Re: MDA vs LLD

[Rick Schoenig <raschoen@SoCA.com>]

ANTHONY F. ARMAGNO wrote:
> 
> HELP!
>     I am trying to revise a lesson plan to teach HPs about counting statistics.
>  I keep getting conflicting definitions for LLD and MDA.  The references seem
> to agree on MDCR, when it is mentioned at all.  Someone, please shed some
> light.
> 
Tony,

Hope this helps and that it is not a waste of bandwidth...but, this
thread comes up every year or so.  I'm also sorry, especially after a 12
hour day, that the formulas have to be in ASCII format.

This may help in understanding some of the published calculations you
may have seen concerning MDA and LLD determinations.  Don't worry too
much about the terms that I've used...it seems everyone has their own
favorites.  It's the formulas, and the assumptions used in there
derivation, that are important.  The following is a brief simplification
of most of the "quick and dirty" formulas that are often quoted as
gospel without qualifications.  To be as inclusive as possible I've
included separate formulas to be used when dealing with the standard
deviations of total counts and for use when dealing in count rates and
count times.  Be careful, this is where a lot of confusion normally
comes into play.  Note that the terms sd and Rd in the formulas are
equivalent to the LLD activity that you are looking for and can best be
approximated only through iteration.  It is normally assumed that this
value is close enough to the background level (by definition you're
looking for the "lowest level above background") that background can be
used negating the need for iteration.  The k2 squared factor (or 2.71 at
the 95% confidence level) algebraically compensates for most of the
error this assumption introduces.  Also note that all of these formulas
are reasonably accurate only if the counting times are sufficient for
the total counts (both background and blank sample) to be above 30. 
Finally note that the LLD value is valid only if all samples above the
MDA value are reported as being "positive" for activity. 

Gosh...This isn't as easy as simple as I thought it was going to be! 


TERMS USED:

MDA (Determistic or Decision Level, Ld) = sample contains activity above
the false positive probability level

LLD (Critical Level, Lc) =predetermined sample activity that could be
detected with both a false positive and a false negative probability
level

k1 = Type I error (false positive) probability factor

k2 = Type II error (false negative) probability factor


FORMULAS FOR LLD AND MDA USING STANDARD DEVIATIONS, where;

sb = standard deviation of the background count,

sd = standard deviation of a blank sample count, and

sc = standard deviation of a sample count containing the "critical
level"


MDA = k1 x (sb + sd)
LLD = (k1 x (sb + sd)) + (k2 x (sb + sc))


If it is assumed that sd ~ sc, then;

MDA = k1 x (sb + sd)
LLD ~ (k1 + k2) x (sb + sd) + (k2 squared)

If k1 and k2 are both assumed to be at the 95% confidence level (1.645),
then;

MDA = 1.645 x (sb + sd)
LLD ~ 3.29 x (sb + sd) + 2.71


If it is assumed that sb and sc are equal (the background and sample
count times are equal), then;

MDA = 2.33 x sb
LLD ~ 4.66 x sb + 2.71



GENERAL FORMULAS FOR LLD AND MDA USING COUNT RATES AND COUNT TIMES,
where;

Rb = background count rate,

Rd = LLD (Critical Level) count rate,

Tb = background count time, and

Ts = sample count time


MDA = k1 x sqrt (Rb/Tb + Rb/Ts)
LLD = (k1 x sqrt (Rb/Tb + Rb/Ts)) + (k2 x sqrt (Rb/Tb + Rd/Rs))


If it is assumed that Rd ~ Rb, then;

MDA = k1 x sqrt (Rb/Tb + Rb/Ts)
LLD ~ (k1 + k2) x sqrt (Rb/Tb + Rb/Ts) + (k2 squared)/Ts


If k1 and k2 are both assumed to be at the 95% confidence level (1.645),
then;

MDA = 1.645 x sqrt (Rb/Tb + Rb/Ts)
LLD ~ 3.29 x sqrt (Rb/Tb + Rb/Ts) + 2.71/Ts


If it is assumed that Tb and Ts are equal, then;

MDA = 2.33 x sqrt (Rb/Tb)
LLD ~ 4.66 x sqrt (Rb/Tb) + 2.71/Ts

E-mail me privately if you have any specific comments or questions.

-- 
Rick Schoenig .... raschoen@SoCA.com