I saw an interesting post by Dr. Roederer on Purdue list archives discussing the topic of negative fluorescence. This question came up in my email as well just last week, so it's nice to get a real scientist's (as opposed to us flow-jocks) perspective:

Recently, I was queried by a user who complained about the new data

transformations (Biexponential, Logicle, Hyperlog, etc.), which show

negative values. The user complained that negative fluorescence is

not biologically meaningful.Indeed, the user is correct. There is no meaning to negative

fluorescence; there is no such thing. However, what we are

displaying on the graphs is no longer fluorescence, it is a corrected

measurement derived from fluorescence. The corrected measurement has

been manipulated by both the instrument firmware, and possibly

software (in the case of compensation), before it is displayed.When the fluorescence is detected off the PMT, the voltage

measurement has "baseline" subtracted from it. Baseline is measured

during the time between pulses. However, there is an error in the

measurement of both the fluorescence as well as the baseline. So,

for "negative" cells (cells with little or no fluorescence), there's

a chance that the measurement for the cell will be less than the

measurement for the baseline, because of the errors involved. After

subtraction, the measurement is less than zero.Likewise, for compensation, additional measurement errors from all of

the compensation channels are aggregated into the final computed

value; thus, frequently this value is less than zero.Look at it another way: there is a distribution of measurements

around the true value; that distribution is dictated by the error in

the measurement (i.e., the SD). If the true fluorescence is nearly

zero, and the error is fairly large, then of course we expect the

distribution to rise to the positive values -- with 95% of the values

being within a few SD of the "true" value. But measurement errors

are always symmetrically distributed--so the distribution has to

reach into the negative values just as far. It would be incorrect to

set these negative values to zero, because they you would

systematically increase the mean of the distribution (you are always

increasing negative values, but you are not decreasing any other

values).But the bottom line is that we are not displaying measured

fluorescence values -- those of course are always positive. We are

displaying corrected measurements. Notably, the corrected

measurements are still linearly related to the fluorescence of the

original event.

*update 5/13/08* There was another question posted about this on the list, and Dr. Roederer added..

negative values are not a consequence of the biexponential scaling --

that's only a visualization too. Rather, negative values are a

consequence of two different operations: (1) baseline restore -- a

subtraction process that instruments do on a event-by-event basis

before reporting the data to the computer; on the DiVa, this can

result in a value less than zero for events that have essentially no

fluorescence, because the error in the estimation of the background

can exceed the absolute magnitude of the background itself; and (2)

compensation, which subtracts values from the fluorescence signal

based on the fluorescence in other channels -- and once again, the

error in the estimation of the amount to subtract can exceed by a

large amount the absolute magnitude of the value after subtraction.

With both of these operations, you will end up with a distribution of

events near zero, but with a standard deviation such that the tails of

the distributions go negative or positive.

If you've over-compensated, or if you've gated on events to select

primarily events that are below zero, then any estimate of the central

tendency of this population (median, mean, geometric mean) will be

negative. If the value is substantially less than zero, then you have

overcompensation problems. If the value is near zero, then it's

probably just random luck that it wasn't above zero... In the former

case, I would be cautious about any MFI values to begin with, since

with improper compensation settings you cannot rely on the MFI. In

the latter case, whether you use the actual value or you use "zero"

probably won't change your answer, so it becomes academic.

Finally, a purist may note that the geometric mean fluorescence should

never be less than zero (because of how geometric mean is

computationally defined). Strictly speaking, this is correct -- a

geometric mean can never be negative. The software, however, is

attempting to compute a central tendency that is weighted similarly to

the geometric mean for these distributions, where negative values can

exist. Thus, it uses the same biexponential transformation to aid in

defining a statistic that gives you this value -- the "geometric" mean

has the same functionality in terms of providing an estimate of the

center of the population as it is drawn in your graphs (after all,

this is origin of the term "geometric") -- and thus, a population that

is centered below zero would have an "undefined" geometric mean by the

original algorithm, but a meaningful and applicable geometric mean in

biexponential geometry.

mr

(Note: here's a trivial example of why the "standard" geometric mean

is not a good estimate of central tendency when you have values near

zero. If you have a single cell with zero fluorescence, and a million

cells with fluorescence of 10^5, then the geometric mean of this

population of 1,000,001 cells is.... zero! Not a very good estimate.

However, software that uses the biexponential geometry to calculate

the gMFI will give the much better value of 10^5, as it would for the

mean and the median...)

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