Diagnostics in Birnbaum–Saunders accelerated life models with an application to fatigue data

Authors


Abstract

In industrial statistics, there is great interest in predicting with precision lifetimes of specimens that operate under stress. For example, a bad estimation of the lower percentiles of a life distribution can produce significant monetary losses to organizations due to an excessive amount of warranty claims. The Birnbaum–Saunders distribution is useful for modeling lifetime data. This is because such a distribution allows us to relate the total time until the failure occurs to some type of cumulative damage produced by stress. In this paper, we propose a methodology for detecting influence of atypical data in accelerated life models on the basis of the Birnbaum–Saunders distribution. The methodology developed in this study should be considered in the design of structures and in the prediction of warranty claims. We conclude this work with an application of the proposed methodology on the basis of real fatigue life data, which illustrates its importance in a warranty claim problem. Copyright © 2012 John Wiley & Sons, Ltd.

1 Introduction

In business and industry, there is great interest in predicting lifetimes of specimens that operate under stress because of safe life and warranty problems [1, 2]. The detection of potentially influential observations and their handling are relevant subjects in the calculation of percentiles for diverse types of analysis, for instance, in warranty. A bad estimation of the lower percentiles of a life distribution can produce significant monetary losses in commercial organizations because of an excessive amount of warranty claims. Then, the potential influence of a case on the estimation of lower percentiles should be considered in the design of structures and in the prediction of warranty claims.

To carry out life tests at levels of low stress can be very time consuming. To avoid this, failures of specimens at levels of high stress are observed, and then the failure times at low levels are predicted. This kind of testing is known as accelerated life (AL) test, and the models used in these tests are named AL models. Generally, these models are based on the physical or chemical nature that is behind the failure mechanism. To establish a parametric AL model, the following questions arise: (A1) What distribution can be assumed under regular stress conditions? (A2) Does the supposed distribution remain at stress levels different from the normal (regular) level? (A3) If the distribution remains, how are its parameters modified? (A4) What is the functional relationship between stressors and parameters? The relationship mentioned in (A4) can be described by an AL model, such that the interest parameter vector (θ) and the stressors of durability (x) are linked by a function θ(x). Habitually, the stressors x identify load, pressure, temperature, vibration or voltage. Usually, the parameter θ represents the mean or the scale of the distribution. This is because often, an increment in the stress level modifies the location or the scale but not the shape of the life distribution, making the distributional family remain unalterable; for more details about life distributions, see [3]. Thus, it is possible to predict the lifetime (T) at different stress levels by using θ(x). This functional relationship involves unknown additional parameters that need to be estimated. The usual AL models are: (B1) Arrhenius; (B2) exponential; (B3) Eyring; (B4) linear; and (B5) power law; for more details, see [4-6].

One distribution that has been used in reliability and quality studies, and notably in material fatigue, is the Birnbaum–Saunders (BS) distribution; see the seminal articles in [7, 8] and the more recent works on quality studies in [9, 10], and [11]. The BS model corresponds to the distribution of the random variable inline image, when Z ∼ N(0,1). We use the notation T ∼ BS(α,β), where α is a shape parameter and β is both a scale parameter and the median of the distribution. The BS model shows that the failure of a specimen is due to the development and growth of a dominant crack produced by stress. The BS distribution is a natural model in many situations where an accumulation of damage produced by stress forces a quantity to exceed a critical threshold provoking the failure of a specimen. For more details about the BS distribution, see [12], pp. 651–658 and [2]. Several generalizations and extensions of the BS distribution have been made by [13-26]. Implementations in the R software of BS distributions can be found in [27-29]. These packages are available from the authors upon request. Although the BS distribution has its origin in engineering, it has been applied successfully to business, environmental and medical data [20-23, 26, 30-37].

