With respect to the underlying statistical technique, whether the duration is shorter or longer than the warranty seems immaterial. You are just making inference on a proportion and you can indeed use the same technique for another point in time (your question 1).
That said, the time points need...
thank you for the reply. I’m assuming that you’ve confirmed my 4 sample calculations or at least 1 of them allowing us to be on the “same computational page”.
I’m encouraged when you say that I can use the same technique for another point in time but I don’t see quite how to do that in regards to the two questions I posted. The technique is able to handle varying “points in time” but I believe only if those points in time are equivalent. My questions involve nonequivalent time points. [I've reached the character limit, more to come]
The technique as you know takes 3 inputs and produces 2 outputs:
Input 1: Confidence Level (only interested in 0.90 for all analyses) Input 2: The Sample Size Input 3: Number of defectives or test failures (this has to assume a test duration that is consistent with the interpretation of the outputs) Output 1: The 90% Lower Confidence Limit associated with the time duration embodied in Input #3 Output 2: The 90% Upper Confidence Limit associated with the time duration embodied in Input #3 [More to come]
I have trust in the internal consistency of my 4 sample calculations above because for each calculation the time durations assumed in Input #3 (number of failures) is equivalent to that required by the interpretation of the two outputs, namely 1 year; that is a simulated 1 year test duration and a 1 year warranty requirement.
However, the questions I have involve the case where the two time durations are not equal as in my Question 1.[More to come]
Here we have the warranty requirement (associated with the confidence limit outputs) as remaining at 1 year but the time duration applied during the test (and associated with the inputs) is 1.5 years. How can the technique deal with this time inconsistency? Don’t we have apples and oranges?
If I’m not seeing something here can you run the numbers and show your results (inputs and outputs) for my Questions 2 & 3 ? That way I should be able to reproduce them.
Many thanks.
Typo in my last posting, sorry. Not "Questions 2 & 3" but Questions 1 & 2 . . . .
@Steve Yes, the numbers seem right. However, I don't understand your issue with the warranty. The length of the warranty or the duration do not figure in the computation of the confidence limits at all. You observe the proportion of failed products under some conditions and make some inference on that proportion. It happens to be a proportion of products that failed after a one-year simulation but the exact same technique can be used for proportions that have nothing to do with on-going processes.
Great that you've confirmed my 4 calculations, establishing a common computational baseline is always useful. But I still don't follow what you are saying. I believe the length of the warranty does figure in (implicitly) and must be equivalent to the test duration to have apples and apples from input to interpretation of output. Can you run your numbers for my original Questions 1 & 2 ? What are the upper and lower limits of the 90% CI for these 2 scenarios ? That should get to the heart of my question. Thanks.
The CI for question 1 is [0%, 95%]. It's indeed exactly the same than if you had no failure after 1 year, 1 day or 1 century. Whether you believe it or not, the duration does not figure in the computation. Simply, if you have observed one unit for one year, then you are making inference on the proportion after one year. If you have observed it for 1.5 year, then you are making inference on the proportion after 1.5 year. If you pick some books randomly from my bookshelf and check the color of their cover, then you are making inference on the color of books in my bookshelf. As simple as that.
Question 2 is a little unclear to me. As I said, I am not sure it should be addressed in the same way because implicitly you seem to be waiting for the units to fail. But assuming you are looking at this after 1.5 years, you would have one failure for two units, CI is [2.5%, 97%]. After 4 years, it would be [22%, 100%].
The only solution would seem to go back to the other approach mentioned by John and myself, namely model the time-to-failure.
As I said, one way or another, the duration just does not figure in the computation. It's not there when making inference after one year, it can't be used to turn a proportion observed after 1.5 year into a proportion of defect after 1 year.
The meaning/interpretation of the result depend entirely on the nature of the observation you are using, the technique does not “know” about duration, warranty, etc. it's just a basic inference technique for proportions of any kind.
I see. This is why I asked the original question. I wasn't sure if the "Binomial Approach" could be extended to solve what I need to solve. Also, let me say that I really do appreciate your time and knowledge on this.
I think you hit it on the head when you say "The meaning/interpretation of the result depend entirely on the nature of the observation you are using"
I was hoping to run a test for say, 1.5 years and then interpret the results thru a "1 year lens" using the "Binomial Approach". What you are saying is that cannot be done, right ?
Got it. And yes, make whatever edits you feel would clarify. Since binomial can't work for what I need, are there other methods (besides tradidional distributions) that might work ?
You do know that units that did not fail after 1.5 years were also still OK after 1 year so you do have an upper bound for the failure rate but that's a trivial point.
I can't think of any other approach, no.
But like I said, I am not a big expert on all this.
What I can tell you, is that (1) I don't see how a binomial confidence interval could easily be extended in that direction (2) the statistical literature on this technique (Alan Agresti's publications on categorical data analysis, etc.) do not mention anything like this.
I agree, but there are awfully smart people out there that I'm sure have dealt with this, perhaps some flavor of Monte Carlo. Anyway, I'll let you go. And thanks again for the help, I do appreciate it. --Steve
Discussion between Gaël Laurans and S…
Discussion between Gaël Laurans and S…