Anabst, April, 1979, Vol. 104, $9. 299-312 299 Statistical Appraisal of Interference Effects in the Determination of Trace Elements by Atomic-a bsorption Spectrop hotometry in Applied Geochemistry Michael Thompson, Stephen J. Walton and Shirley J. Wood Applied Geochemistry Research Group, Department of Geology, Imperial College, London, S W7 2BP Interference effects in the determination of trace elements by atomic- absorption spectrophotometry in applied geochemistry have been studied by a statistical appraisal and described in terms of a simple two-parameter mathematical model. The experimental design incorporated features that would allow possible deviations from the model to be detected, but no serious deviations were detected except in a deliberately chosen example. The study led to the identification of some important interference effects that could affect interpretation of geochemical data and also provided a formula that could be applied to correct the crude results.Keywords : Applied geochemistry ; mineral exploration ; atomic-absorption spectrophotometry ; interferences ; chemometrics Flame atomic-absorption spectrophotometry is by far the most frequently used analytical method in applied geochemistry, whether in the field of mineral exploration or of environ- mental studies. Although sample media and dissolution techniques vary, most commonly a rock, soil or sediment sample is treated with a mineral acid or mixture of acids to release into solution the trace elements of interest, notably copper, lead, zinc and nickel.1,2 Depending on the mineralogy of the sample and the acids used, considerable concentrations of the major metallic constituents of such samples are present in the solution presented for analysis.These elements (calcium, magnesium, iron, aluminium, sodium and potassium) are known to cause a variety of interferences, but apart from background correction by means of a continuum source, the effects are largely ignored in applied geochemical work.334 This is because, to be cost-effective, geochemical analyses have to be simple, rapid and cheap. Procedures such as solvent extraction, which separate the trace elements of interest from the major constituents, are usually inadmissible for this reason. Matrix matching is not usually practicable because wide variation in bulk composition within a batch of samples is common.In view of this widespread use of atomic-absorption spectrophotometry in applied geo- chemistry it is desirable to have a comprehensive and readily applicable account of the interference effects, so that circumstances where the interference is intolerable (i.e., likely to produce an incorrect interpretation of the data) can be identified. Govett and Whitehead5 have studied the effects of the major constituents on the deter- mination of trace amounts of copper, zinc, cobalt and nickel. At low concentrations of the major elements, it was found that the main effects were enhancements due to background absorption. At higher concentrations the enhancements were diminished or even reversed. The effects were found to depend on the concentrations both of the major elements and of the trace elements.However, no clear conclusion was drawn (except the need for caution), because of the difficulty of generalising the results obtained in the absence of any simple mathematical model. A further difficulty in interference studies is the possibility of complex interactions between the major constituents, producing effects on the apparent concentration of the trace analyte that are not predictable from results obtained with each major constituent separately. This difficulty has been pointed out by Woodis et aL6 and Thompson et al.' Usually the possi- bility of complex interactions is ignored because of the difficulty in elucidating them. A simple way to detect such interactions for two interfering elements would be to obtain a three-dimensional response surface at a fixed concentration of the trace analyte, representing300 THOMPSON et al.: STATISTICAL APPRAISAL OF INTERFERENCE Analyst, VoE. 104 the apparent level of the element as a function of the two interferents. This procedure would require at least a 10 x 10 matrix of points to obtain a reasonable representation of the surface, and the experiment would have to be repeated at several analyte concentra- tions. However, this approach would scarcely be possible for more than two interferents because of the large number of possible combinations and the difficulty of representing a surface in more than three dimensions. In this study, we used a simple mathematical model of interference, which greatly facilitated the interpretation of the results obtained and led to straightforward decisions as to whether interference was likely to be important.The model is the same as one that has been widely and successfully used in X-ray fluorescence analysis,* but has not, apparently, been fully developed in