Reliability analysis is commonly thought of as an approach to model failures of existing products. The usual reliability analysis involves characterization of failures of the products using distributions such as exponential, Weibull and lognormal. Based on the fitted distribution, failures are mitigated, or warranty returns are predicted, or maintenance actions are planned. However, reliability analysis can also be used as a powerful tool to design robust products that operate with minimal failures, by adopting the methodology of Design for Reliability (DFR). In DFR, reliability analysis is carried out in conjunction with physics of failure and experiment design techniques. Under this approach, Design of Experiments (DOE) uses life data to "build" reliability into the products, not just to quantify the existing reliability. Such an approach, if properly implemented, can result in significant cost savings, especially in terms of fewer warranty returns or repair and maintenance actions. Although DOE techniques can be used to improve product reliability and also make this reliability robust to noise factors, the discussion in this chapter is focused on reliability improvement.
This section is divided into the following subsections:
Reliability DOE (R-DOE) analysis is fairly similar to the analysis of
other designed experiments except that the response is the life of the
product in the respective units (e.g.
for an automobile component the units of life may be miles, for a mechanical
component this may be cycles, and for a pharmaceutical product this may
be months or years). However, two important differences exist that make
R-DOE analysis unique. The first is that life data of most products are
typically well modeled by either the lognormal, Weibull or exponential
distribution, but usually do not follow the normal distribution. Traditional
DOE techniques follow the assumption that response values at any treatment
level follow the normal distribution and therefore, the error terms, 
, can be assumed to be normally and
independently distributed. This assumption may not be valid for the response
data used in most of the R-DOE analyses. Further, the life data obtained
may either be complete or censored and in this case standard regression
techniques applicable to the response data in traditional DOEs can no
longer be used.
Stresses affecting the life of the product may also be investigated using R-DOE analysis. In this case, the primary purpose of any R-DOE analysis is to identify which of the investigated stresses affect the life of the product (by investigating if change in the level of any stress leads to a significant change in the life of the product). Once the important stresses affecting the life of the product have been identified, detailed analyses can be carried out using ReliaSoft's ALTA software. ALTA includes a number of life-stress relationships (LSRs) to model the relation between life and the stress affecting the life of the product. [Note]
Assume that the life, 
, for a certain product has been found
to be lognormally distributed. The probability density function for the
lognormal distribution is:
(1)
where 
represents the mean of the natural
logarithm of the times-to-failure and 
represents the standard deviation
of the natural logarithms of the times-to-failure [19].
If the analyst wants to investigate a single two level factor that may
affect the life, 
, then the following model may be used:
(2)where:
represents the times-to-failure at
the 
th treatment level of the factor.
represents the mean value of 
for the 
th treatment.
is the random error term.
the subscript 
represent the treatment level of the
factor with 
for a two level factor.
The model of Eqn. (2) is analogous to the ANOVA
model, 
, used in Chapter
6 for traditional DOE analyses. Note, however, that the random error
term, 
, is not normally distributed here
because the response, 
, is lognormally distributed. It is
known that the logarithmic value of a lognormally distributed random variable
follows the normal distribution. Therefore, if the logarithmic transformation
of 
, 
, is used in Eqn. (2),
the model will be identical to the ANOVA model, 
, used in Chapter
6. Thus, using the logarithmic failure times, the model can be written
as:
(3)where:
represents the logarithmic times-to-failure
at the 
th treatment.
represents the mean of the natural
logarithm of the times-to-failure at the 
th treatment.
represents the standard deviation
of the natural logarithms of the times-to-failure.
The random error term, 
, is normally distributed because the
response, 
, is normally distributed. Since the
model of Eqn. (3) is identical to the ANOVA model
used in traditional DOE analysis, regression techniques can be applied
here and the R-DOE analysis can be carried out similar to the traditional
DOE analyses. Recall from Chapter
7 that if the factor(s) affecting the response has only two levels,
then the notation of the regression model can be applied to the ANOVA
model. Therefore, the model of Eqn. (3) can be
written using a single indicator variable, 
, to represent the two level factor
as: [Note]
(4)
where 
is the intercept term
and 
is the effect coefficient for the
investigated factor. Setting Eqns. (3) and (4) equal to each other returns:
(5)
The natural logarithm of the times-to-failure at any factor level, 
, is referred to as the life characteristic
because it represents a characteristic point of the underlying life distribution.
The life characteristic used in the R-DOE analysis will change based on
the underlying distribution assumed for the life data. If the analyst
wants to investigate the effect of two factors (each at two levels) on
the life of the product, then the life characteristic equation can be
easily expanded as follows:
where 
is the effect coefficient for the
second factor and 
is the indicator variable representing
the second factor. If the interaction effect is also to be investigated,
then the following equation can be used:
In general the model to investigate a given number of factors can be
expressed as:
(6)
Based on the model equations mentioned thus far, the analyst can easily conduct an R-DOE analysis for the lognormally distributed life data using standard regression techniques. However this is no longer true once the data also includes censored observations. In the case of censored data, the analysis has to be carried out using maximum likelihood estimation (MLE) techniques.
The maximum likelihood estimation method can be used to estimate parameters
in R-DOE analyses when censored data are present. The likelihood function
is calculated for each observed time to failure, 
, and the parameters of the model are
obtained by maximizing the log-likelihood function. The likelihood function
for complete data following the lognormal distribution is given as:
where:
is the total number of observed times-to-failure.
is the life characteristic and has
been substituted based on Eqn. (6).
is the time of the 
th failure.
