Fisher Matrix vs. Likelihood Ratio Bounds

[Aurora Tiffany-Davis]

I have a life test with 6 samples, 5 of which have failed. I am determining the proper analytic techniques for this test.

Based on my training and research, I consider 6 samples to clearly be a "small" sample size. I am also under the impression that the Likelihood Ratio Bounds method is widely considered to be preferable to the Fisher Matrix method for small sample sizes.

However, when I perform analysis using LRB, the lower confidence bound appears extremely erratic at larger x-axis values. This does not occur with the FM method.

For this reason I am considering using the FM method, however I am concerned due to the fact that I wish to report "worst case" scenarios. The LRB method is more conservative. That is to say, the LRB "worst case" is worse than the FM "worst case".

Do you have any suggestions for me as I attempt to create an accurate picture of unreliability over time?

Thanks.

[David]

I have a life test with 6 samples, 5 of which have failed. I am determining the proper analytic techniques for this test.

Based on my training and research, I consider 6 samples to clearly be a "small" sample size. I am also under the impression that the Likelihood Ratio Bounds method is widely considered to be preferable to the Fisher Matrix method for small sample sizes.

However, when I perform analysis using LRB, the lower confidence bound appears extremely erratic at larger x-axis values. This does not occur with the FM method.

For this reason I am considering using the FM method, however I am concerned due to the fact that I wish to report "worst case" scenarios. The LRB method is more conservative. That is to say, the LRB "worst case" is worse than the FM "worst case".

Do you have any suggestions for me as I attempt to create an accurate picture of unreliability over time?

Thanks.

Hi Aurora,

In regards to the eratic behavior that you are seeing, it is difficult to say without looking at it first hand. What plot are you trying to create? We can take a closer look if you can send the file into support at support@reliasoft.com. In regards to the differences between LRB and Fisher Matrix, you are correct that LRB tend to be more realistic for small sample sizes. You do not want to choose one method simply because you did not like the results using another. However, keep in mind to use that which you feel comfortable. In regards to the larger x-values, just be aware of how far you are extrapolating in regards to the data points that you do have. Be careful to not extrapolate too far beyond the data.

I hope this helps.

[Aurora Tiffany-Davis]

Thanks David,

I've just submitted the file.

- Aurora

[DrDave]

One other thing to consider is that we seldom need to use confidence bounds for large X (time) values. Most inference is about small time values and percentiles.

For example, if I am selling a truck, I don't really care about my reliability / probability of failure at 20 years, since it will be well past the warranty period. I do want accuracy for early life up to, say, the three year point (if that is when my warranty ends).

So I would not panic about erratic behavior for the large values. It is likely due to numerical precision issues, anyway, not a defect with the method.

[Steve]

It would seem that if one does not know the sample size prior to the analysis (say with a large number of data sets to be analyzed by an automated process) that LRB would be the preferred method since it "works" for small samples sizes where FM is not as accurate and large sample sizes where the difference between FM and LRB is not significant. Is that correct?

[David]

Hi Steve,

You are correct that as the sample size increases the difference between the confidence bound methods decrease. However, keep in mind that the Likelihood Ratio Bounds (LRB) are conceptually easier than Fisher Matrix (FM), but the LRB are computationally more intensive. So the LRB will most likely take longer to calculate. This is just something to consider if you are running a batch process.

I hope this helps.

[Steve]

Thanks David.

Would you expect to have less chance of convergence issues using LRB for smaller sample sizes or even for larger sample sizes?

[David]

For LRB bounds we have to solve a non-linear equation. So as the sample size increases, the possibility of not converging decreases.

I hope this helps.