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Weibull accelerated life testing analysis with several variables using multiple linear regression
DOI:
https://doi.org/10.15446/dyna.v82n191.43533Palabras clave:
ALT analysis, Weibull analysis, multiple linear regression, experiment design. (es)Descargas
DOI: https://doi.org/10.15446/dyna.v82n191.43533
Weibull accelerated life testing analysis with several variables using multiple linear regression
Análisis de pruebas de vida acelerada Weibull con varias variables utilizando regresión lineal múltiple
Manuel R. Piña-Monarrez a, Carlos A. Ávila-Chávez b & Carlos D. Márquez-Luévano c
a Industrial and Manufacturing department of the IIT Institute,
Universidad Autónoma de Ciudad Juárez, Chihuahua , México. manuel.pina@uacj.mx
b Industrial and manufacturing deparment of the IIT Institute,
Universidad Autónoma de Ciudad Juárez, Chihuahua , México. carlos.avila@uacj.mx
c Reliability Engineering Department at
Stoneridge Electronics North America. carlos.marquez@stoneridge.com
Received: May 16th, 2014. Received in revised form: February 24th, 2015. Accepted: March 04th, 2015
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Abstract
In Weibull
accelerated life test analysis (ALT) with two or more variables , we estimated, in joint form, the parameters
of the life stress model
and
one shape parameter
. These were then used to extrapolate the
conclusions to the operational level. However, these conclusions are biased
because in the experiment design (DOE) used, each combination of the variables
presents its own Weibull family
. Thus the estimated
is not
representative. On the other hand, since
is
determined by the variance of the logarithm of the lifetime data
, the response variance
and the
correlation coefficient
, which increases when variables are added to
the analysis,
is
always overestimated. In this paper, the problem is statistically addressed and
based on the Weibull families
a
vector
is
estimated and used to determine the parameters of
. Finally, based on the variance
of each
level, the variance of the operational level
is
estimated and used to determine the operational shape parameter
. The efficiency of the proposed method is
shown by numerical applications and by comparing its results with those of the maximum
likelihood method (ML).
Keywords: ALT analysis; Weibull analysis; multiple linear regression; experiment design.
Resumen
En el análisis de pruebas de vida acelerada
Weibull con dos o más variables aceleradas , estimamos en forma conjunta los parámetros
del modelo de relación vida esfuerzo
y un
parámetro de forma
. Después estos parámetros son utilizados para
extrapolar las conclusiones al nivel operacional. Como sea, estas conclusiones
están sesgadas debido a que dentro del diseño de experimentos (DOE) utilizado,
cada combinación de las variables presenta su propia familia Weibull
. De esa forma la
estimada
no es representativa. Por otro lado, dado que
está
determinada por la varianza del logaritmo de los tiempos de vida
, por la varianza de la respuesta
y por el
coeficiente de correlación
, el cual crece cuando se agregan variables al
análisis,
es
siempre sobre estimada. En éste artículo, el problema es estadísticamente
identificado y basado sobre las familias Weibull
un
vector
es
estimado y utilizado para determinar los parámetros de
. Finalmente, basado en la varianza
de cada
nivel, la varianza del nivel operacional
es
estimada y utilizada para determinar el
parámetro de forma
del
nivel operacional. La eficiencia del método propuesto es mostrada a través de
aplicaciones numéricas y por la comparación de sus resultados con los del
método de máxima verosimilitud (ML).
Palabras clave: ALT análisis; análisis Weibull; regresión lineal múltiple; diseño de experimentos.
1. Introduction
In Accelerated Life Testing analysis (ALT) with constant
over time and interval-valued variables ,
the standard approach of the analysis consists in using higher levels of the
stress variables and a life-stress model
,
the lifetime data are obtained as quickly as possible [6]. In this approach,
the function
,
which relates the lifetime data to the stress variables, is parametrized as
Where is a vector of unknown parameters, and
is a vector of specified functions
with
.
