Journal of Geochemical Exploration, 44 ( 1992 ) 2 3 - 3 6
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Elsevier Science Publishers B.V., A m s t e r d a m
Objective evaluation of precision requirements for geochemical analysis using robust analysis of variance M i c h a e l H . R a m s e y a, M i c h a e l T h o m p s o n
b and Martin Hale c
aDepartment of Geology, Imperial College, London SW7 2BP, UK bDepartment of Chemistry, Birkbeck College, Gordon House, 29 Gordon Square, London WC1H OPP, UK Clnternational Institute for Aerospace Survey and Earth Sciences, Kanaalweg 3, 2628 EB Delft, Netherlands (Received 17 September 1990; accepted after revision 15 March 1991 )
ABSTRACT Ramsey, M.H., Thompson, M. and Hale, M., 1992. Objective evaluation of precision requirements for geochemical analysis using robust analysis of variance. In: G.E.M. Hall (Editor), Geoanalysis. J. Geochem. Explor., 44: 23-36. The pursuit of high precision in geochemical analysis has no inherent limit. An appropriate analytical precision requirement can be set, however, by comparing the "analytical variance" with the other sources of variance in geochemical data. The purpose of a geochemical survey is to give a description of the geochemical variation of a region. Numerically this can be expressed in terms of the natural "geochemical variance" of the area. The information content is diminished by the two processes of measurement: the act of taking a sample adds a random error with "sampling variance"; and the act of chemical analysis adds another random error with "analytical variance". In order to optimise the analytical variance for cost-effectiveness, rather than simply to minimize it, all three variances must be estimated. This requires that traditional analytical quality control be extended to include the total measurement process, rather than only the "analytical" portion. Such a Sampling and Analytical Quality Control Scheme (SAX) requires some duplication of field samples and the duplicate analysis of each field duplicate. A robust Analysis of Variance (ANOVA) is then used to separate the three components of the total variance. Robust statistics can accommodate outlying values that have undue influence on classical ANOVA. For a clear description of the natural geochemical variance, the combined sampling and analytical variances for the data should comprise not more than, say, 20% of the total variance. If this figure is exceeded then these extraneous variances should be reduced. The decision as to whether this reduction requires improved sampling or improved analytical precision can also be based on the ANOVA results. As a general rule, we suggest that the analytical variance should comprise not more than 4% of the total variance. If, on the other hand, the analytical variance is less than 1% of the total variance, then needless expense has probably been incurred and the natural geochemical variation can be adequately described with less precise methods of analysis. Similar arguments can be applied to sampling variance. An application of SAX with robust ANOVA to a stream sediment survey for Cu, Pb and Zn demonstrates the advantages of the technique. For the analytical geochemist it provides a realistic target
0 3 7 5 - 6 7 4 2 / 9 2 / $ 0 5 . 0 0 © 1992 Elsevier Science Publishers B.V. All rights reserved.
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M.H. RAMSEY ET AL.
for analytical precision. For the field geochemist it provides a quantitative tool for the design of geochemical surveys. It facilitates the optimisation of both sampling and chemical analysis for a particular region to reveal geochemical patterns at minimal expense.
