Sunday, November 7, 2010

1. Introduction

1.1  Purpose of this blog  variantie-uitbijter-test   variantie-uitschieter-test 
February 2010, I published a variance outlier test, which I called the "G test", in Analytica Chimica Acta.  This journal is well appreciated by analytical chemists.  However, the G test is relevant to other disciplines as well.  The G test is an extension of Cochran's C test [23] and is of potential interest to anyone who needs to decide if one or more variance (or standard deviation) estimates are extreme in comparison with a group with which they are supposed to be comparable.  I am aiming for a broader audience by posting a personal manuscript version of my article in this blog.  The publisher of the original article [33] allows such practice provided that the posting includes a link to the article's Digital Object Identifier (DOI) as well as a complete citation for the original article [34].  Please find the link and the citation immediately below:

        doi:10.1016/j.aca.2009.11.032                               R.U.E. 't Lam, Analytica Chimica Acta 659 (2010) 68–84.

Alternatively, you may visit my manuscript.  In the manuscript I have corrected some errata and inconsistencies. I also have extended the appendices of the original article to supply critical values for a broader selection of designs. Besides, I have added explanatory notes on performing a G test as Chapters 2 to 4 of the Home page. In priciple, a blog is not intended to be permanent. For reference purposes you are therefore advised to include a citation of my original article. If you have questions, suggestions or concerns regarding the original article, the blog or the G test in general, you are encouraged to leave me a message, contact me at or contact me through my page at LinkedIn.

1.2  Abstract
The abstract of the article in Analytica Chimica Acta [34] reads:
ISO Standard 5725 “Accuracy (trueness and precision) of measurement methods and results” recommends Cochran’s C test to numerically verify if three or more normally distributed data sets show “homogeneity of variances” or “homoscedasticity”. The C test is a one-sided outlier test that will identify deviant standard deviations. It can be run on summary data using a pocket calculator. However, the C test has limitations. It only applies to data sets of equal size. It uses critical values that are only available for the upper tail of the variance distribution, at selected numbers of data sets, selected numbers of replicates per set and only at two significance levels. Cochran’s C test will not identify an outlying low variance, but may mistake a high variance for an outlier instead. We transform the C test into a more general “G test”.
Expressions are derived to calculate upper limit as well as lower limit critical values for data sets of equal and unequal size at any significance level. The expressions are validated against literature values and through simulations in Excel. Representative critical values are tabulated for those who prefer to work from tables. The power of the G test is verified for data sets of equal and unequal size. The G test appears superior to the C test in detecting effects from low variances. The G test allows positive identification of exceptionally low variances. The application of the G test is illustrated with a numerical example.

1.3  Errata
1. In § 3.3 equation (18) was corrected to read:
        "ζLL(1) = 1 − ζUL(1) = 1 α1/L" in stead of "ζLL(1) = 1 − ζUL(1) = (1 α1)/L"
2. In § 3.5 the order of equations (30) and (31) was reversed to match the remainder of the manuscript.
3. Some typo's and rounding errors in Tables 6-8 were corrected.  
4. The lower left corner of Table 8 was changed to read "L= 6", instead of "L= 7".

1.4  Application Areas
Variance outlier testing is routinely applied in interlaboratory studies for determining the precision of analytical test methods according to ISO 5725.  However, variance outlier testing is also practiced outside (analytical) chemistry.  A search in Google Scholar on "Cochran's C test" + additional keyword, resulted in the numbers of hits listed below.  Key words with less than ten hits are not shown.  

As Cochran's C test is fully comprised in the G test, the G test is at least as widely applicable as Cochran's C test. Wikipedia provides a short background article on Cochran's C test [36].

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