Among the most common models of
we have the generalized Eyring model, the
temperature-humidity model (T-H), the temperature-non-thermal model, the
proportional hazard model and the generalized log-linear model [9] and [13]. On
the other hand, in ALT Weibull analysis no matter which model we use, all of
them are used to estimate the scale parameter
under different levels of the significant
variables. For example if the (T-H) model is used, then in the Weibull
probability density function (pdf) ([19] and [16] Chapter 1), given by
by replacing with the
model the Weibull/(T-H) pdf is given by
Unfortunately, since in (3) only one shape parameter is estimated and used to represent all the
level combinations of the variables, and because the maximum likelihood (ML) or
multiple linear regression (MLR) methods perform the estimation as linear
combination of
with the coefficients of the vector
is always overestimated. As a consequence, the
related reliability
is overestimated too. In order to show this
problem, in section 2 the generalities of ALT analysis are given. Section 3 presents
the problem statement. In section 4 the problem is statistically addressed. Section
five details the proposed method and, finally, in section 6 the conclusions are
presented.
2. Generalities of Weibull ALT analysis
In ALT analysis, the objective
is to obtain life data as quickly as possible. Data are obtained by observing a
set of n units functioning under various levels of the explanatory variables .
These levels are chosen to be higher than the normal one. With these life data,
we draw conclusions and then, the conclusions are extrapolated to the normal
level. Models used to perform the extrapolation are known as life-stress models
.
Among the most common
models we have the parametric accelerated
failure time models (AFT) (e.g. Arrhenius model) [8], and the proportional
hazard model (PH) (e.g. Weibull proportional hazard model) [5] and [1]. On the
other hand, in the analysis, the lifetime data T is a nonnegative and absolute continuous random variable. Thus,
the survival function is
.
And based on this, the corresponding probability density function is
and the hazard rate function is
.
Additionally, it is important to note that in ALT, the effect that the
covariates
have over T,
is modeled by
,
which as in (3) is included in
.
On the other hand, for constant over time and interval valued variables, the
cumulative risk function is
with
parametrized as
where
and Z are as they were defined in (1), and
represents the base risk when all the
covariates are zero
.
For example, based on this formulation and on the physical principle of
Sedyakin [1], pp. 20), the survival function for two different levels of the
variables
is related by
.
Implying that
which in terms of
mean that
,
and since the survival function
does not depend on
,
then the random variable
does not depend on X either. In particular
observe that, since the expected value of
is
and its variance is
,
then its variation coefficient
does not depend on
either. Thus, for any two stress levels (or variables combination), the
distribution is the same, implying that only the scale changes (see [1] sec.
2.3). Observe the fact that the scale only changing in the Weibull analysis
implies that
On the other hand, R(t) under any two levels is related by
where
and that it is called the acceleration factor.
Finally, it is important to note that by setting
or
,
and because
and
do not depend on
,
then as in (5), the variance of
does not depend on
either.
Despite of this, because in the estimation of the Weibull
life-stress relationship as in (3), we estimate only one value as a linear combination of the
variables
,
is always overestimated as in the following
section.
3. Problem statement
Statement 1: Because in ALT failure
time data is obtained by increasing the level of the variables ,
the variance of the logarithm of the lifetimes
defined in (5) is diminished (the time to
failure is shorted) and as a consequence,
is overestimated.
Statement 2: In multivariate ALT
analysis, when significant variables are added into
,
the relation between the logarithm of the scale parameter
and
tends to be one, thus the corresponding sum
square error is diminished increasing
,
and as a consequence,
is overestimated.
Considering these statements, firstly note that is intrinsically related to the strength
characteristics of the product; thus, if the levels of the significant
variables
are selected in such that, for their higher
effect combination, they do not generate a foolish failure mode (the effect of
the level combinations is lower than the effect that reaches the destructive
limits),
must be
constant, and its value must represent the variance of the strength
characteristic
which in the estimation processes with
constant over time and interval-valued variables is not used (or measured). And
second, we can observe that in the estimation process, the shape parameter
is determined by the variance of the logarithm
of the lifetime data
,
the response variance
and the correlation coefficient
,
and that neither of them represent
.
Thus, due to the dependence of
on
,
and
,
adding significant variables (increases R2) and/or overstressing
their levels (diminishes
)
always overestimates
as in the following section.
