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## Revista mexicana de ciencias geológicas

*versión impresa* ISSN 1026-8774

### Rev. mex. cienc. geol v.25 n.1 México 2008

**Critical values for 33 discordancy test variants for outliers in normal samples up to sizes 1000, and applications in quality control in Earth Sciences**

**Valores críticos de 33 variantes de pruebas de discordancia para los datos desviados en muestras normales con tamaños de hasta 1000 y aplicaciones en control de calidad en las Ciencias de la Tierra **

**Surendra P. Verma ^{1,*}, Alfredo Quiroz–Ruiz^{1}, and Lorena Díaz–González^{2}**

^{1} Centro de Investigación en Energía, Universidad Nacional Autónoma de México, Priv. Xochicalco s/n, Col. Centro, Apartado Postal 34, Temixco 62580, Mexico.* spv@cie.unam.mx

^{2} Posgrado en Ingeniería (Energía), sede Centro de Investigación en Energía, Universidad Nacional Autónoma de México, Priv. Xochicalco s/n, Col. Centro, Apartado Postal 34, Temixco 62580, Mexico.

Manuscript received: June 22, 2007

Corrected manuscript received: August 10, 2007

Manuscript accepted: August 12, 2007

**ABSTRACT**

In two earlier papers (Verma and Quiroz–Ruiz, 2006, Rev. Mex. dene. Geol, 23,133–161, 302–319) precise critical values for normal univariate samples of sizes n up to 100 have been reported. However, for greater n, critical values are available only for a few tests: Nl for n up to 147, N4k2 for nuptol 49, N6, N14 and N15 (for the latter three tests, critical values were reported for only n= 200, 500, and 1000). This clearly demonstrates the need for proposing new critical values for n >100 through an adequate statistical methodology. Therefore, modifications of our earlier simulation procedure as well as new, precise, and accurate critical values or percentage points (with four to eight decimal places; average standard error of the mean 0.00000003–0.0039) of 15 discordancy tests with 33 test variants, and each with seven significance levels a = 0.30, 0.20, 0.10, 0.05, 0.02, 0.01, and 0.005, for normal samples of sizes n up to 1000, viz., n_{min}(1)100(5)200(10)500(20)1000, are reported. For the first time in the literature, the standard error of the mean is also reported explicitly and individually for each critical value. Similarly, a new methodology involving artificial neural network (ANN) was used, for the first time in published literature, to obtain interpolation equations for all 33 discordancy test variants and for each of the seven significance levels. Each equation was fitted using 76 simulated data for n from 100 to 1000 for a given test and significance level. Extremely small sums of squared residuals (5.5 × 10 ^{–8} –8.4 ×10^{–5}; generally <10^{–5}) in the ANN equations fitted for n=100 to 1,000 were obtained. As a result, the applicability of these discordancy tests is now extended up to 1000 observations of a particular parameter in a statistical sample. The new most precise and accurate critical values will result in more reliable applications of these discordancy tests than have been possible so far in various scientific and engineering fields, particularly for quality control in Earth Sciences. The multiple–test method with new critical values was shown to perform better than both the box–and–whisker plot and the "two standard deviation " methods used by some researchers, and is therefore the recommended procedure for handling experimental data.

**Key words:** outlier methods, normal sample, two standard deviation method, 2s method, reference materials, Monte Carlo simulation, critical values, Dixon tests, skewness, kurtosis, artificial neural network, ANN, statistics, petroleum hydrocarbon, Nd isotopes, BCR–1.

**RESUMEN**

En dos trabajos anteriores (Verma and Quiroz–Ruiz, 2006, Rev. Mex. dene. Geol, 23, 133–161, 302–319) se han reportado valores críticos precisos para pruebas de discordancia en muestras normales univariadas n hasta 100. Sin embargo, para n >100, se dispone solamente de valores críticos para las pruebas: NI para n hasta 147, N4k2 para n hasta 149, N6, NI4 y NI5 (para las últimas tres pruebas, valores críticos han sido reportados solamente para n=200, 500 y 1000). Esto demuestra claramente la necesidad de proponer nuevos valores críticos para n >100 mediante una metodología estadística apropiada. Por lo tanto, se reportan las modificaciones del procedimiento de la simulación así como valores críticos o puntos porcentuales nuevos y más precisos y exactos (con cuatro hasta ocho puntos decimales; el error estándar de la media 0.00000003 – 0.0039) para 15 pruebas de discordancia con 33 variantes, y cada una con siete niveles de significancia a = 0.30, 0.20, 0.10, 0.05, 0.02, 0.01 y 0.005, para muestras normales con tamaño n hasta 1000, viz., n_{min} (1)100(5)200(10)500(20)1000. Por primera vez en la literatura, se reporta el error estándar de la media explícitamente y en forma individual para cada valor crítico. De igual manera, una nueva metodología que consiste en la aplicación de redes neuronales artificiales (ANN, por sus siglas en inglés) fue usada, por primera vez en la literatura publicada, para obtener ecuaciones de interpolación para las 33 variantes de las pruebas de discordancia y para cada uno de los siete niveles de significancia. Cada ecuaciónfue ajustada con los 76 datos de las simulaciones para n desde 100 hasta 1,000 correspondientes a cada prueba y cada nivel de significancia. Sumas de cuadrados de los residuales extremadamente pequeñas (5.5 × 10 ^{–8} –8.4 ×10^{–5}; generalmente <10^{–5}) fueron obtenidas en el ajuste de las ecuaciones por ANN para n =100 a 1,000. Como consecuencia, la aplicabilidad de las pruebas de discordancia ha sido extendida hasta 1,000 observaciones de un determinado parámetro en una muestra estadística. Los valores críticos nuevos y mucho más precisos y exactos resultarán en aplicaciones más confiables de las pruebas de discordancia que han sido posibles hasta ahora en una variedad de campos de las ciencias e ingenierías, particularmente para el control de calidad en Ciencias de la Tierra. El método de pruebas múltiples con nuevos valores críticos proporcionó mejores resultados que los métodos de la gráfica de "boxy whisker" y de "dos desviaciones estándar" usados por algunos investigadores y, por lo tanto, el presente método estadístico es el más recomendado para el manejo de datos experimentales.

