## Welcome to Dijemeric Visualizations

Where photography and mathematics intersect with some photography, some math, some math of photography, and an occasional tutorial.

## Thursday, April 08, 2010

### Calculating Detection Limits - Some Math Required

See Detection What is it and What Does it Mean for Photography for an introduction and Detection in the Testing Laboratory for a discussion of the concepts of detection, quantification, false positives, and false negatives.

Detection Concepts Chart: The blue curve is a distribution of measurements on a sample with nothing in it (a very clean blank).  The red horizontal line represents the detection level.  Only 1% of the measurements of an uncontaminated blank will exceed the detection level.  The 1% level protects against false positives 99% of the time.  The orange line represents a level that should be detected all of the time if the detection level is truly achievable.  It protects against false negatives.

MDL Defined by Code
The standard for the calculation of detection limits (MDLs) is the US EPA Code of Federal Regulations (40 CFR 136).    MDLs are calculated from the variance of the replicate analysis of either a ultra-clean sample (blank) or a quality control sample made from a blank by adding the substance(s) of interest at a level above but near detection. Replicates can be analyzed either in a single batch (usually with 7 replicates) or over time (with 20 or more replicates). If the former approach is used, the calculated detection limit ignores the contribution of errors of time-variance. While both approaches are allowed, there are differences in the results. Single-batch limits tend to be less than comparable time-variance determined limits. Some, but not all, laboratories, include one or more low level quality control samples in each day's batch for the purpose of controlling against false negatives and to generate a data base for the calculation of detection limits and/or confirmation of detection capability.

Problems with the Traditional Approach and a Robust Alternative
The analysis of seven replicates in a single batch generates a detection limit that is artificially low and not achievable on a routine basis.  The MDL as does not account for false negatives: an analytical result below detection has a 50% chance of being a false negative.  Performing a single batch detection limit study requires reassigning staff out of the routine production stream.  The aperiodic approach to confirming detection limit capability on a more-or-less annual basis sidelines detection limits from the routine batch based Quality Control.  By adding a routine QC sample to the analytical stream, it is possible to avoid these problems and develop detection limits that are robust, achievable on a routine basis, account for false negatives, and do not require to reassign staff off of production.

False Negatives
The inability to detect an analyte at two to three times the stated detection limit indicates a false negative. Such a failure may be due to chemical interference, instrument malfunction, or analyst error. In addition to a determination that the LCS is routinely detectable with each preparatory batch, accuracy is calculated and reported as an indicator of agreement between the known (i.e., true) and measured concentrations on an on-going basis and is used to generate control charts for LCS results.

A Routine Check for Capability
Setting the LCS at two to three times detection insures that the specified detection limit is achieved on a routine basis and not just at the time of a single-batch determination. The relative standard deviation (RSD) in this region is approximately 20%, or one-half that of the RSD at detection. Using a Student’s-t multiplier of 2.5 for the 99% confidence limits and twenty samples, yields control limits at +/-50% above and below the mean LCS concentration (2.5x20%= 50%). Using a mean concentration centered at 100%, gives limits of 50% (lower control limit/LCL) to 150% (upper control limit/UCL). The average RSD tracked over time is a robust metric for on-going detection capability for a given LCS mix. Deviations above 20% RSD indicate a concentration that may have been set unrealistically low and conversely deviations below 20% provide an opportunity to achieve even lower detection limits if required.

A Derivation
A derivation of the relationship between the LCS and detection starts with defining the FNQS (False Negative Quality Control Sample) as a fortified method blank at which the relative standard deviation is 20%. That an LCS at such a concentration would be approximately twice the concentration of the detection limit can be derived as follows:

1) RSD(LCS) = S(LCS)/(LCS) where LCS = Laboratory Control Sample,
RSD(LCS) = relative standard deviation of the LCS,
S(LCS) = standard deviation of LCS,
(LCS) = concentration of LCS
2) RSD(LCS) = 20% = 0.20
3) S(LCS) = RSD(LCS)*(LCS)
4) MDL = t *S(LCS) where t = Student's-t(N=20, alpha = 0.01) = 2.5
5) MDL = t*RSD(LCS)*(LCS)
6) MDL = 2.5*0.2*LCS = 0.5*LCS
7) LCS = 2*MDL

Follow the algebra in reverse order to show that at a concentration twice MDL, the RSD equals 20%. This provides the rationale for using the 20% RSD statistic to monitor on-going detection capability based on the routine use of a low-level LCS.

A Simple Equation
By including an LCS in every preparatory batch, data can be collected over time for the determination of method detection limits (MDL). It is necessary to carry the LCS through the complete analytical process, including any preparatory steps. An MDL must include all sources of analytical error, including sample preparatory steps, such as digestion for metals or extraction for organic constituents. After twenty or more LCS results have been collected, the MDL is calculated as:

MDL = t*S(LCS)

When large databases are available and commercial standard concentration ranges may vary slightly between lots, an equivalent calculation formula can be used:

MDL = t*S(Rec)*Ave(LCS)/Ave(Rec)
= t*RSD(Accuracy)*Ave(LCS)

By using standard deviation of the accuracy results rather than the standard deviation of the LCS results, the effects of outlier results can be minimized without censoring the database. This takes advantage of the relative independence of accuracy and concentration over small concentration ranges. Thus a 10% change in a commercially prepared standard mix would have non-measurable effect on recoveries but a dramatic effect on the standard deviation of the LCS results. Since Accuracy = 100*(Measured/True), it follows that t*(AveAccuracy(LSC)/100)*RSD(LCS) = 100t*[(LCS)/True/100)*S(LCS)/(LCS)] = t*S(LCS) = MDL.

Next Time
In the next installment I'll show you how to use some of these concepts to reduce digital noise in your images.