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.

Detection and Quantification in the Testing Laboratory

For a previous discussion on the meaning of detection and quantification see Detection - What is it and What Does it Mean for Photography?.

Your Drinking Water
A drinking water laboratory tests water samples to determine what is in them. The ultimate objective is to ensure the safety and reliability of drinking water supplies. But every analytical measurement has error. To control error, laboratories must include check samples to distinguish the known from the unknown. Sometimes the quantity of a substance is too low for the method to distinguish from a sample with nothing in it (a blank). The lowest threshold at which a method can determine the presence of a substance versus its absence is called the detection limit. In other words, detection occurs when a test method confirms the presence of a substance (analyte).

Detection and Quantification
A method detection limit (aka MDL in United States and Lc in most other countries) represents the detection capability of a method including all analytical and preparatory steps. While meaningful in a statistical context, an analytical result at or slightly above detection is not quantified and by itself cannot generally be used to make decisions about the quantity of analyte present in the analyzed sample other than the fact that it is present. The IUPAC (International Union of Pure and Applied Chemistry) term for detection is the Critical Value (Lc).

Quantification is when the method establishes that the amount of the substance is sufficiently large that noise is a small fraction enabling scientific, legal, and process decisions. A quantified result is one that can be assigned a number with a known precision and accuracy while a detected result is one where it is only possible to state that a substance exists at some quantity above zero but the exact number cannot be known.

A quantified result is one with an acceptably low error so that a decision can be made about the quantity of analyte. Reporting Limit (RL), Minimum Level (ML), Detection Limit for Reporting (DLR), FNQS (False Negative Quality Control Sample), and Practical Quantitation Limit (PQL) are some of the terms used to describe the minimum concentration necessary to make a quantified decsion. The IUPAC term for quantification is the Limit of Quantification (Lq). Another term, Limit of Detection (LOD) is an IUPAC quantity between the Lc and Lq that minimizes false negatives.

Measurement Error and False Decisions
All analytical results include measurement error so that any result has a specified probability of being either falsely positive or falsely negative. An substance detected when it is not present is a false positive; a substance not detected when it is present is a false negative. The detection limit (MDL/Lc) is a statistic that is defined to limit the odds of a false positive to 1% while a quantification limit (FNQS, ML, DLR, Lq) is generally defined to limit the odds of a false negative to 5%. Thus a measurement that barely exceeds the MDL could be in error 1 time out of 100 and a measurement that is non-detect at the quantification level could be in error 5 times out of 100. In the former case, the the analyzed substance would be falsely concluded as being present and in the latter it would be falsely concluded as being absent.

Next
In my next blog, I'll show how to calculate detection limits in a way that avoids taking staff out of routine production.

What is Detection? What's it Mean for Photography?

Noise, Observations, Detection
Noise in our daily observations, in analytical measurements, and in photography obscures informational content.  That information can be in a conversation, a test result on a water sample, or detail in a photograph.  Whether we can extract information from a measurement or observation depends upon the sensitivity and selectivity of the measurement method otherwise known as detection and quantitation capabilities.  In this series on detection and quantitation, post 1 will introduce the concepts, post 2 will demonstrate the use of detection and quantitation in a laboratory setting, and post 3 will develop the mathematical concepts.  Posts 4-6 will provide examples as applied to photography.

Detection and Quantitation in an Everyday Experience
Let me start with an example based on this scene from the bazaar in Cairo.  As the two women walk through the gate, the woman to the right turns and asks her friend about the wares of the merchant ahead of them.  The other woman knows her friend is talking to her but cannot hear her above the crowd so asks her to speak up.  She can now hear her friend but cannot understand her words. This is detection.  An observation or measurement distinguishable from the noise is a detection.  She asks her to speak up some more and finally can hear her words and understands her question.  This is quantification; an observation or measurement sufficiently different from the noise to allow decisions to be made.

In the next two blogs I will discuss more of the theoretical concepts before presenting some applications to photography.  I hope to show how the concepts of detection and quantitation relate to photography and are the theoretical basis for dealing with digital noise, optimization of shadow details, and building models to distinguish unaltered vs modified digital images.


The Promise for Photography
Here's the proposal: a photograph with maximum DOF (i.e., minimum aperture), great shadow detail, shot in low light, at low ISO, and exposed for the low end of the highlights (zone 8).  Using the theories on digital noise and detection, I will show you how it can be done.

Next
Detection in the Testing Laboratory for a discussion of detection and quantitation concepts in the testing laboratory and how laboratories can check to ensure that the test results are valid.

Footnote
For those curious about the photo. It is a composite of five photographs all taken within a one hour period in the Cairo souk (market place). Western elements in the main image were replaced with more traditional subjects: two men in jeans walking under the arch were replaced with the two women; the apartments in the background were replaced with a scene from a nearby street; the women talking to the merchant were from a nearby street talking to another merchant and replaced a group of tourists. The turbaned merchant and man entering from the right are as they appeared in the primary photo.