If T ∼ BS(α,β), then:

  • (C1)The PDF of T is
    display math
    where ϕ( ⋅ ) denotes the N(0,1) PDF;
  • (C2)inline image so that Z2 ∼ χ2(1);
  • (C3)bT ∼ BS(α,bβ), with b > 0;
  • (C4)1 ∕ T ∼ BS(α,1 ∕ β);
  • (C5)The CDF of T is inline image, where Φ( ⋅ ) denotes the N(0,1) CDF;
  • (C6)The quantile function (QF) of T is inline image, where z(q) is the N(0,1) QF and inline image is the inverse CDF of T;
  • (C7)The mean and variance of T are E[T] = β[1 + α2 ∕ 2] and V[T] = β2α2[1 + 5 α2 ∕ 4], respectively, and
  • (C8)The failure rate of T is given by
    display math

It is possible to note that hT(t) as defined in (C8) is unimodal for any α, that is, it increases until its change point (tc) and decreases for t > tc. In addition, hT(t) approaches 1 ∕ [2α2β] as t → ∞ , and it tends to be increasing as α → 0, such as shown in [7]. BS and lognormal models have similar properties and shape characteristics, but (i) the lognormal PDF has lighter tails than the BS PDF and (ii) the lognormal failure rate decreases until zero and it does not become stabilized at a positive constant greater than zero, as in the BS case. Then, there are reasons for preferring one of these models instead of the other one. For instance, [7] mentioned that fatigue life data usually have unimodal failure rates, which also occurs with environmental, financial duration and survival data [22, 34, 38-40]. In consequence, if one is interested in a warranty analysis on the basis of fatigue life data, as is the case of our application example, one should postulate the BS distribution as a candidate model to describe this type of data.

Accelerated life models have been considered for the BS distribution, but only some particular cases have been analyzed. For instance, [41] developed a BS power law AL (BSPAL) model. It is possible to present a more general methodology to develop an AL model on the basis of the BS distribution, which we call BS accelerated life (BSAL) models.

Influence diagnostic analysis is an important step in data modeling, which is carried out after the parameter estimation step for assessing the stability of this estimation. The analysis can be conducted by global and/or local influence diagnostic methods. These methods have been frequently employed in normal linear regression models and largely studied in the statistical literature. Global influence methods give emphasis to studying the effect of eliminating observations on the results from the fitted model; for more details on global influence [42, 43]. Cook [44] proposed an alternative method, known as local influence, for detecting the effect of small perturbations in the model and/or data on the maximum likelihood (ML) estimates. For more details about the use of the global and local influence methods in more general models than the normal one, see [30, 36, 45-49]. Diagnostic methods are useful in complex statistical models. However, these methods, specially the concept of generalized leverage (GL), are also useful in simpler models, such as in the case of the experimental design model employed in our application example [43, 50]. Specifically, in this example, we use a real fatigue life data set presented in [8] for illustrating the proposed methodology by means of which we conduct exploratory, confirmatory and diagnostic analyses. Once we choose a suitable model for these data, we estimate lower percentiles to predict warranty claims. The obtained results are implemented in the R software and available from the authors upon request.

The aim of this article is to introduce a methodology based on BSAL models that includes GL and influence diagnostic analyses. BSAL models and their estimation and inference based on likelihood methods are presented in Section 2. The global and local influence and GL methods are derived in Section 3. An application example with real data is discussed in Section 4. Some conclusions and an appendix with technical results are provided in the final sections.

2 Birnbaum–Saunders accelerated life models

The shape parameter α, the scale parameter β or the mean μ = β[1 + α2 ∕ 2] of the BS distribution can be described by an AL model. Because the shape of the BS distribution is not usually modified by any stressor (x), then we can model its scale or its mean. However, in this case, to describe β by an AL model is equivalent, but simpler, to modeling the mean. Thus, the parameter β of the BS distribution as function of a stressor x can be described by an AL model as

display math(1)

where

display math

Then, if T ∼ BS(α,β) and β is described by (1), T ∼ BS(α,β(x)) ≡ BS(α,β0,β1). Therefore, (1) produces a BSAL model corresponding to a three-parameter BS distribution, whose PDF, CDF and QF are

display math(2)
display math(3)
display math(4)

respectively, where β(x) is as given in (1). Note that if h(x) = log(1 ∕ x) in (1), then β(x) = γx − η, with log(γ) = β0 > 0 and η = β1 > 0, which is [13]'s BSPAL model. Figure 1 shows the behavior of the PDF given in (2) for the BSPAL model at levels x1 = 1.0, x2 = 1.5, x3 = 2.5, and x4 = 4.5 of a stressor x and for α = 0.2, γ = 28 and η = 0.4. From this figure, we observe that if the stress level is increasing, the lifetimes are decreasing, which corresponds to a scale change, but the shape of the life distribution does not change, as mentioned.