atomic-absorption work. An experimental design, based on the model, was devised to enable the results to be obtained with the minimum of effort, to allow the data to be tested for possible deviations from the mathematical model and, for the elements studied, to confirm the absence of complex interactions. Where an interference effect was found to be important, the coefficients derived from the data provided correction factors that could be used to improve considerably the accuracy of the raw result. While our conclusions would be expected to have broad generality in geochemical work, the magnitude of the effects are liable to vary with different instrumental arrangements and operating conditions.Consequently, the data presented in this work should be regarded as illustrating the method of investigation. Theoretical Model Interference of a major constituent on a trace analyte can be considered to consist of two components, one independent of the trace analyte concentration and the other dependent on it. In terms of the effect on a trace element calibration graph, these can be considered as a translational effect and a rotational effect as shown in Fig. 1. The translational effect True concentration True concentration Fig.1. Results of (a) the rotational effect and (b) the translational effect on a calibration line. The full lines indicate no interference and the broken lines a constant concentration of interferent. corresponds, in atomic-absorption spectrophotometry, to “background” interferences, such as molecular absorption and light scattering, and the rotational effects to physical, chemical and ionisation interferences. In general, the extent of the effect will be a function of the concentration (X) of the major constituent, so that the rotational effect can be formulated as The two components can operate in combination. where C, and C, are, respectively, the apparent concentration and the true concentration of the analyte. In this instance f(0) = 1, as C, = CT when X = 0, and we have A simple assumption is that C,/C, is a linear function of X.April, 1979 EFFECTS IN THE DETERMINATION OF TRACE ELEMENTS BY AAS where a is the constant coefficient for the rotational effect. In a similar way, the translational effect can be formulated as 301 Again, the assumption is made that the effect is a linear function of X.f(0) = 0 and we obtain the equation In this instance C, = C T + bX . . .. . . .. - * (4) where b is the coefficient for the translational effect. together we have When the two effects are operating C, = C, (1 + a x ) + bX . . .. .. - . (5) or where d, = CTa + b. For a given value of CT, determine the value of dc [= (dC,/dX),,] by plotting the apparent concentration at various levels of X. The slope will have a constant value if the initial assumptions are correct.In a similar manner values of d, at various levels of C, are obtained. As d, = C,a + b, a plot of d, against C, will give a line with a slope of a and intercept of b, as illustrated in Fig. 3. With the values of a and b determined, for known X the true concentration can be obtained from the apparent by Experimental quantification of a and b can be undertaken in the following manner. This is illustrated in Fig. 2. inverting equation (5) : CT = (C, - bX)/(1 + a x ) . . .. .. r C e Cl 2 L_J interferent concentration ( X I (7) Fig. 2. Effect of increasing inter- ferent concentration ( X ) on the appa- Fig. 3. Relationship between d , rent analyte concentration (CA) for a and CT, showing how to obtain values fixed level of analyte (CT).of a and b. If more than one interfering major constituent is present and complex interactions do not occur, the interference can be represented by the equation * (8) C, = CT (1 + CaiXi) + CbiX, . . .. .. where ai, bi and X i refer to values for the ith interfering element, and all the effects are independent and additive. In that instance the individual ai and bi can be obtained as302 THOMPSON et al. : STATISTICAL APPRAISAL OF INTERFERENCE Analyst, VoZ. 104 previously, by studying the effect on C, of varying each Xi separately, or by allowing all of the Xi to vary simultaneously, separating the effects by means of multiple regression techniques. There are no a priori reasons for believing that the simplifying assumptions (of independent, linear, additive effects) are correct.The experimental design must therefore contain features that enable these assumptions to be tested. If the system under study conforms fairly closely to the model, as demonstrated in the results presented here, the interference effects can be characterised simply and quantified quickly. If there are serious deviations, a much more extensive programme of work may be called for. Experimental Apparatus meter. linearisation where appropriate. handbook, the response being optimised for flame height and nebuliser adjustment. mental conditions are shown in Table I. hydrochloric acid solution. All measurements were made on a Perkin-Elmer 403 atomic-absorption spectrophoto- Readings were made without background correction in the concentration mode, with Gas flow-rates were those recommended in the instrument Instru- Single element standards were prepared in a 1 M TABLE I INSTRUMENTAL CONDITIONS Element Cadmium ..Cobalt . . Copper . . Lithium . . Manganese . . Nickel . . Lead .. Titanium . . Zinc. . .. Wavelength/nm .. 