-defective welded studs

The example here reported concerns the influence on the fatigue behaviour of porosity in the weld zoneof M12 studs welded, by means of a welding gun, on circular plates 30 mm diameter and 5 mm thick. Allspecimens have been controlled by ultrasonics adopting the immersion technique, and were classified asdefective or non-defective.

Fatigue tests have been carried out on a resonant machine (AMSLER 10HFP 422) applying a meanstress values of 400 MPa (approximately corresponding to 70% of the yield stress). The testing frequencywas 190 Hz.

For both the samples the fatigue limit (alternate component) has been evaluated by means of the stair-case procedure [26,27], assuming as threshold for infinite life 5( 106 cycles and a step d ! 10 MPa; for eachsample 15 specimens have been used.

For both samples the estimate of the fatigue limit was SD50% ! 26:4 MPa and its 95% confidence interval{21.1–31.7 MPa}, the standard deviation r ! 5:3 MPa, so it is obvious that defects have no influence onfatigue limit. (The precision adopted in reporting the results (10$1 MPa) is consistent with that achievedwhen setting the load on the testing machine. Obviously, from the technical viewpoint it would be enoughto round values to the closest integer or even five. However, data have been kept in this form to allow forfurther comparison.)

This result may not be directly applied to whole Whler curve. For this reason tests at stress higher thanthe fatigue limits have been carried out separately for defective and non-defective specimens (at 110–80–60MPa).

Considering also the data from staircase the non-defective sample has 28 data (seven runouts) anddefective sample 30 data (seven runouts).

These data were elaborated using the developed !Maximum Likelihood" program considering semilogdiagrams and compared using the methods described above. Resulting log likelihood for the two samplesare reported in Tables 1 and 2.

Applying the LR(D) test for both samples and for all models result Twl < 0, except for model as of thedefective sample. In this case, considering a confidence level of 95%, result Twl ! 2:48 < 3:84 ! v2#0:95; 1%.Therefore for all cases log-normal distribution has been chosen.

To state the most adequate model the LR(M) test was applied (Table 3).


Observing these data the most adequate model for the ‘‘non-defective’’ sample is dsl, whereas that for the‘‘defective’’ sample is csl. It is worth of note that the di!erence between the two models is the constant ornon-constant standard deviation.

In addition, it can be noted that, in this case, the observation of the confidence intervals of the coe"-cients (reported in Tables 4 and 5) leads to the same conclusions.

The di!erence in the model is not su"cient to conclude that the populations present di!erent fatiguebehaviour, in fact the di!erences could be due to the low number of points.

The graphical comparison can help to decide about the di!erence between the models with engineeringjudgement.

In our case the comparison of the confidence interval of the B10 curves obtained using the two models(Fig. 1) leads to state that it is not possible, from a statistical point of view, to distinguish between the twomodels, so it is possible to continue in the comparison procedure using the higher order model dsl for bothsamples (this can be convenient but it is not compulsory).

The fatigue strengths at 5( 106 cycles calculated using model dsl is 24.6 MPa for non-defective sampleand 27.1 MPa for defective sample, both in the 95% confidence interval of the fatigue limit at 5( 106 cyclesobtained from staircases (SD50% ! 26:4 MPa {21.1–31.7 MPa}). It is worth of note that models asl and bsl

(for non-defective sample) give fatigue strengths out of this confidence interval. This fact confirm that theuse of a wrong model gives non-reliable results.

To compare graphically the two samples B50 and B10 curves and their 95% confidence interval are su-perposed (Figs. 2 and 3 respectively); it is possible to note only a small di!erence at lower stress levels,where the staircases indicate an equal fatigue behaviour. This fact can be due to the presence at these levelsof all the runouts [13].

Table 1Log likelihood (£) for non-defective sample

Model Log-normal Weibull Twl

as )290.74 )291.31 )1.14bs )285.00 )287.44 )4.88cs )280.88 )283.63 )5.50ds )277.99 )281.56 )7.14

Table 2Log likelihood (£) for defective sample


as )315.38 )314.14 2.48bs )312.52 )313.75 )2.46cs )298.99 )301.43 )4.88ds )297.90 )300.33 )4.86

Table 3LR test to chose models

Sample Tba 1 (D2) Tca 1 (C3) Tdc 1 (D2) Tdb 1 (C3) Tda 2 (C3;D2)

Non-defective 11.46 19.72 5.78 14.02 25.50Defective 5.72 32.78 2.18 29.24 34.96

Bold if Tpq < v2#1$ a;G% (no influence of coef.): 1 ! limit value v2#0:95; 1% ! 3:84, 2 ! limit value v2#0:95; 2% ! 3:84.


To apply the LR(C) test the maximum log likelihood £0 for the pooled data have to be computed: theresults are reported in Table 6. Applying the LR(D) test to the pooled data and considering a limit value forTwl ! v2#0:95; 1% ! 3:84 the most suitable distribution for the pooled data is again the log-normal.

