Valores críticos de 33 variantes de pruebas de discordancia para los datos desviados en muestras normales con tamaños muy grandes de 1,000 a 30,000 y evaluación de diferentes modelos de regresión para la interpolación y extrapolación de valores críticos

Surendra P. Verma* and Alfredo Quiroz–Ruiz

* Centro de Investigación en Energía, Universidad Nacional Autónoma de México, Priv. Xochicalco s/no., Col Centro, Apartado Postal 34, Temixco 62580, México. spv@cie.unam.mx

Manuscript received: March 14, 2008
Corrected manuscript received: May 14, 2008 ]]> Manuscript accepted: May 14, 2008

ABSTRACT

In this final paper of a series of four, using our well–tested simulation procedure we report new, precise, and accurate critical values or percentage points (with four to eight decimal places) of 15 discordancy tests with 33 test variants, and each with seven significance levels α = 0.30, 0.20, 0.10, 0.05, 0.02, 0.01, and 0.005, for normal samples of very large sizes n from 1,000 to 30,000, viz., 1,000(50) 1,500(100)2,000(500)5,000(1,000)10,000(10,000)30,000, i.e., 1,000 (steps of 50) 1,500 (steps of 100) 2,000 (steps of 500) 5,000 (steps of 1,000) 10,000 (steps of 10,000) 30,000. The standard error of the mean is also reported explicitly and individually for each critical value. As a result, the applicability of these discordancy tests is now extended to practically all sample sizes (up to 30,000 observations or even greater). This final set of critical values for very large sample sizes would cover any present or future needs for the application of these discordancy tests in all fields of science and engineering. Because the critical values were simulated for only a few sample sizes between 1,000 and 30,000, six different regression models were evaluated for the interpolation and extrapolation purposes, and a combined natural logarithm–cubic model was shown to be the most appropriate. This is the first time in the literature that a log–transformation of the sample size n before a polynomial fit is shown to perform better than the conventional linear to polynomial regressions hitherto used. We also use 1,402 unpublished dataseis from quantitative proteomics to show that our multiple–test method works more efficiently than the MAD_Z robust outlier method used for processing these data and to illustrate thus the usefulness of our final work on these lines.

Key words: outlier methods, normal sample, Monte Carlo simulations, critical value tables, Dixon tests, Grubbs tests, skewness, kurtosis, statistics, regression equations, log–transformation, proteomics.

RESUMEN

En este trabajo final de una serie de cuatro, usando nuestro procedimiento de simulación bien establecido reportamos nuevos valores críticos o puntos porcentuales, precisos y exactos (con cuatro a ocho puntos decimales) de 15 pruebas de discordancia con 33 variantes y cada uno con siete niveles de significancia α = 0.30, 0.20, 0.10, 0.05, 0.02, 0.01 y 0.005, para muestras normales de tamaños muy grandes n de 1,000 a 30,000, viz., 1,000 (50)1, 500(100)2,000 (500) 5,000(1,000)10,000(10,000)30,000, esto es, 1,000 (pasos de 50) 1,500 (pasos de 100) 2,000 (pasos de 500) 5,000 (pasos de 1,000) 10,000 (pasos de 10,000) 30,000. Se reporta también el error estándar de la media en forma explícita e individual para cada valor critico. Como consecuencia, la aplicabilidad de estas pruebas de discordancia ha sido extendida a prácticamente cualquier tamaño de muestra estadística (hasta 30,000 observaciones o aún mayores). Este conjunto final de valores críticos para tamaños muy grandes cubrirá cualquier necesidad presente o futura de aplicación de estas pruebas de discordancia en todos los campos de las ciencias e ingenierías. Dado que los valores críticos fueron simulados para pocos tamaños de muestra entre 1,000 y 30,000, seis modelos de regresión diferentes fueron evaluados para la interpolación y extrapolación de los datos y se demostró que un modelo combinado de logaritmo natural–cúbico es el más apropiado. Es la primera vez en la literatura mundial que se demuestra que una transformación logarítmica del tamaño de muestra n antes de un ajuste polinomial resulta mejor que los ajustes convencionales desde lineal hasta polinomial de tercer grado usados a la fecha. Finalmente, usamos 1,402 conjuntos de datos de laproteómica cuantitativa con el fin de demostrar que nuestro método de pruebas múltiples funciona más eficientemente que el método robusto MAD_Z usado para procesar estos datos y, de esta manera, ilustrar la utilidad de nuestro trabajo final en estas líneas.