228.8 .. 240.7 .. 324.7 .. 335.4 .. 279.5 .. 232.0 .. 283.3 .. 365.3 .. 213.8 Lamp current/mA Slit width 10 4 20 3 15 4 15 4 20 3 20 3 10 4 25 3 20 4 Reagents The trace metal solutions were prepared from BDH atomic-absorption standards, except titanium, which was prepared from potassium titanium oxalate. The major element solutions were prepared from specially purified samples of the metal chlorides. De-ionised water and AnalaR hydrochloric acid were used throughout. Computing All computing was carried out on Imperial College Computer Centre’s CDC 6400/7314 facility. Multiple regression was performed by a slightly modified version of the routine STEPR from the IBM Scientific Subroutine Package.Regression with analysis of variance for lack of fit, and weighted linear regression were carried out, respectively, by means of the FORTRAN routines REPELF and WAYLIN written by one of the authors (MT). Experimental Design One hundred solutions were prepared in 1 M hydrochloric acid, each of which contained the trace analytes (vix., cadmium, cobalt, copper, lithium, manganese, nickel, lead and zinc) and the major constituents (vix., aluminium, calcium, iron, potassium, magnesium and sodium) at a pre-determined level. The concentration levels selected for the analytes spanned the useful calibration range for each element, while those of the major constituents covered the range likely to be encountered in solutions derived from rocks, soils and sedi- ments.The 100 solutions were divided into five sets of 20. Within each set all of the trace analytes were present at only one of the five concentration levels, the five sets thus covering the whole range of the trace con- centrations. For each solution within a set, the concentration of each major element was The concentrations are shown in Tables I1 and 111.A@?’&?, 1979 EFFECTS IN THE DETERMINATION OF TRACE ELEMENTS BY AAS 303 TABLE I1 CONCENTRATIONS OF ANALYTES (pg ml-l) USED IN INTERFERENCE STUDY r Analyte 1 Cadmium . . . . 0.00 Cobalt ... . 0.00 Copper .. . . 0.00 Lithium . . . . 0.00 Manganese . . . . 0.00 Nickel . . . . . . 0.00 Lead .. .. . . 0.00 Zinc . . .. . . 0.00 2 0.48 1.19 1.19 1.19 2.38 1.19 2.38 0.48 Level 3 0.88 2.19 2.19 2.19 4.38 2.19 4.38 0.88 4 1.43 3.57 3.57 3.57 7.14 3.57 7.14 1.43 - 5 - 1.90 4.76 4.76 4.76 9.52 4.76 9.52 1.90 individually selected at random from its ten possible levels in such a way that each level was selected exactly twice. An example of such a randomised scheme is shown in Table IV. The apparent concentration of trace analytes in each solution was then determined by atomic- absorption spectrophotometry, each solution being analysed twice in a random sequence. TABLE I11 CONCENTRATIONS OF INTERFERENTS (yo m/V) USED IN INTERFERENCE STUDY Level r 1 Interferent 1 2 3 4 5 6 7 8 9 10 Aluminium .. 0 0.0190 0.0380 0.0570 0.0760 0.0952 0.1143 0.1333 0.1524 0.1905 Calcium. . . . 0 0.0952 0.1905 0.2857 0.3810 0.4762 0.5714 0.6667 0.7619 0.9524 Iron . . . . 0 0.0190 0.0380 0.0570 0.0760 0.0952 0.1143 0.1333 0.1524 0.1905 Potassium . . 0 0.0190 0.0380 0.0570 0.0760 0.0952 0.1143 0.1333 0.1524 0.1905 Magnesium . . 0 0.0190 0.0380 0.0570 0.0760 0.0952 0.1143 0.1333 0.1524 0.1905 Sodium . . . . 0 0.0190 0.0380 0.0570 0.0760 0.0952 0.1143 0.1333 0.1524 0.1905 The types of sample preparation generally used in applied geochemical work do not lead to the presence of silicon in solution except sometimes as a trace constituent, and silicon was therefore not included among the interferents. Other non-metallic interferents (e.g., sulphur and phosphorus) are likely to be present only in minor concentrations compared with the metals, and were excluded from this study.The solutions used are thus reasonably representative of those derived from real samples. The randomisation scheme is necessary in order to avoid any systematic effects, which are thereby converted into random effects. In addition, complex interactions may be obscured if there is any correlation between the concentrations of the major elements. The duplication of major element concentrations is required in order to make a statistical test for lack of fit (caused in this instance by non-linearity). The duplication of the measure- ments is undertaken to obtain an estimate of instrumental variance, to provide a base-level against which other variances can be compared.The experimental design does not allow for the possibility of interference from trace element interactions. These are assumed to be unimportant at such concentrations. Results