INTRODUCTION
Much has been written on the precision that can be achieved by various methods of geochemical analysis. The analytical precision that is required for any particular geochemical interpretation, however, is not well characterised. The pursuit of high precision seems a worthwhile goal for the analytical geochemist, but finite resources require that targets for the analytical precision should be justified by an objective assessment of the particular requirement. The difficulty in setting targets for analytical precision has its origins in a lack of communication between analytical and field geochemists. The analytical geochemist may ask the question "What precision is required?" but it is often difficult for the field geochemist to answer before seeing the analytical results in the context of the field problem. Alternatively, the field geochemist may ask the analytical geochemist "Is the precision good enough for a geochemical interpretation ?", but the analytical geochemist finds it hard to judge without knowing the magnitude of the elemental variation that is to be interpreted. Neither person has enough information to answer the questions posed. A new integrated approach is needed. A further difficulty is the uncertainty generated in the process of sampling. Usually the analytical geochemist confines himself to producing a result that is representative of the contents of a sample submitted for analysis. The field geochemist may attempt to take a large enough sample to provide sample representativity, but rarely is any attempt made to estimate the measurement uncertainty that remains due to the sampling process. Without the knowledge of the error caused by the acts of both sampling and analysis, the field geochemist cannot judge whether the analytical measurements can justifiably be interpreted, or whether more resources should be allocated for improving the methods of sampling and/or analysis. The solution to these problems that we propose is the extension of analytical quality control into field sampling procedures and the application of analysis of variance (ANOVA). This allows the estimation of the random errors produced by both the sampling and the analysis. The sum of these two variances, here termed the "technical variance", can be compared with the geochemical variance (which is a parameter of the geochemical variability). A precise description of this geochemical variability is the ultimate objective of geochemical analysis. The advantage of variance as a measure of random error, i.e., precision, is that variances (o~ ) are additive whereas standard deviations (tT) are not.
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Analytical quality control is becoming widely adopted for geochemical analysis but the details of the schemes vary. In the scheme proposed by Ramsey et al. ( 1987), estimates of bias (systematic error) are made using reference materials and reagent blanks. The analytical precision (or random error) is estimated by means of analytical duplicates. These consist of two weighings of the same test material, subsequently analysed as if they were two different test materials. The analytical precision can then be estimated from the means and differences of the duplicated pairs as a function of concentration by regression (when there are > 50 duplicates pairs), or tested against a model of precision (e.g., 2 10%) (Thompson and Howarth, 1976). However, analytical quality control ignores the random error introduced in the process of field sampling. This can be estimated by including sampling duplicates in a Sampling and Analytical Quality Control Scheme (SAQCS = “SAX”). Duplicate samples are two distinct field samples that are taken to represent the same location, so that in the field they are separated by a distance that reflects the uncertainty in the location of the site on the map. The random error caused by the sampling procedure, expressed as the sampling variance, cannot be estimated directly from the analyses of the sample duplicates because it is overprinted by the analytical variance. The statistical technique used to separate the analytical variance from the sampling variance is ANOVA. CLASSICAL
ANOVA
A single measurement of analyte concentration (X) on one sample from a particular location can be considered to comprise the following independent components:
x=x,,,,,+t, +t, where J&.,,, is the true average concentration of the analyte at the site, with a variance of dg; E, is the total error due to sampling (both systematic and random), with variance of c?,; and E, is the total analytical error, with variance c?,. If the sources of variation are independent, the total variance $t is given by
The sampling and analytical variances combined, or technical variance, is given by
(2) hence
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M.H. RAMSEY ET AL.
Analysis of Variance can be used to separate the three sources of variance and give Sg, Ss and Sa which are estimates of ag, as and aa respectively. This technique was first applied to geochemical surveys by Miesch (1964) and Garrett ( 1969 ).
