Life data analysis involves the prediction of lifetimes of people or products in a population using a representative sample drawn from the population.  By fitting a statistical distribution to such a sample, we attempt to estimate key life characteristics of the people or products, such as:

• Average/expected lifespan
• Failure rate as a function of the passage of time
• Reliability, or probability of failure at a particular point in time 

Also, depending on the type of unit under consideration, we might measure lifetime in terms of either time or distance.  For example, for mechanical components we might use hours or operating cycles; for automobile tires we might use miles traveled; and so on.

When considering people in a medical context, this type of analysis is often referred to as survival analysis.  In a marketing context, it is often referred to as churn or loyalty analysis.  Although our focus here will be on mechanical failure in a business operations context, to see a medical example of patient survival analysis, please visit our Survival Analysis page.  And to see a marketing example of customer churn in the telecommunications industry, please visit our Cox Regression page in the Marketing Analytics section of the website.

 
Reliability Analysis

There are several probability distributions that are commonly used in reliability analysis.  We will not cover them all, but will instead focus most of our attention on one particular distribution that is probably the most commonly used: the Weibull distribution.  (Some of the other distributions are actually special cases of the Weibull.)
 

The Weibull Distribution

The two-parameter Weibull distribution is usually sufficient for most reliability analysis.  In the two-parameter distribution, the Beta parameter (β) represents the slope or shape parameter of the distribution, and the Eta parameter (η) represents the scale parameter of the distribution (its height on the vertical axis).  Here are three examples of Weibull distributions differing in their Beta values:

Readers having a statistical background will notice that in the examples above, when Beta = 1 the Weibull distribution takes the form of the exponential distribution; and when Beta = 3, the Weibull takes the form of the normal distribution.  This flexible ability of the Weibull distribution to "morph" into other well-known distributions is part of what makes it a preferred distribution for reliability analysis, allowing it to solve a wide variety of problems.

Although not shown above, when Beta is less than one, the exponential shape becomes more steeply convex, and this is typical of high infant mortality rates; or is reflective of problems with high mechanical or production-line "burn-in" failure rates, where relatively many failures occur early on, but then the remaining observations show a more gradual failure rate as defective units are removed or production line operation problems are resolved.

When Beta = 1, the failure rate is constant, which is typical of random/chance events/factors that cause failure, such as with automobile tire failures.  When Beta is greater than one, we see a pattern typical of wear, in which a product or component, through usage, gradually falls increasingly out of specification and finally wears out (fails).  This would be typical of mechanical systems, for example.

For most reliability/failure analyses, the two-parameter Weibull distribution is adequate.  However, in some situations, we must add a third parameter, Gamma (γ) to define the location of the distribution along the horizontal axis.  (In the two-parameter analysis, Gamma = 0 and the distribution begins at the left side at zero on the horizontal axis.)

In addition, some lifetime distributions reflect multiple phenomena affecting different units in the population differently; i.e., some combination of burn-in problems, random/chance influences and unit wearout.  These situations are modeled using what is called mixed Weibull analysis.  Here the goal is to isolate the various components of the mixed distribution and model them separately to arrive eventually at a final, combined analysis of these mixed populations.

To make things even more complicated, reliability analysis data can be either complete (all units in the analysis are monitored until failure) or censored (some units do not fail during the observation period).  In addition, there are various types of censoring:

• Right-censored (suspended) data occur when a unit under observation either is removed from the test before the observation period ends, or does not fail before the end of the observation period
  
• Interval-censored data occur when units are not observed continuously, but only at intervals, e.g., every 50 or 100 hours; or every 10,000 miles; etc.; in which case we cannot determine the exact time of failure, but only the time interval in which failure occurred

• Left-censored data are a special case of interval-censored data in which we do not know the exact point at which a unit failed at the beginning of an observation period; all we know is that it failed sometime before the first observation point

These various types of censoring require us to use special types of analysis beyond what we would use for complete data; but that is beyond the scope of this basic discussion.
 

A Weibull-Based Reliability Analysis Example

Here is a hypothetical example of a reliability analysis for a mechanical component using complete (i.e., uncensored) data.  For this component, the table below shows the number of operating cycles at which each of 10 observed units failed:

First we do some basic calculations to prepare the input data for the Weibull analysis:

The first column contains the data from the first table.  The second column simply ranks the units in order of failure.  The third column contains median ranks.  And the remaining columns perform various mathematical transformations of the data for input into the Weibull analysis.  These transformations allow us to use familiar least-squares regression modeling to analyze the data and build a predictive survival model for future planning purposes.  For example, we might use the model to decide on a reasonable warranty period to offer on the component, based on what percent of units are expected to survive after a given number of operating cycles.

Here is the summary output from the regression analysis:

And here is the goodness-of-fit graph that indicates that the Weibull distribution is a good fit to the data in our example:

The blue dots represent our actual data points, and the red line is the model's prediction.  We can see that most of the data points fall on or very close to the line.  And although we do not show it here, we can also create standard statistical confidence intervals around our estimates to give us a good idea of how accurate our predictions will be.

In addition, we create a table showing estimated cumulative failure (cumulative Survival Probability) as a function of number of operating cycles:

Among other things, we can see that the first failure is not expected to occur until after 50,000 cycles; and after 100,000 cycles we would expect about 99.93% of units to still be operating.  By about 1,000,000 cycles we would expect all units to have failed.

And here is the survival graph of these data:

Such an analysis could help us with many planning issues, such as figuring out when to reorder replacement parts, and in what quantities; or estimating the best warranty period to offer on a mechanical component in order to remain competitive while also keeping warranty fulfillment costs within reasonable limits.

Weibull analysis has many applications in fields other than industrial failure analysis.  A few of these applications include:

• Medical/biological survival analysis and comparative drug effectiveness evaluation
  
• Industrial engineering (e.g., estimating production and delivery times)
  
• Weather forecasting (e.g., predicting the expected frequency of wind speeds in the vicinity of wind turbines connected to the electric power grid)
  
• Extreme Value forecasting, which involves predicting rare, extreme events (e.g., predicting annual maximum one-day rainfalls and river discharges, to assist with flood control and resource planning)
  
• Wireless telecommunications signal fading characteristics
  
• Intangible asset valuation, e.g.:
• • Estimating reinsurance claims
  • Modeling credit card default rates
  • Predicting subscriber behavior (e.g., newspaper or magazine subscriptions, season ticket sales, etc.)

The number of applications is virtually endless, and SmartDrill can assist management to better anticipate events and manage costs and risks.