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Revista mexicana de ciencias geológicas

versión impresa ISSN 1026-8774

Rev. mex. cienc. geol v.26 n.1 México abr. 2009


Evaluation of polynomial regression models for the Student t and Fisher F critical values, the best interpolation equations from double and triple natural logarithm transformation of degrees of freedom up to 1000, and their applications to quality control in science and engineering


Evaluación de modelos polinomiales de regresión para los valores críticos de t de Student y F de Fisher, las mejores ecuaciones de interpolación con transformación logarítmica natural de tipo doble y triple de grados de libertad hasta 1000, y su aplicación en el control de calidad en ciencias e ingenierías


Surendra P. Verma


Centro de Investigación en Energía, Universidad Nacional Autónoma de México, Priv. Xochicalco s/no., Apartado Postal 34, Temixco, Mor. 62580, Mexico.


Manuscript received: June 17, 2008
Corrected manuscript received: September 4, 2008
Manuscript accepted: September 4, 2008



Serious gaps exist in the present critical value tables for the Student t and Fisher F or ANOVA significance tests. Statistically correct applications of these tests to the experimental data therefore become difficult. A total of 18 different regression models were evaluated for the Student t and 24 for the Fisher F critical values. These models varied from simple polynomial (quadratic to 7th order) to the combined single (ln), double (lnln), or triple (lnlnln) natural–logarithm– (ln–) transformed polynomial models. The advantage of ln–, lnln– or lnlnln–transformations of the degrees of freedom for interpolating the Student t and Fisher F critical values has been documented for the first time in the published literature. The use of critical value equations applicable in the range of 1–1000 degrees of freedom for ln–transformation, 2–1000 for lnln–transformation, or 3–1000 for lnlnln–transformation, instead of the tables, is proposed as a 21st century innovation for the computer programming of these significance tests. A number of application examples are pointed out to illustrate the usefulness of this work.

Key words: F–ratio, ANOVA, critical value, degrees of freedom, reference material, significance tests.



Las tablas de valores críticos para las pruebas de significado de t de Student y F de Fisher o ANOVA, se caracterizan por serias deficiencias. Aplicaciones estadísticamente correctas de estas pruebas a los datos experimentales, por lo tanto, se hacen difíciles. Se evaluaron un total de 18 modelos de regresión para los valores críticos de t de Student y 24 modelos para los de F de Fisher. Estos modelos varían de modelos polinomiales (de tipo cuadrático hasta la 7ª. potencia) sencillos hasta los polinomiales combinados con la transformación logarítmica natural (ln) de tipo sencillo (ln), doble (lnln), o triple (lnlnln). Las ventajas de la transformaciones ln, lnln o lnlnln de los grados de libertad para la interpolación de los valores críticos de t de Student y F de Fisher se demuestran por primera vez en la literatura publicada. Para la programación computacional de dichas pruebas de significado se propone, en lugar de las tablas, como la innovación del siglo XXI, el uso de las ecuaciones de valores críticos aplicables en el intervalo de 1–1000 grados de libertad para la transformación ln, de 2–1000 para la transformación lnln, o de 3–1000 para la transformación lnlnln. Se presentan una serie de ejemplos de aplicación con el fin de ilustrar la utilidad de este trabajo.

Palabras clave: Relación F, ANOVA, valor crítico, grados de libertad, materiales de referencia, pruebas de significado o significancia.



Quality control in all branches of science and engineering demands the application of significance tests, such as the Student t, Fisher F or F–ratio, and ANOVA or analysis of variance (e.g., Anderson, 1987; Ebdon, 1988; Otto, 1999; Jensen et al., 2000; Miller and Miller, 2000; Bevington and Robinson, 2003; Verma, 2005; Walker and Maddan, 2005). It is customary to apply these tests at a given pre–determined confidence level (CL) such as 95% (e.g., Miller and Miller, 2000) or 99% (e.g., Verma, 1998, 2005; Verma and Quiroz–Ruiz, 2008). Such critical values or percentage points should thus be available for all degrees of freedom (dF or v) required for their statistically correct application. An examination of the published literature readily reveals that this is not the case.