Implicit assumptions of classical ANOVA The rigorous application of ANOVA depends on three assumptions: 1. The variances should be independent. In this application, for example, the sampling variance should not vary as a function of the geochemistry. 2. Each level of variance should be homogeneous, that is it should not vary systematically within one level. For example, the analytical variance should not vary between different rock types. 3. The distributions of errors within each level of variance should be approximately Gaussian. The strict adherence of geochemical data to these criteria is hardly ever encountered. For example, it is well established that analytical variance varies as a function of concentration. This invalidates the assumption of homogeneity for most geochemical applications. The assumption of an approximately Gaussian distribution has been substantiated for analytical error (Thompson and Howarth, 1980). However, geochemical distributions frequently depart significantly from normality, often being either roughly lognormal or bimodal (Rose et al., 1979). Despite these theoretical limitations, the applicability of ANOVA to separating sources of variation within geochemical data has been widely reported. The problems of applying ANOVA to data which are not normally distributed has long been recognised, and the suggested remedy has been to log transform the data (Garrett, 1969). A more recent approach to describing distributions which are predominantly normal, but contain a small fraction of outlying or anomalous values, is to use a robust statistical procedure. ROBUST STATISTICS
The traditional treatment of outlying values rests on their identification, by a test such as that proposed by Dixon ( 1951 ), and their subsequent removal. Robust statistics rely on the accommodation of outlying values rather than their rejection. They provide estimates of mean, standard deviation etc., that are almost unaffected by a minor population of outliers among the data. Many robust methods have been proposed, and we are following procedures recently advocated for analytical chemistry (Analytical Methods Committee, 1989a). Briefly, robust estimates (2r and s,-) of the mean (p) and standard deviation (~) used in ANOVA are made by an iterative process. In this process,
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data that exceed a certain distance from X (such as outliers) are assigned a new value equal to that distance. The distance is the product of a constant (c) and the standard deviation, and the value of c is chosen to suit the expected proportion of outliers. A value of c = 1.5 is normally regarded as optimal. Initial values of X and s can be obtained by classical statistics. These values are then used to adjust values falling outside the range X + cs to be equal to X + c s or X - c s , as appropriate. Modified mean and standard deviation estimates (X2, s2) are then calculated from the modified population. Iteration continues until the calculated values of Xr and sr converge, at which point they are called the robust estimates.
EVALUATION OF ROBUST ANOVA
A FORTRAN program for nested robust ANOVA (by B.D. Ripley, Analytical Methods Committee, 1989b) was evaluated against classical ANOVA using four simulated sets of data of known means and variances. The data were taken at random from populations with the parameters given in Table 1. Trial data set A represents 100 sites with duplicated geological samples and duplicated analysis on each sample, represented diagrammatically in Fig. 1. For TABLE 1
Comparison of classical and robust ANOVA for four simulated data sets ( A - D ) mean
(/~g g - J)
A--Classical
standard deviations (gg g - t ) analytical
sampling
geochem.
Sa
Ss
Sg
100.35 100.13
1.890 1.918
4.912 5.241
9.706 9.615
102.85 102.00
16.015 2.047
6.519 5.799
8.078 10.869
C--Robust
105.35 102.67
1.890 1.918
23.744 5.821
4.994 11.875
D--Classical D--Robust
189.28 102.28
17.700 2.233
22.889 6.018
307.002 11.830
Simulation parameters of parent populations
100
5.0
10.0
A--Robust
B--Classical B--Robust
C--Classical
2.0
The robust estimates are largely unaffected by the outlying values, and correspond well to the simulation parameters.
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M.H. RAMSEY ET AL.
sile
sample 1
\\ anal l anal 2
sample 2
'X anai 1 anal 2
Fig. 1. The experimental design used in the geochemical survey that is suitable for analysis of variance (ANOVA).
trial B, 10% of the analytical duplicates were given spuriously high variances. For trial C, 10% of the sampling duplicates were given spuriously high variance. In trial D, 10% of geochemical values were set to high values ( 500-1800 ~tg g- 1) compared with the overall mean ( 100/zg g- I ). The results are summarized in Table 1. Set A (no outlying values) shows little difference between the estimates of the statistics made by classical ANOVA or robust ANOVA; both are similar to the parameters of the parent populations. For set B the classical ANOVA gives a very high estimate of the analytical standard deviation ( 16.02 ), but the robust ANOVA gives a value (2.05) close to that simulated (2.0). The classical estimate of the sampling standard deviation is similarly affected for set C, but the robust estimate is affected to a much lesser extent. For set D, all the estimates from the classical ANOVA have been severely affected by the outlying values, but again the robust estimates vary little from the parameters of the parent populations. The robust ANOVA was therefore considered validated for data sets with up to 10% of outlying values, whether the outliers consist of gross errors in analysis or sampling, or are due to real anomalies among the background values.
