## Limit of detectionThis applet provides an interactive demonstration of some of the principles involved in the concept of a Consider a measurement,
subject to measurement uncertainty ("error"), that is made in the
presence of an irreducible "background" (or, "noise"), which
is also random in nature. If the background were constant, we could
just subtract it, but a major part of the detection problem is that the background One way to define LLD (lower limit of detection) is to say that it is By convention the Type I
error rate (denoted by alpha) is usually set at 0.05, the complement of
which is the famous "95% level". The Type II rate (beta) is, in some
definitions of LLD, also set to 0.05. There is no good reason for this
number, it just has become accepted over the years. These error rates
(and the "operating characteristic" curve, which shows the probability
of detection as a function of signal strength) for any form of
hypothesis test should be defined with careful consideration of the What
happens in practice is that we characterize the background, and then
define a "threshold" level based on that background level. With a
measurement having been made (the "a posteriori" situation) there are
four possibilities. The measurement (of signal plus background) might
be Here is the procedure for setting up an LLD problem in this applet. (1) The background mean is fixed at 10 units, for graphing convenience; adjust the width (sigma) of the background PDF as desired. (2) Adjust the "threshold" slider at the upper right so that the alpha probability is 0.05 (use the arrow keys for finer control). (3) Adjust the signal distribution's width (sigma). (4) Adjust the signal distribution's mean until the indicated beta level is also (near) 0.05. When the alpha and beta levels are equal, read off the mean of the signal PDF; this is the minimum detectable level for the signal measurement. There is a display of the LLD that is activated only when the alpha and beta levels are very nearly equal, since that is the conventional way of defining an LLD (note that the alpha and beta levels do not have to be 0.05 although this is very commonly used). Note that (1 - beta) is the probability of correctly concluding that a signal is present when in fact it is present; in statistical practice this is called the "power" of the test. Much, much more can be said about this problem- see any statistics text, or signal processing text, or quality control text, etc. This applet should help students to see the basic ideas of the detection / hypothesis test problem.
WCEvans (01/10), Created with GeoGebra |