Palabras clave: métodos de valores desviados, muestra normal, simulaciones Monte Carlo, tablas de valores críticos, pruebas de Dixon, pruebas de Grubbs, sesgo, curtosis, estadística, ecuaciones de regresión, transformación–log, proteómica.

Three recent papers (Verma and Quiroz–Ruiz, 2006a, 2006b; Verma et al., 2008a) 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 7 significance levels α = 0.30, 0.20,0.10,0.05,0.02,0.01, and 0.005, and for sample sizes n up to 1,000 for 15 discordancy tests with 33 variants. These tests were summarizedby Verma et al. (2008a) and therefore will not be repeated here. For greater n (>1,000), practically no critical values are available in the literature for any of these tests (Barnett and Lewis, 1994; Verma, 2005).

It may be pointed out that the critical values simulated by Verma and Quiroz–Ruiz (2006a, 2006b) and by Verma et al. (2008a) are for testing the discordancy of outliers in normal statistical samples under the assumption of some kind of a contamination model (see Barnett and Lewis, 1994 and Verma et al., 2008b for details on the possible contamination models). The outliers are simply extreme observations, irrespective of their discordancy, for example, an upper outlier x_(n) or a lower outlier x₍₁₎ in an ordered sample array of n observations or data x₍₁₎, x₍₂₎, x₍₃₎, x_(n–2) , x_(n–1) , x_(n) . In an "uncontaminated" normal sample these outlying observations will ideally not be discordant whereas in a "contaminated" normal sample they are likely to be identified as discordant. A "statistical" sample (without any assumption for the population from which it was drawn) can actually come from any distribution such as a beta or a gamma distribution and not necessarily from a normal distribution. Only under the assumption that a statistical sample was drawn from a normal population and was probably contaminated in some way, it is true that the outliers in this sample should be tested using the discordancy tests that have been especially designed for normal samples (Barnett and Lewis, 1994). As an example, a statistical sample of experimental data (such as geochemical data) is most likely drawn from a normal population (Verma, 2005), in which the outliers can be tested as discordant (or not discordant) using the discordancy tests along with the critical values for normal samples(e.g., Verma and Quiroz–Ruiz, 2006a, 2006b; Verma et al., 2008a). For a statistical sample drawn from a different distribution the discordancy tests especially designed for that particular distribution along with the corresponding critical values, if available, will have to be used. Thus, the critical values for 33 discordancy test variants have been simulated for outliers in normal samples, with the possibility of their application for discordancy of outliers in statistical samples assumed to be drawn from a normal population.

In inter–laboratory analytical studies for quality control purposes, the number of individual data (n) for a given chemical element in a reference material (RM) seldom exceeds 1,000, but this might become more common in future. In those cases, at present the multiple–test method (see Verma et al., 2008a and references therein) is not likely to be appropriately applicable due to the unavailability of precise critical values for n > 1,000 for these discordancy tests. New critical values could therefore be proposed for n> 1,000 through an adequate statistical methodology. Requirements of critical values for very large n (> 1,000) also exist in an altogether different field of molecular and cellular proteomics (Murray Hackett, written communication, June 2007 and February 2008).

