9.3.5. Weibull Analysis
Two or three parameter Weibull models can be estimated employing maximum likelihood or ordinary least squares (OLS) regression methods. The program reports the estimated parameters, their confidence intervals and covariance matrix and features an interpolation facility.
It is sufficient to select one data column to run a Weibull Analysis. Multisample data can be selected either in the form of multiple columns (not necessarily of equal length) or data columns classified by one or more factor columns (see 6.0.4. Multisample Tests). If at least one factor column is selected, then a further dialogue will pop up asking for the combination of factor levels to be included.
9.3.5.1. Weibull Inputs
An Intermediate Inputs dialogue enables you to control the following parameters:
Model: This can be one of twoparameter maximum likelihood or two or threeparameter OLS regressions.
Method: If the method is OLS, then you can choose to regress Y on X (the default) or X on Y.
Population size: If the population size is known, you can enter it here. This is the number N used in generating the median ranks as described below. If this box is left as 0, then the program assumes that N is equal to the number of cases in the sample n.
Median Ranks: In order to run a regression and to plot a chart, Weibull Analysis needs to generate median ranks as Yaxis values. Median ranks can be computed in one of the following ways.
· Exact Binomial
· (i – 0.3) / (N + 0.4)
· i / (N + 1)
· (i – 0.5) / N
· (i – 0.375) / (N + 0.25)
where N is the population size supplied by the user as described above. If N is not known, it is assumed to be equal to the sample size n. The default method adopted by UNISTAT is Exact Binomial, where the exact median ranks are determined from the binomial distribution as:
where R_{i} is the median rank of the i^{th} observation. Other options in this list are various approximations to median ranks. For large values of N not all exact median ranks can be computed. When this is the case, the program cannot proceed with the analysis. However, you can still run your model by selecting one of the approximation formulas.
In StandAlone Mode, you can generate a data column containing median ranks using the Data Processor’s MdRk(N) function (see 3.4.2.5. Statistical Functions).
9.3.5.2. Weibull Output Options
9.3.5.2.1. Weibull Parameter Estimates
Parameters of the Weibull distribution can be estimated by one of the following three methods.
Twoparameter maximum likelihood estimation: The probability density function for the twoparameter Weibull distribution is given as:
and the log likelihood function is:
where n is the sample size. Differentiating L with respect to β and η and setting equal to zero, we obtain the following two equations used in estimation:
Twoparameter OLS estimation: The cumulative distribution function for the twoparameter Weibull distribution is given as:
where T is the failure time, β is the shape parameter, η is the scale parameter. Taking the natural logarithm of both sides and rearranging we obtain:
This is clearly a line equation of the form:
where, conveniently, the left hand side is the gompit (or cloglog) function:
and F(T) is the median rank function, with the slope:
and the intercept:
Threeparameter OLS estimation: The cumulative distribution function for the threeparameter Weibull distribution is given as:
where β is the shape parameter, η is the scale parameter, and γ is the location parameter. This equation is linearised as above and all three parameters are estimated simultaneously using the iterative least squares method.
Mean failure time: This is found using the gamma function:
9.3.5.2.2. Weibull Covariance Matrix
For all estimation methods, the variancecovariance matrix is defined as the inverse of the second partial derivatives matrix of the log likelihood function:
9.3.5.2.3. Weibull Interpolation
In the Output Options Dialogue, clicking on the [Opt] button situated to the left of the Interpolation check box, the interpolation dialogue can be accessed.
You can enter a failure probability to estimate the failure time or enter a failure time to estimate the probability of failure. Each time you will need to enter an asterisk for the parameter to be estimated. The program will display the estimated parameter together with its confidence interval.
9.3.5.2.4. Weibull Chart
If you use UNISTAT’s XY Plots procedure and set the Yaxis scale as gompit and the Xaxis scale as logarithmic, you instantly have a Weibull probability paper. If you select median ranks as the Yaxis and failure times as the Xaxis variable and fit a trend line, you would have already estimated the twoparameter Weibull model. This is precisely the way we generate the Weibull chart here.
In StandAlone Mode, you can generate a data column containing median ranks using the Data Processor’s MdRk(N) function (see 3.4.2.5. Statistical Functions).
If the data lies on a nearstraight line, then it is said to conform to the Weibull distribution.
Clicking the [Opt] button situated to the left of the Weibull Chart option will place the graph in UNISTAT’s Graphics Editor. In the Edit → Data Series dialogue, two check boxes allow you to show or hide the eta and mean failure time lines on the chart.
9.3.5.3. Examples
Example 1
Example 16.15 on p. 580 from Banks, Jerry (1989). Failure times of 8 components are given. The population size is known to be 100.
Open TIMESER and select Statistics 2 → Quality Control → Weibull Analysis. From the Variable Selection Dialogue select Failure time (C17) as [Variable]. In the next dialogue, enter 100 for the population size and select (i – 0.5) / N for median rank approximation. The following output is obtained:
Weibull Analysis
Parameter Estimates
Data variable: Failure time
Number of Cases: 8
Population Size: 100
Median Rank Method: (i – 0.5) / N
2Parameter OLS Estimation
Regress Y on X

Value 
Lower 95% 
Upper 95% 
Beta 
0.8962 
0.5791 
1.3870 
Eta 
3689.4178 
2098.0830 
6487.7337 
Mean Failure Time 
3890.9191 


Correlation Coefficient = 
0.9956 
Rsquared = 
0.9911 
Covariance Matrix

Beta 
Eta 
Beta 
0.0399 
192.7873 
Eta 
192.7873 
1128917.0915 
Interpolation
Probability 
Time 
Lower 95% 
Upper 95% 
0.632120558828558 
3689.4178 
2098.0830 
6487.7337 
Example 2
Continuing from the last example, this time select 3parameter model with population size 0 (unknown) and Exact Binomial median ranks:
Weibull Analysis
Parameter Estimates
Data variable: Failure time
Number of Cases: 8
Median Rank Method: Exact Binomial
3Parameter OLS Estimation
Regress Y on X

Value 
Lower 95% 
Upper 95% 
Beta 
1.4466 
0.8149 
2.5682 
Eta 
143.7028 
85.1939 
242.3940 
Gamma 
15.9995 


Mean Failure Time 
130.3404 


Correlation Coefficient = 
0.9976 
Rsquared = 
0.9952 
Covariance Matrix

Beta 
Eta 
Beta 
0.1795 
1.7235 
Eta 
1.7235 
1469.3800 
Interpolation
Probability 
Time 
Lower 95% 
Upper 95% 
0.632120558828558 
143.7028 
85.1939 
242.3940 