Figure 1.

PDF of the BSPAL model for the indicated stress level.

The parameters of the BSAL model based on (1) can be estimated by the ML method. However, numerical methods must be used for maximizing the likelihood function. To use these methods, starting values that allow for the iteration of the algorithm are required. For the BSPAL model, [13] proposed two techniques to find these starting values, one on the basis of the BS CDF and another that we summarize next; see also [51] for a third technique. If we consider the model given in (1) and if ϵ ∼ BS(α,1), then by using Property (C3), a log-linear regression model of the form T = γx − ηϵ ∼ BS(α,γx − η) can be derived. Now, applying logarithm, we obtain the linear model

display math(5)

where the term log(ϵ) follows a logarithmic version of the BS distribution [52]. Thus, estimates of γ and η can be obtained by the least square method.

Consider lifetimes inline image of ni specimens subjected to the kth level of the stressor x, which correspond to observations of a random sample T = [T1, … ,Tk] ⊤ , with inline image, from Tij ∼ BS(α,β(xi)) for inline image, with i = 1, … ,k and j = 1, … ,ni. Then, the log-likelihood function for θ = [α,γ,η] ⊤  is expressed as

display math(6)

where inline image is the total number of observations. Thus, the corresponding ML estimate of θ = [α,γ,η] ⊤  is found maximizing (6). For simplicity's sake, inline image can be considered, where

display math(7)

with c being a constant that does not depend on θ. The derivatives of (θ) with respect to γ and η produce non-linear equations. However, α can be analytically expressed in terms of γ and η. Hence, the equations for γ and η must be numerically solved. Thus,

display math(8)

To solve the equations given in (8), starting values are needed, which can be obtained by means of (5). The corresponding score vector U(θ) and the Hessian matrix H(θ) are given in Appendix A.

The expected Fisher information matrix (EFIM) for the BSPAL model on the basis of the N observations is expressed as inline image. However, this expected value must be approximated using numerical integration. Thus, the corresponding 3 × 3 EFIM is given by

display math(9)

Asymptotic inference for θ can be based on the normal approximation of the distribution of the ML estimators inline image given by

display math(10)

where I3 is the 3 × 3 identity matrix and inline image denotes the asymptotic variance–covariance matrix of inline image, which can be obtained by inline image, where inline image is given in (9).

Asymptotic inference for the BSAL QF, namely t(q; x), may be based on the asymptotic distribution of inline image given in (10) by using the delta method; see [[53], pp. 387–388]. Thus,

display math(11)

where τ(θ) = t(q; x). By substituting the ML estimates of α, γ and η in

display math(12)

we obtain its ML estimate by using the invariance property of the ML estimators. The result in (11) can be used to obtain confidence bounds for t(q; x).

3 Influence diagnostics in Birnbaum–Saunders accelerated life models

Several diagnostic methods have been proposed for detecting observations that could have a potential influence on the estimates of the model parameters. These methods are divided in two: (i) elimination of cases for assessing global influence and (ii) the incorporation of different schemes of perturbations for assessing local influence.

3.1 The generalized Cook distance

The Cook distance allows us to study the change in the estimated parameters when a case is dropped (global influence). It is possible to generalize this distance to the BSAL model for the jth specimen in the jth stress level by

display math

where inline image is an estimate of the asymptotic variance-covariance matrix of inline image and inline image is the ML estimate of inline image without considering the ijth case. As mentioned, inline image can be obtained by inline image. If we use a first-order approach of the type inline image, we get

display math(13)

which is a generalized Cook distance for the BSAL model with p = 3, where the vector U(θ) and the matrix H(θ) are given in Appendix A. In addition, inline image and inline image are computed in an analogous way to inline image. Then, large values of Dij indicate the ijth case has a high impact on the ML estimate of θ.

