They are constants used to estimate the sample standard devaition based on the observed sample range.
When the sample size is small (<10 or 15) the range-based method is fairly efficient compared to use of the more typical standard deviation formula.
Thanks Doc, but why are they used in that particular formula? The standard deviation of the repeatability is calculated using d2* and so already has the correction for the small sample size.
I would have thought that the standard deviation of the bias was already the best estimate and could be used directly in the t test.
Obviously d2/d2* is some sort of correction factor for the t test, but I can’t see why it is there. The values of d2 and d2* are pretty similar with the sample size that is used in bias calculations (> 10) and so it doesn’t make much difference, but I am curious.
I can’t find too much that compares d2 and d2* directly. I did find an article on the web coauthored by Doug Montgomery. The best way to find it is to search Google with “using ranges to estimate variability”. The article does address some of the differences between d2 and d2*, though not sufficiently for my curiosity.
The appropriate use of d2 and d2* is dependent on the size of the sample in terms of the number of subgroups (g) and the number of units within subgroups (n). Recall, d2 is based only on the size of (n) where as d2* brings into account the number of subgroups and as the number of subgroups approaches infinity, d2* approaches d2. Studies involving smaller number of subgroups should use d2* as the small subgroup size correction is included. This would include any gage R&R study that is using a range approximation of the test-retest standard deviation. As a rule, 15 subgroups are required for the use of d2. It is correct that the bias correction is similar to the degrees of freedom correction in the t-statistic. Actually it is based on the c2 distribution (but of course so is the t-statistic). An excellent reference is Quality Control and Industrial Statistics, fifth addition by Acheson J. Duncan. The bible for anyone involved in statistical application in industrial processes.
The paper by Woodall and Montgomery, that Doc mentioned, is interesting. It gives a different perspective on the distinction between d2 and d2* and when they should be used. It also discusses how to derive d2* from d2 which satisfies another curiosity.
It also proposes a third version which is d2 divided by d2*squared which it claims minimises the MSE
This may be a clue to the answer to my original question about the formula for the confidence bounds in the bias formula in the MSA Manual because multiplying a standard deviation obtained using R/d2* by d2/d2* gives this third version.
I’m still intrigued by why it crops up in the bias formula though, is it based on Woodall and Montgomery’s views or did they arrive at it by some other route? Also why doesn’t if follow through into the confidence limits for the linearity study?
Sorry, I needed to go back to your original message to remind myself as to what the discussion was about. What I believe your original inquiry was about was the use of both d2 and d2* in a significance test comparing two means in a gage R&R study when the test-retest error was estimated using the range. I originally responded as to the difference between d2 and d2* which probably didnt help you much.
After reading the paper by Woodall and Montgomery, one must wonder why the third constant for estimating sigma from ranges is not used exclusively as a replacement for d2*. All references that I was aware of prior to this discussion used d2* for limited subgroup analysis. This paper clearly shows the reduction in estimation error using the constant c
Where sigma = c*Rbar and c= d2/( d2*)**2
Since d2* approaches d2 and the number of subgroups (g) approaches infinity,
c*Rbar approaches Rbar/ d2 as (g) approaches infinity.
Your MSA manual must be a very new update. It appears the original work on the constant c was published in 1996.
As for why doesn’t it follow through into the confidence limits for the linearity study is probably that the linearity analysis is using least squares rather than ranges.
My MSA Manual is the third edition published March 2002. It is an extensive rewrite from the original edition (I don’t have the second).
It has some very significant changes, for example in the Gauge R&R studies the ‘K’ factors they use have all been changed by a factor of 5.15 so that repeatability variation is now in standard deviations instead of 99% confidence intervals. I know of one company that was puzzled as to why their spreadsheets were all wrong!
You are correct about the linearity study, the same argument doesn’t apply because they don’t use the range method in the standard deviation calculation.
I am looking for metodos leaves or with which it can make the MSA studies, (bias, Stability, Linearity, etc).
Could you please help me???
Plasticos Flambeau, Saltillo Coahuila Mexico.