Theory behind decision rule simply explained – Part B
There are a few approaches for decision making leading to a conformity statement after testing.
Simple approaches for binary decision rule involving comments of pass/fail, compliant/non-compliant:
- A result implies non-compliance with an upper limit if the measured value exceeds the limit by the expanded uncertainty. See Figure 1A. This is a clear-cut case.
- A result equal to or above the upper limit implies non-compliance and a result below the limit implies compliance – provided that uncertainty is below a specified value or assumed zero. This is normally used where the uncertainty is so small compared with the limit that the risk of making a wrong decision is acceptable. For example, the relative uncertainty of the measured value is 1-2% whilst the Type I error that you are prepared to take is 5%. See Figure 1B.
However, to use such a rule without specifying the maximum permitted value of the uncertainty would mean the risk (probability) of making a wrong decision would not be known.
More complicated approaches for decision rule by use of Guard Bands:
Many learned organizations like ILAC, Eurachem, etc. have suggested to consider incorporating some tolerance limits (or interval) or guard bands (say, +g) added to the nominal specification for risk decision. In this instance, a rejection zone can be defined as starting from the specification limit L plus or minus an amountg (the Guard Band). The purpose of establishing such an “expanded” or “conservative” error on the specification value is to draw “safe” conclusions concerning whether measurement errors are within acceptable limits with a calculated risk as agreed by both the customers and the laboratory concerned.
The value of g is chosen so that for a measurement result greater than or equal to L + g, the probability of false rejection is less than or equal to alpha (Type I error) which is the accepted risk level.
In general, g will be a multiple of the standard uncertainty of the test parameter, u. The multiplying factor can be 1.645 or 1.65 (95% confidence) or 3.3 (>99% confidence). That is to say that the amount of uncertainty in the measurement process and where the measurement result lies with respect to the tolerance limit set help to determine the probability of an incorrect decision.
A situation is, for example, when you set your guard band g to be the expanded uncertainty of the measurement, that is U = 2u above the upper limit of specification. In this case, your estimated critical measurement result plus 1.645u with 95% confidence is well inside the L + g zone, and hence, your risk of making a wrong decision is at 5%. This is shown graphically in Figure 2A below:
Often it is the customer who would specify such a tolerance limit as in Figure 2A, indicating that he would be happy to accept when such tolerance level or guard band is above the upper specification limit or below the lower specification limit in the rejection zones. Hence, the risk is at the customer’s side. It is also known as ‘relax rejection zone’ which covers the Type II (beta) error.
However, if the laboratory operator is to set his own risk limit, it is best for him to set the tolerance limit or guard band below the upper specification level or above the lower specification level to safeguard his own interest. It is known as ‘conservative or stringent acceptance zone’, leading to the Type I (alpha) error.
How to estimate the critical value for acceptance?
Let’s illustrate it via a worked example.
One of the toxic elements in soil is cadmium (Cd). Let the upper acceptable limit on the total Cd in soil required by the environmental consultant client as 2.0 mg/kg on dried matter. The measurand is therefore the total Cd content in soil by ICP-OES method.
Upon analysis, the average value of Cd content in soil samples, say, was found to be 1.81 mg/kg on dried basis, and the uncertainty of measurement U was 0.20 mg/kg with a coverage factor of 2 (95% confidence). Hence, the standard uncertainty of the measurement = 0.20 / 2 = 0.10 mg/kg. This standard uncertainty included both sampling and analytical uncertainties.
Our Decision rule:The critical value or the decision limit was the Cd concentration where it could be decided with a confidence of approximately 95% (alpha=0.05) that the sample batch had a concentration below the set upper limit of 2 mg/kg.
The guard band g is then calculated as:
1.645 x u = 1.645 x 0.10 = 0.165 mg/kg
where k = z = 1.645 for one-tailed value of normal probability distribution at 95% confidence.
The client would then be duly informed and agreed that all reported values below this critical limit value of 1.84 mg/kg were in the acceptance zone. Hence, the test result of 1.81 mg/kg in this study was in compliance with the Cd specification limit of 2.0 mg/kg maximum.
