How to Perform a Weibull Analysis – Validation of Results and Reliability Improvement (Part 3 of 3)

Statistical Functions       

Reference: http://www.reliawiki.org/index.php/Basic_Statistical_Background   

Step 8: Indicate Confidence Bounds  

Given the limitation of time and resources, we can only select relatively small but representative samples of units to understand the life characteristics of all products (i.e. the probability of failure) in the population. To quantify the uncertainty due to sampling error, we use Confidence Bounds (aka Confident Interval) to estimate the precision of an estimation. 

The Confidence Bound gives an estimated range of values that is likely to include an unknown population parameter. It is calculated from the set of sample life data. 

1-sided and 2-sided Confidence Bounds

• 1-sided confidence bounds:  One-sided bounds are used to indicate that the quantity of interest is above the lower bound or below the upper bound with a specific confidence. 

• 2-sided confidence bounds:  Two-sided bounds are used to indicate that the quantity of interest is contained within the bounds with a specific confidence.  

Tips: 

The appropriate type of bounds depends on the application.  

• 1-sided lower bound on reliability;
• 1-sided upper bound for percent failing under warranty;
• 2-sided bounds on the parameters of the distribution;

Confidence Bounds Methods 

In this post, we just list the methods of calculating Confidence Bounds. If you want to review the methodologies comprehensively, read the “Confidence Bounds” chapter in ReliaSoft Weibull Analysis eTextbook. 

(1) Fisher Matrix Confidence Bounds (FM): These bounds are employed in many statistical and life data analysis packages, as well as most ReliaSoft applications. In general, these bounds tend to be more optimistic (tighter) than the non-parametric beta-binomial or likelihood ratio bounds.  

(2) Beta Binomial Confidence Bounds (BB): A non-parametric approach to confidence interval calculations involves the use of rank tables.  

(3) Likelihood Ratio Confidence Bounds (LR): LR and FM are both commonly used in Weibull Analysis to calculate reliability confidence bounds for different life distributions. Here are the differences between LR and FM: 

• LR is much simpler than FM. 
• LR is computationally intensive and needs a much longer time to plot.  
• LR is more conservative than those calculated with the FM method.  
• FM relies on a normality assumption, while LR relies on the assumption that follows a Chi-Square distribution. 

(4) Bayesian Confidence Bounds (BSN): Can be used when one has some prior knowledge about the reliability of the component with adequate historical data and/or engineering judgment. 

Order of Preference for Confidence Bound Methods for Small Samples: BSN > LR > FM >BB

The SimuMatic tool in Weibull++ can be used to perform many reliability analyses on data sets that have been created using Monte Carlo simulation.  

Functions: 

• Better understand life data analysis concepts 
• Experiment with the influences of sample sizes and censoring schemes on analysis methods 
• Construct simulation-based confidence bounds 
• Better understand the concepts behind confidence bounds 
• Design reliability tests 

Display confidence bounds on time (Type I) or on reliability (Type II)  

When drawing a probability plot, confidence bounds (except Beta Binomial Confidence Bounds) can be displayed in two ways: 1) on time (Type I) or 2) on reliability (Type II). Type I is to read values from the x-axis (time), while Type II is to read values from the y-axis (probability of failure). 

The rule of thumb is: display confidence bounds on the value that you do not know (i.e., the value that you are trying to estimate).  

• Type I: given an unreliability value, what is the corresponding time? For example, if you want to determine the time by which 8% of the units have failed (i.e. 92% reliability) then you would use confidence bounds on time. 
• Type II: given a time value, what is the corresponding unreliability? For example, if you want to determine the probability of failure at 1500 hours, then you would use confidence bounds on reliability. 

Step 9: Analysis Review  

Before taking actions, you need to review the entire Weibull Analysis. Basically, you should consider 4 aspects: Practical, Graphical, Analytical, and Confidence. 

Practical

In terms of the practical aspect, ask yourself: 

• Does the data show trends or clues that are of practical importance?
• What does your in-house subject matter expert think about the data based on their expert and previous experience?
• Is the variation in data associate with some outside influences such as:  
  • changes from shift-to-shift,
  • weather related variation caused by temperature or humidity, 
  • part-to-part variation (items may not be identical due to new models or versions).

Graphical

In terms of the graphical aspect, ask yourself: 

• Are your data points reasonably spaced along the line, or are some points far from the line? 
• Does your Weibull plot contain an “S” or “dogleg” bend in the data? (it is a clue to the potential of multiple failure modes, which means you need to have a more diligent review) 
• Does your Weibull plot appear to curve downward at early life? (it is a clue that your assumption of origin time might be incorrect. Time does not start at T=0. It may be physically impossible for the failure mode to produce failures instantaneously, or at early life.) 

Analytical

In terms of the analytical aspect, ask yourself: 

• Are your Rho (ρ) or Likelihood values too low? (it means your data collection might have some issues.) 
• Do the parameters match expectations? (In Weibull distribution if the slope is too high it may be an indication of poor data sampling, you may have a very narrow window of extreme use.) 
• Can you apply historic information to improve your estimate? (One common way to do this is to impose a known Weibull slope based on multiple test results that engineering judgement has determined to best represent the failure mode under study.) 

Confidence

In terms of the confidence aspect, ask yourself: 

• Is the confidence bound adequate to cover the variation risk? 
• Are there any outliers that lie an abnormal distance from other values? Why do they appear? After eliminating them, how likely it is that similar values will continue to appear? (remember, always document the justification and subsequent removal of any outliers) 
• Is the width of the confidence bound reasonable? (A very wide interval may indicate that more data should be collected before anything very definite can be said about the parameter) 

Step 10: Determine and Implement Appropriate Strategies  

Now you have everything you need to understand the life characteristics of all products in the population, what do you then do with these results? What actions do you take to improve the reliability and cost performance of products?  

Choose the Optimal Strategies based on Beta Value, β  

You can determine the best maintenance strategies based on the types of the failure patterns of the component, which are represented by the beta value, β. The beta value is simply a measure of the slope of the probability plot. 

The “Reliability Bathtub Curve” in the Failure Rate vs Time plot (see image below) is a graphical representation that comprises all of the 3 failure patterns: infant mortality failures (β <1) with a decreasing failure rate, random failures (β = 1) with a low, relatively constant failure rate, and wear-out failures (β >1) that shows an increasing failure rate. 

Run Simulation to Determine Your Optimal Strategies   

Alternatively, you can use the Weibull results by putting the data into your RBD Blocks and running the full system simulation of a period of time, you will be able to accurately define the failure profile of the component and system, and forecast the best strategies to meet your reliability and cost needs.