**Palabras clave:** métodos de valores desviados, muestra normal, prueba de dos desviaciones estándar, 2s, materiales de referencia, simulación Monte Carlo, valores críticos, pruebas de Dixon, sesgo, curtosis, redes neuronales artificiales, RNA, estadística, hidrocarburos de petróleo, isótopos de Nd, BCR–1.

**INTRODUCTION**

Two recent papers (Verma and Quiroz–Ruiz, 2006a, 2006b) have reported a highly precise and accurate Monte Carlo type simulation procedure for N(0,1) random normal variates and presented new, precise, and accurate critical values for seven significance levels *a = *0.30,0.20, 0.10, 0.05, 0.02,0.01, and 0.005, andfor sample sizes *n *up to 100 for 15 discordancy tests with 33 variants. Table 1 summarizes these tests. However, for greater *n, *only a few critical values are available in the literature (Barnett and Lewis, 1994; Verma, 2005). These values are for tests: Nl *(n *up to 147); N4k2 *in *up to 149); N6, N14 and N15 (for the latter three tests, critical values with only two decimal places were reported for only *n *=200, 500, and 1000).

Reference materials (RMs) are routinely used for quality control in Earth Sciences (*e.g., *Verma, 1997,1998, 2005; Velasco–Tapia *et al*., 2001; M.P. Verma, 2004; Lozano and Bernal, 2005; Guevara *et al, *2005; Sang *et al, *2006; Santoyo *et al, *2006; Papadakis *et al, *2007). In other fields of science and engineering also, quality control through RMs has become mandatory, for example, in biology and medicine (Okamoto *et al, *1996; Dybczynski *et al, *1998; Patriarca *et al, *2005); environmental sciences (Gill *et al, *2004; Graybeal *et al, *2004; Farre *et al, *2006); and food research (In't Veld, 1998; Langton *et al.*, 2002; Gabrovská *et al, *2006).

When a large number of laboratories around the world participate in a cooperative study of a RM, the number of individual data (*n*) for a given chemical element in that RM can exceed 100. In these cases, at present the multiple–test method initially proposed by Verma (1997) and practiced by Verma (1998,2005) and Verma and Quiroz–Ruiz (2006a, 2006b), among others, is not likely to be appropriately applicable due to the unavailability of precise critical values for *n *>100 for most discordancy tests (Table 1). This clearly demonstrates the need for proposing new critical values for *n *>100 through an adequate statistical methodology. Requirements of critical values for large *n *(>100) also exist in an altogether different field of molecular and cellular proteomics (Xia *et al.*, 2006; Murray Hackett, written communication, June 2007).

For the present work, we have included most discordancy tests for normal univariate samples (15 tests with 33 test variants; see Table 1) for simulating new, precise, and accurate critical values for the same seven significance levels (a = 0.30 to 0.005) and for *n *up to 1000, viz., *n*_{min} (1)100(5)200(10)500(20)1000 (where *n*_{min }is the minimum number of data that could be tested by a given statistical test; see Table 1), using a simulation procedure slightly modified after Verma and Quiroz–Ruiz (2006a, 2006b). Further, a novel approach is followed, for the first time in the literature, for presenting these new critical values along with the respective standard errors and for interpolating the simulated critical values using artificial neural network (ANN). These results are useful in all fields of science and engineering, especially in quality control in Earth Sciences. We present a few examples of the application of all normal univariate tests (Table 1) for which we have reported new, most precise critical values in this paper.

**DISCORDANCY TESTS**

We will not repeat the explanation of discordancy tests; the reader is referred to Barnett and Lewis (1994), Verma (2005), or the recent papers by Verma and Quiroz–Ruiz (2006a, 2006b). The 15 tests with their 33 variants for which critical values were simulated are listed in Table 1.

**SIMULATION PROCEDURE FOR MOST PRECISE AND ACCURATE CRITICAL VALUES**

Our highly precise and accurate Monte Carlo type simulation procedure has already been described in detail (Verma and Quiroz–Ruiz, 2006a, 2006b) and, therefore, will not be repeated here. However, some required changes will be mentioned.

