Principles of Fluorescence Spectroscopy

132 TIME-DOMAIN LIFETIME MEASUREMENTS
predict the probability for obtaining a value of χ R 2 because
of random errors. These values can be found in standard
mathematical tables of the χ R 2 distribution. Selected values
are shown in Table 4.2 for various probabilities and numbers
of degrees of freedom. Suppose you have over 200 datapoints
and the value of χ R 2 = 1.25. There is only a 1%
chance (P = 0.01) that random errors could result in this
value. Then the model yielding χ R 2 = 1.25 can be rejected,
assuming the data are free of systematic errors. If the value
of χ R 2 is 1.08, then there is a 20% chance that this value is
due to random deviations in the data. While this may seem
like a small probability, it is not advisable to reject a model
if the probability exceeds 5%, which corresponds to χ R 2 =
1.17 (Table 4.2). In our experience, systematic errors in the
data can easily result in a 10–20% elevation in χ R 2. For
example, suppose the data does contain systematic errors
and a two-component fit returns a value of χ R 2 = 1.17. The
data are then fit to three components, which results in a
decreased χ R 2. If the three-component model is accepted as
the correct model, then an incorrect conclusion is reached.
The systematic errors in the data will have resulted in addition
of a third component that does not exist in the experimental
system.
While we stated that assumptions 1 through 6 (above)
are generally true for TCSPC, we are not convinced that
number 5 is true. Based on our experience we have the
impression that the TCSPC data have fewer independent
observations (degrees of freedom) in the TCSPC data than
the number of actual observations (channels). This is not a
criticism of NLLS analysis or TCSPC. However, if the
effective number of independent datapoints is less than the
number of datapoints, then small changes in χ R 2 may not be
as significant as understood from the mathematical tables.
Complete reliance on mathematical tables can lead to
overinterpretation of the data. The absolute value of χ R 2 is
often of less significance than its relative values. Systematic
errors in the data can easily result in χ R 2 values in excess
Table 4.2. χ R 2 Distribution a
Probability (P)/
degrees of freedom 0.2 0.1 0.05 0.02 0.01 0.001
10 1.344 1.599 1.831 2.116 2.321 2.959
20 1.252 1.421 1.571 1.751 1.878 2.266
50 1.163 1.263 1.350 1.452 1.523 1.733
100 1.117 1.185 1.243 1.311 1.358 1.494
200 1.083 b 1.131 1.170 1.216 1.247 1.338
aFrom [1, Table C-4].
b Mentioned in text.
of 1.5. This small elevation in χ R 2 does not mean the model
is incorrect. We find that for systematic errors the χ R 2 value
is not significantly decreased using the next more complex
model. Hence, if the value of χ R 2 does not decrease significantly
when the data are analyzed with a more complex
model, the value of χ R 2 probably reflects the poor quality of
the data. In general we consider decreases in χ R 2 significant
if the ratio decreases by twofold or more. Smaller changes
in χ R 2 are interpreted with caution, typically based on some
prior understanding of the system.
4.9.4. Autocorrelation Function Advanced Topic
Another diagnostic for the goodness of fit is the autocorrelation
function. 3 For a correct model, and the absence of
systematic errors, the deviations are expected to be randomly
distributed around zero. The randomness of the deviations
can be judged from the autocorrelation function. Calculation
of the correlation function is moderately complex,
and does not need to be understood in detail to interpret
these plots. The autocorrelation function C(t j ) is the extent
of correlation between deviations in the k and k+jth channel.
The values of C(t j ) are calculated using
C(tj) ( 1
m ∑m DkDkj ) / (
k1
1
n ∑n D
k1
2 k )
(4.23)
where D k is the deviation in the kth datapoint and D k+j is the
deviation in the k+jth datapoint (eq. 4.23). This function
measures whether a deviation at one datapoint (time channel)
predicts that the deviation in the jth higher channel will
have the same or opposite sign. For example, the probability
is higher that channels 50 and 51 have the same sign
than channels 50 and 251. The calculation is usually
extended to test for correlations across half of the data
channels (m = n/2) because the order of multiplication in