Cinturón Volcánico Mexicano Application of discordancy and significance statistical tests for the comparison of dacitic volcanism from the central part of the Mexican Volcanic Belt

Our aim is to show a statistical procedure along with two new computer programs (DODESSYS and UDASYS). For this task we compiled a database of 249 samples of dacite coming from four closely located Mexican Volcanic Belt (MVB) areas: monogenetic volcanoes from the Sierra de Chichinautzin and Valle de Mexico, the Nevado de Toluca stratovolcano, the Iztaccihuatl stratovolcano and the Popocatepetl stratovolcano. The discordancy and significance (ANOVA – ANalysis Of Variance–, Fishers´ F and Student´s t) statistical tests were applied at 99% confidence level. The final statistical was calculated for 98 geochemical parameters, these include major oxides, rare earth elements, trace elements and additional parameters, as well as log-ratio parameters used in new tectonic discrimination diagrams. These geochemical parameters were treated as univariate statistical samples and were classified according with the four MVB regions. Discordancy statistical tests detected discordant outliers in 124 (amount to about 35%) statistical samples. ANOVA tests showed significant differences among all groups in 32 parameters. The similarities and differences between the log-ratios parameters elements may eventually be useful in future to propose tectonic discrimination diagrams from a representative database.


Introduction
Recently, a new computer programs has been developed, UDASYS (Univariate Data Analysis SYStem) [1].UDASYS is freely available from any of the authors to any scientist interested in correctly processing experimental data.This program, written in Java [2], provides statistical tools pertaining to both robust and outlierbased methods for univariate data.UDASYS also incorporates an updated version of the DODESSYS software [3].Whereas DODESSYS allowed the application of discordancy tests ( [3] for more details on these tests see Table S1 in online Supplementary Material) for statistical sample sizes up to 1000, all discordancy tests can now be applied to statistical sample sizes as large as 30000.Computer programs to enable the application of discordancy tests were practically nonexistent as documented by Barnett and Lewis (1994).
Later about 12 years ago, a computer program (SIPVADE) was published by Verma et al.
(1998), but it is now outdated for several reasons.The most important among them are that SIPVADE uses old, less precise and sometimes even inaccurate critical values then available in the literature (Barnett and Lewis 1994;Verma 2005) and relies on linear interpolation of these values when for a given statistical sample size n, the corresponding critical values were not tabulated.Both of these aspects have been shown to cause errors in the final statistical inferences.
More importantly, unlike all available software to date (e.g., [4]), UDASYS allows a highly efficient use of significance tests of Fisher's F, Student's t, and ANOVA.This work illustrates the application of statistical discordancy and significance tests using geochemical data.A geochemical database of major-elements in rocks from the Mexican volcanic belt (MVB) was established long ago by Pal et al. [5].These authors used their database to objectively characterise for the first time the nature of volcanism in the MVB.This work was later extended by including more analyses of MVB rocks in this database which permitted to highlight the complexity of magmas in the MVB (e.g., [6]).Mean and standard deviation estimates of compositional data were presented by these authors, but this was done without the application of discordancy tests [7].Similarly in local geochemical studies from this volcanic province (MVB), these two statistical parameters for laboratory analytical data were specifically reported by Verma [8][9][10] and Verma et al. [11].Other researchers have used mean and standard deviation estimates for geochemical interpretation [12].
In this work geochemical data are compiled for dacitic rocks from four nearby areas of the MVB.The geochemical parameters are compared through the significance tests such as Fisher´s F and Student´s t [13] without and with the application of discordancy tests [14][15][16][17].The results highlight the importance of these statistical tests in geosciences.
We searched the published geoscience literature for specific applications of discordancy and significance tests and found that it is not a common practice to apply them in geoscientific studies.Below we list some the reports found that made use of either one of these statistical methodologies.
Rice and Church [18] presented a statistical study on the variability in grain size of sediment from two confluent rivers in northeastern British Columbia (Canada).They stated that it was not appropriate to apply ANOVA test because the statistical samples did not show a normal distribution and their variances were unequal.However, the validity of the first condition can be checked by discordancy tests, whereas the second condition (equal variances) is not a requisite for ANOVA.They applied tests, such as Brown-Forsythe and chi-square, for comparing statistical sample means when sample variances are unequal.
Takano et al. [19] made a statistical comparison of inter-laboratory analytical data of fluid samples from crater lake of Maly Semiachik volcano, located in the central part of the Eastern Volcanic Belt of Kamchatka (Japan), obtained from eight different institutions.They used different analytical techniques (atomic absorption spectrometry, atomic emission spectrometry, mass spectrometry, ion chromatography, high performance liquid chromatography, colorimetry, and titrimetry) to compare the measured isotopic data coming from elements such as hydrogen, sulfur, and oxygen.Their comparison consisted of simply calculating the central tendency (mean) and dispersion (coefficient of variation) parameters for each one of these techniques.Experience shows that it would have been worthwhile to apply the discordancy and significance tests for such inter-laboratory evaluations as suggested earlier by several authors [20][21].
Wani and Mondal [22] carried out a geochemical study of shale samples from the Mesoproterozoic-Neoproterozoic Chhattisgarh and Indrāvati basins.They compared chemical compositions of the calcareous and non-calcareous shales of the Chhattisgarh and Indrāvati basins applying only the Student's t at 95% confidence level.They should have applied Fisher's F test prior to the application of the t test since this significance test is sensitive to the presence of discordant outliers.We emphasize once again that discordancy tests should be applied to detect anomalous data in individual statistical samples previous to the comparison and use of significance tests.
The correct statistic application, such as the work we are reporting, tries to promote the evolution of geochemistry towards geochemometrics, where statistics is an essential part of experimental data treatment.In general, in the area of geochemistry is not customary to apply a correct statistics methodology in the processing of databases.For example, Takano et al. [19] assessed the statistic differences in their experimental databases, but failed to apply the methodology based on significance tests and discordancy.However, recently some authors applied successfully this complete methodology in processing geochemical data [17,23].