Acceptable levels of "technical variance" Despite the considerable use that has been made of ANOVA in geochemical surveys, relatively little attention has been given to assessing the levels of sampling and analytical variance that are acceptable, and thus no explicit use has been made of ANOVA to set precision targets for geochemical analysis. Garrett ( 1969 ) suggested that the ratio of the total variance to the technical variance should exceed 4.0. This implies that the technical variance should not contribute more than 25% to the total variance. Any value is somewhat arbitrary; our experience suggests that beyond 20% the contribution of the technical variance tends to obscure a geochemical interpretation. A graphical presentation (Fig. 2) shows that lower levels of technical variance have a rapidly diminishing effect, due to the properties of eq. (3). Figure 2 uses a
PRECISION REQUIREMENTSFOR GEOCHEMICALANALYSISUSING ROBUSTANALYSISOF VARIANCE 29
particular example to show the effect o f increasing technical variance on the total variance. The example taken is a synthetic geochemical data set of mean 100/tg g - ~ and standard deviation of 10 #g g - ~. At low levels of technical variance ( < 1%) the contribution to the total variance is negligible, but as the technical variance increases beyond 5% then the proportion increases rapidly. When the technical standard deviation reaches half of the geochemical standard deviation ( 5/tg g - l ) then it contributes 20% to the total variance. This is taken as the practical limit for the technical variance, beyond which significant geochemical information is lost. Because a reduction of the technical variance below 1% of the total variance therefore produces no significant decrease in the total variance, a reduction of technical variance below this level is not cost effective.
A target for analyticalprecision Traditionally the target for precision of geochemical analysis is set at an arbitrary level. For example, trace element analysis for mineral exploration is often tested against a precision model of _+ 10% (2s) ( T h o m p s o n and Howarth, 1976 ). This level of precision m a y be better than that required for the
30 25 °t
cut off limit of 20% , ~
20
ot
..°
o
15 C~
10
contribution above 5%
J
~ 0
negligible effect below 1% ! 1
! 2 S tech
! 3
! 4
i 5
(~.1.g g " 1 )
Fig. 2. The effect of increasing technical variance (from sampling and analysis) on the total variance of a hypothetical geochemical data set of mean 100/zg g- 1and standard deviation of 10/tg g-1 .
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M.H. RAMSEY ET AL.
detection of high contrast anomalies but insufficient for discerning more subtle geochemical variations. A more effective target for the analytical precision is one related to the precision requirement of a particular geochemical circumstance, in this example, the detection of a given anomaly contrast. The analytical variance ideally should not contribute significantly to the total variance, and therefore not obscure the description of the true geochemical distribution of the element. However, because the effect of the analytical variance is often masked by the sampling variance, a more useful target for the analytical variance is that it should not contribute significantly to the technical variance. Just as the technical variance should not exceed 20% of the total variance, then analogous calculations suggest that the analytical variance should not exceed 20% of the technical variance, otherwise the technical variance becomes limited by the analytical uncertainty. When the technical variance is approaching 20% of the total, the analytical variance should, therefore, be less than 4% of the total variance. If, however, the sampling variance is a small proportion of the total (e.g., 10%) then even an analytical variance of 4% of the total may be a limiting factor. Again, by analogy with the higher level of variance, if the analytical variance contributes less than 1% of the technical variance then the analytical precision is better than that required. A less precise, and consequently less expensive analytical method could be employed. This target for analytical precision therefore depends on the particular geochemical contrast that is being investigated, rather than an arbitrary fixed value.