In our present work, the simulations were of sizes 500,000 for tests N3–N5 and N7–N13; 1,000,000 for N14; and 2,000,000 for Nl, N2, N6, and N15. They were repeated ten times (each using a different set of 500,000,000 to 2,000,000,000 random normal variates). Different simulation sizes (500,000 to 2,000,000) were appropriate to optimize the simulation time required for the use of personal computers and to obtain, at the same time, "acceptable" simulation errors for all tests. For tests N2, N5–N8, N14 and N15, the final mean critical value or percentage point and its standard error * *for each *n *and *α *were estimated from ten repetitions. However, for tests such as Nl (Table 1) two independent test statistics (one for an upper and the other for a lower outlier) were simulated and thus 20 independent results could be obtained from the same simulation scheme as reported earlier (Verma and Quiroz–Ruiz, 2006a, 2006b). Besides test Nl, because of the existence of the upper and lower versions of the statistic (Table 1), 20 results of critical values and their error estimates were also obtained for tests N3–k=2,3,4, N4–k=l,2,3,4, and N9–N13. For all these tests, therefore, and * *calculations were based on 20 independent results.

**RESULTS OF NEW CRITICAL VALUES**

Both * *and data for 33 discordancy test variants (Table 1), for *n* from *n*_{min} (3,4, 5,6,7, 8, or 9, depending on the type of statistic to be calculated) up to 1000, viz., *n*_{min}(1)100(5)200(10)500(20)1000, and * >α = *0.30, 0.20, 0.10, 0.05, 0.02, 0.01, and 0.005 (corresponding to confidence level of 70% to 99.5%, or equivalently significance level of 30% to 0.5%), are summarized in Tables A1–A40 (40 tables in the electronic supplement; 20 odd–numbered tables for

*and 20 even–numbered tables for*

*.*Thus, our data presentation approach is novel because, for the first time in the literature, the precision estimates are explicitly tabulated for each critical value. For example, in Table Al the rounded

*values are presented individually for each*

*n*and

*, whereas in Table A2 the rounded values are similarly listed for test Nl; the rounding procedure follows the guidelines suggested by Verma (2005). Similarly,*

*α**and values are presented consecutively for the remaining tests N2 to N15 in Tables A3–A40.*

For all cases, our present values are more reliable (error is given by a small number on the third up to the eighth decimal place) than the earlier literature values (compiled by Barnett and Lewis, 1994; Verma, 2005), including those reported by Verma and Quiroz–Ruiz (2006a, 2006b). In fact, the errors of these literature critical values, except those by Verma and Quiroz–Ruiz (2006a, 2006b), are not precisely known. A synthesis of standard errors of the mean for all tests is presented in Table A41 of the electronic supplement. The errors of the present critical values for *n *up to 1,000 (Table A41) range as follows: 0.00000009–0.0007 for test N1 (see also Table A1); –0.00000003–0.0009 for test N2 (Table A3), 0.00005–0.0019 for N3–k=2 (Table A5); 0.00009–0.0020 for N3–k=3 (Table A7); 0.00010–0.0021 for N3–k=4 (Table A9); 0.00000023–0.00040 for N4–k=1 (Table A11); 0.00000007–0.00025 for N4–k=2 (Table A13); 0.0000017–0.00021 for N4–k=3 (Table A15); 0.0000021–0.00018 for N4–k=4 (Table A17); –0.00000005–0.00035 for N5–k=2 (Table A19); 0.00000005–0.0012 for N6–k=2 (Table A21);–0.000016–0.0005 for N7 (Table A23); 0.000015–0.0006 for N8 (Table A25); 0.000015–0.00028 for N9 (Table A27); 0.000016–0.00032 for N10 (Table A29); –0.000015–0.00028 for N11–k=2 (Table A31); 0.000008–0.00025 for N12–k=2 (Table A33); –0.000011–0.00024 for N13–k=2 (Table A35); 0.000023–0.0012 for N14 (Table A37); and 0.000015–0.0039 for N15 (Table A39).

The much greater precision (and accuracy) of critical values simulated by Verma and Quiroz–Ruiz (2006a, 2006b) as compared to the literature values was already documented. Here, we compare the mean values of the standard errors for *n *up to 100 for all tests obtained in the present work with those obtained by Verma and Quiroz–Ruiz (2006a, 2006b) and show that the most precise critical values than ever attempted in the literature are now being reported (Table A41; see also Figure 1). And this improvement is due to the fact that much larger simulation sizes of 500,000 to 2,000,000 are used in the present work, which have resulted in smaller standard errors than was earlier possible from sizes of 100,000 to 500,000. Further, when in the present work the sample sizes were exactly the same as those in Verma and Quiroz–Ruiz (2006b), for example, 500,000 for tests N3–k=2, 3, and 4 (see footnote of Table A41 for a correction and explanation), the errors were exactly the same (see diamond symbols that plot right at the diagonal line in Figure 1). For these cases, not only the errors were exactly the same, but also the critical values were identical in both simulations (this work and Verma and Quiroz–Ruiz, 2006b). This is an interesting observation and testifies the high reproducibility of our simulation procedure because the earlier simulations (Verma and Quiroz–Ruiz, 2006b) were programmed in C whereas the present simulations were programmed in a different language (Java) and were run on a different and faster personal computer equipped with a different processor than our earlier work.