and Discussion The essential feature in the interpretation of the data is the analysis of variance at each level of a trace analyte. If there were no interference then each solution would produce an identical result (apart from instrumental and volumetric variations). In fact, all of the solutions give different results owing to the interferences and this is quantified by the total variance. The analysis of variance distributes this total variance between (i) that which can be accounted for by linear relationships with the major constituent concentrations, i.e., by linear regressions; (ii) that due to “lack of fit,” i.e., by a failure of the initial304 THOMPSON et al.: STATISTICAL APPRAISAL OF INTERFERENCE Analyst, VoZ. 104 TABLE IV EXAMPLE OF A RANDOMISED DUPLICATED SCHEME FOR THE Solution number 1 2 3 4 6 6 7 8 9 10 11 12 13 14 16 16 17 18 19 20 CONCENTRATIONS OF THE MAJOR ELEMENTS Concentration level* A \ Aluminium 1 8 3 2 6 6 8 4 1 9 7 9 7 2 5 4 10 3 10 6 Calcium 4 8 6 7 6 10 4 3 1 2 6 5 2 8 3 9 7 10 9 1 Iron 7 10 8 8 1 2 6 6 1 3 7 9 4 6 10 9 2 6 3 4 Potassium 6 2 3 1 7 2 1 6 4 8 3 4 10 7 8 6 6 10 9 9 Magnesium 8 3 6 7 2 4 9 10 3 2 6 6 1 7 8 9 6 1 4 10 Sodium' 1 4 9 3 7 6 3 6 8 1 9 8 7 6 5 2 2 4 10 10 * For actual concentrations corresponding to these levels, see Table 111. assumption that the interference effects are linear functions of the major element concentra- tions; (iii) that due to complex interactions, ie., by a failure of the second assumption; and (iv) that due to instrumental variations and volumetric errors. Each of these features is illustrated in the ensuing discussion.The diagrams are not usually necessary for the statistical analysis but serve here to illustrate several of its features, and are sometimes helpful in interpreting dubious examples. A comprehensive account of regression methods is given by Draper and Smith.9 Copper The first stage of the procedure is the multiple regression, which is carried out separately for each level of the trace analytes. Thus, for copper at level 1 (k, zero concentration) STEPR produced the results summarised in Table V.The regressions are given in decreasing order of significance as shown by the values of Student's t. Only the regressions on calcium and iron are significant at the 95% confidence limits. The intercept (0.002) is not signifi- cantly different from the true value of zero. The standard error of the estimate is equivalent to the standard deviation of the distances between the experimental points and the calculated TABLE V MULTIPLE REGRESSION OF COPPER ON MAJOR CONSTITUENTS Results at level 1 (zero analyte concentration). Regression Major constituent coefficient Calcium .. .. 0.1569 Iron . . .. .. 0.1088 Aluminium . . .. 0.0294 Potassium . . . . 0.027 2 Sodium . . .. .. 0.009 1 Magnesium . . .. 0.0109 Standard error of coefficient 0.0090 0.047 3 0.043 4 0.045 7 0.0484 0.043 7 t-value 17.36 2.30 0.68 0.60 0.23 0.21 Intercept = 0.0020.Standard error of estimate = 0.013. Explained variance = 96.5%.A$riZ, 1979 EFFECTS IN THE DETERMINATION OF TRACE ELEMENTS BY AAS 305 lines. In this instance it is not significantly greater than normal instrumental noise levels (0.01 pgml-l) as judged from the duplicated measurements. The implication is that all of the measurable interference can be attributed to calcium and iron by linear relationships with their concentrations. However, consideration of the standard error of the estimate is not a certain test of good regression, as will be shown below, and other statistics should be considered at the same time. In the context of this work, the percentage variance explained and an examination of residuals are useful supplementary tests.The multiple regression is repeated at each of the analyte concentration levels. The next stage of the procedure is illustrated by the results obtained for the interference effects of calcium on copper, but the same procedure is followed for each analyte - interferent combina- tion. The regression data extracted for the copper -calcium pair are given in Table VI. TABLE VI STATISTICS FOR THE REGRESSION OF APPARENT CONCENTRATION OF COPPER ON CALCIUM CONCENTRATION, AT VARIOUS CONCENTRATIONS O F COPPER Analyte Copper concentration/ Regression Standard error Standard error level pg ml-l coefficient of coefficient t-value of estimate* 1 0.00 0.156 9 0.0090 17.36 0.013 2 1.19 0.042 9 0.01 7 2 2.50 0.024 3 2.19 -0.013 8 0.018 1 -0.76 0.026 4 3.67 - 0.089 7 0.0309 -2.91 0.043 5 4.76 - 0.145 4 0.0462 -3.14 0.065 * This value is the residual after all the regressions are included, not only calcium.These data and statistics are illustrated in Fig. 4, which shows the apparent concentrations of copper as a function of calcium concentration. (It must be noted that the visible differ- ences between the duplicated results at each level of calcium is not variance caused by analytical error but by the different levels of all of the other major