Graphical presentation of SAX information One problem encountered in the discussion of the contributions of random measurement error to geochemical interpretation is that of clear communication of the statistical information. A clear graphical presentation is potentially more accessible to the users of the analytical data than tabulated variance ratios, for example. A pie chart conveniently displays the m a x i m u m proportions of the three components of the total variance (Fig. 3 ). Visual comparison of experimental data against this pie chart reveals immediately whether either the sampling or analytical procedures have contributed undue uncertainty to the geochemical information. A P P L I C A T I O N O F SAX T O A S T R E A M S E D I M E N T S U R V E Y
A stream sediment orientation survey for Cu, Pb and Zn was conducted in the Truro area of Cornwall in S.W. England to evaluate the use of SAX. The
PRECISION REQUIREMENTS FOR GEOCHEMICAL ANALYSIS USING ROBUST ANALYSIS OF VARIANCE
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Variance increasing •"
' sis
Maximum technical variance (20%)
ochemical variance Dss than 80% Of total []
Geochemical
[] []
Sampling Analytical
>Technical
Fig. 3. Visual representation of data quality using a pie chart to improve the accessibilityof the precision information. The maximum proportions for technical and analytical variance are indicated for comparison with data from geochemicalanalysis. area is underlain by predominantly unmineralized Devonian metavolcanic rocks, known locally as killas. The study was designed to be typical of routine surveys in that no exceptional measures were taken to improve the skill of the samplers or the analyst engaged in the project. Field duplicate samples were taken at all 58 sites. The separation between the location of these duplicates was approximately 5 m, which represented the likely error in location for the scale of field map being used. The experimental design consisted of a balanced, three level ANOVA, in which both field duplicates were also m a d e analytical duplicates (Fig. 1 ). The unbalanced design advocated by Garrett and Goss ( 1980 ) was not applied in this case in order to maximize the n u m b e r of analytical duplicates in this relatively small study. The chemical analyses involved decomposition of the < 80 mesh fraction of samples with concentrated nitric acid at 100 °C for 1 hour ( T h o m p s o n and Wood, 1982 ). The diluted solutions were analysed by flame atomic ab-
32
M . H . R A M S E Y E T AL.
sorption spectrophometry (AAS) for Cu, Pb and Zn under standard operating conditions in a field laboratory. The instrumental analysis was conducted blind in a random sequence. The bias and precision of the analytical measurements were initially evaluated and passed using a traditional analytical quality control scheme (Ramsey et al., 1987). The frequency distribution of each element was inspected using histograms and cumulative frequency curves. The distributions were all positively skewed due to the presence of a small proportion of geochemically anomalous samples. When the anomalous samples were removed, the background populations all showed Gaussian distributions. The objective of the ANOVA was to characterize the variance of the background population, so it was carried out on the un-edited, un-transformed element concentration measurements. Heteroscedasticity in the analytical variance was expected as the analytical standard deviation is known to increase as a function of concentration (Ramsey et al., 1987). However, the concentration values of the background population range over less than one order of magnitude for all three elements. This will therefore limit any problems with heteroscedasticity in the ANOVA of un-transformed concentration values.
Results Classical ANOVA on the Cu data produced a very large estimate of the analytical standard deviation (Table 2), approximately equal to the mean. Inspection of the data revealed two very poor analytical duplicates, 1884/ 213.6 and 124.0/16.0 #g g- ~. An obvious explanation is that in the field laboratory there had been a transcription error of a factor of ten. The correction of these two analytical duplicates (to 1884/2136 and 124.0/160.0 #g g- ~) reduces the estimate of the analytical standard deviation by more than a factor of two (Table 2). In this relatively small data set, such gross mistakes would normally be identified and corrected before reporting the analyses. In TABLE 2 Classical ANOVA of all determinations of Cu, before and after the correction of two poor analytical duplicates mean (#gg-J)