For the present work, we have included most discordancy tests for normal univariate samples (15 tests with 33 test variants; see Table 1 in Verma et al., 2008a) for simulating new, precise, and accurate critical values for the same 7 significance levels (α = 0.30 to 0.005) and for very large sample sizes n, viz., 1,000(50)1,500(100) 2,000(500)5,000(1,000)10,000(10,000)30,000, using a highly precise and accurate simulation procedure described earlier (Verma and Quiroz–Ruiz, 2006a, 2006b; Verma et al., 2008a). The above is a rather standard nomenclature to express the availability of critical values (see, for example, Barnett and Lewis, 1994) and has been used by us in the past(e.g., Verma and Quiroz–Ruiz, 2006a, 2006b). As an example, the expression "1,000(50)1,500" actually means that the critical values were simulated for the sample sizes of 1,000 to 1,500, with the sample size steps of 50, i.e., for n=1,000; 1,050; 1,100; 1,150; 1,200; 1,250; 1,300; 1,350; 1,400; 1,450; and 1,500. Therefore, our present simulation was forthe sample sizes of: 1,000 (steps of 50) 1,500 (steps of 100) 2,000 (steps of 500) 5,000 (steps of 1,000) 10,000 (steps of 10,000) 30,000.

The importance of our present work resides in the fact that to date, precise critical values are available for sample sizes only up to 1,000 (Verma and Quiroz–Ruiz, 2006a, 2006b; Verma et al, 2008a). Even the sophisticated regression equations obtained by the artificial neural network (ANN) methodology presented by Verma et al. (2008a) are only valid for interpolation purposes, i.e., for sample sizes n < 1,000, and are not recommended to be used for extrapolation purposes, i.e., not for sample sizes n > 1,000, because the extrapolation is always a less accurate operation than the interpolation. Furthermore, no critical values for n > 1,000 were available to test the quality of these regression equations for extrapolation purposes. Therefore, the present work fulfills the gap of the still much needed critical values for very large sample sizes (1,000 to 30,000).

These results would be useful in all fields of science and engineering, especially in molecular and cellular proteomics and also for quality control purposes. For the first time, we evaluate six different regression models for the interpolation and extrapolation of critical values and show that the natural log–transformation (the In function) of sample size (n) combined with a polynomial fit provides the best regression model for interpolating (and also for extrapolating) the critical value (CV) data. We close our final paper in this series by demonstrating that the multiple–test method works more efficiently than the MAD_Z robust outlier method hitherto practiced for processing quantitative data on proteins.

DISCORDANCY TESTS AND THE SIMULATION PROCEDURE

Details on discordancy tests can be found in Barnett and Lewis (1994), Verma (2005), or the papers by Verma and Quiroz–Ruiz (2006a, 2006b) and Verma et al. (2008a). The 15 tests with their 33 variants for which critical values were simulated include the Dixon tests, the Grubbs tests, and the skewness and kurtosis tests (see tab. 1 in Verma et al., 2008a).

RESULTS OF NEW CRITICAL VALUES (CV)

Both the standard error of the mean and the mean critical values for 33 discordancy test variants, forn 1,000(50)1,500(100)2,000(500)5,000(1,000)10,000( 10,000)30,000 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 ). The present values cannot be compared with literature data because the latter are not available for such large sample sizes.

As for our earlier tables for sample sizes up to 1,000 (Verma and Quiroz–Ruiz, 2006a, 2006b; Verma et al, 2008a), 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, or A. Quiroz–Ruiz, aqr@cie.unam.mx). Similarly, the regression equations (see below) can also be obtained in a doc file with plain text format.

EVALUATION OF REGRESSION MODELS

(n–CV and In(n)–CV axes)

Six different regression models were fitted to obtain regression equations for the interpolation and extrapolation purposes. These include: three models of the (n–CV) type (linear; quadratic; and cubic) and three of the combined natural logarithm–transformed–n (In(n)–CV) type (combined In–linear; combined In–quadratic; and combined In–cubic). Other models, such as those based on logarithm–base 10, are likely to provide similar results as the natural logarithm–transformed models, and therefore are not evaluated here. Similarly, more complex models based on the ANN methodology were already successfully applied by Verma et al. (2008a) and are also not evaluated in the present paper. A comparison of these ANN and more complex higher order models should be the subject of a separate paper.