3.2 Local influence

Consider the likelihood displacement (LD) given by inline image, where inline image denotes the ML estimate of θ under a perturbed model. Then, the normal curvature for θ in the direction vector l, with | | l | | = 1, is given by inline image, where Δ is a 3 × N perturbation matrix and, as mentioned, inline image. The elements of Δ must be evaluated at inline image and at the non-perturbed vector inline image, for r = 1,2,3 and inline image. For the BSPAL model, as mentioned, H(θ) is given in Appendix A, with θ being evaluated at inline image. Local influence diagnostics are generally based on several index plots, for example, of the characteristic vector lmax. It corresponds to the greatest eigenvalue of the matrix inline image, named inline image, and it can reveal those cases that, under small perturbations, produce an obvious influence on LD(ω). Another direction of interest is ls = esN, which is the sth unit vector of inline image. In this case, the normal curvature is given by Cs = 2 | bss | , where bss is the sth element of the diagonal of B, for inline image, called total local influence. Also, we can calculate the normal curvature for each parameter, that is, Cl(α), Cl(γ), or Cl(η). The perturbation matrix for the different schemes is given in general by Δ = [Δrs]3 × N = [2(θ | ω) ∕ ∂θ r∂ωs]3 × N, for r = 1,2,3 and s = 1, … ,N, considering the BSPAL model and its log-likelihood function given in (7).

3.2.1 Case-weight perturbation

With this perturbation scheme, we wish to evaluate whether the contribution of the cases with different weights affects the ML estimate of θ. This is the most commonly used scheme for evaluating local influence on a statistical model. The N × 1 weight vector inline image, where inline image represents the perturbation at the ith stress level. Then, when the jth specimen is perturbed at the ith stress level, the log-likelihood function is expressed as

display math(14)

where fT( ⋅ ) is as given in (2). Then, deriving (14) with respect to ω, we obtain

display math(15)

with inline image representing an N × 1 vector that has an one at the ijth position, and zeros in the other positions. If (15) is derived with respect to θ = [α,γ,η] ⊤ , we obtain the perturbation matrix given in (A.3) of Appendix A.

3.2.2 Perturbation of the response

This perturbation scheme can be carried out in several ways. An additive perturbation is given by

display math(16)

where ωij represents the perturbation at the ith stress level of the jth specimen. In addition, 0 ⩽ ωij 1 and inline image. The log-likelihood function for this scheme is expressed as

display math(17)

where tijω is defined in (16) and c is again a constant that does not depend on θ. Before deriving (17), consider that inline image. On the basis of (17), the first partial derivatives with respect to the parameters are given by

display math

Then, we obtain the 3 × N perturbation matrix, which is given in (A.5) of Appendix A.

3.2.3 Perturbation of the stressor

As in the perturbation scheme of the response, the stressor can also be perturbed in several ways. An additive perturbation of the ith stress level is given by

display math(18)

where inline image is a k × 1 vector. The log-likelihood function is now expressed by

display math(19)

with c being once again a constant that does not depend on θ. Before deriving (19), consider the following result. When the stressor is additively perturbed, xi is transformed into (18). Then, inline image, where inline image is an N × 1 vector of zeros (0) with a vector of ones (inline image) of size ni associated with the ith stress level.

Based on (19), the first partial derivatives with respect to the parameters are

display math

Then, we obtain the 3 × k perturbation matrix, which is given in (A.6) of Appendix A.

3.3 Generalized leverage

The primary idea of the concept of leverage is to evaluate the influence of the observed response tij on its own estimated value inline image [42, 54]. This influence can be represented by the derivative inline image, which is equal to pii in the case of normal linear regression, where pii is the ith element of the main diagonal of the projection matrix P = X[X ⊤ X] − 1X ⊤ , with X being the design matrix of the model. Extensions of this result to more general regression models have been studied, for example, by [47]. If (θ) is the log-likelihood function given in (6), then inline image is the ML estimate of θ = [α,γ,η] ⊤  and inline image is the expected value of inline image, with inline image, for i = 1, … ,k. According to the BS mean, for the BSPAL model, inline image is obtained, for inline image and inline image. Then, inline image is the predicted response vector. Thus, the N × N GL matrix inline image can be expressed as GL(θ) = Dθ[ − H(θ)] − 1H θt, which must be evaluated at inline image, where H(θ) is as given in Appendix A for the BSPAL model presented in (1),

display math

with

display math

being N × 1 vectors, inline image a k × 1 vector of stress levels, and J = [E1m, … ,Ekm] an N × k matrix, where EiN is as given in the perturbation scheme of the stressor. Thus, once the N × N GL matrix is constructed, potentially influential cases can be revealed through the index plot of the case number (i,j) versus the corresponding element in the diagonal of the leverage matrix GL (θ).