elements.) Both the data and Fig. 4 show the regression coefficient changing from positive to negative as the copper concentration increases. At approximately 2 pg ml-1 of copper the coefficient is zero, showing that at this concentration there is no apparent effect of calcium on copper, 0 0.2 0,4 0.6 0.8 1.0 Calcium concentration, % Fig.4. Effect of calcium concentration on the apparent concentration of copper a t various true concentrations (CT) of the analyte. Open points, single observations; and closed points, two coincident observations. This system shows a significant change in slope with CT.306 THOMPSON et aZ. : STATISTICAL APPRAISAL OF INTERFERENCE Analyst, VoZ. 104 even though there is an effect at both higher and lower copper concentrations. This high- lights the erroneous conclusions that can be drawn from interference studies carried out at a single trace element concentration. The standard error of the estimate is satisfactorily low at each copper concentration level, suggesting that there is no measurable non-linearity or complex interactions for copper.The values of the regression coefficients correspond to variable dc in equation (6) while the analyte concentrations correspond to C,. The values of a and b can now be obtained by regression of d, on C,. It is necessary to obtain also the standard errors of the intercept (b) and the slope (a) so that the values can be tested as to whether they are significantly greater than zero. However, normal regression can give misleading results for the standard errors, as the variance at higher trace levels, necessarily greater than at lower levels, has too much influence on the regression. In this technique, each observation is weighted by a factor inversely proportional to its variance. The variance in this instance is the square of the standard error of the coefficient.The necessity for this procedure is illustrated in Fig. 5. The error bars on each point represent the 95% confidence limits, i.e. , twice the standard error. It is clear that the regression lines produced by weighted regression and by simple regression are very close and not significantly different. However, the standard errors of the intercepts are very different. The 95% confidence limits for the simple regression include zero, implying that the intercept is not significant. The same confidence limits for the weighted regression, which are clearly more realistic, show the intercept to be highly significant. The values extracted for copper were a = -0.071 (t = 11.2) and b = 0.150 (t = 16.1).Hence, the measurable interference from calcium on copper under the conditions studied can be expressed by C, = (C, -0.15OX)/(l -0.071X) where C, and C, are expressed in micrograms per millilitre and X as a percentage. Weighted regression therefore has to be used. 0.3 - f--,95% confidence region for intercept - normal regression 1 I I I I 0 1 2 3 4 5 -0.3 I True copper concentration (CT)/pg ml-' Fig. 5. Regression of dc on CT for copper/calcium. The error Broken line, bars are the 95% confidence regions for the dc values. normal regression ; and solid line, weighted regression. The Other Analytes The results are summarised in Table VII, which shows all of the a and b values that were more signifi- cant than 95%. Most of the values were also more significant than 99% and those are shown in italic type.All but two of the a values were negative, representing suppression of the analytical signal by the major component. The exceptions were sodium and potassium on lithium. Most of the b values are positive, representing positive changes in background absorption, which in most instances accounts for the greater part of the interference. Broadly, the effects found were as expected from experience of atomic-absorption spectrometry, in confirmation of the adequacy of the model. An example of an analyte - interferent pair (cadmium - calcium) , which shows only background interference, is illustrated in Fig. 6. As before, non-adherence to the theoretical model 'was tested by examination of the standard errors of the estimate, and the percentage variance explained, at each level for each analyte.The procedure outlined for copper was repeated for the other seven analytes.April, 1979 EFFECTS IN THE DETERMINATION OF TRACE ELEMENTS BY AAS 307 TABLE VII SIGNIFICANT VALUES OF a AND b FOUND IN THE INTERFERENCE STUDY “important” effects. Figures in italic type are significant at 99% level, others a t 95% level. Asterisks indicate Major r constituent Cd co Aluminium . . a -0.114 -0.320* b 1.026* b 0.213* 0.852* Iron . . .. a b Potassium . . a b Magnesium . . a b Sodium . . .. a b 0.085 0.547 Calcium. . .. a - 0.051 Trace analyte c u Li Mn 0.044 0.150* 0.036 0.193 -0.168 -0.224* -0.162 -0.071 -0.262* -0.066 - 0.281 -0.016 0.065 -0.111 -0.118 0.019 -0.011 > Ni Pb Zn - 0.290* -0.111 0.068 -0.062 1.277* 1.601* 0.133 0.831 0.360 0.114 - 0.315 0.559 Of the analytes only lithium showed