As our earlier tables for sample sizes up to 100 (Verma and Quiroz–Ruiz, 2006a, 2006b), these new critical value data, along with their individual uncertainty estimates, are available in other formats such as *txt, Excel, *or *Statistica. *on request from the authors (S.R Verma spv@cie.unam.mx, A. Quiroz–Ruiz aqr@cie.unam.mx, or L. Díaz–González ldg@cie.unam.mx). Similarly, the interpolation equations (see below) can also be obtained in a *doc *file with plain text format.

**RESULTS OF INTERPOLATIONS OF CRITICAL VALUES USING ARTIFICIAL NEURAL NETWORK ****(ANN)**

A new methodology was developed that involved the use of ANN for obtaining the best–fitted interpolation equations. This was actually required because, for 100 __<__ *n *__<__1000, critical values were not simulated for all *n *(see Table 1 for tests and Tables A1–A40 for information on simulated *n). *No attempt was made in the present work to fit equations to critical values for *n *<100 mainly because precise and accurate critical values for all *n *<100 have already been simulated (Verma and Quiroz–Ruiz, 2006a, 2006b; this work). Therefore, interpolation equations were actually not required for small *n. *Prior to our work, different kinds of interpolation or fitting (Bugner and Rutledge, 1990; Rorabacher, 1991; Verma *et al, *1998) to low precision critical values available in the literature (see Barnett and Lewis, 1994; Verma, 2005; Verma and Quiroz–Ruiz, 2006a, 2006b) were used for this purpose.

This is the first time in published literature that ANN was used for fitting highly sophisticated equations to the most precise and accurate simulated critical value data for *n *between 100 and 1000, with extremely small sums of squares of residuals and thence for predicting interpolated critical values with the smallest error. Details on the ANN can be found in Hassoun (1995) or Haykin (1999).

The fitted equations for test N1 using all critical values listed for *n *from 100 to 1000 (Table A2) for each *a *(from 0.30 to 0.005) are presented in Table 2. The values of the sum of squared residuals of each equation Σ(*SIM–ANN) ^{2} *for the 76 simulated critical values, corresponding to

*n =*100(5)200(10)500(20)1000 and

*α =*0.30, 0.20, 0.10, 0.05, 0.02, 0.01, and 0.005, used for equation fitting are also included in Table 2. For the remaining tests (N2 to N15), the equations are summarized in Tables A42–A60 of the electronic supplement.

The fitting quality parameter Σ* *(*SIM–ANN*)* ^{2} *for the interpolation equations for n=100(5)200(10)500(20)1000 (Tables 2 and A42–A60) range as follows: ~2.4 × 10

^{-7}~ 4.1 × 10

^{-6}for test N1 (see Table 2); ~4.1 × 10

^{-7}– 7.8 × 10

^{–}

^{6}for test N2 (Table A42), ~1.8 × 10

^{}

^{–}

^{6}– 3.0 × 10

^{}

^{–}

^{5}for N3–k=2 (Table A43); ~4.1 × 10

^{–6}– 7.6 × 10

^{–5 }for N3–k=3 (Table A44); ~7.7 × 10

^{–6}– 8.4 × 10

^{}

^{–5}for N3–k=4 (Table A45); ~1.9 × 10

^{–8}– 4.8 × 10

^{–7 }for N4–k= 1 (Table A46); ~1.0 × 10

^{–8}– 8.2 × 10

^{–7}for N4–k=2 (Table A47); –9.3 × 10

^{–8}– 8.7 × 10

^{–7 }for N4–k=3 (Table A48); ~1.9 × 10

^{–7}– 8.8 × 10

^{–7}for N4–k=4 (Table A49); ~7.5 × 10

^{–8}– 5.3 × 10

^{}

^{–7}for N5–k=2 (Table A50); ~6.7 × 10

^{–7}– 2.2 × 10

^{–5 }for N6–k=2 (Table A51); ~4.2 × 10

^{–7}–1.3 × 10

^{–6 }for N7 (Table A52); –2.7 × 10

^{}

^{–7}– 9.7 × 10

^{–7}for N8 (Table A53); –5.1 × 10

^{–7}– 2.0 × 10

^{–}

^{5}for N9 (Table A54); ~5.7 × 10

^{–7}– 9.6 × 10

^{–6}for N10 (Table A55); ~1.4 × 10

^{–7}–8.0 × 10

^{–7}for N11–k=2 (Table A56); ~1.9 × 10

^{–7}– 7.2 × 10

^{–6}for N12–k=2 (Table A57); ~2.8 × 10

^{–6}–2.9 × 10

^{}

^{–}

^{5 }for N13–k=2 (Table A58); ~5.7 × 10

^{–7}– 3.0 × 10

^{–}

^{5}for N14 (Table A59); and ~1.8 × 10

^{–7}– 2.8 × 10

^{–5 }for N15 (Table A60). Thus, the fitting quality parameter Σ (

*SIM–ANN*)

^{}

*for the interpolation equations was generally <10*

^{2}^{5}(andalways <10

^{4}).

These equations can be used to compute precisely the interpolated critical values for all *n *between 100 and 1000, for which such values are not listed in Tables A2–A40 (see even–numbered tables). Thus, precise critical values can be made available for all *n *between *n*_{min} and 1000, viz., *n*_{min} (1)1000 (see Table 1 for more information on all tests and their *n*_{min} values). Figure 2 shows, as an example, the simulated critical values for test Nl *(CV _{TN1}) *for all values of a (0.30 to 0.005) as a function of

*n*(from 100 to 1000). The respective interpolation equations are also plotted using dotted or dashed curves. Note these equations very closely match the simulated critical values to such an extent that the curves cannot be properly observed in Figure 2.