For illustration purposes, we selected two powerful single–outlier tests (skewness N14 and kurtosis N15; see e.g., Velasco and Verma, 1998; Velasco et al, 2000; Verma, 2005). The results for the skewness test N14 are presented in Table 1, whereas those for the kurtosis test N15 are included in Table A41 (see electronic supplement). The 27 simulated critical values (n from 1,000 to 10,000; see Table A38 for N14 and Table A40 for N15 in the electronic supplement) for a given test variant and significance level (α) were used to obtain six different fitted equations for each significance level.

To evaluate the equations for the interpolation and extrapolation purposes, the parameters {ΣSIM–FitEq)²}_int and {ΣSIM–FitEq)²}_ext , respectively were computed and reported in Table 1 for test N14 and Table A41 for test N15. These equations can be used to compute the interpolated critical values for all n between 1,000 and 10,000 and if required, the extrapolated critical values for greater n, although the latter values will be, as expected, more approximate than the interpolated values.

However, to evaluate the six regression models and to decide which of the fitted equation would be better to use for the interpolation or extrapolation purposes, we have plotted the R² parameter (Figure 1a,b), the interpolation residuals (Figure 1c,d), and the extrapolation residuals (Figure 1e,f) for these six models. Similarly, the simulated critical values for these two tests (N14 and N15) along with the interpolated and extrapolated critical values for the simpler models (linear to cubic equations) are plotted in Figures 2a,b and 2c,d, respectively. The results for the natural log–transformed linearte cubic models are presented in Figures 2e,f and 2g,h, respectively.

From the examination of Figures 1 and 2 as well as of Tables 1 and A41, the following conclusions can be drawn:

(1)  The simple linear model is probably the worst because this model provides the lowest R² values and the highest interpolation residuals (see the results for the Fitting Model L in Figure 1a–d), although the linear extrapolation residuals are smaller than for the quadratic and cubic models (compare L, Q and C models in Figure 1e,f). Linear models are sometimes employed to interpolate critical values, especially using only two simulated data for n adjacent to the missing critical value(e.g., Verma et al., 1998). The values of linear correlation coefficients (r, which is equivalent to R for linear regressions), although statistically significant at the 99% confidence level (see Bevington and Robinson, 2003; Verma, 2005), are certainly relatively low (R²< < 1). Further, strictly speaking the behavior of critical values as a function of n is certainly not linear (Figure 2a,b). This can be easily demonstrated by the application of proper statistical tests such as those practiced by Andaverde et al. (2005) for bottom hole temperature data.

(2)  The more complex quadratic and cubic models used by some researchers(e.g., Rorabacher, 1991) are also not recommended because of relatively low R² (< 1) and fairly large interpolation and extrapolation errors (see Figure 1c–f; also Figure 2a–d). Infact, these models provide totally unrealistic extrapolated critical values (Figure 2c,d).

(3) All linear to cubic models (without the log–transformation) are, therefore, inadequate for interpolation (Figure 2a,b) and absurd for extrapolation (Figure 2c–d) purposes.

(4) Undoubtedly, as judged by these criteria significantly better results are obtained with the combined natural logarithm–linear or polynomial models than the respective simple linear or polynomial models, and the best ones are for the combined logarithm–cubic model (compare the lnL, lnQ and lnC models respectively with the L, Q and C models in Figure 1a–f and Figure 2a–h). The R² approaches 1 (this being theoretically the maximum possible value for R²; see Figure 1a,b), the interpolation errors are lower than those for the simpler models (Figure 1c,d), the extrapolation errors are the lowest (Figure 1e,f), and the interpolation and extrapolation curves better fit the simulated data (compare Figure 2c,d with Figure 2a,b for interpolation and Figure 2g,h with Figure 2c,d for extrapolation). Infact, the natural log–transformed cubic fitted equations (Tables 1 and A41) are the best for both interpolation (Figure 2e,f) and extrapolation (Figure 2g,h) purposes.