4. Statistical analysis
Let us first show that is completely determined by
,
and
.
To see this, let us use the Weibull reliability function given by
Which in linear form is given by
Where ,
,
,
, and
is the cumulative failure function of t given by
.
based on the median rank approach is estimated
as [14].
For another possible approximation of F(t), see [3], [4]
and [21]. On the other hand, observe from (7a) that is a critical parameter (see [10] and [16]
sec. 2.3) and thus, the analysis depends on the accuracy by which it is
estimated. Also, observe that (7b) is in function of the sample size n and that
for
the F(t) percentile is greater than 90% [2].
Regardless of this, note that
represents the 0.367879 reliability percentile
which corresponds to
implying from (7a) that for center response Y
Thus, under multiple linear regression the coefficients of (7a) are estimated as
From (9b) observe that its denominator is the variance of the logarithm of the lifetime data defined in (5), which in terms of the covariates is given by.
On the other hand, the goodness of fit of the polynomial given in (7a) is performed by the anova analysis where its sources of variation are
The goodness of fit index is given by
Finally from (9c), (10), (11) and (13), is given by
On
the other hand, to see that increasing (or decreasing)
affects
as in Statement1, note from (8) that
increasing (or decreasing)
is equivalent to increasing (or decreasing)
in (9b). That is to say, shortening the time in
which the lifetime occurs, decreases their variance
and thus, according to (14), is
overestimated.
In the case of Statement 2, from (11) to (14), it is clear
that although the levels of the variables are not stressed, by adding
significant variables, since and
,
are fixed as in (5), then in (14)
is increased, and as a consequence,
is always overestimated.
5. Proposed Method
To see numerically that is overestimated as in section 4, first note
that each combination of the variables, as in Fig. 1, presents its own Weibull
family, and that data are gathered by using a replicated experiment design DOE
as presented in Fig. 2 (see [15] and [20] Chapter 13).
Second, to illustrate this, let us use the DOE data from Table 1, which corresponds to twelve electronic devices. Data were published by [17] p.11.
From these data, the Weibull/(T-H) parameters defined in
(3), by using ML are ,
,
and
(the ALTA Pro software was used). In addition,
observe that although in Table 1 there are three level combinations among the
variables, which as a consequence lead to three Weibull families in this DOE,
regardless of this, in the standard approach [eq. (3)], only one shape
parameter
was estimated.
Thus, it is not representative of the whole set of data.
To see this, in Table 2, the scale and shape parameters ,
estimated by ML, and their associated reliability
for t=150 are given.
In order to compare the standard results of Table 2, with those found in the DOE, Table 3 presents the Weibull family and R(t) for each DOE combination, using (8) and (9b) with centered response (Y).
By comparing these
results, we observe that the estimated in Table
2, in contrast to the estimated
from Table
3, does not represent the expected 0.367879 percentile as defined in (8). And
that
does not
represent the shape parameter of the levels found in the DOE. Thus the proposed
method to avoid this issue, using MLR, is as in the following section.
5.1. Regression approach for statement 1.
In ALT with one interval valued and constant over time variables, as is the case of Weibull/Arrhenius, Weibull/Inverse power law and Weibull/Eyring, it is possible to estimate their parameters by applying (8), (9a) and (9b) by following the next steps.
Step 1. For each replicated level of the stress variable
(We must have almost 4 replicates, although 10 are recommended), determine the
corresponding and
parameters
by using (7a), (8), (9a) y (9b). (In this one variable approach,
is
generally constant). If
is not
constant, proceed as in section 5.2.
Step 2. Take the effect of the
corresponding linear transformation (see next section) of the time/stress model defined in (1) as
(e.g. in Arrhenius
)
and the corresponding logarithm of the scale parameter
of the i-th
level of the variable estimated in step 1 as
.
Step 3. Using (9a) and (9b),
estimate by regression between the variables and
defined in step 2, the parameters of the life/stress
model
.
Note: In the Eyring case, do not forget to subtract the
logarithm of the reciprocal of the temperature from the logarithm of
before you perform the regression.