**APPLICATIONS IN SCIENCE AND ENGINEERING**

The tests (Table 1) after extending their applicability to samples of sizes up to 1,000, can be applied to all examples earlier summarized by Verma and Quiroz–Ruiz (2006a, 2006b). These include all the following fields (but are not limited to them): Agricultural and Soil Sciences; Aquatic Environmental Research; Astronomy; Biology; Biomedicine and Biotechnology; Chemistry; Electronics; Ecology; Geochronology; Geodesy; Geochemistry; Isotope Geology; Medical Science and Technology; Meteorology; Paleontology; Petroleum Hydrocarbons and Organic Compounds in Sediment Samples; Quality Assurance and Assessment Programs in Biology and Biomedicine, in Cement Industry, in Food Science and Technology, in Environmental and Pollution Research, in Nuclear Science, in Rock Chemistry, in Soil Science, and in Water Research; Structural Geology; Water Resources; and Zoology. Further, our new critical values for *n *up to 1,000 will be equally useful for applying these discordancy tests to identify outliers in linear regressions, such as those recently employed by Verma *et al. *(2006).

**APPLICATIONS IN QUALITY CONTROL IN EARTH SCIENCES**

As stated earlier in the Introduction section, the most important requirement for simulating critical values for *n *>100 was for processing interlaboratory data for RMs. This is confirmed from the information synthesized in Table 3. Thus, the data for all chemical elements, without exception, in all RMs (Table 3) can now be processed for possible outliers and thence for correctly computing central tendency (or location) and dispersion (or scale) parameters (see Verma, 2005, for more details) using outlier–based statistical methods.

**Specific examples**

We now present examples or case histories to illustrate the use of all discordancy tests for which new critical values have been obtained in this work. For these applications, we chose the strict confidence level of 99% (*i.e., *we used the 99% CL, or 1% SL, or 0.01 *a *column; see the respective critical values in Tables A2 to A40 – even–numbered tables in the electronic, supplementary data file).

*Example 1. Comparison of multiple–test method with box–and–whisker plot method: Chlorinated pesticides and petroleum hydrocarbons in a sediment reference sample*

Verma and Quiroz–Ruiz (2006b) used interlaboratory data for one sediment RM (IAEA–417; IAEA–International Atomic Energy Agency) to show that, for detecting outliers, the multiple–test method of Verma (1997) performed better than the box–and–whisker plot method used by Villeneuve *et al. *(2002). Here, we use, as the first example, a different sediment RM (IAEA–408) from an interlaboratory study by Villeneuve *et al. *(1999) to highlight the use of the multiple–test method and compare its performance with the box–and–whisker plot method used by the original authors. The individual data for nine selected compounds (six chlorinated pesticides and three petroleum hydrocarbons) were compiled in Table A61 (see electronic supplement) from the original report by Villeneuve *et al. *(1999). The multiple–test method (Table 1) consisted in applying all nine single–outlier (with 13 test variants) and seven multiple–outlier tests (with 20 test variants) at the strict confidence level of 99% to a given set of data although, if desired, a lesser number of tests or a less strict confidence level, such as 95%, could instead be used. The final concentration data along with the basic statistical information are summarized in Table 4. A greater number of discordant outliers were detected in the data for most compounds listed in Table A61, by the multiple–test method than the box–and–whisker plot method used by the original authors (Villeneuve *et al., *1999). Consequently, smaller standard deviations (and somewhat different mean values, although probably not statistically different) were obtained from the multiple–test method as compared to the box–and–whisker plot method for all cases except one compound (PCB101) for which none of the two methods detected any outliers. Note also that, because of the presence of outliers, the initial mean and standard deviation data strongly differ from the final statistical parameters. Finally, had we applied the multiple–test method at a less strict confidence level of 95% (which will probably be consistent with the box–and–whisker plot method although details on the respective confidence level were not provided by Villeneuve *et al., *1999), a greater number of outliers would have been identified in most cases than those obtained at 99% confidence level, with the consequent reduction of the standard deviation values obtained by our method and possible changes in the mean values (Table 4).

Application of significance tests, such as F–test and Student–t test (see Verma, 2005 for more details on significance tests), to evaluate the performance of the multiple–test method versus the box–and–whisker plot method is not advisable using the present data, because rather small number of degrees of freedom (7–33; the final number of data remaining after outlier elimination vary from 8 to 31 for *n _{f}* and from 13 to 34 for

*n*in Table 4) are involved. We consider these numbers too small for quality control purposes. Therefore, an objective statistical comparison of the multiple–test method and the box–and–whisker plot method should be done using larger datasets.

_{l }Nevertheless, as in Verma and Quiroz–Ruiz (2006b), but using different datasets, we now conclude that the multiple–test method exemplified in this work can be advantageously used in future to arrive at the final statistical parameters in such interlaboratory studies.

