New critical values for F and their use in the ANOVA and Fisher’s F tests for evaluating geochemical reference material granite G-2 (U.S.A.) and igneous rocks from the Eastern Alkaline Province (Mexico)

Abstract

Monte Carlo method was used to simulate precise and accurate critical values (with 2 to 7 decimal places) of Fisher´s F test for degrees of freedom (ν1 and ν2) of up to 2000 and confidence levels of 20% to 99.9% for two-sided or two-tailed test (or equivalently to 60% to 99.95% for one-sided or one-tailed test). Unlike the existing literature, we present 15 new critical value tables along with 15 additional tables of the error estimates of individual values. For the two degrees of freedom – horizontal (ν1) as well as vertical (ν2) –the critical values orrespond to 1(1)30(5)100(10)160(20)200(50)400(100)1000 and 6(1)30(5)100(10)160(20)200(50)400 (100)1000(200)2000, respectively, where the numbers before and after the parenthesis are the initial and final values for which critical values were simulated and the numbers in parenthesis indicate the step size how the initial value of degrees of freedom approached to the final one. Thus these critical values consist of 62x62 values for each of the eleven confidence levels. Even though a large number (3844 for each confidence level, amounting to a total of 42284 values for all confidence levels) of new values of F were simulated, values do not exist for many different degrees of freedom in the range of 1-2000. This problem of unavailable critical values was resolved by evaluating regression models based on simple polynomial functions as well as those involving logtransformation of the independent variable. New best-fit equations were thus proposed to estimate the not-tabulated critical values as well as for the estimation of probability of calculated F value. This methodology of log transformations should be useful also for other kinds of applications involving polynomial fitting. We compared these simulated results with commercial as well as freely available software. The new precise and accurate critical values were used to illustrate the application of analysis of variance (ANOVA) and Fisher´s F tests to geochemical data for international geochemical reference material granite G-2 from U.S.A. We also present the methodology to statistically compare geochemical data for ultrabasic, basic and intermediate magmas from eight regions of the Eastern Alkaline Province of Mexico and U.S.A. Finally, we use these as well as the discordant outlier-free data for ultrabasic and basic rocks in multi-dimensional discrimination diagrams to explore the tectonic setting for this volcanic province, which was inferred to be mainly an extensional zone or a continental rift.