The critical values for the Student t test are available in most literature sources (e.g., Verma, 2005) as a total of 42 critical values corresponding to v = 1(1)30(5)50(10)100 (100)200(300)500(500)1000 (each set being for seven two–sided CL of 60% to 99.8%, or equivalently one–sided CL of 80% to 99.9%), and for the Fisher F test as 20 values of horizontal dF (HdF) v1=1(1)12(3)15(5)30(10)50 (50)100(900)1000 and 39 values of vertical dF (VdF) v2= 1(1)30(5)40(10)60(20)100(100)200(300)500(500)1000 (the Fisher F values are generally available for 95% and 99% CL). This shows that serious gaps exist in the critical value tables for these very frequently used significance tests, e.g., within the dF range of 1–1000 and for any given CL, 958 values out of 1000 are missing for the Student t and a total of 980×961 values for the Fisher F. Note that the critical value corresponding to the dF of ∞ is not considered here because the ∞ is an undefined number in mathematical terms and refers to the population (and not to a statistical sample).

I present a new methodology for the interpolation of the existing critical values, evaluate 18 and 24 different regression models for the Student t and Fisher F, respectively, and propose the new best–fitted polynomial double or triple natural logarithm–transformed equations (defined as lnln and lnlnln functions, respectively) that allow us to extend the availability of critical values for all dF (v for the Student t; v1 and v2 for the Fisher F) from 1 up to 1000, i.e., 1(1)1000.



For the manipulation of critical values, linear to cubic regressions have been used in the literature (e.g., Bugner and Rutledge, 1990; Rorabacher, 1991; Verma et al., 1998). In fact, I tried several polynomial fits (from quadratic up to 7th order) to obtain new equations for the Student t and Fisher F critical values, but to my surprise none of them performed satisfactorily for interpolation purposes (see Figure 1 and the explanation below in this section). The failure of the polynomial fits motivated me to perform some kind of data transformation before undertaking the polynomial regressions. Single natural–logarithm (ln) transformation for statistically correct handling of compositional data was proposed long ago by Aitchison (1986) and was used a couple of years ago by Verma et al. (2006) for proposing new discriminant function diagrams. To my pleasant surprise, this methodology has also recently provided excellent interpolations of critical values of discordancy tests (Verma and Quiroz–Ruiz, 2008).

Although representing an incomplete treatment of compositional data, log–transformation has been traditionally used for fitting a linear function to a log–transformed compositional ratio variable (Na/K) in geothermal fluid geothermometry (Fournier, 1979; Verma and Santoyo, 1997; Díaz–González et al., 2008). For this example of geothermics to be comparable to my present work, quadratic and higher–order regression fits should have been evaluated. Furthermore, a correct log–ratio transformation would be to use more than two compositional variables and a common denominator for log–ratios as suggested by Aitchison (1986, 1989), Verma et al. (2006), and Agrawal et al. (2008); dealing with just one such ratio (Na/K) is not sufficient to recognize the multivariate nature of the compositional data (Aitchison, 1989; Agrawal and Verma, 2007).

Prior to the polynomial regressions, three types of natural–logarithm transformations of the dF (v, v1, and v2) –called here as the ln, lnln, and lnlnln functions– were carried out and evaluated for the first time in the published literature. These three ln–transformations mean that one uses the ln(v), ln(ln(v)), and ln(ln(ln(v))) variables, respectively, instead of the raw v for the Student t, or the raw v1 or v2 for the Fisher F, in the theoretical regression function. The results of the evaluation of the quadratic to 7th order fits are graphically presented in Figure 1.

First, the quality parameter R2 called the multiple–correlation coefficient (Bevington and Robinson, 2003) was used (Figures 1a, 1c, 1e). R2 is simply an extrapolation of the well–known concept of the linear–correlation coefficient r, which characterizes the correlation between two variables at a time, to include multiple correlations, such as polynomial correlations, between groups of variables taken simultaneously. The parameter r is useful for testing whether one particular variable should be included in the theoretical function that is fitted to the data, whereas the parameter R2 characterizes the fit of the data to the entire function (Bevington and Robinson, 2003; Verma and Quiroz–Ruiz, 2008; Verma et al., 2009). Thus, a comparison of the R2 for different functions is useful in optimizing the theoretical functional forms such as those evaluated in the present work (Figures 1a, 1c, 1e).