*Example 2. Comparison of multiple–test method with "two standard deviation" (2s) method: Two chemical elements and one isotopic ratio in a geochemical reference material (BCR–1) from U.S.G.S.*

We present the example of just two elements (petro–genetically important trace elements Sm and Nd) and one widely used radiogenic isotopic ratio (^{143}Nd/^{144}Nd) in a rock RM (Columbia River basalt) BCR–1 from the United States Geological Survey (U.S.G.S.), U.S.A. This RM was extensively used four decades ago because it was recommended as *the *RM for all studies of lunar rocks, *i.e., *researchers reporting data on lunar rocks had to evaluate their data quality by reporting the analysis of this particular RM. Since then, this RM has been widely used in geochemical laboratories, including isotope laboratories; most Nd isotope studies, even today, report ^{143}Nd/^{144}Nd in this RM. However, because B CR–1 is no longer available for distribution by the U.S.G.S., it is now replaced by BCR–2 (a sample collected at the same site as BCR–1).

The individual data for BCR–1 for Sm, Nd, and ^{143}Nd/^{144}Nd were compiled from Gladney *et al. *(1990) and are presented in supplementary Tables A62, A63, and A64, respectively. No attempt was made to complement them with more recent data on this widely used RM because our main aim was to compare the performance of the multiple–test method with the "two standard deviation" (2s) method used by Gladney *et al. *(1990) to process their compiled data. Although Gladney *et al. *(1990) is a relatively old compilation reference, this is the latest one available on this particular RM (BCR–1) in published literature.

The results of the application of the multiple–test method, along with those from the 2s method, are also summarized in Table 4. For Sm, the same number (10) of outliers were detected by both methods (multiple–test versus 2s) whereas for Nd the multiple–test method detected more outliers (22) than the 2s method (11). However, note that the multiple–test method was applied at the strict confidence level of 99% (and not at the less strict level of 95%, which would, in theory, correspond to the 2s method, and is likely to detect more outliers than the 99% level).

Nevertheless, the statistically correct procedure for applying discordancy tests to such analytical datasets (*e.g., *Sm and Nd data obtained by different types of analytical methods) would be to: (i) construct statistical samples for each method–group and process them separately for outliers using the multiple–test method; (ii) apply ANOVA ("ANalysis Of VAriance") test to decipher any statistically significant differences among different method–groups, and isolate a particular group if significantly different from the remaining ones at a predetermined confidence level; (iii) combine the data from different method–groups showing no significant differences and process them again for possible outliers; and (iv) calculate the final statistical parameters from the normally distributed final data. We suggest that ANOVA be applied at the strict confidence level of 99% (see Verma, 2005, for the respective tabulated critical values). The results from such a statistically correct procedure are also presented for Sm and Nd in rows marked by an asterisk (*) in Table 4. Gladney *et al. *(1990) did not present such method group–based results for their 2s method although they did so for individual techniques. A clear advantage of the multiple–test method as compared to the 2s method is observed for detecting outlying observations for both Sm and Nd concentration data if we compare our results (see rows marked by an asterisk in Table 4) with the results of "all–groups" presented by Gladney *et al. *(1990).

We applied the F–test and Student–t test to the two sets of Sm and Nd concentration data in order to evaluate if there were significant differences between the final results obtained by the multiple–test method and the 2s method (Table 4). Significant differences (at 95% confidence level for both Sm and Nd, and even at 99% confidence level for Nd) in standard deviation or variance were observed between these two sets of data when statistically correct procedure involving method groups was used for the multiple–test method (see rows marked by an asterisk in Table 4). Variance estimates of Sm and Nd data processed by the multiple–test method were significantly lower than those obtained by the 2s method. Such results will have important implications for instrumental calibrations (using weighted regression techniques such as those used by Guevara *et al.*, 2005) and data quality assessments (using significance tests). More details can be found in Verma (2005). The comparison of the Nd isotope data is discussed below.

First of all, it is very important to note that for ^{143}Nd/ ^{144}Nd from California Institute of Technology (CalTec)–type laboratories that normalize, during data acquisition, the Nd isotopic composition to ^{148}Nd/^{144}Nd = 0.243082 (see DePaolo and Wasserburg, 1976, for details on CalTec–type laboratories), the data have to be converted according to the following Equation (1), in order to make them consistent with the numerous laboratories around the world that are Lamont–type and use for normalization ^{146}Nd/^{144}Nd = 0.7219 (see O'Nions *et al.*, 1977, for more details on Lamont–type laboratories).

Almost concurrently with the CalTec and Lamont laboratories, Richard *et al. *(1976) from the University of Paris also discovered the utility of the Sm–Nd isotope sys–tematics in Earth Sciences, but they used ^{143}Nd/^{146}Nd instead of ^{143}Nd/^{144}Nd to show the usefulness of their work.

The existence of two types of laboratories was probably one of the main reasons, among others, why the CalTec researchers introduced the *ε _{Nd} *notation (for the definition of

*see DePaolo and Wasserburg, 1976,1977). Although the actual values of*

*ε*_{Nd}^{143}Nd/

^{144}Nd from these two different types of laboratories are drastically different (see the nine lower values in the compilation by Gladney

*et al.*, 1990, that are totally distinct from the rest of the compiled data; these values are identified by an asterisk and listed in Table A64

*after*their conversion according to Equation (1) and therefore, do not significantly differ from the rest of the data), the use of

*makes the Nd isotope data from these two types of laboratories directly comparable and fully consistent. It has so happened that the Lamont–type laboratories have become much more numerous than the CalTec–type laboratories (for example, see the compilation by Gladney*

*ε*_{Nd}*et al,*1990, in which only nine values, out of a total of 102, were from the CalTec–type laboratories; see also Table A64 in which CalTec–type data are identified by an asterisk).