Database and procedures
Geochemical data for 249 Neogene dacitic rock samples from four closely located areas of the MVB were compiled.The literature sources were as follows: [9,21,.Data are identifiqued as group numbers Gr1 to Gr4 (see locations of these regions on a map presented in TAS (Total Alkalis vs Silica) diagram was generated by IgRocs sofware [61]; see Figure 2.
Geochemical data are concentrated in classification area for dacite rocks.The statistical central tendency (mean) and dispersion (standard deviation) parameters were calculated for several conventional variables, which were 11 major oxides (adjusted values) from (SiO 2 ) Adj to (P 2 O 5 ) Adj , selected normative minerals, Mg-number (or Mg-value), and 6 other indices detailed by [61], followed by 14 rare earth elements from La to Lu, and 22 trace elements from Ba to Zr.In addition to these conventional chemical data, 30 additional parameters were computed and evaluated.These include two ratio parameters defined by Verma [62] called Nbanomaly with respect to Ba and La and Ta-anomaly with respect to Ba and La, as well as 28 log-ratio parameters of elements used in new multi-dimensional tectonic discrimination diagrams [63][64][65].
Figure 3 shows the flow diagram of statistical methodology applied in this work.
Conventionally, significant test are applied without prior application of discordancy tests.
However, because these tests should be applied to normally distributed statistical samples, data for each variable from all individual groups (Gr1-Gr4) were first processed for discordant outliers by single-outlier type discordancy tests (see Table S1 in [1]) at 99% confidence level, and the discordant outlier-free groups were evaluated from the two-sided ANOVA-test and t-test at 99% confidence level (see Geological implications in [1] for more details on application of two-sided version of significant tests).The statistical parameters of mean and standard deviation were simply calculated from the discordant outlier-free individual groups.We note that, ANOVA test can only be applied to three or more groups or statistical samples [7], therefore this significant test was applied to the data from each group (Gr1-Gr4).The application of ANOVA would result in any of the following: (i) no statistically significant differences among the four regions (Gr1-Gr4); (ii) one -e.g., Gr1-of the four regions showing a statistically significant difference as compared to the other three regions -e.g., Gr2-Gr4-; and (iii) statistically significant differences among the four regions (Gr1-Gr4), which will have to be resolved by Fisher´s F and Student´s t significance tests.When ANOVA detects significant differences among the four regions, the data should be processed thorough the combination of Fisher´s F and Student´s t tests, which are applicable to only two groups at a time [63,64].The Fisher´s F test compares the two variances and could result in either the two variances are equal or the two are different.Depending on the result of the F test, the appropriate version of the t test should be applied.The Fisher's F and Student's t tests were applied to each one of the combinations Gr1-Gr2, Gr1-Gr3, Gr1-Gr4, Gr2-Gr3, Gr2-Gr4 and Gr3-Gr4.
It has been suggested that the data from different groups or regions should only be combined after ascertaining that no statistically significant differences exist among them [1].
Thus, for a given chemical parameter or variable, the groups that showed no significant differences were combined and statistical information was obtained for the combined data.
Finally, these combined data were once again processed for discordant outliers, and the discordant outlier-free data were used to obtain final statistical (mean and standard deviation).

