1208 Volume 54, Number 8, 2000 APPLIED SPECTROSCOPY 0003-7028 / 00 / 5408-1208$2.00 / 0 q 2000 Society for Applied Spectroscopy On the Role of Statistical Weighting in the Least-Squares Analysis of UV-Visible Spectrophotometric Data JOEL TELLINGHUISEN Department of Chemistry, Vanderbilt University, Nashville, Tennessee 37235 Spectrophotometric data are inherently heteroscedastic, which means that least-squares component analyses of absorbance spectra should properly employ weighted ts. The effects of neglecting weights (the common practice) is examined through Monte Carlo calculations on a three-peak model having two closely overlapping components of comparable strength and a third component that appears as a weak shoulder. The results show statistically signicant loss of precision in all parameters; however the magnitude of this loss is &30% for realistic conditions. For comparison, experimental spectra of I 2 in CCl 4 (which was the basis for the Monte Carlo test model) are similarly analyzed. These results suggest that model in- adequacy is likely to be a greater practical problem than neglect of weights, because the great precision of spectrophotometric data places extreme demands on the t model. In the present instance, for example, incorporation of a correction term for the sinusoidal error in the spectrometer wavelength signicantly reduces the t chi-square. Index Headings: UV-visible spectrophotometry; Statistical errors; Calibration; Least-squares weights; Spectral tting. INTRODUCTION In analytical applications, spectrophotometry has tra- ditionally been a ‘‘selected wavelength’’ method, in which absorbance data are recorded at one or more wave- lengths chosen to optimize the sensitivity for the target analyte(s). However, the ease with which an abundance of very high quality spectrophotometric data can be col- lected in the laboratory, coupled with the power and so- phistication of modern computational methods, has led to an increasing use of data analysis techniques in which entire spectra or sets of spectra are modeled as sums of component spectral bands. 1–6 Nonlinear least-squares ts to such models yield information about the spectral com- ponents in addition to the usual analyte concentrations. Such analyses have generally been carried out by using unweighted regression methods, 6 even though spectro- photometric absorbances are strongly heteroscedastic. 7–9 Unweighted analysis is satisfactory in many cases, be- cause sample preparation errors often dwarf the instru- mental errors. However, there are also cases when the spectral information is the primary goal of the analysis, 3 and in these situations the dependence of the statistical error in the absorbance (A) on wavelength and on A itself should properly demand a weighted t, since minimum- variance estimates of the parameters in the model are obtained only when the data are weighted as their recip- rocal variances. 10,11 To carry out a properly weighted analysis of absor- bance spectra, one must rst obtain reliable information about the statistical errors of measurement. In a recent Received 7 February 2000; accepted 3 April 2000. study I described a practical procedure for achieving such a statistical error calibration. 12 In the present work I ad- dress the question: What difference does it make? To an- swer this question I have carried out Monte Carlo (MC) calculations on a realistic model based on the observed absorption spectrum of I 2 in inert solvents. To put these results in perspective, I have also analyzed some exper- imental spectra of I 2 in CCl 4 . The results show that, while the loss of precision through neglect of weights is sig- nicant in a statistical sense, it may still be insignicant in a practical sense. The reason for this apparent contra- diction is that modern spectrophotometers yield absor- bances of such great precision as to place extreme de- mands on the t model used to analyze the spectra. To illustrate this point by anticipating the results to be dis- cussed below, the x 2 values for the nonlinear ts of the experimental spectra dropped by a factor of ; 2 when a correction for the sinusoidal ‘‘wobble’’ error in the wave- length was added to the t model—even though the am- plitude of this wobble error was only ; 0.05 nm, as com- pared with typical component bandwidths of ; 100 nm! EXPERIMENTAL The I 2 /CCl 4 spectra were recorded over the spectral range 400–850 nm, with the same instrument that was used in the recent error calibration study 12 (Shimadzu Model UV-2101PC) and the same instrumental parame- ters (1.0 nm slit widths, 1.0 nm recording interval, me- dium scan speed). As was noted in the earlier study, the statistical errors can depend on the instrumental param- eters in unanticipated ways, so it is important that the instrumental settings be the same for the error determi- nation as for the recording of the spectra to be analyzed. The I 2 concentrations ranged from 5.7 3 10 24 mol/L to 1.6 3 10 23 mol/L, giving peak absorbances (near 520 nm) between 0.52 and 1.50 for 1 cm cuvettes. The instrument was wavelength calibrated with the aid of atomic discharge lamps (Hg, Ar, Ne), the light from which was admitted through the source entrance slit, by simply removing the D 2 lamp and lining the discharge lamp up on axis. These atomic spectra were recorded in ‘‘Energy’’ mode, using the narrowest slits (0.1 nm), slow- est scan speed, and highest recording density (0.05 nm). The peaks were located by least-squares tting the data in the vicinity of a line to a Gaussian plus a background. MONTE CARLO CALCULATIONS To check the effects of proper vs. improper weighting of spectrophotometric data on the outcome of a least- squares (LS) spectral component analysis, I have carried out Monte Carlo calculations using a model based on preliminary results from a study of the absorption spec-