Secondly, the sum of the squared residuals SSR = {∑(cvtable – cvcalc)2}int was investigated as the other quality parameter (Figures 1b, 1d, 1f), where cvtable is the value listed in a table for the Student t or Fisher F test for any given CL, and cvcalc is the value calculated from the corresponding regression equation; the subscript int emphasizes that the regression equations are for the interpolation purposes only, and should not be generally used for the extrapolation of the data, although Verma and Quiroz–Ruiz (2008) have shown that such ln–transformed equations may as well be useful for the extrapolation purpose. No attempt was made to normalize this quality parameter (SSR) with respect to the number of tabulated critical values for a given case, nor with respect to some other variable such as the mean critical value, because the main interest was to use it for the visual comparison of the different (18 for the Student t and 24 for the Fisher F) regression models (Figures 1b, 1d, 1f). Nevertheless, the use of normalized SSR values will only change the vertical scale in Figure 1, without any modification in the observed trend.

It is readily seen that in all cases for 99% CL (Figure 1), the R2 parameter for purely polynomial fits from the quadratic (q) to the 7th order (p7) is consistently small (0.08044–0.44374 in Figure 1a; 0.36301–0.55106 in Figure 1c; and 0.37629–0.59237 in Figure 1e) and the corresponding SSR parameter is unduly large (60–34 in Figure 1b; 10,500–8,600 in Figure 1d; and 5.7–4.0 in Figure 1f) to be of any use in such interpolations. The improvement from any of the ln–, lnln–, and lnlnln–transformations preceding the polynomial fits of the q to 7th order is highly significant because the fitting quality parameter R2 varied, respectively, from 0.71096 to 0.99845, 0.97063 to 1.00000, and not reported (because lnln–transformation already reaches the theoretical maximum value of 1) in Figure 1a; from 0.94694 to 0.99761, 0.97097 to 0.99424, and 0.93975 to 0.98627 in Figure 1c, and from 0.95395 to 1.00000, 0.9380 to 0.99999, and 0.99365 to 0.99999 in Figure 1e. When for a polynomial model involving ln–transformation the R2 practically approaches the theoretical maximum value of 1, any further improvement in the fitting–quality is impossible to attain even if one uses a higher–order polynomial or a more complex ln–transformation. The SSR parameter for the ln–transformed models correspondingly is extremely small (17.7–0.1, 1.8–(2.8×10–6), and not reported in Figure 1a; 2,100–150, 700–150, and 880–200 in Figure 1d; and 0.9–9×10–5, 0.07–8×10–5, and 0.05–8×10–5 in Figure 1f) as compared to the respective simpler polynomial models (see above). The relatively large squared residuals in Figure 1d as compared to those in Figure 1f are due to the fact that the 99% CL critical values for the Fisher F corresponding to VdF=1 and HdF=1–1,000 are much greater (405–637) than those for VdF=1,000 and HdF=1–1,000 (6.66–1.16; for critical values see any standard textbook on the subject; e.g., Anderson, 1987; Urbina–Medal and Valencia–Ramírez, 1987; Verma, 2005).

Similarly, also for purely polynomial fits for the Student t and Fisher F tests corresponding to 95% CL critical values (plots not shown) the R2 parameter was consistently very small (0.1070–0.5272 and 0.0180–0.5053, respectively) and the SSR parameter was consistently unreasonably large (about 840–6 and 19,900–1,000, respectively). At this CL (95%) the ln–, lnln–, and lnlnln–transformations prior to the polynomial fits provided much greater R2 values, respectively, from 0.4937–1.0000, 0.9659–1.0000, and 0.9768–1.0000 (for HdF=1 and VdF=1–1000, or VdF=1 and HdF=1–1000). The SSR parameter correspondingly was extremely small for the ln–, lnln–, and lnlnln– transformed models (as low as 0.024, 0.008, and 0.021, respectively).

These kinds of results and trends were shown by all other critical value sets as well, i.e., the superiority of the ln–transformed polynomial models as compared to the simpler polynomial models (without ln–transformation) has been demonstrated beyond any doubt.



I decided to call the best–fit equation as the one that: (i) provided R2 close to 1 (in fact, practically equal to 1); (ii) showed small sum of absolute (SAR=∑Abs (cvcalc–cvtable), or squared (SSR defined above) residuals; and (iii) was based on the smallest number of regression terms and the less complex ln–transformation, i.e., under similar circumstances, the ln function was preferred as compared to the lnln function and the latter as compared to the lnlnln function.

For the Student t critical values, the lnln–transformed 5th order best–fit equation for v =2–1000 is: 

In equation (1) of the 5th order polynomial regression involving lnln–transformation, I is the intercept term and F1, F2, F3, F4, and F5 are the coefficients of the linear, quadratic, 3rd, 4th, and 5th order terms, respectively. All these coefficients of the best–fit equation (1) have been summarized in Table 1. This equation can be used to compute any critical value for v=2–1000 and for any desired CL (see Table 1).