significantly large standard errors. Non-linear effects accounted for the deviation in the lithium results.These effects were tested for by performing a regression with analysis of variance for “pure error” and “lack of fit” at each analyte level. In this technique, the variance caused by the difference between the pairs of results for each level of the interferent (which is due to the effect of different concentrations of the other major elements) is compared with that related to the mean distance of the two points from the regression line. This procedure is illustrated in Fig. 7, which shows the results of the regression of the lithium results on calcium. The individual results are plotted together with the calculated linear regression lines.For the higher concentrations of lithium the diagram clearly shows a non-linear effect, which the linear regression underestimates in some ranges and overestimates elsewhere. All of the significant regressions were tested 2‘5r- CT = 1.90 CT = 1.43 CT = 0.00 0 0.2 0.4 0.6 0.8 1 .o Calcium concentration, % Fig. 6. Effect of calcium on the apparent concentration This system shows only of cadmium at various levels. “background” interference. Symbols as in Fig. 4.308 THOMPSON et al. : STATISTICAL APPRAISAL OF INTERFERENCE Analyst, VoZ. 104 e &\. h c, F ,CT = 3.57 c,= 1.19 A . _ . _ P ' . O t z l c, = 0.00 A A &~- A A ! ' 5 ' 0 0.2 0.4 0.6 0.8 1 .o Calcium concentration, % Fig.7. Effect of calcium concentration on the apparent concentration of lithium at various levels. This shows the significant lack of fit of the linear regressions (solid lines) at the higher levels. Symbols as in Fig. 4. individually in this way, but only lithium (at four levels) and cobalt (at one level) were found to have significant lack of fit, as judged by the value of the F (variance ratio) test. To obtain an adequate fit for such data a more elaborate mathematical relationship is required. Generally, polynomial fits do not give satisfactory results. However, it is clear from Fig. 7 that even a linear fit would provide a substantial improvement in accuracy. For cobalt, inspection of the residuals strongly suggested that the isolated significant lack of fit was caused by a random fluctuation rather than a non-linear effect at one level.The statistical test alone cannot distinguish between the two possibilities. The success of the two-parameter model in describing the interferences was tested by applying to the experimental data corrections based on the inverse of equation (€9, i.e., C, = (C, - CbiXi)/(l + ZaiXi) .. .. ' (9) TABLE TUII CORRECTION OF EXPERIMENTAL RESULTS BY USE OF TWO-PARAMETER MODEL The results of all significant corrections have been applied to the uncorrected (raw) data to give the corrected value (con.) for comparison with the true value. Means are given for each level (standard deviations in parentheses). Analyte concentration/pg ml-I A Trace 7 \ level Result Cadmium Cobalt Copper Lithium hlanganese Nickel Lead Zinc 1 Raw 0.12 (0.07) 0.63 (0.27) 0.09 (0.05) 0.02 (0.01) 0.11 (0.07) 0.83 (0.42) 1.02 (0.49) 0.12 (0.05) Corr.0.03 (0.01) 0.17 (0.09) 0.02 (0.01) 0.00 (0.00) 0.03 (0.04) 0.25 (0.09) 0.25 (0.09) 0.01 (0.01) True 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2 Raw 0.61 (0.06) 1.85 (0.26) 1.21 (0.03) 1.03 (0.09) 2.43 (0.05) 1.96 (0.39) 3.42 (0.44) 0.59 (0.04) Corr. 0.52 (0.01) 1.43 (0.10) 1.20 (0.02) 1.23 (0.02) 2.42 (0.05) 1.42 (0.08) 2.71 (0.09) 0.49 (0.01) True 0.48 1.19 1.19 1.19 2.38 1.19 2.38 0.48 3 Raw 0.98 (0.06) 2.79 (0.25) 2.09 (0.03) 1.82 (0.18) 4.22 (0.06) 2.87 (0.40) 5.28 (0.41) 0.95 (0.05) Corr. 0.89 (0.01) 2.40 (0.10) 2.13 (0.03) 2.17 (0.06) 4.25 (0.06) 2.35 (0.09) 4.63 (0.08) 0.85 (0.02) True 0.88 2.19 2.19 2.19 4.38 2.19 4.38 0.88 4 Raw 1.56 (0.06) 4.15 (0.23) 3 42 (0.05) 3.02 (0.29) t.99 (0.12) 4.30 (0.35) 8.18 (0.36) 1.66 (0.06) Corr.1.46 (0.02) 3.79 (0.17) 3.32 (0.04) 3.62 (0.06) r.11 (0.08) 3.81 (0.11) 7.61 (0.10) 1.45 (0.03) True 1.43 3.57 3.57 3.57 7.14 3.57 7.14 1.43 5 Raw 2.01 (0.07) 5.32 (0.23) 4.48 (0.09) 3.97 (0.39) 9.17 (0.21) 5.43 (0.38) 10.47 (0.37) 2.02 (0.06) Corr. 1.92 (0.03) 5.00 (0.16) 4.62 10.06) 4.76 (0.11) 9.35 (0.17) 4.98 (0.14) 9.97 (0.17) 1.93 (0.02) True 1.90 4.76 4.76 4.76 9.62 4.78 9.62 1.90Apt‘&?, 1979 EFFECTS I N THE DETERMINATION OF TRACE ELEMENTS BY AAS 309 for all significant values of a and b. The results obtained are summarised in Table VIII, which shows, for each level of each trace analyte, the mean and standard deviation of all the solutions of the uncorrected (raw) results, the corrected results and the true concentra- tion present.In virtually all instances the mean result was brought substantially closer to the true result and the range (as indicated by the standard deviation) was markedly reduced, showing a considerable degree of success with the two-parameter model. Some of the data are illustrated in Fig. 8 for two different types of interference. D Concentration