Comparing the sample applying the LR(C) test, considering the model dsl (number of coe"cients Q ! 5)and a 95% confidence interval results:

T ! 2#£1 " £2 $ £0% ! 2#$277:99$ 297:90" 577:82% ! 3:866 11:07 ! v2#0:95; 5%

Table 4Non-defective sample: 95% interval confidence of the models coe"cients

M12––Sm 400 MPa non-defective Lower Upper

Model asl C1 6.0553 6.0043 6.1064C2 )0.0199 )0.0216 )0.0183D1 )2.0458 )2.3625 )1.7291

Model bsl C1 6.0172 5.9545 6.0798C2 )0.0180 )0.0192 )0.0168D1 )2.0619 )2.4044 )1.7194D2 )0.0279 )0.0397 )0.0160

Model csl C1 6.1959 5.7495 6.6422C2 )0.0124 )0.0161 )0.0087C3 19.8400 8.9864 30.6940D1 )2.3322 )2.6352 )2.0292

Model dsl C1 6.3549 6.0172 6.6926C2 )0.0138 )0.0164 )0.0112C3 16.7770 8.0588 25.4950D1 )1.5306 )2.3333 )0.7278D2 )0.0157 )0.0282 )0.0032

Table 5Defective sample: 95% interval confidence of the models coe"cients

M12––Sm 400 MPa defective Lower Upper

Model asl C1 6.0308 5.9823 6.0792C2 )0.0205 )0.0220 )0.0190D1 )2.0723 )2.3665 )1.7781

Model bsl C1 5.9384 5.8609 6.0159C2 )0.0170 )0.0185 )0.0154D1 )1.9411 )2.2668 )1.6154D2 )0.0324 )0.0442 )0.0206

Model csl C1 5.8674 5.5186 6.2162C2 )0.0101 )0.0129 )0.0073C3 30.2940 21.7240 38.8640D1 )2.6517 )2.9444 )2.3590

Model dsl C1 5.9953 5.6394 6.3511C2 )0.0111 )0.0138 )0.0085C3 27.2460 17.8690 36.6220D1 )2.0161 )2.9923 )1.0400D2 $0:0104 $0:0245 0:0038

Bold if the 95% confidence interval includes zero.


so, the two Q pairs of coe"cients of the samples do not di!er significantly and it is possible to state thatthe samples are extracted from the same population (at least with respect to the fatigue behaviour).

It is worth of note that the same holds if the model csl is considered, in fact in this case it is

T ! 2#£1 " £2 $ £0% ! 2#$280:88$ 298:99" 583:31% ! 6:886 11:07 ! v2#0:95; 5%

On the basis of the above considerations and the result of the LR(C) test it is possible to state that thetwo samples are extracted from the same population (with respect to the fatigue behaviour). The LR(M)tests for the pooled data (Table 7) confirm the choice of the dsl model. The fatigue strength at 5( 106 cyclescomputed from this model for the pooled data is 25.8 within the 95% confidence interval of the fatigue limitat 5( 106 cycles obtained from staircases. Fig. 4 reports the B10, B50 and B90 curves obtained with thismodel for the pooled data.

Fig. 1. Defective sample: comparison of B10 curves obtained using csl and dsl models.

Fig. 2. Comparison of defective and non-defective samples: B50 curves.


Table 6Log likelihood £0 for pooled data


as )606.67 )605.94 1.46bs )599.06 )602.37 )6.62cs )583.31 )586.60 )6.58ds )577.82 )583.22 )10.80

Fig. 3. Comparison of defective and non-defective samples: B10 curves.

Table 7LR test to chose models

Tba 1 (D2) Tca 1 (C3) Tdc 1 (D2) Tdb 1 (C3) Tda 2 (C3;D2)

Pooled data 15.22 46.72 10.98 42.48 57.70

Bold if Tpq < v2#1$ a;G% (no influence of coef.) 1 ! limit value v2#0:95; 1% ! 3:84, 2 ! limit value v2#0:95; 2% ! 3:84.

Fig. 4. SNP curves of the pooled data.


5. Conclusions

The two procedures described in the paper allow to compare the fatigue behaviour of two populationcorresponding to batches that di!er for a factor on the basis of fatigue data of two samples; the first is basedon the likelihood ratio test, the second one is an empirical graphical procedure.

In both cases the fatigue data have to be elaborated using the maximum likelihood method, and pro-cedures to identify the most adequate distribution and model to fit the data and to perform correctly thecomparison are stated.

In the reported example is pointed out that the two procedures can be used complementary to performa correct comparison with engineering judgement.

Acknowledgements

This research was supported by the Italian Research Council (CNR Grants. 96.02437.CT07 and97.03106.CT07).

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Comparison of Fatigue Data Using the Maximum Likelihood Method

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