4 Application example

We use a real fatigue life data set presented in [8] to illustrate the proposed methodology. These data are provided in Table A.I (see Appendix A) and correspond to fatigue life (T) of 6061-T6 aluminum pieces expressed by cycles ( ×  10  − 3) until the failure occurs. These pieces were cut parallel to the direction of rolling and oscillating at 18 cycles/s at a maximum stress levels of x1 = 2.1, x2 = 2.6 and x3 = 3.1 psi ( ×  10 4). The sample sizes of these data groups are n1 = 101, n2 = 102, and n3 = 101 for stress levels x1, x2 and x3, respectively. Thus, the total number of observations is inline image and the stress levels are k = 3. All pieces were tested until they failed.

In this example, we conduct exploratory, confirmatory and influence diagnostic analyses, and estimate the lower percentiles of the life distribution. The results are obtained using the R computer language.

4.1 Exploratory data analysis

In Table 1, we present a descriptive summary of the data in [8]. We observe that as the stress level increases, the median and mean fatigue life, as well as its variability, decrease considerably. Thus, we note that the scale parameter β should decrease as the stress level increases. We recall that β is the scale parameter and also the median, but the mean is also in direct relation to it. We should detect that α, the shape parameter, it is not modified as the stress level increases.

Table 1. Descriptive statistics of fatigue life of aluminum pieces 6061-T6 (in cycles ×  10  − 3) for the indicated stress level.
Stress level (psi)Min.First quartileMedianMeanThird quartileMax.St. dev.ni
3.1 × 10470120133133.7314621222.36101
2.6 × 104233352400397.8843956062.32102
2.1 × 104370111514161400.8416422440391.01101

Figure 3 shows histograms and boxplots of the fatigue life data. Note that there are some atypical observations, a certain asymmetry in the distribution of the data, and that the scale of the distribution decreases as the stress level increases. Notice that the shape of the distribution is not substantially altered in the different stress levels.

4.2 Confirmatory analysis

The confirmatory analysis is composed by three parts: (i) identification of the life distribution, (ii) selection of the AL model, and (iii) inference in the BSPAL model.

4.2.1 Identification of the life distribution

Two life distributions that have a close connection are the BS and inverse Gaussian (IG) models [2, 55]. To analyze the fatigue life T, the BS and IG distributions are proposed. We apply the Kolmogorov–Smirnov (KS) test and Schwartz's Bayesian information criterion (BIC) to decide which of these distributions fits the data better (Table 2). The parameters of each model are estimated by the ML method. From Table 2, we detect that the BS distribution fits the data slightly better than the IG distribution. In addition, by Property (C2), we construct QQ plots with envelopes, which are shown in Figure 2. From this figure, we note a good fit of the BS distribution to the data. However, some cases are not accommodated well. This point is retaken in the diagnostic analysis.

Table 2. Kolmogorov–Smirnov test p-values and Bayesian information criterion values of the indicated models for the fatigue life data.
  1. KS, Kolmogorov–Smirnov test; BIC, Bayesian information criterion; BS, Birnbaum–Saunders distribution; IG, inverse Gaussian models.

3.1 × 104 psi2.1 × 104 psi2.6 × 104 psi
ModelKSBICModelKSBICModelKSBIC
BS0.459923.77BS0.5541511.97BS0.9191144.65
IG0.457923.80IG0.5421512.36IG0.9181144.68
Figure 2.

QQ plots with simulated envelopes on the basis of the BS distribution for fatigue life data at the stress levels (a) 3.1 × 104, (b) 2.6 × 104 and (c) 2.1 × 104 psi.

