Quality Control for Multi-variable Problems
Analytical methods were many elements are determined simultaneously
present special difficulties in quality control. The use of individual
control charts for each variable (element) is generally not effective
due to the combinative aspect of the variation. If there are n elements
and the error on each is independent, the probability that all the
results will fall within their respective 95% confidence limits is
(0.95)n. If n = 30 (multi-element analytical packages typically have 25
to 40 elements) this probability is equal to (0.95)30 or 0.2146. In
other words there will be one or more elements out of bounds in 78.53%
of all batches, which on a simplistic interpretation suggests that about
two-thirds of all batches will be rejected when nothing is actually
wrong with the sample data.
In practice there will normally be both correlated error, (due to
instrument variations that affect all elements or groups of elements)
and independent error which is due to variations in the individual
element measurement channels.
Some of the solutions that are being used or tested are listed below.
- Widen Control Limits
The general approach has been to widen the control limits for
multi-element analyses to ( 3s. This has the effect of reducing the
probability of false batch rejection for purely independent error to 1 -
(0.9974)n which is 0.075 (7.5%).
This disadvantages of this approach are a sacrifice in data quality for
each individual element and as more and more elements are included in
data packages (50 + ) the probability of false batch rejection is
relatively high - greater than 10% for 50 elements.
- Multivariate Quality Control
Principle component analysis has been examined. R.J. Howarth, M.H.
Ramsey and B.J. Coles presented ``The potential of multivariate quality
control as a diagnostic tool in Geoanalysis'' at Geoanalysis 97 (Vail,
CO, June 97). Data from a 35 element multi-element analytical package
was reduced to four principle components. They then suggested control
be done by using the five control charts - one for each of the four
components and a fifth for Hotelling's T - representing the multivariate
This is a sophisticated approach that requires large training sets and
significant calculations for each test. Ideally the QC control can be
done `on-line' during the analytical sequence (the instrument software
recognizes a control sample and subroutine testing for in / out of
control is down before the next sample is analyzed - in less than a
minute - so that time is not wasted doing out of specification
analysis). The calculations may be too complex for on-line control.
More problematic is the `components' that are identified. Once a sample
is identified as unacceptable the instrument operator needs to take
corrective actions to resolve the problem before proceeding with further
analyses. If the components are not closely related to physical
properties (or are all combinations of a number of properties),
identification of the `fix' to the out of control problem may be
As well, multiple control charts, one for each component, are still
- Binomial Probabilities
Tables of the binomial probability of M out of N points falling above
the 95th (90th or 99th as well) can be constructed. Control limits can
then set for 95% (( 2s) and the number of elements failing versus the
total number determined be evaluated. If the probability of this event
is less than a predetermined level (i.e. 0.5) the sample can be
determined to be a failure.