F-statistics

Berendsen, H.J.C., Postma, J.P.M., Van Gunsteren, W.F. Statistical mechanics and molecular dynamics The calculation of free energy, in Molecular Dynamics and Protein Structure, J. Hermans, ed.. Polycrystal Book Service, PO Box 27, Western Springs, 111., USA, (1985) 43-46. 

The F statistic, along with the z, t, and statistics, constitute the group that are thought of as fundamental statistics. Collectively they describe all the relationships that can exist between means and standard deviations. To perform an F test, we must first verify the randomness and independence of the errors. If erf = cr, then s ls2 will be distributed properly as the F statistic. If the calculated F is outside the confidence interval chosen for that statistic, then this is evidence that a F 2. 

The F statistic describes the distribution of the ratios of variances of two sets of samples. It requires three table labels the probability level and the two degrees of freedom. Since the F distribution requires a three-dimensional table which is effectively unknown, the F tables are presented as large sets of two-dimensional tables. The F distribution in Table 2.29 has the different numbers of degrees of freedom for the denominator variance placed along the vertical axis, while in each table the two horizontal axes represent the numerator degrees of freedom and the probability level. Only two probability levels are given in Table 2.29 the upper 5% points (F0 95) and the upper 1% points (Fq 99). More extensive tables of statistics will list additional probability levels, and they should be consulted when needed. 

It is possible to compare the means of two relatively small sets of observations when the variances within the sets can be regarded as the same, as indicated by the F test. One can consider the distribution involving estimates of the true variance. With sj determined from a group of observations and S2 from a second group of N2 observations, the distribution of the ratio of the sample variances is given by the F statistic ... 

F statistic, 239, 241 False negatives, 152—153 False positives, 152—153 Fenoximone, 188 First-order kinetics, 167 Fluorescence resonance energy transfer, 182... 

Starting with the next to the last eigenvalue, we can then compare the F statistic calculated for that eigenvalue to the F(l, s - j) value in the statistical... 

Nys, G. G., Rekker, R. F. Statistical analysis of a series of partition coefficients with special reference to the predictability of folding of drug molecules. Introduction of hydrophobic fragmental constants (/-values). Chim. Therap. 1973, 8, 521-535. 

In Table 31.9 we represent the results of Malinowski s F-test as computed from the eigenvalues of the transformed retention times in Table 31.2. The second eigenvector produces an F-value which still exceeds the critical F-statistic (6.6 with 1 and 5 degrees of freedom) at the 0.05 level of probability. Hence, from this evidence we conclude again that there are two, possibly three, structural eigenvectors in this data set. 

The OUTSTAT= output data set pvalue contains the p-value in the PROB variable. If you have multiple predictor variables, you need to use the PROC GLM ODS data set OverallANOVA to get the overall model p-value from the ProbF variable. These output data sets contain other variables, such as the degrees of freedom, sum of squares, mean square, and F statistic, if you need them for an ANOVA table presentation. 

The test to determine whether the bias is significant incorporates the Student s /-test. The method for calculating the t-test statistic is shown in equation 38-10 using MathCad symbolic notation. Equations 38-8 and 38-9 are used to calculate the standard deviation of the differences between the sums of X and Y for both analytical methods A and B, whereas equation 38-10 is used to calculate the standard deviation of the mean. The /-table statistic for comparison of the test statistic is given in equations 38-11 and 38-12. The F-statistic and f-statistic tables can be found in standard statistical texts such as references [1-3]. The null hypothesis (H0) states that there is no systematic difference between the two methods, whereas the alternate hypothesis (Hf) states that there is a significant systematic difference between the methods. It can be seen from these results that the bias is significant between these two methods and that METHOD B has results biased by 0.084 above the results obtained by METHOD A. The estimated bias is given by the Mean Difference calculation. 

For this chapter we continue to describe the use of confidence limits for comparison of X, Y data pairs. This subject has been addressed in Chapters 58-60 first published as a set of articles in Spectroscopy [1-3]. A MathCad Worksheet ( 1986-2001 MathSoft Engineering Education, Inc., 101 Main Street Cambridge, MA 02142-1521) provides the computations for interested readers. This will be covered in a subsequent chapter or can be obtained in MathCad format by contacting the authors with your e-mail address. The Worksheet allows the direct calculation of the f-statistic by entering the desired confidence levels. In addition the confidence limits for the calculated slope and intercept are computed from the original data table. The lower limits for the slope and the intercept are displayed using two different sets of equations (and are identical). The intercept confidence limits are also calculated and displayed. 

Narasaka T, Watanabe H, Yamagata S, Muna-kata A, Tajima T, Matatsunaga F Statistical analysis of diverticulosis of the colon. Tohoku J Exp Med 1975 115 271-275. 

F statistic A statistic which is used as a measure of the goodness of fit of a data set to a correlation equation. The larger the value of F, the better the fit. Confidence levels can be assigned by comparing the F value calculated with the values in an F table for the Np and DF values of the data set. 

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