For the parameters α and β of the BS distribution, Table 3 presents their ML estimates, 95% asymptotic confidence intervals (CI), say CI inline image, and relative change (RC) given by inline image, for r = 1,2, with θ1 = α and θ2 = β, and k ≠ l = 1,2,3, where inline image denotes the ML estimate of θr for the lth level of the stressor. From Table 3, we detect that the greatest changes are produced in the scale parameter β through the different stress levels. As the stress level increases from 2.1 to 3.1, a change of the 90% in inline image is detected, whereas inline image changes around 45%. With these results, we describe the scale parameter (β) of the BS distribution by two AL models.

Table 3. Maximum likelihood estimate, CI 95% and relative change of α and β for the fatigue life data.
Stress level (psi)lαinline imageuαRC (%)lβinline imageuβRC (%)
  1. RC, relative change; IC, confidence interval

3.1 × 1040.1690.1700.17145.06127.455131.819136.18390.14
2.6 × 1040.2200.2240.22823.85367.719384.303400.88765.70
2.1 × 1040.2670.3100.35327.841256.521336.3801416.24071.25

4.2.2 Selection of the accelerated life model

As the BS distribution is an appropriate model at the different stress levels (Table 2) and the parameter that is mainly modified is β (Table 3), we postulate two AL models to relate this parameter to the stressor of durability. This relationship is incorporated in the life distribution so that the BS distribution also depends on the stressor x. The AL models proposed for this example are based on models of continuous cumulative damage under different stress levels by increasing load until the failure of the pieces occurs. The exponential and power AL models given in (1) have been already considered to describe fatigue life of specimens subjected to stress [13, 41]. To find the ML estimates for the exponential and power AL models, the parameter β must be replaced by β(x) = γ exp(ηx) and β(x) = γx − η, respectively, inside the corresponding PDF. To carry out this estimation with the fatigue life data, we need starting values, which we obtain from the linearization of the model given in (5) and are inline image for the exponential AL model and inline image for the power AL model. The parameter α is estimated in an analytical way by means of the ML estimates of γ and η, such as shown in (8). Also, the BIC criterion is calculated to compare the fit of the proposed AL models to the fatigue life data and to select the model that reduces the loss of information. The results of estimation and fit are shown in Table 4. According to the values of the model selection criterion presented in this table, the AL model for the BS distribution that best fits the analyzed fatigue life data is the BSPAL model conducting to the lower loss of information (smaller BIC), that is, the power BSAL model.

Table 4. Maximum likelihood estimate for the indicated Birnbaum–Saunders accelerated life model parameters and their values of log-likelihood and Bayesian information criterion for the fatigue life data.
BSALinline imageinline imageinline imageinline imageBIC
  1. BSAL, Birnbaum–Saunders accelerated life model; BIC, Bayesian information criterion.

Exponential0.2271696.568 − 2.317 − 408.249823.95
Power0.2261109.5665.938 − 406.706820.86

4.2.3 Inference on the BSPAL model

We first infer on the BSPAL model and then conduct diagnostics. Using the estimates of the BSPAL parameters shown in Table 4, we calculate the EFIM given in (9) obtaining

display math(20)

From (20), the variances of the ML estimators can be obtained from the diagonal elements of inline image and then their respective standard errors (SE) can be computed. In addition, with the approximation given in (10), we can construct asymptotic CIs. These results appear in Table 5. The estimates of the BSPAL model parameters can be used for estimating the PDF of the BS distribution. Figure 3 shows the histogram of the data with the estimated BS PDF. From this figure, we observe that the distribution remains practically unalterable, only its scale varies. Before using the prediction model for the percentiles, we perform a diagnostic analysis to detect potentially influential observations.

Table 5. Maximum likelihood estimate, estimated SE and CI of the indicated Birnbaum–Saunders power law accelerated life model parameter for the fatigue life data.
Parameter (θr)inline imageSECI 95%(θr)
  1. SE, standard errors; IC, confidence interval

θ1 = α0.2260.009[0.208, 0.244]
θ2 = γ1109.56685.849[941.305, 1277.827]
θ3 = η5.9380.081[5.780, 6.097]
Figure 3.

Histograms with estimated PDFs of the Birnbaum–Saunders power law accelerated life model and boxplots for fatigue life data at the stress levels (a) 3.1 × 104, (b) 2.6 × 104 and (c) 2.1 × 104 psi.

