As a Director of AMP I get asked a multitude of questions, most are simple and somewhat repetitive. But I often find the diamond in the rough, a question that if resolved has the potential to advance reliability. I received an inquiry that had me puzzled; a hazards model originally developed for human health applied to reliability. This question is a challenge. I have my opinion but let’s see who can take a deep dive. Remember, it is not as simple as just as answer. Let’s stand in the inquiry and advance reliability.

 Dear Mr. Smith,

I read the book “The Reliability Handbook” by John D. Campbell in which Mr. Campbell explains how to implement the Cox’s Proportional Hazards Model for condition monitoring analysis. I have been studying this model on my own, however literature references on this subject are scarce and not quite comprehensible.

 I would be very grateful if you could give me some advice on the following questions:

 1. What is the best method to estimate the covariate parameters vector γ? I have read literature indicating Cox’s partial likelihood method as the method of choice, however other authors mention multiple (linear) regression.

 2. How do you estimate the baseline hazard? One of Cox PHM advantages is that covariate parameters γ are invariant to the baseline hazard, however from a time-to-failure perspective, it might be necessary to quantify it.

 3. Is Cox PHM suitable for heterogeneous populations? I understand one should try to perform reliability analysis on homogeneous populations of equipment, however Cox PHM is widely used in biostatistics where populations are quite heterogeneous.

I would really appreciate if you could give me some advice on these questions and I look forward to hearing from you.

 Best regards,

Alejandro Celemín

 Now I would like to add some questions of my own.

1) Should you use the hazards model how would you apply the findings?

2) Could this provide the link to correlate risk profiles to renewal programs

3) Who has experience with this model and how did you apply it?

 Jeff Smith, CRL-Bb

Last edited by Erin Thewlis
Original Post

Good morning!

Let me have a go at Alejandro's questions:

1) Cox's partial likelihood has always worked for me. It is reasonably easy to set up the goal function and just about any numerical optimization method can find the regression parameters that maximize the goal function. I normally use a Simplex-search method but many others work just as fine;

2) The quantification of the baseline hazard function (I prefer to call this Force of Mortality) is done simultaneously with the estimation of the regression coefficients;

3) Hazard proportions work fine for heterogeneous populations as long as you separate different strata through one or more covariates. More covariates place additional demands on the required samples size which is not ideal since we often work with small data samples in reliability.

This case study on using the PHM to optimize the replacement interval on pumps might help you with the theory and application:

As for Jeff's questions:

1) Describing failure data with the PHM simply models mortality in a sophisticated way -- you still have to decide what to use as the counter balance for mortality or risk. More often than not in reliability, the other side of the equation is the cost and consequence of an unplanned event (failure). Hazard proportions should be used to trade off the waste of intervening too soon in a life-limiting maintenance strategy versus the often hugely costly consequence of unplanned downtime. Something else that I find even more attractive is to use hazard proportions to estimate the useful remaining life of equipment to aid decision making;

2) It certainly provides that link and the quality of information is so much better than a basic straight line MBTF which could be very misleading;

3) I have used it over the last 20 years in a variety of applications inside reliability environments to optimize decision instants, estimate residual life, quantify risk, identify significant covariates in failure mechanisms to name a few.

There are many publications on PHM in reliability in academic literature. Unfortunately many of those are of a highly theoretical nature and difficult to follow if you start from scratch. A good place to find material to start learning about PHM is in medical research where the model is used to estimate the time to reoccurrence of cancer amongst other things.

Hope this helps!



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