1. User’s Guide to the ‘QDE Toolkit Pro’
Ch 7: 94
April 16, 2002
National Research Conseil national
Council Canada de recherches
User’s Guide to the ‘QDE Toolkit Pro’
Excel Tools for Presenting Metrological Comparisons by B.M. Wood, R.J. Douglas & A.G. Steele
Chapter 7. Covariance and Correlation Coefficients
This chapter reviews some basic information about covariance and correlation coefficients, discusses how the default options are set by the QDE Toolkit Pro, and how easily it can cope with a covariance as is suggested in the ISO Guide to the expression of Uncertainty in Measurement’. Type B methods for establishing covariances are discussed.

2. Covariance and Correlation Coefficients for the QDE Toolkit Pro
Ch 7: 95
Covariance and Correlation Coefficients for the QDE Toolkit Pro
Covariance: For N repetitions of a particular inter-laboratory comparison, between Lab 1 and Lab 2 the covariance < (x1 - < x1 >) (x2 - < x2 >) > Note, in general: <x1> <x2> KCRV
q might be determined by an ISO Guide “Type A method” as
i=1N (x1,i - (j=1N x1,j) / N) (x2,i - (j=1N x2,j) / N).
When, as is usually the case, this method is not feasible we
q might identify covariances by inspecting the detailed uncertainty budgets of the two laboratories. For this ISO Guide “Type B method”, it is easiest to identify intrinsically identical corrections whose uncertainties are exactly shared by the two Labs. This uncertainty component is a fully covariant uncertainty component.

3. Covariance and the adjectives “Systematic” and “Random”
Ch 7: 96
Covariance and the adjectives “Systematic” and “Random”
The ISO Guide to the Expression of Uncertainty in Measurement explicitly deprecates the terms “systematic uncertainty” and “random uncertainty”, requiring a combined “standard uncertainty”…
In the context of a bilateral comparison measurement, the “systematic effects” are covariant, and the “random effects” are independent. Note that these terms are not deprecated by the ISO Guide (see sections and 3.2.3).
The formal consequences of our knowledge of effects that affect Lab 1 and Lab 2 randomly or systematically can be placed in the covariance: zero for random effects, and the full variance as the covariance for fully systematic effects. A fully systematic effect might be a correction which has used the same value and the same uncertainty in Labs 1 and 2 (and, for similar measurements would continue to use them). This is a “Type B method”.

4. Covariance and the Pair Uncertainty Budget
Ch 7: 97
Covariance and the Pair Uncertainty Budget
Effect Lab1 only Both Labs Lab 2 only
Random Effect1 u1, u2,12
Systematic Effect2 ub,22
Random Effect3 u1, u2,32
Systematic Effect4 ub,42
In this case the variance for Lab 1 is: u1,12 + ub,22 + u1,32 + ub,42 and
the variance for Lab 2 is: u2,12 + ub,22 + u2,32 + ub,42 and
the covariance of Lab 1 and Lab 2 is: ub,22 + ub,42. (This is a “Type B method”.)
Having a correlation coefficient r1,2 = r2,1 equal to
ub,22 + ub,42 / [(u1,12 + ub,22 + u1,32 + ub,42 )(u2,12 + ub,22 + u2,32 + ub,42 )]1/2.

5. Covariance and a Shared Sensitivity Coefficient: Theory
Ch 7: 98
Covariance and a Shared Sensitivity Coefficient: Theory
Consider a shared effect such as the temperature coefficient of an artifact that is measured in N Labs at N (possibly) different temperatures {Ti}. If the temperature coefficient  and its uncertainty u() are determined only once, and the same  and u() are used for all Labs, then the correction for Labi is
 [Ti - T0] [1]
with uncertainty
 u(Ti - T0) + u() [Ti - T0], [2]
and giving a covariance between Labi and Labj of
u2() [Ti - T0] [Tj - T0]. [3]
The correlation coefficient is, if ui and uj are the complete standard uncertainties of Labi and Labj, including [2].
u2() [Ti - T0] [Tj - T0] / (ui uj). [4] NB

6. Covariance and a Shared Sensitivity Coefficient: Theory
Ch 7: 99
Covariance and a Shared Sensitivity Coefficient: Theory
If this shared effect has not yet been included in the uncertainty for Labi and Labj, ui’ and uj’, then things are calculated a little differently:
the covariance between Labi and Labj would still be
u2() [Ti - T0] [Tj - T0].
but we have to remember to include the shared effect in ui and uj, the complete standard uncertainties of Labi and Labj ui2 = ui’2 + 2u2(Ti - T0) + u2() [Ti - T0]2
uj2 = uj’2 + 2u2(Tj - T0) + u2() [Tj - T0]2
and then the correlation coefficient of Labi and Labj is
u2() [Ti - T0] [Tj - T0] / (ui uj).

7. Covariance and a Shared Sensitivity Coefficient: Example
Ch 7: 100
Covariance and a Shared Sensitivity Coefficient: Example
Consider this real example where the correlation effects are very small. From u(a), the {Ti - T0} and the {ui} we can quickly set up Excel (sorry, since there are usually more than one effect to consider we do not have a universal macro!) to calculate a correlation coefficient matrix (note that values are both + and -), which we can paste into our comparison Table

8. Copying a correlation coefficient matrix
Ch 7: 101
Copying a correlation coefficient matrix
Check that the diagonal of the correlation coefficient matrix you are copying are all 1’s: QDE Toolkit Pro macros will overwrite any matrix which is not all 1’s on the diagonal. Select it on the source worksheet, copy it (Control-C), select the range on the destination worksheet and Edit|Paste Special select Value and OK, and your new matrix is in place...
The Degrees of Equivalence Tables, QDE, Automated RV uncertainties, etc. can now be re-calculated with correlations:

9. A Comparison Conundrum Concerning Covariance
Ch 7: 102
A Comparison Conundrum Concerning Covariance
Who should identify covariances in the uncertainty budgets of inter-laboratory comparisons?
The ISO Guide envisages that sufficient information be included with the uncertainty budget from each Lab that the Pilot Lab (for example) can identify all important covariances. Perhaps a Key Comparison should be testing this completeness.
We think that covariances could be still more convincing if the Pilot and each pair of Labs would agree on the the covariance to be assigned. If this is done, we think that it must be done when the comparison results are still in escrow, and with the constraint that no Lab’s overall uncertainty can be adjusted. Perhaps a Key Comparison should be seeking the most convincing description of covariances.
Whichever way is chosen, we think that this issue is best decided beforehand, in the protocol for the Key Comparison.