Multilevel analysis of variance used to partition components of optical imprecision in an analyser system with disposable cuvettes

ACKNOWLEDGEMENTS The authors thank the Department of Health and Social Security for providing the system to be tested, Dr Burnett of St Albans City Hospital and Dr Cheng of Watford General Hospital for their help with the comparative studies and the staff of Instrumentation Laboratory Ltd for their co-operation. The advice of Dr S S Brown of the Clinical Research Centre, Harrow and Dr W Percy-Robb of Edinburgh Royal Infirmary is also gratefully acknowledged. REFERENCES [1] Gibson PF, Calzi C, Musetti A, and Caliri R. X International Congress of Clinical Chemistry, Mexico City, 1978. 2 Broughton P M G, Gowenlock A H, McCormack J J, and Neill DW (1974) Annals of Clinical Biochemistry, 11,207-218.


Introduction
With the increasing use of disposable cuvettes in modern spectrophotometric instrumentation it is vital for the analyst to be aware of the various types of errors that can be introduced into the analytical process. Other investigators have described these errors and their propagation in spectrometric systems [1][2][3][4][5] or have examined random errors in various specific components of their systems [6][7][8][9][10]. On the basis of these studies, various professional organisations have proposed guidelines for spectrometric instruments [11][12][13][14]. However, for an analytical system using a disposable rotor containing a large number of cuvettes which is used only once and discarded, a statistical technique must be implemented to quantitate random optical error, to check actual instrument performance against manufacturer's specifications, to assess the quality of incoming supplies for the centrifugal analyser, and to provide criteria for explicit operational practices in the use of the analyser.
Analysis of variance (ANOVA) has previously been used in assigning magnitudes of error to sources within a multiunit instrument or method [15], but has been usually limited to an examination of only two or three variables. This approach has been extended to a three-level nested ANOVA which separates the random optical noise component from errors in absorbance associated with possible changes in the physical parameters of the disposable cuvettes. These errors include variations in the absorbance within-cuvettes, betweencuvettes, between rotors, and between manufacturer's lots of rotors.

Materials and methods
Instrument The centrifugal analyser system evaluated was the Multistat III micro centrifugal analyser [3] (Instrumentation Laboratory, Lexington, MA 02173 USA). The  consists of two modules, an automated loader and the spectrophotometric analyser. The analyser uses disposable plastic cuvettes (20 per rotor) for incubation, mixing, and measurement of absorbance ( Figure 1). The 0.5 cm optical cell is formed by moulding clear windows in the cuvette top and bottom surfaces. Of the 20 cuvettes in the rotor, the first is used as the reference cuvette, and the remaining 19 are used for any combination of standards and samples necessary. Narrow bandpass interference filters are used in the photometer to isolate the spectral range of' interest. Transmitted radiation is measured by using a photomultiplier tube and an autoranging gain circuit so that the signal from the reference cuvette is always set to yield the maximum output from the analog-to-digital converter no matter how much the intensity of the reference beam varies as different filters are used. This study was concerned only with the photometric module error; consequently, no evaluation of the loader module was made. Photometric performance was evaluated at the wavelength most used in this laboratory, 340 rim.

Reagents and solutions
The solution whose absorbance was measured at 340 nm was prepared by dissolving 19. Mathematical model Any single absorbance measurement (Yijkl) can be described as Yijkl =/z + Li + RiO) + CVi(jk) + eijkl (1) where /.t is the true value of the absorbance, Li is the lot effect, RiO is the rotor-within-lot effect, CUi(jk)is the cuvette-within-rotor-within-lot effect, and eijkl is the random error associated with the ijkl-th measurement. The number of lots (4), rotors in a lot (3), cuvettes in a rotor (19), and measurements in a cuvette (12) are assigned as nl, n2, ns, and n4, respectively. The total number of measurements made, N, is defined as" N (nl) (n2) (n3) (n4) (2) A corrected mean (CM) is defined [15]  The total sum of squares, SST, is defined as: SST SSL + SS(ROT in L) + SS(CU in ROT) + SS(RDG in CU) (7) where SSL is the sum of squares due to the lots, SS(ROT in L) is the sum of squares due to the rotors in the lots, Results and Discussion Table 3 shows the results of the three-level nested ANOVA performed on 2736 absorbance measurements. The grand mean was an absorbance (A) of 0.4194. The null hypotheses that variances rCU z rROT 2, and eL 2 equal zero were rejected (a 0.001) by F-testing, giving statistical evidence of variation between lots, between rotors within lots, and between cuvettes within rotors. The mean square value for "Readings-within-Cuvettes" was calculated as 5.18 x 10 -6 ( Table 3). The previous description of the mathematical model showed that this computed value is really the variance of a mean of 32 measurements (s). The "noise" of the photometer unit (i.e., the expected variance for a single photometric sampling) is 1.66 x 10 -4 (SD 1.29 x 10-2 A, CV 3.1%). Though the SD of the single sampling (12.9 mA) seems relatively large, the absorbance value reported to the user is in fact the mean of 32 of these readings. Because of this averaging, the mean value reported will provide a 95% confidence interval of + 4.5 mA. Table 3 now shows that the total variance is then partitioned as follows: between-lot (46.2%); readings-within-cuvettes (40.3%); between-cuvette (9.8%); and between-rotor (3.7%). If one assumes that S OTO R"' R Oll0   [17], the overall standard deviation (SD) of a single photometric sampling is an absorbance of 0.0203 and the coefficient of variation (CV) is 4.8%.