Identification and separation of discordant outliers
Geochemical data for a total of 96 variables o parameters from each group (Gr1-Gr4) were processed in this work.Single-outlier type discordancy tests at a very strict 99% confidence level were then applied to individual groups, outlying observations were separated, and statistical parameters were calculated from discordant outlier-free data.These statistical parameters are reported in Table 1; the first column gives the name of the chemical or ratio parameter, the next columns gives statistical parameters such as statistical sample size (n), mean and standard deviation from all individual groups (Gr1-Gr4); i.e. columns 2-4 show stastistical parameters from Sierra de Chichinautzin and Valle de Mexico monogenetic volcanoes.The second column gives the final statistical sample size (n) after discordant outlier detection and separation, the third column reports the mean, and the fourth one provided the standard deviation.The number of discordant outliers is represented by a symbol as superscript: α -one-; β -two-; γ -three-; δfour -; £ -five-; ζ -seven-; η -eight-; λ -ten-.For the total of 350 statiscal samples processed in this work, 124 (35%) samples showed discordant outliers.

Application of discordancy tests after combining data (significance tests)
Single-outlier type discordancy tests were applied to the combined groups, outlying observations were separated, and statistical parameters were calculated from discordant outlier-free data (see Table 2 in appendix).These discordant outliers were rejected (or separated) and final statistical were calculated and shown in Table 2. Discordant outliers were represented by a symbol as superscript: α -one-; β -two-; γ -three-; δ -four -; £ -five-; ζ -seven-; η -eight-; λ -ten-.

Conclusions
In this work, we have shown a statistical procedure to decipher mean compositions and uncertainty estimates including various regions.For this, geochemical data are compiled for 249 Neogene dacitic rock samples from the four MVB regions.

All single-outlier type discordancy tests and significance (ANOVA -ANalysis Of Variance-,
Fishers´ F and Student´s t) statistical tests were applied at the very strict 99% confidence level.
These statistical tests were applied to each one of the 98 geochemical parameters, which were major oxides, selected normative minerals, rare earth, trace elements, two ratio parameters called Nb-anomaly and Ta-anomaly, as well as 28 log-ratio parameters of elements used in new multidimensional tectonic discrimination diagrams.
All geochemical parameters were treated as univariate statistical samples.Final statistical parameters were calculated from discordant outlier-free data.We suggest that the final mean compositions could be used to compare statistically the geochemical data for the same type of igneous rocks, i.e., dacite type, sampled around the world.
Furthermore, significance statistical tests determined significant differences and similarities among various geochemical parameters from the four MVB regions.Particularly, the similarities and differences among the log-ratios parameters could be useful to propose new diagrams to discriminate tectonic settings, with a more representative database.

Figure 2 .
Figure 2.This figure shows a diagram of discrimination TAS.Geochemical data are concentrated in classification area for dacite rocks.

Figure 3 .
Figure 3. Schematic flow diagram of statistical methodology applied in this work.

not show statistically significant differences among all groups, hence
they were combined; e.g., (Na 2 O) Adj , (K 2 O) Adj , or Norm , ANOVA test determined that 32 variables did

Table 1 .
Final statistical of samples of dacitic rocks from four nearby regions of the Mexican volcanic belt.

Table 1 (
continuation).Final statistical of samples of dacitic rocks from four nearby regions of the Mexican volcanic belt.