presendpg mI-’ Fig. 8. Range of experimental results for (a) lithium and (b) lead, both uncorrected (narrow bars) and corrected with all significant parameters (thick bars). The Magnitude of the Interferences While the procedure described above enabled all of the statistically significant (k, measurable) effects to be quantified, it did not in itself provide information as to whether the magnitude of the effects was likely to be “important” ( i e ., likely to affect interpretation of geochemical data). This information can be obtained by the application of empirical criteria appropriate to the data usage. For geochemical work we have applied the following criteria: an effect is “important” if la1 > o.05/Mg, or Ibl > T,/Mgg, where Mgg is the 99th percentile concentration (in per cent.) of the major element in the population of samples to be analysed, and T , is the fifth percentile of the trace element concentration (in micrograms per millilitre). Thus, when the factor a is equal to the criterion, only for one sample in 100 will the interference effect exceed a relative change of &5y0 in the apparent analyte con- centration. When b is equal to the criterion, the interference will equal the analyte concentration only when the interferent exceeds its 99th percentile at the same time as the trace analyte falls below its 5th percentile, i.e., a probability of 0.0005 if the concentrations of the two elements are independent.The “importance” of the b value thus depends on the concentration range of the analyte. For example, the b value for cadmium - calcium of 0.213 is graded “important” : a sample of pure limestone (which contains 40% calcium) would produce an apparent cadmium level of 40 x 0.213 or about 8 p.p.m. compared with the normal level of <1 p.p.m. This is clearly important. The corresponding effect for zinc would amount to 40 x 0.133 or 5.3 p.p.m.However, the normal level of zinc is in the range 50-100 p.p.m. so the effect is not important. The “important” effects identified in this way are shown in Table VII with an asterisk. The percentiles were taken from a major geochemical survey involving 50000 ~amples.~ Only aluminium and calcium produce “important” effects in the context of geochemical analysis. Aluminium produces mainly rotational type effects (factor a) , on cobalt, lithium and nickel. Calcium produces translational effects (factor b) on cadmium, cobalt, copper, nickel and lead. While the background interference can be at least partly corrected by means of absorption measurements with a continuum source, there is no universally applicable method for removing rotational effects. For lesser major concentrations a smaller interference will be found.310 THOMPSON et d.: STATISTICAL APPRAISAL OF INTERFERENCE Analyst, VOZ. 104 An Example of Complex Interaction The interference effects studied could be effectively accounted for by a simple model of linear, independent, additive effects. Only in one instance (calcium on lithium) was a significant non-linear effect identified, and even here the linear approximation was an adequate model. In these circumstances the methodological approach suggested in this work is capable of elucidating the interference effects. However, it is often suspected and sometimes demonstrated that more complex types of interference are occurring. We have examined one such system, the interference of iron and aluminium on titanium, which is known to have non-linear effects and complex interactions.llSl2 The sole purpose of this additional study was to examine the performance of the statistical approach under conditions where the simple assumptions of the method are likely to fail, and to determine whether the failure manifests itself clearly by the statistical tests.Titanium would be only partially extracted by the acid attacks described here. The experimental design was as described above, with titanium at 5 levels (0, 10,20, 30 and 40 pg ml-l) and iron and aluminium within the ranges 0-10000 and 0-2000 pg ml-1, respectively, and with no other interferent present. Multiple regression (the first stage of the data analysis) produced the statistics summarised in Table IX. The percentages of the variances explained were small, the standard errors of the estimates were large compared with the pure analytical error, and the regression co- efficients and values of t were small.In addition, analysis of variance for lack of fit (titanium against aluminium) showed highly significant lack of fit. Thus, in all these ways the statistics show a non-conforming interference system. Only the low regression coefficients might suggest to the unwary that there was no significant interference. The reason for this is the extreme non-linearity of the effect of aluminium. This is illustrated in Fig. 9, which shows the failure of linear regression to represent this type of data. Thus, it is important to take note of all statistics and visually to examine