Principles of Fluorescence Spectroscopy

162 FREQUENCY-DOMAIN LIFETIME MEASUREMENTS
lated values, as indicated by a minimum value for the goodness-of-fit
parameters χ R 2:
χ 2 R 1
ν ∑ ω
[ φω φcω ] δφ
2
1
ν ∑ ω
[ mω mcω ] δm
2
(5.11)
where ν is the number of degrees of freedom. The value of
ν is given by the number of measurements, which is typically
twice the number of frequencies minus the number of
variable parameters. The subscript c is used to indicate calculated
values for assumed values of α i and τ i , and δφ and
δm are the uncertainties in the phase and modulation values,
respectively. Unlike the errors in the photon-counting
experiments (Chapter 4), these errors cannot be estimated
directly from Poisson statistics.
The correctness of a model is judged based on the values
of χ R 2. For an appropriate model and random noise, χ R 2
is expected to be near unity. If χ R 2 is sufficiently greater
than unity, then it may be correct to reject the model. Rejection
is judged from the probability that random noise could
be the origin of the value of χ R 2. 10,11 For instance, a typical
frequency-domain measurement from this laboratory contains
phase and modulation data at 25 frequencies. A double-exponential
model contains three floating parameters
(two τ i and one α i ), resulting in 47 degrees of freedom. A
value of χ R 2 equal to 2 is adequate to reject the model with
a certainty of 99.9% or higher (Table 4.2).
In practice, the values of χ R 2 change depending upon
the values of the uncertainties (δφ and δm) used in its calculation.
The effects of selecting different values of δφ and
δm has been considered in detail. 12–13 The fortunate result is
that the recovered parameter values (α i and τ i ) do not
depend strongly on the chosen values of δφ and δm. The
parameter values can be expected to be sensitive to δφ and
δm if the data are just adequate to determine the parameter
values, that is, at the limits of resolution.
For consistency and ease of day-to-day data interpretation
we use constant values of δφ = 0.2E and δm = 0.005.
While the precise values may vary between experiments,
the χ R 2 values calculated in this way indicate to us the
degree of error in a particular data set. For instance, if a particular
data set has poor signal-to-noise, or systematic
errors, the value of χ R 2 is elevated even for the best fit. The
use of fixed values of δφ and δm does not introduce any
ambiguity in the analysis, as it is the relative values of χ R 2
that are used in accepting or rejecting a model. We typically
compare χ R 2 for the one-, two-, and three-exponential
fits. If χ R 2 decreases twofold or more as the model is incre-
mented, then the data probably justify inclusion of the additional
decay time. According to Table 4.3, a ratio of χ R 2 values
of 2 is adequate to reject the simpler model with a 99%
certainty. It should be remembered that the values of δφ and
δm might depend upon frequency, either as a gradual
increase in random error with frequency, or as higher-thanaverage
uncertainties at discrete frequencies due to interference
or other instrumental effects. In most cases the recovered
parameter values are independent of the chosen values
of δφ and δm. However, caution is needed as one approaches
the resolution limits of the measurements. In these cases
the values of the recovered parameters might depend upon
the values chosen for δφ and δm.
The values of δφ and δm can be adjusted as appropriate
for a particular instrument. For instance, the phase data
may become noisier with increasing modulation frequency
because the phase angle is being measured from a smaller
signal. One can use values of δφ and δm which increase
with frequency to account for this effect. In adjusting the
values of δφ and δm, we try to give equal weight to the
phase and modulation data. This is accomplished by adjusting
the relative values of δφ and δm so that the sum of the
squared deviations (eq. 5.11) is approximately equal for the
phase and modulation data.
Another way to estimate the values of δφ and δm is
from the data itself. The phase and modulation values at
each frequency are typically an average of 10 to 100 individual
measurements. In principle, the values of δφ and δm
are given by the standard deviation of the mean of the phase
and modulation, respectively. In practice we find that the
standard deviation of the mean underestimates the values of
δφ and δm. This probably occurs because the individual
phase and modulation measurements are not independent of
each other. For simplicity and consistency, the use of constant
values of δφ and δm is recommended.
In analyzing frequency-domain data it is advisable to
avoid use of the apparent (τ φ ) or modulation (τ m ) lifetimes.
These values are the lifetimes calculated from the measured
phase and modulation values at a given frequency. These
values can be misleading, and are best avoided. The characteristics
of τ φ and τ m are discussed in Section 5.10.
5.1.2. Global Analysis of Frequency-Domain Data
Resolution of closely spaced parameters can be improved
by global analysis. This applies to the frequency-domain
data as well as the time-domain data. The use of global
analysis is easiest to visualize for a mixture of fluorophores