FROM:N. Ulbrich, Jacobs Technology Inc.,NASA ARC

TO:Ray Rhew, David Cahill

FROM:N. Ulbrich, Jacobs Technology Inc.,NASA ARC

CC:Tom Volden, Mark Kammeyer, Jan van Aken (reviewers)

SUBJECT:Percent Contribution (fourth revision)

DATE:Oct. 28, 2011

Below is a proposed “new”chapterfor the AIAA Recommended Practice on Calibration and Use of Strain-Gage Balances. The “new” chapterdiscussesthe calculation of the “percent contribution” of individual regression coefficients of the gage outputs. The “new” chapter is printed in blue color below.

PLACEMENT OF THE NEW CHAPTER (TO BE ADDED ON PAGE 13):

The “new”chapter should be addedafter the chapter“3.1.2.2Global Regression”.It should be inserted right after the sentence “… Equation (3.1.9) is implicit in the regression analysis techniques that are used to determine the calibration matrix.”

TEXT OF THE NEW CHAPTER:

… Equation (3.1.9) is implicit in the regression analysis techniques that are used to determine the calibration matrix.

3.1.2.3 Percent Contribution

The “percent contribution” is defined as the contribution of each term of the regression model of the gage outputs to the total predicted value, expressed as a percentage of the contribution of the principle diagonal term. It is used in the aerospace testing community to assess the degree of linearity (or lack thereof) of the regression model of the gage outputs. In addition, some analysts use the “empirical”percent contribution threshold of 0.05 % to identify and remove terms in the model that make a very small contribution to the fitted gage outputs. It is important to point out that the global regression analysis of the gage outputs becomes a two-step process whenever the percent contribution threshold of 0.05% is used for term removal. First, a preliminary regression analysis has to be performed to identify terms for removal that may cause “over-fitting” of gage outputs. Then, the final regression analysis of the gage outputs has to be performed using the reduced regression model. Balance designers are also interested in the percent contribution because it shows them how to make a balance more linear through physical design.

In principle, the percent contribution indicates whether a term of the regression model of the gage outputs is “significant” in an “engineering” sense. The divergence of the load iteration process, for example, can often be traced back to an off-diagonal or non-linear term in the regression model that has a large percent contribution. The percent contribution does, however, not showwhether a term is “significant” in a “statistical” sense. More advanced metrics like the p-value of the t-statistic of a regression coefficient have to be used for that purpose (see, e.g., the discussion of tests on individual regression coefficients in Montgomery, Peck, Vining, “Introduction to Linear Regression Analysis,” 4th ed., John Wiley & Sons, 2006, pp.84-89). Nevertheless, some agreement between conclusions drawn from the percent contribution and the p-value of the t-statisticcan be observed. Comparisons performed at NASA ARC, for example,have shown that the insignificantterm setidentified using the 0.05% percent contribution threshold is a subset of the insignificant term set identified using the p-value of the t-statistic of a regression coefficient.

The calculation of the percent contribution of individual coefficients of the regression model of the gage outputs is a simple algebraic operation. It only depends on (a) the known capacities of the load components of the balance and (b) the regression coefficients that are the result of the regression analysis of the balance calibration data.The percent contribution is defined as the ratio of two numerical values. For convenience, this ratiois expressed as a percentage. The first numerical value is the product of the regression coefficient of the primary linear term (b1i,i)of the regression model of the gage output with the capacity (CAPi) of the related balance load component. This product can be written as follows:

Qi= b1i,i. CAPi (3.1.10)

The product defined in equation (3.1.10) is the reference value that is used to investigate the linearity of the regression model of the gage output. It is the denominator of the ratio that defines the percent contribution. The numerator of the ratio, on the other hand, is the product of the investigated regression coefficient with the related regressor variable value that is obtained from the load capacities. Now, the percent contribution of the ten math term type groups defined in equation 3.1.3 can be summarized as follows:

PCi,b1= 100 [%] . { b1i,j. CAPj } / {Qi} (3.1.11)

PCi,b2 = 100 [%] . { b2i,j. |CAPj| } / {Qi}(3.1.12)

PCi,c1 = 100 [%] . { c1i,j. (CAPj)2} / {Qi}(3.1.13)

PCi,c2 = 100 [%] . { c2i,j. CAPj.|CAPj| } / {Qi} (3.1.14)

PCi,c3 = 100 [%] . { c3i,j,k. CAPj.CAPk } / {Qi} (3.1.15)

PCi,c4 = 100 [%] . { c4i,j,k. | CAPj. CAPk| } / {Qi}(3.1.16)

PCi,c5 = 100 [%] . { c5i,j,k. CAPj. | CAPk| } / {Qi}(3.1.17)

PCi,c6 = 100 [%] . { c6i,j,k. | CAPj | . CAPk } / {Qi}(3.1.18)

PCi,d1 = 100 [%] . { d1i,j. (CAPj)3 } / {Qi}(3.1.19)

PCi,d2 = 100 [%] . { d2i,j. | (CAPj)3 | } / {Qi}(3.1.20)

where

j = 1, …, n and k = j+1, …, n

The reference value Qidepends on the gage output indexi. It changes whenever the index of the gage output changes during the calculation of the percent contribution.

The use of the percent contributionfor the assessment of the “significance”of individual terms of the regression model of the gage outputs has an advantage if compared with a statistical metriclike the p-value of the t-statistic of a regression coefficient:the percent contribution is easily understood and implemented. The percent contribution, however, has the disadvantage that itwasprimarily designed to investigate the linearity of the regression model of the gage outputs. Therefore,the empirical percent contribution threshold of 0.05 % can only identify a subset of the “statistically” insignificant terms of the regression model whenever term reduction for the prevention of “over-fitting” of balance calibration data needs to be performed.