For right censored data the likelihood function is:[19]
where:
is the total number of observed suspensions.
is the time of 
th suspension.
For interval data the likelihood function is:[19]
where:
is the total number of interval data.
is the beginning time of the 
th interval.
is the end time of the 
th interval.
The complete likelihood function when all types of data (complete, right
censored and interval) are present is:
(7)
Then the log-likelihood function is:
(8)
The MLE estimates are obtained by solving for parameters 
so that:
Once the estimates are obtained, the significance of any parameter,
, can be assessed using the likelihood
ratio test.
Hypothesis testing in R-DOE analyses is carried out using the likelihood
ratio test. To test the significance of a factor, the corresponding
effect coefficient(s), 
, is tested. The following statements
are used:
The statistic used for the test is the likelihood ratio, 
. The likelihood ratio for the parameter
is calculated as follows:
(9)where:
is the vector of all parameter estimates
obtained using MLE (i.e. 
...![$]^{/prime }$](graphics/chapter11__73.png){kind=small}
).
is the vector of all parameter estimates
excluding the estimate of 
.
is the value of the likelihood function
when all parameters are included in the model.
is the value of the likelihood function
when all parameters except 
are included in the model.
If the null hypothesis, 
, is true then the ratio, 
, follows the Chi-Squared distribution
with one degree of freedom. Therefore, 
is rejected at a significance level,
, if 
is greater than the critical value
.
The likelihood ratio test can also be used to test the significance
of a number of parameters, 
, at the same time. In this case, 
represents the likelihood value when
all 
parameters to be tested are not included
in the model. In other words, 
would represent the likelihood value
for the reduced model that does not contain the 
parameters under test. Here, the ratio
will follow the Chi-Squared distribution
with 
degrees of freedom if all 
parameters are insignificant (with
representing the number of parameters
in the full model). Thus, if 
, the null hypothesis, 
, is rejected and it can be concluded
that at least one of the 
parameters is significant.
To illustrate the use of MLE in R-DOE analysis, consider the case where
the life of a product is thought to be affected by two factors, 
and 
. The failure of the product has been
found to follow the lognormal distribution. The analyst decides to run
an R-DOE analysis using a single replicate of the 2
design. Previous studies indicate
that the interaction between 
and 
does not affect the life of the product.
The design for this experiment can be set up in DOE++ as shown in Figure 11.1.
The resulting experiment design and the corresponding times-to-failure
data obtained are shown in Figure 11.2. Note that,
although the life data shown in Figure 11.2 is
complete data and regression techniques are applicable, calculations are
shown using MLE. DOE ++ uses MLE for all R-DOE analysis calculations.
|
Figure 11.1: Design properties for the experiment in Example 11.1. |
|
Figure 11.2: The 2 |
(10)
where 
is the mean of the natural logarithm
of the times-to-failure at the 
th treatment combination (
), 
is the effect coefficient for factor
and 
is the effect coefficient for factor
. The analysis for this case is carried
out in DOE++ by dropping the interaction 
using the Select Effects icon in the
Control Panel.
The following hypotheses need to be tested in this example:
This test investigates the main effect of
factor 
. The statistic for this test is:
where 
represents the value of the likelihood
function when all coefficients are included in the model and 
represents the value of the likelihood
function when all coefficients except 
are included in the model.
This test investigates the main effect of
factor 
. The statistic for this test is:
where 
represents the value of the likelihood
function when all coefficients are included in the model and 
represents the value of the likelihood
function when all coefficients except 
are included in the model.
To calculate the test statistics, the maximum likelihood estimates of the parameters must be known. The estimates are obtained next.
Since the life data for the present experiment are complete and follow
the lognormal distribution, the likelihood function can be written as:
Substituting 
from Eqn. (10),
the likelihood function is:
Then the log-likelihood function is:
(11)
To obtain the MLE estimates of the parameters, 
and 
, the log-likelihood function must
be differentiated with respect to these parameters:
Equating the 
terms to zero returns the required
estimates. The coefficients 
, 
and 
are obtained first as these are required
to estimate 
. Setting 
:
Substituting the values of 
, 
and 
from Figure 11.2 and
simplifying:
Thus:
(12)
Setting 
:
Thus:
(13)
Setting 
:
Thus:
(14)
Knowing 
and 
, 
can now be obtained. Setting 
:
Thus:
(15)
Once the estimates have been calculated, the likelihood ratio test can be carried out for the two factors.
The likelihood ratio test for factor 
is conducted by using the likelihood
value corresponding to the full model and the likelihood value when 
is not included in the model. The
likelihood value corresponding to the full model (in this case 
) is:
The corresponding logarithmic value is 
.
The likelihood value for the reduced model that does not contain factor
(in this case 
) is:
The corresponding logarithmic value is 
.
Therefore, the likelihood ratio to test the significance of factor 
is:
(16)
The 
value corresponding to 
is:
Assuming that the desired significance level for the present experiment
is 0.1, since 
, 
cannot be rejected and it can be concluded
that factor 
does not affect the life of the product.
The likelihood ratio to test factor 
can be calculated in a similar way
as shown next:
(17)
The 
value corresponding to 
is:
Since 
, 
is rejected and it is concluded that
factor 
affects the life of the product. The
previous calculation results are displayed as the Likelihood Ratio Test
Table in the results obtained from DOE++ as shown in Figure 11.3.
|
Figure 11.3: Likelihood ratio test results from DOE++ for the experiment in Example 11.1. |
experiment design and the corresponding life
data for Example 11.1.