We emphasize that this is the first time that a natural logarithm–transformation of the sample size variable (n) has been proposed and its effects evaluated. This transformation is shown to perform much better in combination with the respective simpler linear or polynomial model. The latter simpler models without any log–transformation are hitherto practiced in the literature (Bugner and Rutledge, 1990; Rorabacher, 1991; Verma et al, 1998). We suggest that, in future, the natural logarithm–cubic models (or of higher order terms) should be used routinely to obtain the interpolated or extrapolated critical values whenever required. The criterion should be to practically reach the theoretical value of 1 for R² by including all statistically meaningful terms in the natural log–transformed polynomial regression model.

The discordancy tests after extending their applicability to samples of sizes now up to 30,000 (or even greater), can be applied to practically any univarite, bivariate and even multivariate examples (for the latter two, after computing the studentized residuals) in all scientific or engineering fields. Similarly, for applications in studies related to quality control in Earth Sciences, such as those presented by Verma (2004), Guevara et al. (2001), and Velasco–Tapia et al. (2001), the new critical values and regression equations will certainly be useful and will cover all future needs. The examples such as Torres–Alvarado (2002), Colombo et al. (2007), Méndez–Ortiz et al. (2007), Nagarajan et al. (2007,2008), Ramos–Leal et al. (2007), Salleh et al. (2007), and Rodríguez–Ríos et al. (2007) were already pointed out in our earlier paper (Verma et al., 2008a). Another recent example concerns the rainfall data from South India (Yadava et al., 2007), for which the authors presented statistical inferences. This statistical treatment could have been certainly improved if the concepts and methodology presented in our earlier work (Verma, 1997, 2005; Verma and Quiroz–Ruiz, 2006a, 2006b; Verma et al., 2008a) were followed. Similarly, these statistical principles can be (or have been) applied to the petrographic and geochemical data for sandstone samples from an Iranian oil field by Jafarzadeh and Hosseini–Barzi (2008), chemical data for minerals from Argentina by Montenegro and Vattuone (2008) and Vattuone et al. (2008) and from Mexico by Vargas–Rodriguez et al. (2008), geothermal fluid chemistry data from Turkey by Palabiyik and Serpen (2008) and those compiled from all around the world by Díaz–González et al. (2008), nitrate pollution data in water samples from Jordan by Obeidat et al. (2008), and geochemical data for igneous rocks and minerals from the Mexican Volcanic Belt by Meriggi et al. (2008).

In summary, therefore, we emphasize, as in our earlier papers, that the multiple–test method originally proposed by Verma (1997) and exemplified in our four papers (Verma and Quiroz–Ruiz, 2006a, 2006b; Verma et al, 2008a; this work) 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). The multiple–test method was already shown 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 (Verma et al., 2008a). Here in the following, we show that our method performs better than the MAD_Z outlier detection method.

A new set of examples from proteomics

One area where the new critical values and the combined natural logarithm–cubic equations will be useful is molecular and cellular proteomics (Xia et al., 2006; unpublished data from Xia et al. were kindly provided by these researchers, which are used to illustrate this application).