Step 4. Using the regression parameters of estimated
in step 3, estimate the logarithm of
for the
operational (or desired) level (see next section). Finally, form the Weibull
family of the operational (or desired) level W(
) with the shape parameters estimated in step1 and
the scale parameter estimated in this step. And with these Weibull parameters,
determine the desired reliability indexes.
5.1.1 Let
us exemplify the above methodology, through the Weibull/Arrhenius and
Weibull/Eyring relationship, which are parametrized as in (1). In
the case of Arrhenius, the infinitesimal characteristic (see [18]) is given by , thus the primitive (integral)
of
, is given by
. Since
shows the
form in which the variable affects the time, in the Arrhenius model the effect
is
(see step 2
of section 5.1). Thus from (1) and (4), the Arrhenius model is given by: (for
details see [1], Chapter 5).
In (15a), and
are the parameters to be estimated, and T is
the absolute temperature (Kelvin). The linear form of (15a) is given by
Using (15a) the Weibull/Arrhenius pdf is given by
As a numerical application, consider the data in Table 4.
Data were published by [17]. The Weibull parameters of step 1 are given in Table
5a. The effect for step 2, and
are given in Table 5b.
The Weibull/Arrhenius parameters of step 3 using Minitab® and data of Table 5b, are and
with
.
Finally, by using these parameters, the Weibull family mentioned in step 4, for
a level of 323K is
.
5.1.2 In the
case of the Weibull/Eyring relationship the infinitesimal characteristic is
given by ,
with primitive
of
,
given by
,
thus
.
This formulation with
is used in the Eyring model when the
temperature is used. The Eyring model is given by:
In (17), and
are parameters to be estimated and T is the
absolute temperature. The linear relationship of (17) is
And the linear relationship used to estimate the parameters is given by
The Weibull/Eyring pdf using (17) is
Using data of Table 4, the Weibull/Eyring parameters of
step 3 using Minitab® with data of Table 6, are and
with
.
By using these parameters the Weibull family mentioned in step 4, for a level
of 323K is
.
Finally, for the one variable case, when the shape parameter is not constant for all the stress levels proceed as in the multivariate case of the following section.
5.2. Regression approach for statement 2.
For the multivariate ALT analysis, as in Fig. 1, each
covariate combination presents its own Weibull family. Thus, because in the
standard ALT analysis, the estimated value does not represent the whole set of
data, in MLR, we propose to estimate the Weibull/life/stress parameters through
the following steps.
Step 1. For each replicated
combination level of the stress variables (We must have almost 4 replicates; 10
is recommended; see comment below eq. (7b)), determine the corresponding
Weibull family .
This could be performed by ML, but MLR is recommended. (ML is a biased
estimator and n is small).
Step 2. Take the effect of the
corresponding linear transformation of the variables as the independent
variables and the corresponding logarithm of the scale
parameter
of the i-th
Weibull family of step1 as the dependent variable
.
Step 3. Estimate the parameters of the life/stress model by regression between the set of variables
and
defined in step 2. If there are not enough
degrees of freedom to perform the analysis, proceed as follows.
a) Estimate a vector by reordering (7a) as
Estimate the parameters of by performing a regression between
and
. In (20),
is as in (7b), and
and
are the shape parameter and the logarithm of
the lifetime data of the i-th Weibull
families of step 1.
b) Based on (5) and on the fact that where
is the sample variance of the lifetime data,
form the logarithm vector
where
is the variance of the i-th level defined in (9c) and n is the number of replicates of the i-th level of step1.
c) Take the inverse of the effect of the covariates of
step 2, as the independent variables and
as the response variable and perform a
regression between
and
.
Observe that and
are vectors for the complete DOE data (or
families).
Step 4. Using
the regression parameters of estimated in step 3-a), estimate the scale
parameter
for the operational level by applying (4). By
using the regression parameters of step 3-b), estimate the value of
of the operational level, and by applying
(14), with
of step 1 and a desired
index, estimate the corresponding
value.
are the parameters of the Weibull family of
the desired stress level and they could be used to determine any desired
reliability index. Observe that the estimation of
using (14) is robust (almost insensible) to
the selected
index.