Twenty Random Points Resolved

What is Random?
Random to many means 'chaotic and without structure' but random has pattern, predictability, and order. Flip a coin 100 times. Before you do that, see if you can predict the number of heads. You will probably predict 50 and it may or may not be correct. You should be close, however. How close? That's a matter of statistics and why I say random events are ordered events and are predictable. I would predict on a toss of an unbiased coin, one you haven't shaved or weighted, that it will come up heads 40 to 60 times and if you were to repeat the 100 tosses 1000 times, more than 90% of the tosses would be 40 to 60 heads.

Predicting Out of Control Events

BCC Storefront Art Exhibit

BCC Storefront Art Exhibit
Originally uploaded by misterken

The Berkeley Camera Club together with the Berkeley Storefront Art Project presents a selection of work from its members.

On display in the former Ross Department Store at 2190 Shattuck, Berkeley starting March 9, 2010.

Of Gardening and Mice

Spring is the time for wildflowers, planting the vegetable garden, and mice.

Hoping to get an early start on this year's garden, I bought some starts for peas, lettuce, spinach, and artichokes. There are three raised beds with one sprouting chard and parsley from last year, another with a nitrogen-fixing bean nearing harvest, and an empty bed readied for the starts. The garden tools are kept nearby in a converted horse stable with garden gloves in a mouse-proof cupboard. This proved convenient not just for me but also a mother mouse who discovered the cupboard wasn't mouse proof and contained most excellent material for a nest: my gloves.

On opening the cupboard a gray object moves rapidly from behind a box and backs into a corner - with two baby mice still not fully disengaged from mom; she is in the process of giving birth. I changed my plans and ran for camera and surgical gloves. Momma was still hugging the corner when I returned and I was able to take a few shots before she ran and disappeared behind another box. A minute later she reappeared sans babies, scampered up the inside of the cupboard, and out through a very small hole. I then removed the box and ... no babies! I removed every item in the cupboard and looked at the bottom and sides of each object removed... no babies.

It is a mystery to me. I can only presume she was either able to reverse the birthing process or (gasp!), she ate them.

Red Bubble T-Shirts

A little self promotion: I've put some photos on Red Bubble that work with T-shirts.

A Vision for Restoration

"A Vision for Restoration" shows a portion of the former Sakai Brothers Rose Company that the City of Richmond plans to incorporate into the site development as a reminder of its agrarian past. It will be displayed at the Bucci's exhibit with photos from photographer Dale Mead and me.

The Sakai nursery started in 1906 with 2.5 acres in Richmond and a single greenhouse salvaged from Berkeley. The Oishi nursery started shortly after. The combined nurseries grew as more land was acquired and more buildings were added, and were shut down in 1942 during World War II. They resumed operations when the families returned after the war and continued until fairly recently.

"The property, located near Wall Avenue across I-80 from Home Depot, is not only the last vestige of a once thriving
industry that spread along the Richmond-El Cerrito border, but it is arguably the last of the intact pre-WW II nurseries in California"

I have visited the site several times, beginning in January 2007 and most recently in September 2009. Roses still bloom in abundance even though unattended.

Exhibit at Bucci's

Yes, it's official. Dale Mead and I will be exhibiting some of our photos at Bucci's Restaurant in Emeryville (1601 Hollis St) starting with a reception on December 10 (4-7pm) and running through January 15. So if you haven't found that perfect Christmas gift yet, come on by and see if there's something you like.

construction Choreography

Pile driving noise can be bothersome to those who work or live nearby, but build we must. Or, if you have a camera, join the fun!

LOcal time

High Tech Noon

The Annual Laboratory Christmas Party-Luncheon 2008

Big Bang Theory - The Cooper-Hofstadter Polarization

Worth watching - even if you are not a physicist! And if the subtitles work, a great way to practice your Spanish.

On Location Model Shoot Aboard the USS Nokomis

A portrait studio offers great control of lighting and a skilled photographer can create wonderful images, but for a story to go with it, on-location shooting offers greater opportunities and challenges. Together with photographers Jon Bauer, Dave Lund, and Vic Miller, I had the opportunity to learn some techniques for on-location photography using ambient light, reflectors, and off-camera flashes. Of course it helps to have truly elegant models, as we did with Reina Lam and Zarif Sadiqi together with the makeup artist, Erika Zunon.

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Chicago and WEFTEC 08

I love Chicago, in October at least. The rest of the year I'm told the weather in Chicago is either too hot and humid or too cold. But October is great. I went there for the annual Water Environment Federation (WEF) conference and give a talk. with 33K attendees there was a lot to see, hear, and do - especially if one decides to get out and walk and use the transit system, which I did.

The Palmer House is grand, especially so after a very expensive ($300M) remodel job. Fellow presenter Saskia V was on the 20th floor, so I got some night shots from there in addition to the grand hall, ballrooms, staircases, etc. I did get out and walked a bit, taking in the "Bean" (aka Cloud Gate), Museum of Art, and the Field Museum of Natural History. Back at the conference, I caught Mayor Daley's keynote speech and heard how a big city can get 'green.' I found an unoccupied walkway and shot a panorama of one of the exhibits. It's really hard to see the layout when you're on the floor. The Op's Challenge was on the floor, so I got a couple of shots of the action. So on with the show...

Sounds of Paris - A Concert in the Church

http://www.cweb.com/audio/Sounds_of_Paris_-_Concert_at_the_Church

Special District One and the Annual BBQ of 2008