Suggested types of guard bands
The guard band is often based on a multiple, r, of the expanded measurement uncertainty, U where g = rU.
For a binary decision rule, a measurement result below the acceptance limit AL = (L-g) is accepted.
The above example of g = U is quite commonly used, but there may be cases where a multiplier other than 1 is more appropriate. ILAC Guide G08:09/2019 titled “Guidelines on decision rules and statements of conformity” provides a table showing examples of different guard bands to achieve certain levels of specific risks, based on the customer application, as reproduced in Figure 3 Table 1 below. Note that probability of False Accept PFA refers to false positive or Type I error.
It may be noted that the multiplying factor of 0.83 in the guard band of 0.83U as given by ISO 14253-1:2017 is derived by calculation of 1.65/2, where 1.645 has been approximated to 1.65 and 2 is the coverage factor of 1.96 rounded up to the nearest integer, for 95% confidence interval.
A worked example for decision rule on conformity statement
Consider a measurement value y = 2.70ppm with a standard uncertainty of u(y) = 0.20ppm. (Its expanded uncertainty = k x 0.20ppm = 2 x 0.20ppm = 0.40ppm where coverage factor k = 2 at 95% confidence). It is also given that the single tolerance or specification upper limit of Tu = 3.0ppm.
Assuming the normal probability distribution data and a type I error alpha = 0.05 (5%), we are to make a statement of specification conformity at probability of (1-alpha) or 0.95 (95%).
Our decision rule is that :“Acceptance if the hypothesis Ho: P(y< 3.0ppm)> 0.95”is true.
Use Microsoft Excel spreadsheet function:“= 1-NORM.DIST(2.7,3.0,0.2,TRUE)” to calculate P(y< 3.0ppm) to get 0.933 or 93.3%. Note that the function “=NORM.DIST(2.7,3.0,0.2,TRUE)” gives the cumulative area under the curve from far left to right for a value of 0.067 approximately.
Alternatively, we can also calculate a normalized z -value as (2.7 – 3.0)/0.2 = – 1.50, and look up the one-tailed normal distribution table for cumulative probability under the curve with z =|1.5| which gives 0.5000 + 0.4332 = 0.9332, as a normal distribution curve is symmetrical in shape. See Appendix A for the normal distribution cumulative table. In fact, we would get the same answer if we were to use the Excel function “=1- NORM.DIST(-1.5,0,1,TRUE)” as well.
Since 93.3% < 95.0%, the Ho is rejected, i.e. the sample result of 2.70ppm can be declared non-compliant with the specification limit, or put it more mildly, “not possible to state compliance” or “conditional pass” or some other qualification wordings!
If, for discussion sake, the measured value was 2.60ppm, instead. Would it be within the upper specification limit of 3.0ppm by the above evaluation?
Indeed, by following the above reasoning, we would find that the normalized z-value as (2.6-3.0)/0.2 = – 2.0 and the cumulative area under the curve was 0.5000 + 0.4772 = 0.977 which is larger than 0.950. Therefore, the Ho is not rejected, i.e. the sample or test item is declared in compliant with the specification limit.
What is the critical acceptable value Xppm in order not to get Ho rejected?
The task will be simple if we know how to find the critical z -value in a normal distribution curve where the area under the curve on the right tail is 0.05 out of 1.00, or 5%, as we have fixed our Type I (alpha) risk as 5%.
Reading from the normal distribution cumulative table in Appendix A, we note that when z = 1.645, the area under the curve is 0.5000 + 0.4500 = 0.9500. Similarly, the absolute value of Excel function “=NORM.INV(0.05,0,1)” also gives a |z|-value 0f 1.645.
The critical acceptable value X is then calculated as below:
which gives X = 2.67ppm.
The conclusion therefore is that any test value found to be less than or equal to 2.67ppm will be declared as in compliance with the specification of 3.0ppm maximum with 95% confidence (or 5% error risk). Any value found larger than 2.67ppm will be assessed for compliant by considering the higher than 5% risk that the test laboratory is willing to undertake, probably based on some commercial reason. In other words, where a confidence level of less than 95% is acceptable to the laboratory, a compliance statement may be possible. Decision is entirely yours!
Appendix A