As a numerical application consider the data in Table 7. Data were published in [17].
On the other hand, data of step 3 using (20) are given in Table
8. By using Minitab, the parameters of W(T-H) model by regression between and
are
,
,
and
with
.
The parameters of the regression between
and
are
,
and
with
.
To show the method, suppose that the operational level is
with
,
then, by using the above parameters as in step 4,
,
and by taking
,
and
,
.
Thus, the operational Weibull family is
.
On the other hand, the ML parameters using the ALTA
routine are ,
,
with
and
.
With operational Weibull Family given by
.
A comparison of the Weibull parameters and reliability index of the ML and the
proposed Method is given in Table 9.
In Table 9, we can see that the shape parameter is not representative of the observed Weibull families as it is in the proposed method. The same occurs with the estimated reliability. In particular, it is important to note that the proposed method is based on the observed variance and thus it is directly related to the operational factors of the process.
6. Conclusions
In Weibull multivariate ALT analysis, each combination of
the significant variables presents its own behavior, thus the standard approach
of estimating only one shape parameter to represent all the Weibull families is
suboptimal. Since depends on
,
which increases when variables are added to the analysis, in the multivariate
case
is always overestimated. Clearly, since the
change in the scale parameter
is reflected in
,
thus the proposed method could easily be generalized to the right censured case
by reflecting the censured data on
and by substituting
for
in (9b) where
is the number of failure. Although the
proposed method depends greatly on the accuracy in which
is estimated, because
stabilize the variance as defined in step 3-b,
the proposed method could be considered robust for this issue. It is important
to mention that
in (14) is not highly sensitive to the
selected
index. Knowing (14), it seems to be possible
to generalize the proposed method to the ML approach by formulating a
log-likelihood function based on the b values of the Weibull families, but
more research must be undertaken. Since the shape parameter
is inversely related to
,
and because
is the standard deviation of the lognormal
distribution, which presents a flexible behavior and similar analysis to the Weibull distribution [11], it seems
to be possible to extend the present method to the lognormal analysis. On the
other hand, although the proposed method is practical and its application could
easily be performed by using a standard software routine, as Minitab does, a
more detailed method could be proposed by using a copula to modeling in joint
form the Weibull families behavior, but because the Weibull distribution is
determined by an non-homogeneous Poison processes [7] and its convolutions do
not have a closed form [12], more research must be undertaken.
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M.R. Piña-Monarrez, is a Researcher-Professor at the Autonomous University of Ciudad Juarez, Mexico. He completed his PhD degree in Science in Industrial Engineering in 2006 at the Technological Institute of Ciudad Juarez, Mexico. He had conducted research on system design methods including robust design, design of experiments, linear regression, reliability and multivariate process control. He is member of the National Research System (SNI-1), of the National Council of Science and Technology (CONACYT) in Mexico.
C.A. Ávila-Chavez, is a PhD student on the Science in Engineering Doctoral Program (DOCI), at the Autonomous University of Ciudad Juarez, Mexico. He completed his MSc. degree in Science in Industrial Engineering in 2011 at the Technological Institute of Ciudad Juarez, Mexico. His research is based on Accelerated lifetime and Weibull analysis.
C.D. Márquez-Luevano, is a reliability engineering at the Stoneridge Electronics North America El Paso Texas, USA. He completed his MSc. degree in Industrial Engineering in 2013 at the Autonomous University of Ciudad Juarez, Mexico. His research studies on reliability focus on random vibration and Weibull analysis.
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1. Manuel R. Piña-Monarrez, Jesús F. Ortiz-Yañez. (2015). Weibull and lognormal Taguchi analysis using multiple linear regression. Reliability Engineering & System Safety, 144, p.244. https://doi.org/10.1016/j.ress.2015.08.004.
2. Manuel R. Piña‐Monarrez. (2017). Conditional Weibull Control Charts Using Multiple Linear Regression. Quality and Reliability Engineering International, 33(4), p.785. https://doi.org/10.1002/qre.2056.
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