Consensus Values, Regressions, and Weighting Factors.

An extension to the theory of consensus values is presented. Consensus values are calculated from averages obtained from different sources of measurement. Each source may have its own variability. For each average a weighting factor is calculated, consisting of contributions from both the within- and the between-source variability. An iteration procedure is used and calculational details are presented. An outline of a proof for the convergence of the procedure is given. Consensus values are described for both the case of the weighted average and the weighted regression.

1. Introduction

The problem of computing consensus values when the errors of measurement involve both internal (within group) and external (between group) components has been discussed in a number of papers [1-4]. The present authors have studied the case of a simple weighted average, as well as that in which the measured quantity y is a linear function of a known variable x.

In the present paper we extend our results to cases in which the error standard deviations are functions, of known form, of the x-variables. We also provide an outline of a proof for convergence of the iterative process described in reference [1].

While our procedure is entirely reasonable, and results in acceptable values, we have no mathematical proof that the weights, which we calculate from the data, are optimal in any well-defined theoretical sense. The problem has been recognized in the literature [5], but we know of no attempt to provide the proof of optimality.

2. Review

If ω denotes the weight (reciprocal variance) of a quantity, then the general equation for a weighted average is:

If equals the average of n results from group i (i=1 to m), then where Then the weight ω of is equal to:

= the component of standard deviation within group i (the value can be estimated from the n results within each group)

σb = the component of standard deviation between groups.

The weight equation, yields: or

Generally, all σ-values, and consequently the ω values are unknown. The can be estimated (as ) from the replicate measurements. We derive an estimate for and consequently for the ω by using the quantity which we equate to unity. Thus we have

Equation (3) is used in “reverse fashion” to estimate the ω and from the sample data. This is possible if in eq (2), the are estimated from the within-group variability, so that the only unknown is σb. Note in eq (3) that σb is embedded within each weight and therefore within . The estimated and σb can also be used to estimate the standard deviation of the weighted average, which is equal to . Henceforth, we use the symbol ω for the sample estimate of ω.

The same general reasoning holds for the weighted regression case. The variance of a simple weighted average is replaced by the residual mean square from a weighted least squares regression. For a regression with m groups and p coefficients the analogue of eq. (3) is where ω is given by eq (2) and is the fitted value.

We now describe the case of a weighted regression with p =2. The fitted value , for the ith group can be written as follows: or where is a weighted average analogous to the weighted average described by eq (1), and and are weighted least squares estimates of the coefficients, α and β. Again, the only unknown is σb, which can now be estimated from sample data by use of eq (4).

A direct solution for σb in either eq (3) or (4) would be extremely complicated since ω, , and all contain σb. The number of terms m, in both equations will vary depending on the number of groups in a particular sample data set. Furthermore, for the regression case, the and also depend on σb. Therefore an iterative solution was proposed in reference [1]. This iterative procedure is central to the practical solution of either eq (3) or (4). In order that this paper be self-contained, we briefly review the iterative procedure for the regression case using eq (4) with p =2.

3. Iteration Procedure

We define the function:

In view of eqs (2) and (4), the estimate of must be such that . For ease of notation let . Start with an initial value, υ0≈0, and calculate an initial set of weights and then evaluate eq (6). In general, will be different from zero. It is desired to find an adjustment, dυ, such that F(υ0+dυ)=0. Using a truncated Taylor series expansion, one obtains:

Evaluating the partial derivative in this equation, one obtains:

The adjusted (new) value for υ is:

This new value is now used and the procedure is iterated until dυ is satisfactorily close to zero.

The iterative procedure is easily adapted to the computer. The programming steps are as follows:

Evaluate the from the individual groups of data.

Start the iteration process with a value of υ0 just slightly over zero.

Evaluate eq (2) to get estimates of .

Fit eq (5) by a weighted least squares regression of on , and get estimates of the .

Use eq (6) to evaluate F0. If F0<0, then stop the iteration and set υ=0. If not, continue with 6.

Use eq (7) to evaluate dυ.

If dυ is positive and small enough to justify stopping, then stop. If it is positive, but is not small enough, repeat steps 3–7 [using the new υ0 from eq (8)].

The consensus values are the final coefficients of the regression equation. One is also interested in the final value since this is needed to characterize the imprecision of the fit.

For the case of a weighted average [see eq (1)] the above iteration steps are the same, except that in place of step 4, is calculated by eq (1), and steps 5 and 6 use in place of , and unity is used for the p value. The authors have frequently used this procedure for the evaluation of Standard Reference Materials [6].

4. Theoretical Extensions

Once one recognizes the between- as well as the within-group component of variance in the evaluation of consensus values, one can begin to consider functional forms for these components. The within-group component can be of any form, and can be easily handled since the appropriate sample values of the component are simply substituted into the weights described by eq (2). Thereafter, this component does not affect the iteration procedure. See for example reference [7], where the within component of variance refers to a Poisson process. The between-group component, however, affects the iteration procedure and must be handled more carefully. As an example, consider the case where the between-group component of standard deviation is believed to be a linear function of the level of X:

Let us assume that we have preliminary estimates, c and d for the γ and δ coefficients. Suppose further that we wish to adjust the estimated value of the variance by a fixed scale factor, say υ′. The desired between-group component of variance is thus:

The weights estimated by eq (2) would then be:

This newly defined weight can be used in the iteration process. The iteration process proceeds as before, but now the adjustable iteration parameter υ′ is the multiplier needed to make eq (4) true, that is, to make it consistent with the sample data sets. The denominator of eq (7) which is used in iteration step 6 for calculating dυ, needs to be slightly modified since the derivative of F with respect to υ now contains the function described by eq (10).

All other steps in the iteration process are the same. The final between-group components of variance will be described by eq (10).

5. Example

The iteration process will be used to fit the data of table 1 to a straight line. These are real data taken from a large interlaboratory study for the determination of oxygen in silicon wafers.

Table 1

Data used in example of iteration process

X	Y1	Y2	Y3
0.806	2.83	2.85
1.429	4.62	5.35	5.01
1.882	6.89	6.66
2.140	7.56	7.67
2.256	7.94	7.90
2.279	8.42	8.12
2.814	10.04	9.70	10.17
2.957	10.34	10.05
2.961	11.09	11.07
3.108	11.63	11.69
3.124	10.87	11.01
3.403	12.40	12.22
3.466	11.94	12.17	12.92
3.530	12.63	12.41
3.543	12.98	13.27
3.724	12.95	12.56
3.836	13.07	13.69	13.56
3.902	14.54	14.19
4.280	15.59	16.24
4.770	16.62	16.59

A preliminary examination of the data indicates that the within error has a constant standard deviation and that the between error has a standard deviation proportional to X. Thus, the error structure for the example is given by the equation: where υ now stands for the product υ′d2 of eq (2′).

From the replicates, the pooled within standard deviation is readily calculated to be 0.265. The iteration process then yields the following results

Figures 1a and 1b show, respectively, the standard deviations within, and the residuals , as functions of X.

Figure 1

(a) Standard deviations within as a function of X (b) residuals of a function of X.

The figures support the assumpions made concerning the nature of the within and between errors.