4.3 Diagnostic analysis

The primary goal of the diagnostic methods is to detect cases that could produce a potential influence on the ML estimates. The diagnostic analysis that we present is divided into three parts: (i) global influence, (ii) local influence, and (iii) generalized leverage.

4.3.1 Global influence

With Figure 4, we see that the case #204 (370 cycles × 103) has a great influence on the ML estimates of the BSPAL model parameters. In addition, there is a large difference between the ML estimates with all the data and without considering the case #204 (Table 6). A less pronounced influence is detected in the cases #1 (70 cycles  × 103) and #304 (2440 cycles  × 103).

Figure 4.

Index plot of (a) likelihood displacement and (b) generalized Cook distance for the fatigue life data.

Table 6. Detected cases as influenced by global and local influence for the fatigue life data.
CaseFatigue life (in cycles  × 10 − 3)Stress level (psi)
#1703.1 × 104
#2043702.1 × 104
#30424402.1 × 104

4.3.2 Local influence

Case-weight perturbation

From Figure 5, we note that once again the case #204 has a great influence on the ML estimates for this perturbation scheme.

Figure 5.

Index plot for case-weight perturbation of (a) local influence and (b) total local influence for the fatigue life data.

Additive perturbation of the response

From Figure 6, we observe that the case #1 has a great influence on the ML estimate for this scheme. A less pronounced influence is detected in the case #204.

Figure 6.

Index plot for additive perturbation of the response of (a) local influence and (b) total local influence for the fatigue life data.

4.3.3 Generalized leverage

The analysis of GL confirms that the case #204, detected as potentially influential in the analyses of global and local influence (Figure 7), possesses a high leverage value. This indicates that such a case considered as influential has also a high impact on its own predicted value.

Figure 7.

Index plot of generalized leverage for the fatigue life data.

4.3.4 Summary of diagnostics

From the corresponding diagnostic analysis, three cases are detected as obviously influential on the ML estimates of the BSPAL model parameters. These cases, with their respective stress levels, appear in Table 6.

Once the atypical cases are detected, the model parameters are estimated without those cases. ML estimates, estimated SE, RC and value of the log-likelihood function are calculated for the BSPAL model presented in (1). In this case, the relative changes are given by inline image, for r = 1,2,3, θ1 = α, θ2 = γ and θ3 = η, where inline image denotes the ML estimate of θr when the set I of cases is removed. These results appear in Table 7 from where it is possible to observe that when the cases detected as potentially influential are removed, a small change in the estimates occurs. The greatest changes take place mainly in the estimates of the parameters α and γ of the BSPAL model. The case #204 is the one that has a higher potential influence on the estimates of the parameters, because when removed, it produces the greatest change in the value of the log-likelihood function, in addition to a substantial improvement in the BSPAL model reflected in the increase of the value of the log-likelihood function. Next, we study how the cases detected as potentially influential affect the estimation of percentiles.

Table 7. Maximum likelihood estimate, estimated standard errors and relative change of the indicated Birnbaum–Saunders power law accelerated life model parameter and value of the log-likelihood function once the indicated influential cases are removed from the fatigue life data.
Dropped caseinline imageα SERC (%)inline imageγ SERC (%)inline imageη SERC (%)inline imageRC (%)
  1. SE, standard errors; RC, relative change.

None0.2260.0091109.685.855.9380.081 − 406.71
#10.2220.0091.451091.483.231.645.9180.0800.33 − 405.300.35
#2040.2110.0086.561159.183.874.465.9790.0760.68 − 387.664.68
#3040.2230.0091.191087.883.181.965.9200.0800.32 − 402.561.02

4.4 Calculation of percentiles

Often, engineers and manufacturers are interested in the lower percentiles of the life distribution, for instance, to produce a more reliable system and/or to specify warranties. In this application example, the 10th percentile of the life distribution, that is, the t value such that P(T < t) = 0.10, is considered for each stress level using the expression given in (12) and following an approach similar to that proposed by [13]. Table 8 presents the ML estimate, CI and RC for the 10th percentile, once the potentially influential cases have been removed. From Table 8, it is estimated that 90% of aluminum pieces 6061-T6 under a maximum stress level of 2.1 × 104 psi do not fail before 1015.1 cycles  × 103. However, for levels 2.6 and 3.1, this does not happen before the 285.6 and 100.5 cycles × 103, respectively. The case #204 is the one that most influences the ML estimate of the 10th percentile t(0.1; x) (Table 8). The atypical cases producing the greatest effect belong to the same stress level ( 2.1 × 104 psi). The presence of influential cases can overestimate the calculation of percentiles. This aspect should be considered in the design of structures and for predicting warranty claims.