standard deviations (#g g- ~) analytical
sampling
geochemical
~V
Sa
Ss
Sg
l 15 122
119 49
137 92
397--all data 445--outliers edited
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much larger data sets, however, and where the difference within the duplicate pair is not so large, the problem of identification of such "outlying" values is more problematical. Occasional outliers are recurring features of analytical data sets and the way robust statistics handles the problem they pose is readily demonstrated with these stream sediment data. Application of robust ANOVA to stream sediment data Using the data for all 58 sites classical and robust estimates of the mean and the three component variances were made for each of the three elements. The robust ANOVA results for Cu (Table 3) are much lower than those from classical ANOVA, even those using corrected data. (Table 2 ). This indicates that, in addition to the obvious "mistakes" in the data set noted above, there were other less obvious outlying values. In the estimation of the mean and the geochemical variance, the accommodation of the geochemically anomalous values, rather than the accommodation of the obvious analytical "flyers", has a strong lowering effect. The technical variance for Cu was calculated using eq. (2) and accounts for 14.6% of the total variance. Comparison of this variance with that of the target value of 20% shows the total measurement variance to be at an acceptable level. The analytical precision, expressed as a relative standard deviation, is 11.5%, slightly exceeding the traditional quality control target of 10%. As a proportion of the technical variance, the analytical variance is 32% which exceeds the 20% target. Reduction of the analytical variance would therefore be beneficial if it were necessary to reduce the technical variance, but this is already acceptable at 14.6% of the total. As a proportion of the total variance the analytical variance is 4.7% however, which only marginally exceeds the target of 4%. An exact application of a particular target is not justified due to the uncertainty which is inherent in the estimation of variance. Broadly, therefore, the the random error for Cu caused by the processes of sampling TABLE 3 Estimates of statistics using robust ANOVA from SAX data for stream sediment survey Element
Cu Pb Zn
mean (~gg-t)
34 90 243
standard deviations (/tg g - 1) analytical
sampling
technical
geochem,
total
Sa
Ss
Stech
Sg
Slo t
3.9 8.2 13.8
5.6 13.8 19.4
6.9 16.1 23.8
16.7 76.3 154
18.1 78.0 156
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r t H . RAMSEY ET AL.
and analysis is acceptable. This conclusion is confirmed visually by the SAX pie chart (Fig 4a ). For the chemical determination of Pb, the technical variance is only 4.3% of the total variance, which is well within the 20% target (Table 3 ). Similarly the analytical variance, at 1.1% of the total variance is well below the 4% target (Fig 4b). However, the analytical variance is 26% of the technical variance, which indicates that it is contributing significantly, being above the 20% target. An improvement in the analytical precision would therefore reduce appreciably the technical variance, however, the benefits to the overall system may not be worth the cost when the technical variance is already < 5%. The cause of the analytical imprecision (Sa= 8.2/~g g - l ) is the proximity of the analyte concentration to the instrumental detection limit, which, on this field-based AAS system, was estimated to be 6 #g g- 1 (3s). For Zn the pie diagram (Fig. 4c ) immediately shows that the technical variance (2.3% of the total) is not significantly affecting the geochemical description of the area. From a laboratory point of view the analytical variance, although adequate (0.8% of the total variance), is the limiting factor in a technical variance (at 33% ). The cause of the analytical imprecision ( s = 13.8 ~tg g - ~) was due to opting for reduced sensitivity of the AAS measurement in order to obviate the need for the dilution of all the samples. The justification for this procedure is shown by the SAX in the small contribution made to the total variance. Overall in this survey, the random measurement errors from both sampling and analysis have been demonstrated by SAX to be well within the control targets, and therefore of no impediment to the geochemical interpretation of
Copper
Lead
Zinc
[ ] ANALYTICAL [ ] SAMPLING [ ] GEOCHEMICAL
Fig. 4. Visual representation of data quality using pie charts for the stream sediment data for Cu, Pb and Zn.