For the Fisher F tables, two different sets of best–fit equations had to be proposed for any given CL, one for the interpolation of the vertical dF (VdF v2) for a given horizontal dF (HdF v1), and the other for the HdF (v1) for a given VdF (v2) and CL. For example, the following double ln–transformed 6th order best–fit equation applicable for v2= 2–1000 for a given v1 is:

Similarly, the following double ln–transformed 5th order best–fit equation applicable for v1= 2–1000 for a given v2 is:

The values of the coefficients for the 99% CL are listed in Tables 2 and 3, respectively.

Finally, not all combinations of HdF and VdF are covered by equations 2 and 3. For example, critical values will be missing for the combination of v1=13 and v2=31–34, 36–39, 41–49, 51–59, 61–79, 81–99, 101–199, 201–499, and 501–999. First, I tried to evaluate more complex polynomial models involving different kinds of ln–transformations of simultaneously both v1 and v2 in a single equation, but failed to obtain any acceptable solution.

Therefore, a "second–round" of equations had to be proposed to complete the missing Fisher F values. As an example, the following triple ln–transformed 6th order best–fit "second–round" equation applicable for v2= 3–1000 for a given v1 is:

The values of the coefficients for the 99% CL are listed in Table 4 for v1=13–29, thus, completing the Fisher F critical values for v1=1(1)30. Because for the F–ratio and ANOVA tests, v1 (HdF) refers to the dF that correspond to the number of classes or groups, the above equations and Tables 2, 3 y 4 cover most, if not all, applications of these significance tests. However, if there were still needs for greater values of v1, the present method of the triple ln–transformed 6th order fit can be easily extended to include any missing cases.

Another example includes the following double ln–transformed 5th order best–fit "second–round" equation applicable for v1= 2–1000 for a given v2 (note equation 5 is identical to equation 3):

The values of the coefficients for the 99% CL (Table 5) are for v2=31–49. They complete the Fisher F critical values for v2=1(1)50.

As stated earlier, this new methodology involving lnln– or lnlnln–transformations can be easily extended to calculate any other critical value for the F–ratio or ANOVA test. Similar equations and Tables were also generated for the Fisher F 95% CL, but are not included here; these are available by email request to the author.

This methodology of ln–, lnln–, or lnlnln–transformation should be useful to handle all other types of critical value tables if one is interested in precisely estimating interpolated values for their application in significance tests (this work), discordancy tests (Verma and Quiroz–Ruiz, 2008), or any other type of statistical tests. On the other hand, because the best interpolation equations based on these innovative natural logarithm–transformations along with polynomial fits provide ideal solutions with R2 values of practically 1 (maximum theoretical value attainable) and extremely small SSR values, I consider no need to try any other conventional fitting methods such as numerical computational methods or artificial neural network (ANN) methodology (Verma et al., 2008; Díaz–González et al., 2008).

It will be a good idea to abandon the use of the critical value tables; instead, the new critical value equations can be easily programmed in spreadsheets as well as in new computer software. Thus, the use of critical value equations applicable in the range of 1–1000 degrees of freedom for ln–transformation, 2–1000 for lnln–transformation, or 3–1000 for lnlnln–transformation, instead of the tables, can be advantageously proposed as a 21st century innovation for the computer programming of these significance tests. This computer programming work is currently under progress.



I first suggest a number of literature references from different areas of science and engineering, which deal with the kind of research for which the new critical value equations will be useful. Then I provide a few actual application examples from a reference material (RM) in geochemistry.

The new equations presented in this work will be useful for the statistical analysis of data in several different fields. A few of them are: agricultural and food science and technology (Pellegrini et al., 2003; Sauvage et al., 2007); biotechnology (Gonzalez et al., 2002); clay mineralogy of sediment cores (Pandarinath, in press); energy and fuels (Bansal et al., 2008); environmental science and technology (Wang et al., 2006); fluid geothermometry (Palabiyik and Serpen, 2008; Díaz–González et al., 2008); instrumentation (Goodman, 1998); medical science and technology (Zacheis et al., 1999; Cooper et al., 2006); proteomic research (Maurer et al., 2005; Verhoeckx et al., 2005; Xia et al., 2006; Verma and Quiroz–Ruiz, 2008); water–rock interaction (Verma et al., 2005); and zoology (Harcourt et al., 2005).