I C U V E T T E S I
Further treatment of the data by a one-way ANOVA on each rotor in turn disclosed that the three rotors from one lot had within-cuvette s values of 0.705, 2.334, and 7.345 mA. On visual examination, rotors taken from the lot with the largest s's were found to contain particulate material inside the cuvettes. On the basis of this finding, operators of the system have been instructed to ascertain that cuvettes and the cell windows are free from particulate contamination before using a rotor. However, subsequent lots of rotors have not appeared to have this contamination problem.
Eight of the nine rotors from the remaining three lots had within-cuvette s values of <( 0.53 mA and the ninth had a within-cuvette s of 1.03 mA. This data was reanalysed by using the three-level nested ANOVA ( Table 4). The overall coefficient of variation of the optical unit was 3.4% of the 0.4144 absorbance grand mean (n 2052). The between-lot variance accounted for 68.6% of the total variance; the between-rotor variance, for 10.2% of the total variance; and the between-cuvette variance, for 17.2% of the total; but the variance component attributed to a single photometric sampling decreased from 40.3% to 4.0% of the total. The SD of a single photometric reading was now 2.82 mA, a decrease from the 12.9 mA computed for all 12 rotors ( Table 3). The between-cuvette SD was 5.9 mA. The linear combination for the four variance components [17] indicated that the overall SD of a single photometric sampling using these three lots of rotors was an absorbance of 0.0142 (CV 3.4%). The null hypothesis of no variation between lots was accepted (a 0.001), but similar hypotheses of no variation between rotors in lots and of no variation between cuvettes in rotors were rejected (a 0.001). The effects of changing rotors and lots may be a realistic estimate of the between-day precision available for equilibrium measurements with this instrument.
Tiffany et al. 5] reported an uncertainty of 0.14 mA at an absorbance of 0.44 for a prototype Multistat III and they  CFor a single one of the 32 photometric readings averaged to report one absorbance value. a Variance of the mean of 32 photometric readings averaged to report one absorbance value. b Variance of any one of the 32 photometn'c readings averaged to report one absorbance value. Computed as 32 x the Mean Square. CFor a single one of the 32 photometric readings averaged to report one absorbance value. equation predicts an s, x of 0.14 mA. Squaring x and multiplying by 30 (the number of readings he averaged to report one value) yields a variance for a single photometric sampling of 5.8 x 10-7 (SD 0.8 mA). Again, the variance quoted for the GEMSAEC is about an order of magnitude lower than that which we compute for our instrument. In contrast to the Multistat III, the GEMSAEC does not use a disposable rotor and does perform an initial correction on each absorbance reading in an attempt to reduce the cuvette-to-cuvette variability. By using only the three rotor lots for the instrument at an absorbance of 0.4144, an SD of 2.8 mA for a single measurement (CV 0.7%) was calculated. The CV's computed for the same absorbances for the instruments of Tiffany and Maclin are about 0.2%. In this case, it is obvious that the lot of rotors which were eliminated severely affected the estimates of the precision of a single measurement because that lot contained rotors which gave highly imprecise absorbance readings. The data shown in Table 4 are much more representative of the actual precision attainable with the instrument in this laboratory.