The Weibull Distribution is one of the commonly used distributions (in addition to lognormal and exponential) to conduct Reliability DOE analysis to investigate stresses that affect the life of a product.
This section includes the following subsection:
The probability density function for the two parameter Weibull distribution
is:
where 
is the scale parameter of the Weibull
distribution and 
is the shape parameter.[19] To
distinguish the Weibull shape parameter from the effect coefficients,
the shape parameter is represented as 
instead of 
in the remaining chapter.
For data following the two parameter Weibull distribution, the life
characteristic used in R-DOE analysis is the scale parameter, 
.[18] Since 
represents life data that cannot take
negative values, a logarithmic transformation is applied to it. The resulting
model used in the R-DOE analysis for a two factor experiment with each
factor at two levels can be written as follows:
(22)where:
is the value of the scale parameter
at the 
th treatment combination of the two
factors.
is the indicator variable representing
the level of the first factor.
is the indicator variable representing
the level of the second factor.
is the intercept term.
and 
are the effect coefficients for the
two factors.
is the effect coefficient for the
interaction of the two factors.
The model can be easily expanded to include other factors and their
interactions. Note that when any data follows the Weibull distribution,
the logarithmic transformation of the data follows the extreme-value
distribution, whose probability density function is given as follows:
(23)
where the 
s follows the Weibull distribution,
is the location parameter of the extreme-value
distribution and 
is the scale parameter of the extreme-value
distribution. [Note]
Eqns. (22) and (23) show
that for R-DOE analysis of life data that follows the Weibull distribution,
the random error terms, 
, will follow the extreme-value distribution
(and not the normal distribution). Hence, regression techniques are not
applicable even if the data is complete. Therefore, maximum likelihood
estimation has to be used.
The likelihood function for complete data in R-DOE analysis of Weibull
distributed life data is:
where:
is the total number of observed times-to-failure
is the life characteristic at the
th treatment
is the time of the 
th failure
For right censored data, the likelihood function is:
where:
is the total number of observed suspensions
is the time of 
th suspension
For interval data, the likelihood function is:
where:
is the total number of interval data
is the beginning time of the 
th interval
is the end time of the 
th interval
In each of the likelihood functions, 
is substituted based on Eqn. (22) as:
The complete likelihood function when all types of data (complete, right
and left censored) are present is:
Then the log-likelihood function is:
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. Other results can also be obtained as discussed in Chapter
11, Maximum
Likelihood Estimation for the Lognormal Distribution and Chapter 11,
Fisher Matrix Bounds
on Parameters.
See Also:
Fisher Matrix Bounds on Parameter
R-DOE Analysis of Data Following the Exponential Distribution
Maximum Likelihood Estimation for the Lognormal Distribution