The quality control (assurance and assessment programs) using inter–laboratory data on RM is another important research area where these new critical value equations will be of much use. To name a few of these research areas on the study of RMs, they are: biology and biomedicine (Ihnat, 2000; Patriarca et al., 2005), cement industry (Sieber et al., 2002), environmental and pollution Research (Dybczynski et al., 1998; Gill et al., 2004), food science and technology (Langton et al., 2002; Morabito et al., 2004), organochlorinated compounds and petroleum hydrocarbons in sediments (Villeneuve et al., 2002, 2004); rock chemistry (Verma, 1998; Marroquín–Guerra et al., in press), and water research (Holcombe et al., 2004; Verma, 2004).

The example cases were chosen from the RM database recently used to evaluate the performance of single discordant–outlier tests (Verma et al., 2009). It is important to remind that the Student t and Fisher F–ratio tests are sensitive to the presence of discordant outliers (Jensen et al., 2000). For the ANOVA test also, the data to be examined should ideally be free from discordant outliers. Therefore, in order to correctly apply these tests, the individual datasets should first be processed for the possible presence of such outliers using appropriate discordancy tests (Barnett and Lewis, 1994) along with the new, precise, and accurate critical values (Verma and Quiroz–Ruiz, 2006a, 2006b, 2008; Verma et al., 2008).

A large number of examples exist in this extensive RM database that fall in the category of actual gaps in the critical value tables, for example, for the Student t the dF or v as 30< v <1000, but distinct from v = 30(5)50(10)100(100)200 (300)500(500)1000, i.e., different from the tabulated v of 35, 40, 45, 50, 60, 70, 80, 90, 100, 200, 500, or 1,000. Similarly, for the Fisher F tables, such gaps exist for horizontal dF (HdF) v1>12, i.e., v1 different from 12(3)15(5)30(10)50(50)100(900)1000 and for vertical dF (VdF) v2> 30, i.e., v2 different from 30(5)40(10)60(20)100(100)200(300)500(500)1000.

The examples (Table 6) include the major element data in the RM diabase W–1 from the United States Geological Survey (U.S.A.) compiled by Verma et al. (2009) that require the use of newly interpolated critical values. The extensive footnote of Table 6 provides more information on the application of the ANOVA, F–ratio and Student t tests.

For TiO2, Al2O3, MgO, CaO, K2O, and P2O5, the ANOVA test and for H2O+, and H2O, the F–ratio and Student t tests showed that the data from the different analytical method groups can be combined into a single group and an overall mean and standard deviation as well as confidence limits of the mean can be calculated. For the remaining major elements or oxides (SiO2, Fe2O3t, MnO and Na2O), statistically significant differences were observed among the method groups, and the combination of all methods into a single group was therefore not recommended. For the latter cases, the overall statistical parameters were calculated for only those method groups that showed no significant differences among them.

For these significance tests (ANOVA, F–ratio, and Student t), the new equations provided precise interpolated critical values as documented in the earlier section. If we were to calculate the 95% or 99% confidence limits of the mean (not included in Table 6), we would also need precise critical values for the Student t test corresponding to the appropriate degrees of freedom (v).



The criteria of the multiple–correlation coefficient (R2) and the interpolation residuals (SAR and SSR) clearly suggest that simple polynomial regressions are not appropriate for the interpolation of the Student t and Fisher F critical values. The ln–transformation is a required operation to achieve better regression models. The ln–, lnln–, and lnlnln–transformations combined with the polynomial regression models, were shown to perform better than the simple polynomial models. The best interpolation models involved lnln– and lnlnln–transformations prior to the polynomial fits. The use of these new best interpolation equations in spreadsheet calculations or computer programs is recommended for all applications in science and engineering involving these significance tests. Finally, the new interpolated critical values for the Student t test would be useful to calculate more precisely the 95% or 99% confidence limits of the mean.



The writing of a single–author book on statistics during 2004–2005 and the collaboration with A. Quiroz–Ruiz on a simulation procedure to generate new, precise critical values for discordancy tests published in four papers during 2006–2008 permitted me to identify this gap in the published literature and motivated me to fulfill it through the present paper. I am much grateful to two anonymous reviewers for their highly positive evaluation of my work and the co–editor in chief Carlos González León for efficiently handling this manuscript.



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