This essential conversion for handling ^{143}Nd/^{144}Nd from these two types of laboratories was not recognized by Gladney *et al. *(1990) as a requirement to statistically process Nd isotopic data, which resulted in erroneous processing of the isotope data (Table 4). Thus, their 2s method should have certainly, but erroneously, rejected the CalTec–type data (nine data; see Table 4) as outliers although this was not clearly specified by these authors. The multiple–test method, on the other hand, showed that no outlier is present in this dataset at the strict confidence level of 99%.

Verma and Quiroz–Ruiz (2006b) extensively commented on the shortcomings of the 2s method and used a rock RM peridotite JP–1 from Japan to show that just one multiple–outlier test N3 in its three variants (k=2,3,4) applied at 95% confidence level, performed better than the 2s method for detecting outliers. Here, using different datasets (Sm, Nd and ^{143}Nd/^{144}Nd in BCR–1), we conclude that the multiple–test method exemplified in this work can be advantageously used in future to arrive at the final statistical parameters in such interlaboratory studies and the probably statistically erroneous 2s method should be abandoned. Such outlier methods based on "fixed" multiples of standard deviation have also been recently criticized by Hayes *et al *(2007).

*Example 3. Other examples of outlier tests applicable in Earth Sciences*

We now briefly comment on the need of using the above multiple–test method in numerous geoscience studies. Verma and Quiroz–Ruiz (2006b) already applied the multiple–test method to oxygen isotope data for the Los Azufres hydrothermal system reported by Torres–Alvarado (2002). They also pointed out other studies where the multiple–test method would be useful.

Similarly, this multiple–test method with the new critical values can be successfully applied to process and better interpret: (i) effective weight, variation index and other groundwater data discussed by El–Naqa *et al. *(2006); (ii) univariate and bivariate data of ammonites documented by López–Palomino *et al. *(2006), the latter (bivariate) data using studentized residuals from the regression; (iii) bivariate data for naturally fractured reservoirs from Mexico presented by Miranda–Martinez *et al. *(2006); (iv) data acquisition stage of mass spectrometric instrumentation used for ^{40}Ar–^{39}Ar (Molina–Garza and Ortega–Rivera, 2006) or K–Ar geochronology (Solé *et al.*, 2007), including Rb–Sr, Pb–Pb, Sm–Nd, and Re–Os geochronology or isotope geology (see for example, Wang *et al, *1998; Dougherty–Page and Bartlett, 1999, who used just one Dixon test); (v) geochemical data on granitic xenoliths and rocks presented and compiled by Corona–Chavez *et al. *(2006); (vi) micro–probe data recently reported by Colombo *et al *(2007); (vii) chemical data of inactive tailings from the Santa Barbara mineral zone, Chihuahua, documented by Gutiérrez–Ruiz *et al. *(2007); (viii) chemical data for recent and historic tailings of a Pb–Zn–Ag skarn deposit analyzed by Méndez–Ortiz *et al *(2007); (ix) geochemical and stable isotope data for sedimentary rocks obtained by Nagarajan *et al. *(2007, in press); (x) cation and anion composition data for underground water reported by Ramos–Leal *et al. *(2007); (xi) geotechnical variables for oil prospect exploration decision making discussed by Salleh *et al. *(2007); (xii) major and trace element data for metabasic volcanic rocks presented by Shekhawat *et al. *(2007) and for topaz–bearing rhyolites documented by Rodríguez–Ríos *et al. *(2007); and (xiii) mineral composition data for metagabbroic rocks reported by Cruz–Gámez *et al *(2007).

We note that the discordant outliers, if present, are of much value to further understand the geological processes provided they are not caused solely by analytical uncertainty. The remaining normally distributed data (after eliminating outlying observations as judged by the applied discordancy tests) can then be used for correctly calculating the central tendency or location (mean) and dispersion or scale (standard deviation) parameters (see Verma, 2005, for more details).

As an example of the application of multiple–test method, we can mention that Colombo *et al. *(2007) applied five single–outlier tests (Nl, N2, N7, N8 and N9, with their seven test variants) to ascertain the absence of outliers in their geochemical data before calculating the mean and standard deviation values. We suggest that for small datasets such as those mentioned in this section, the multiple–test method could consist of applying consecutively all nine single–outlier tests (Nl, N2, N4–k=l, N7–N10, N14, and N15), with their 13 test variants, to detect possible outlier(s). The multiple–outlier tests (k=2 to k=4 types), with 20 test variants, could additionally be used for larger datasets such as those obtained in interlaboratory studies of RMs.

In summary, therefore, we emphasize that the multiple–test method proposed by Verma (1997) and exemplified in our paper is a recommended procedure to process experimental data under the assumption that the data are drawn from a normal distribution and departure from this assumption due to any contamination or presence of discordant outliers can be properly handled by tests Nl to N15 (all 15 tests with their 33 variants, or only those selected for this purpose).