the residual plots in doubtful instances.A two-way plot of these data showed that there were complex interactions in the system as well as non-linear features. The data produced in the simple experiment were not sufficient to characterise the response surface and a more comprehensive experiment was required. It was found that the effect of iron and aluminium can be represented approximately by the surface shown in Fig. 10, for any level of titanium. It is difficult adequately to express a complex surface like this in terms of polynomial trend analysis. TABLE IX STATISTICS FOR THE MULTIPLE REGRESSION STUDIES ON THE INTERFERENCE OF ALUMINIUM AND IRON ON TITANIUM Analyte Regression Standard error Variance 1 Iron 0.000 :;:i4 ) 0.74 18.5 Aluminium 0.005 2 Iron 0.001 Aluminium 0.002 Aluminium -0.002 -0.25 } 2*8 Aluminium -0.004 -0.42 } 3'8 Aluminium 0.000 -0.06 } 3*8 level Interferent coefficient t-value of estimate explained, % } 1.81 37.5 18.9 3 Iron 0.000 0.07 15.I 4 Iron 0.005 2.55 30.0 5 Iron -0.008 -3.82 Summary and Conclusions It has been shown that a simple two-parameter model of interference with independent, linear, additive effects is adequate to describe the measurable interference in the atomic- absorption determination of cadmium, cobalt, copper, lithium, manganese, nickel, lead and zinc in geochemical samples. The statistically significant parameters have been evaluatedApril, 1979 EFFECTS I N THE DETERMINATION OF TRACE ELEMENTS BY AAS 311 and those of important magnitude have been indicated by an empirical but uniformly applicable criterion.This enables the analyst to identify conditions where interferences might have a noticeable effect on data interpretation, and where appropriate, to make suitable corrections in the form of equation (9). Where large numbers of data are involved the corrections can be applied automatically by computer. This will become increasingly easy as microprocessor-controlled data logging systems become more common. Although an elaborate experimental design and data appraisal are required in order to characterise the interference model, the parameters a and b can be evaluated with only four solutions once it has been established that the system conforms to the model. Naturally the values of a and b will depend on the instrumental settings, the use of background correc- tion and other factors, and should be checked for each analytical batch.In addition, it is good practice to reduce interferences to as low a level as possible before attempting correc- tions, which should generally be regarded as a last resort. Where deviations from the model occur, the experimental design ensures that they are detected and suggests how more elaborate tests can be made to characterise them. In most of these instances informal designs in interference studies would give misleading results, especially with regard to complex interactions in multi-interferent systems, which usually remain undetected. These effects may still remain too complex to elucidate fully, but even so, it is important for the analyst to know when these occur. Where they occur it is necessary to resort to the use of appropriate standard solutions whose major constituents correspond in composition to those of the average sample. CT = 0.0 0 I h 9 : Q G II ”,” a n P 0 - li Fig. 9. Effect of aluminium on the apparent con- centration of titanium in the system Ti- A1 - Fe. This shows failure of the linear system to represent these data. Symbols as in Fig. 4.312 THOMPSON, WALTON AND WOOD 9 Iron concentration, 96 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. Fig. 10. Combined effect of aluminium and iron on the apparent concentration of titanium a t 30 pg ml-1. The contours indicate relative enhancement as a percentage. References Foster, J. R., Can. Min. Metall. Bull., 1973, 66, 85. Webb, J . S., and Thompson, M., Pure Appl. Chem., 1977, 49, 1507. Fletcher, K., Econ. Geol., 1970, 65, 588. Foster, J. R., in Boyle, R. W., and McGerrigle, J. I., Editors, “Proceedings of the Third International Geochemical Exploration Symposium, Toronto, 1970,” Special Volume 11, Canadian Institution of Mining and Metallurgy, Toronto, 1970, pp. 554-560. Govett, G. J. S., and Whitehead, R. E., J . Geochem. Explor., 1973, 2, 121. Woodis, T. C., Hunter, G. B., and Johnson, F. J., Analytica Chim. Acta, 1977, 90, 127. Thompson, M., Pahlavanpour, B., Walton, S. J., and Kirkbright, G. F., Analyst, 1978, 103, 705. Alley, B. J., and Myers, R. H., Analyt. Chem., 1965, 13, 1685. Draper, N. R., and Smith, H., “Applied Regression Analysis,” John Wiley, New York, 1966. Webb, J. S., Thornton, I., Thompson, M., Howarth, R. J., and Lowenstein, P., “The Wolfson Geochemical Atlas of England and Wales,” Oxford University Press, London, 1978. Cobb, W. D., Foster, W. W., and Harrison, T. S . , Analytica Chim. Acta, 1975, 78, 293. Walsh, J. N., Analyst, 1977, 102, 972. Received June 26th, 1978 Accepted November 15th, 1978