Table 8. Maximum likelihood estimate and CI of the 10th percentile (in cycles × 103) of a Birnbaum–Saunders power law accelerated life model, without indicated influential cases for the fatigue life data.
Dropped caseStress level,x (psi)inline imageCI 95%(t(0.1; x))RC (%)
  1. RC, relative change; IC, confidence interval

None3.1 × 104100.5[ 0.982, 1.028]
#13.1 × 104101.5[ 0.994, 1.036]1.00
None2.6 × 104285.6[ 2.802, 2.910]
None2.1 × 1041015.1[10.010, 10.292]
#2042.1 × 1041048.5[10.347, 10.623]3.41
#3042.1 × 1041012.7[ 9.988, 10.266]0.24

Conclusions

In this paper, we have proposed a methodology that can be useful for detecting potentially influential cases in a Birnbaum-Saunders accelerated life model model. We have presented an application example of this methodology using real fatigue life data. The example has shown the importance that such a methodology could have in a warranty problem. Because in industrial statistics there is great interest in predicting lifetimes at low stress levels, engineers should have available tools that allow them to detect the precision of the estimates of these percentiles. A bad estimation could produce significant monetary losses to organizations owing to an excessive amount of warranty claims. In consequence, the methodology presented in this study should be considered in the design of structures and the prediction of warranty claims.

Appendix A

Data

In Table A.I, the data are identified by rows from the case #1 (70 cycles × 10 − 3) to the case #101 (212 cycles × 10 − 3) for a stress level = 3.1 × 104 psi, from the case #102 (233 cycles  × 10 − 3) to the case #203 (560 cycles × 10 − 3) for a stress level = 2.6 × 104 psi, and from the case #204 (370 cycles × 10 − 3) to the case #304 (2440 cycles × 10 − 3) for a stress level = 2.1 × 104 psi.

Table A.I. Fatigue life (in cycles × 10 − 3) of aluminum pieces submitted to the maximum indicated stress level provided by [8].
3.1 × 104 psi2.6 × 104 psi2.1 × 104 psi
7090969799233258268276290370706716746785
100103104104105310312315318321797844855858886
107108108108109321329335336338886930960988999
10911211211311433834234234234410001010101610181020
11411411611912034935035035135110551085110211021108
12012012112112335235235635835811151120113411401199
12412412412412436036236336636712001200120312221235
12812812912913037037037237237412381252125812621269
13013013113113137537637937938012701290129313001310
13113113213213238238938939539613131315133013551390
13313413413413440040040040340414161419142014201450
13413613613713840640840841041214521475147814811485
13813813913914141441641641642015021505151315221522
14114214214214242242342642843215301540156015671578
14214214414414543243343343743815941602160416081630
14614814814915143943944344544516421674173017501750
15115215515615745245645646046417631768178117821792
15715715715815946646847047047318201868188118901893
16216316316416647447647648648818951910192319241945
16616817017419648949049150351720232100213022152268
212    540560   2440    

Score vector U(θ)

display math(A.1)

where

display math

Hessian matrix H(θ)

display math(A.2)

where

display math

where Hγα = Hαγ, Hηα = Hαη, and Hηγ = Hγη.

Perturbation matrices

Case-weight perturbation
display math(A.3)

where

display math(A.4)
Perturbation of the response
display math(A.5)

where

display math
Perturbation of the stressor
display math(A.6)

where

display math

Acknowledgements

The authors wish to thank the editor-in-chief, Dr. Fabrizo Ruggeri, the associate editor and two anonymous referees for their constructive comments on an earlier version of this manuscript that resulted in this improved version. This study was partially supported by FONDECYT 1090265 (A. Sanhueza), 1110318 (M. Galea) and 1120879 (V. Leiva) grants from the Chilean government.

Ancillary