Principles of Fluorescence Spectroscopy

134 TIME-DOMAIN LIFETIME MEASUREMENTS
ues. This ratio depends on the number of degrees of freedom
(ν) for each fit, and this number will depend on the
model. The values of the χ R 2 ratios that are statistically significant
at various levels of probability are available in statistical
tables, of which a few values are listed in Table
4.3. In practice, there are usually many more datapoints
than parameters, so that ν is almost the same for both
fits. For this reason we did not consider the slightly different
number of degrees of freedom in the numerator and
denominator.
For the mixture of p-terphenyl and indole the residuals
of the single-decay-time fit oscillate across the time axis,
which is characteristic of an incorrect model. The value of
χ R 2 = 16.7 is obviously much greater than unity, and according
to Table 4.2 there is a less than 0.1% chance that random
error could result in such an elevated value of χ R 2.
Additionally, the χ R 2 ratio of 17.6 is much larger than the
values of the F statistic in Table 4.3. Hence, the singledecay-time
model is easily rejected for this sample. There is
a potential problem with the use of an F statistic to compare
two χ R 2 values originating from different mathematical
models and the same data set. 189 The use of the F statistic
requires that the residuals for each analysis be independent
from each other. It is not clear if this assumption is correct
when analyzing the same set of data with different mathematical
models.
4.10.3. Parameter Uncertainty:
Confidence Intervals
The NLLS analysis using the multi-exponential model
returns a set of α i and τ i values. It is important to recognize
that there can be considerable uncertainty in these values,
particularly for closely spaced decay times. Estimation of
Table 4.3. F Statistic for Comparison of χ R 2 Values a
Degrees of Probability (P)
freedom
(ν) 0.32 0.25 0.10 0.05 0.025 0.01
10 1.36 1.55 2.32 2.98 3.96 4.85
15 1.28 1.43 1.97 2.40 2.86 3.52
20 1.24 1.36 1.79 2.12 2.46 2.94
40 1.16 1.24 1.51 1.69 1.88 2.11
60 1.13 1.19 1.40 1.53 1.67 1.84
120 1.09 1.13 1.26 1.35 1.43 1.53
4 1.00 1.00 1.00 1.00 1.00 1.00
a From [188, Table A-4]. In general, the F values are computed for different degrees of freedom for each χR 2 value.
For TCSPC data and for FD data, the degrees of freedom are usually similar in the numerator and in the denominator.
Hence, F values are listed for only one value of ν. Additional F values can be found in [1, Tables C-6 and
C-7].
the uncertainties in the recovered parameters is an important
but often-ignored problem. With nonlinear leastsquares
analysis there are no general methods for estimating
the range of parameter values that are consistent with
the data. Uncertainties are reported by almost all the data
analysis programs, but these estimates are invariably smaller
than the actual uncertainties. Most software for nonlinear
least-square analysis reports uncertainties that are based on
the assumption of no correlation between parameters.
These are called the asymptotic standard errors (ASEs). 185
As shown below for a mixture having more closely spaced
lifetimes (Section 4.10.5), the errors in the recovered
parameters often exceed the ASEs. The ASEs usually
underestimate the actual uncertainties in the parameter values.
In our opinion the best way to determine the range of
parameters consistent with the data is to examine the χ R 2
surfaces, which is also called a support plane analysis. 186
The procedure is to change one parameter value from its
value where χ R 2 is a minimum, and then rerun the leastsquares
fit, keeping this parameter value constant at the
selected value. By rerunning the fit, the other parameters
can adjust to again minimize χ R 2. If χ R 2 can be reduced to
an acceptable value, then the offset parameter value is said
to be consistent with the data. The parameter value is
changed again by a larger amount until the χ R 2 value
exceeds an acceptable value, as judged by the F χ statistic
(eq. 4.25) appropriate for p and ν degrees of freedom. 190
This procedure is then repeated for the other parameter values.
The range of parameter values consistent with the data
can be obtained using P = 0.32, where P is the probability
that the value of F χ is due to random errors in the data.
When the value of P exceeds 0.32 there is less than a 32%