**APPLICABILITY AND PERFORMANCE OF DISCORDANCY TESTS**

There has been considerable confusion regarding the applicability of discordarcy tests. Miller and Miller (2000, p. 54–57), among others, have expressed the view that Dixon tests are applicable to only small data sets, without actually providing any reasons for this limitation. Surprisingly, these authors intuitively refer to Dixon tests as a Dixon test (Dixon's Q test). Dixon's Q test (N7 by Dixon, 1951; N8 by King, 1953; see Table 1) is said to be "valid" for small *n = *3 to 7 (Miller and Miller, 2000, p. 54). Unfortunately, this kind of view has plagued the literature in chemistry. Dixon (1951) presented approximate critical values independently for all tests and for *n *up to 30. Then, why should Dixon tests be limited to *n = *3 to 7? It is true that Dixon tests are especially vulnerable to possible masking effects (Dixon, 1950; Barnett and Lewis, 1994), but this refers to their power and *not *their applicability. In fact, the power of tests as inferred by Dixon (1950) could have been seriously affected by the approximate nature of the critical values by Dixon (1951) – see the large differences between these values and those simulated by Verma and Quiroz–Ruiz (2006a), or those presented in the present work. This performance evaluation (Dixon, 1950) should also be considered rather incomplete from the modern statistical point of view presented by Barnett and Lewis (1994) and Hayes and Kinsella (2003). Therefore, the performance of Dixon and other tests should await further detailed work, which is already in progress by Verma and collaborators.

Nevertheless, statisticians specializing in outlier theory (*e.g.*, Barnett and Lewis, 1994, in their authoritative book, p. 218–234) have pointed out no such "applicability" limitations of single–outlier tests, including Dixon tests, the only limitation being the availability of critical values (see *e.g., *Verma, 1997; Velasco *et al, *2000; Verma and Quiroz–Ruiz, 2006a, 2006b) and the efficiency of discordancy tests. Barnett and Lewis (1994, p. 126) have, in fact, suggested that single–outlier tests should be classified as consecutive tests, and therefore, these tests can be used for identifying multiple (*i.e., *more than one) outliers (see p. 125–127 in Barnett and Lewis, 1994). The null and alternate hypotheses can thus be repeatedly postulated to test a given dataset for several outliers using single–outlier tests in a consecutive way. Of course, multiple–outlier tests or block procedures (see Barnett and Lewis, 1994) could be more suited for large datasets, but more work is needed to quantitatively evaluate the performance of all single– and multiple–outlier tests. For this reason we believe that new, precise and accurate critical values should be generated not only for small *n *but also for very large *n, *as has been accomplished in the present work. We plan to address these questions of utmost importance in our future work; the first paper in this direction is already in preparation by S.R Verma, L. Díaz–González, and R. González–Ramírez.

The performance of a discordancy test is an important "quality" factor that should be properly evaluated (Barnett and Lewis, 1994; Velasco and Verma, 1998; Velasco *et al., *2000; Hayes and Kinsella, 2003; Efstathiou, 2006). During the last decade, Barnett and Lewis (1994, p. 131) pointed out that much remains to be done in the area of performance of the various available procedures in the presence of outliers. Even today, this statement seems to be true. Thus, appropriate statistical procedures for handling and interpretation of experimental data should be the focus of future work. Masking and swamping effects are also important factors that require special attention and evaluation (Barnett and Lewis, 1994). Finally, because the performance of discordancy tests is not properly known at present although some empirical results were reported by Velasco and Verma (1998) and Velasco *et al. *(2000). Verma and collaborators (Verma, 1997,1998,2005; Verma *et al, *1998; Guevara *et al., *2001; Velasco–Tapia *et al., *2001; Verma and Quiroz–Ruiz, 2006a, 2006b; this work) have proposed and practiced the use of the multiple–test method and *not *just some selected test(s). The computation of new, precise and accurate critical values as carried out in the present work should facilitate in future to empirically better evaluate the performance of discordancy tests than attempted by Velasco *et al. *(2000).

**CONCLUSIONS**

We have used our established and well–tested Monte Carlo–type simulation procedure for generating new, precise and accurate critical values for 15 discordancy tests with 3 3 test variants for sample sizes up to 1000. For the first time in the published literature, these critical values , along with their individual uncertainty estimates* *as well as new interpolation equations obtained by ANN, are also tabulated. These new critical values will be very useful in many diverse fields of science and engineering, including in quality control in Earth Sciences. Specific examples are presented to highlight the use of these new critical values for quality control. The multiple–test method outlined in the present work seems to perform better than both the box–and–whisker plot and the "two standard deviation" (2s) methods used for processing interlaboratory data on RMs for quality control purposes. Much work is still needed to evaluate the performance of discordancy tests. The new critical values for all samples sizes up to 1000 simulated and interpolated in this work should certainly facilitate the performance evaluation.

**ACKNOWLEDGEMENTS**

This third manuscript was prepared as a result of the invitation to the first author extended during 2006 by the editor–in–chief Susana Alaniz–Álvarez. We are also grateful to Constantinos E. Efstathiou and an anonymous reviewer for a highly positive evaluation of our work. In spite of this appreciation, both of them pointed out several shortcomings in our earlier manuscript. Their critical comments helped us significantly improve our presentation.

**APPENDIX A. SUPPLEMENTARY DATA**

Tables A1–A64 can be found at the journal web site <http://satori.geociencias.unam.mx/>, in the table of contents of this issue (electronic supplement 25–1–01).

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