PRECISION REQUIREMENTS FOR GEOCHEMICAL ANALYSIS USING ROBUST ANALYSIS OF VARIANCE
35
the data. The analytical precision from this field laboratory would be marginal when judged by the traditional criteria of a fixed target ( R S D = 5%). However, the use of objective targets shows that for this application the analytical variance caused no significant loss of information in the description of geochemical element distributions from the chemical analyses. CONCLUSION
Analytical geochemists need targets for analytical precision. This enables resources to be concentrated on those analyses that require the greatest precision. The targets need to be related to the geochemical objective of the survey, not fixed at arbitrary values. The random measurement error introduced by the process of field sampling should be estimated by the analytical geochemist, and communicated to the field geochemist. A Sampling and Analytical Quality Control Scheme (SAX) extends traditional analytical quality control into the field environment to measure and provide targets for both analytical and sampling precision. This utilises duplicate field samples that are further duplicated during the analytical procedure, in addition to the other quality control materials, such as reference materials and procedural blanks. Analysis of Variance (ANOVA) is used within the SAX procedure to estimate the proportions of total variance contributed by analysis, sampling and geochemistry. Robust statistics are required because of the high sensitivity of classical ANOVA to small numbers of outlying values, due to either technical or geochemical causes. In addition, the robust procedure accommodates any enhanced measurement errors, especially those due to sampling that are associated with anomalous sites. Thus the method provides valid estimates of analytical and sampling variance at an average background level, regardless of the presence of a minor proportion of samples from anomalous sites. The precision requirement of geochemical analysis is that it should contribute less than 4% to the total variance. Furthermore the combined sampling and analytical (or technical) variance should contribute less than 20% to the total variance. Ideally, the analytical variance should be less than 20% of this technical variance. The presentation of these contributions to the total variance as a pie chart provides a rapid means of communicating the quality of the geochemical information. For the analytical geochemist SAX provides information for method development and selection by identifying the need for improvement in technical precision, either in the sampling or in analytical procedure. For the field geochemist the SAX system is particularly appropriate for orientation surveys in which a high level of duplication can be used to optimize sampling procedures (such as collection protocol, sample size, field processing) and the selection of appropriate analytical techniques and laboratories.
36
M.H. RAMSEY ET AL.
When implemented at a lower density (say, 10% field duplicates) in a subsequent large scale survey, SAX provides confirmation of the contribution to random measurement error made by the sampling and analytical procedures, and therefore confidence in the suitability of the data for geochemical interpretation. Potentially useful applications for SAX include orientation surveys for gold exploration, where sampling variability is a well recognised problem. Similarly, in environmental geochemistry sampling is often the limiting factor in the description of pollution, rather than imprecision in chemical analysis. ACKNOWLEDGEMENTS
The authors would like to thank Prof. B.D. Ripley, Department of Statistics, University of Oxford, for the program for nested robust ANOVA.
REFERENCES Analytical Methods Committee, 1989a. Robust statistics--how not to reject outliers--Part 1. Basic concepts. Analyst, 114: 1693-1697. Analytical Methods Committee, 1989b. Robust statistics--how not to reject outliers--Part 2. Inter-laboratory trials. Analyst, 114:1699-1705. Dixon, W.J., 1951. Ratios involving extreme values. Ann. Math. Statist., 22: 68-78. Garrett, R.G., 1969. The determination of sampling and analytical errors in exploration geochemistry. Econ Geol., 64: 568-569. Garrett, R.G. and Goss, T.I., 1980. UANOVA: A Fortran 1V program lbr unbalanced nested analysis of variance. Comput. Geosci., 6: 35-60. Miesch, A.T., 1964. Effects of sampling and analytical error in geochemical prospecting. In: G.A. Parks (Editor), Computers in the Mineral Industry (Part 1 ). Stanford Univ. Publ. Geol. Sci., 9( 1 ): 156-170. Ramsey, M.H., Thompson, M. and Banerjee, E.K., 1987. A realistic assessment of analytical data quality from inductively coupled plasma atomic emission spectrometry. Anal. Proc., 24: 260-265. Rose, A.W., Hawkes, H.E. and Webb, J.S., 1979. Geochemistry in Mineral Exploration. Academic Press, New York, pp. 33-36. Thompson, M. and Howarth, R.J., 1976. Duplicate analysis in geochemical practice. Analyst, 101: 690-698. Thompson, M. and Howarth, R.J., 1980. The frequency distribution of analytical error. Analyst, 105:1188-1195. Thompson, M. and Wood, S.J., 1982. Atomic absorption methods in applied geochemistry. In: J.C. Cantle (Editor), Atomic Absorption Spectrometry. Elsevier, Amsterdam.