We processed 1,402 protein datasets for the application of our multiple–test method for the sample sizes n between 3 and 23,773. The datasets were divided into three groups according to n as follows: (a) very large sizes 1,001–30,000 (49 cases of protein data with sizes between 1,002 and 23,773); (b) medium sizes 101–1,000 (451 cases of protein data with sizes between 101 and 999); and (c) small sizes 3–100 (902 cases of protein data with sizes between 3 and 100). The criterion for this selection depended directly on the sample sizes for which new, precise critical values were simulated in different papers (see, respectively, this paper for very large sample sizes >1000; Verma et al., 2008a for medium sample sizes 101–1,000; and Verma and Quiroz–Ruiz, 2006a, 2006b for small sample sizes up to 100). All 33 discordancy tests were applied to cases in (a) and (b) using the most precise critical values and equations (presented in Verma et al., 2008a and this study) whereas only 13 single–outlier test variants were applied to cases in (c) using the critical values published earlier (Verma and Quiroz–Ruiz, 2006a, 2006b). It is important to note that our multiple–test method was applied at the strict 99% confidence level as originally proposed by Verma (1997). The numbers of discordant outliers detected were registered individually for all 1,402 cases. Similarly, the corresponding numbers for MAD_Z robust outlier method were also counted in the unpublished database by Xia et al. No attempt was made to present and compare the central tendency and dispersion parameters from these two statistical methods simply because the protein data made available to us are still unpublished. This can be done in Xia et al. paper itself (if those authors would decide to do so), or after they publish their results of the MAD_Z outlier method.

In Figure 3(a–c) we schematically present an objective comparison of our multiple–test method (MTM) with the MAD_Z method practiced by Xia et al. (unpublished database). In most cases (Figure 3a–c), especially for medium (Figure 3b) and very large sizes (Figure 3a), our method detects a greater number of discordant outliers than the MAD_Z method. For very small datasets (n < 5), both methods detect practically no discordant outliers in protein databases.

Thus, the multiple–test method initially proposed by Verma (1997) along with the new, precise critical values and relevant interpolation and extrapolation equations (Verma and Quiroz–Ruiz, 2006a, 2006b; Verma et al. 2008a; this work) can be advantageously used to process such large databases as the 1,402 protein cases presented here.

FUTURE WORK

For such large n (several thousands!) these tests with k=1 to k=4, at first glance, may not appear to be of much use, because a much greater number of discordant outliers (> > 4) may actually be present in such datasets, and new test statistics (with significantly greater k) may have to be proposed and investigated. Surprising, however, these tests seem to perform satisfactorily well for the proteomics data (see Figure 3a for very large sample sizes).

A suitable computer program that could facilitate the application of these discordancy tests on a routine basis would be much useful in all areas of science and engineering, including the geosciences. A preliminary version of this program is already available although its improvements are underway for enabling its publication (Verma and Díaz–González, in preparation).

Nevertheless, a proper evaluation of efficiencies(e.g., Velasco and Verma, 1998; Velasco et al, 2000) or probabilities as well as of type I or type II errors may have to be first undertaken (see, e.g., Barnett and Lewis, 1994; Hays and Kinsella, 2003; Efstathiou, 2006). This type of work, already in progress, will constitute a great step forward in the application of discordancy tests and a proper handling of experimental data, especially those involving large sample sizes. In fact, the first paper on these lines is already in press (Verma et al, 2008b). Finally, a proper evaluation of the two broad groups of methods – robust methods and outlier–based methods – for the statistical handling of experimental data (Verma, 2005) would be of interest to the scientific and engineering community.

And some of the ideas expressed in this final section should constitute a new series of papers of great interest to all those involved in the handling and interpretation of experimental data.

CONCLUSIONS

In this final paper, 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 33 test variants for sample sizes up to 30,000. These new critical values will cover any future needs of many diverse fields of science and engineering, including molecular and cellular proteomics and quality control in Earth Sciences. In fact, the example set for proteomics presented here demonstrates the usefulness of the multiple–test method. Much work remains to be done to evaluate the performance of discordancy tests and to propose new test statistics, especially for very large sample sizes.

ACKNOWLEDGEMENTS

APPENDIX A. SUPPLEMENTARY DATA

Tables A1–A59 can be found at the journal web site <http://satori.geociencias.unam.mx/>, in the table of contents of this issue (electronic supplement 25–3–01).

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