Tables of Variance Ratio

Abridged from Table V of Statistical Tables for Biological, Agricultural and Medical Research. (R. A. Fisher and F, Yates Oliver and Boyd). 

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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. 

F Distribution In reference to the tensile-strength table, the successive pairs of daily standard deviations could be ratioed and squared. These ratios of variance would represent a sample from a distribution called the F distribution or F ratio. In general, the F ratio is defined by the identity... 

The last column of Table 10-5 shows (1) that the long-term source of variation clearly overshadows the short-term, the ratio of variances exceeding 70 and (2) that the short-term variance is comparable with Arx, which is, of course, the standard counting error squared (Equation 10-8). 

This is, then, the regression sum of squares due to the first-order terms of Eq. (69). Then, we calculate the regression sum of squares using the complete second-order model of Eq. (69). The difference between these two sums of squares is the extra regression sum of squares due to the second-order terms. The residual sum of squares is calculated as before using the second-order model of Eq. (69) the lack-of-fit and pure-error sums of squares are thus the same as in Table IV. The ratio contained in Eq. (68) still tests the adequacy of Eq. (69). Since the ratio of lack-of-fit to pure-error mean squares in Table VII is smaller than the F statistic, there is no evidence of lack of fit hence, the residual mean square can be considered to be an estimate of the experimental error variance. The ratio... 

These factors are used in the equations given in Table I. The computation requires only that the variance ratios be accurately known. The absolute precision of the method may change from day to day without affecting the validity of either the least-squares curve-of-best fit procedure or the confidence band calculations. (It is not practical to regularly monitor local variances, and errors may develop in variance ratios. Eowever, the error due to incorrect ratios will almost always be much less than the error due to assuming constant variance. Even guessed values of, say, S a concentration are likely to yield more precise data.)... 

Tables la to Id represent the value of the ratio of the variances which would occur owing to chance if the two variances were in fact from normal distributions which have the same variance. The probabilities given in the tables (0 20 for la, 0 05 for lb, etc.) cover what is called the one-tailed test situation. In using the table with this probability level, we are interested only in determining whether the F ratio obtained is larger than that which could be attributed to chance.

This table represents the upper and lower tails of the distribution of the ratios of ranges of samples taken from a normal distribution of variance oz. If the ratio R / R2 of the ranges R and R2, with size and N2 respectively, is calculated, the probability that it... 

The variance ratio (F value) is not readily calculated because replicated data are not available to allow the residual error term to be evaluated. However, it is usual practice to use the interaction data in such instances if the normal probability plot has shown them to be on the linear portion of the graph. By grouping the interaction terms from Table 7 as an estimate of the residual error,... 

If it is assumed that the pooled run variance is a reasonable estimate for the residual variance. Table 7 can be reworked and the variance ratios (F values) calculated for each of the effects. The results of this rework are shown in Table 10. This approach confirms that the methanol effect is the largest by a very long way. The F value (1,8 df) is 5.32. Whilst this confirms that A is not significant. 

The answers need an analysis of the s/n ratio. Table 2.17 summarises the Pareto analysis of variance (see Figure 2.7 for the calculations involved) and Figure 2.8 displays the factor plots associated with the calculations. 

Fig. 26. Spectral condition number B 301 cond(K) of the inverse operator related to Eqs. (47a-c) and applied in the calculations 30) of the MWD from measured PDC-elution curves having the ratio of variances ctd/ o from Table 10...

We note from Table 1.19 that the sums of squares between rows and between columns do not add up to the defined total sum of squares. The difference is called the sum of squares for error, since it arises from the experimental error present in each observation. Statistical theory shows that this error term is an unbiased estimate of the population variance, regardless of whether the hypotheses are true or not. Therefore, we construct an F-ratio using the between-rows mean square divided by the mean square for error. Similarly, to test the column effects, the F-ratio is the be-tween-columns mean square divided by the mean square for error. We will reject the hypothesis of no difference in means when these F-ratios become too much greater than 1. The ratios would be 1 if all the means were identical and the assumptions of normality and random sampling hold. Now let us try the following example that illustrates two-way analysis of variance. 

Analysis of variance has also been applied in testing three factors of composite rocket propellant burning rate in Crawford s-bomb at lOObar and 20 °C temperature. The analyzed factors are contents of fine fraction in bimodal oxidizer mixture (C) contents ratio of oxidizer mixture-aluminum powder (B) and contents of burning rate catalyst (A). The obtained burning rate values are given in the table in mm/s ... 

Table 2.1 summarizes the single factor ANOVA calculations. The test for the equality of means is a one-tailed variance ratio test, where the groups MS is placed in the numerator so as to inquire whether it is significantly larger than the error MS ... 

As an example, consider the data on day-to-day and interindividual variability of fruit growers respiratory and dermal exposure to captan shown in Table 7.3 (de Cock et al., 1998a). The ratio of the 97.5th percentile to the 2.5th percentile of the exposure distribution R95) is usually larger for the intraindividual or day-to-day variability, when compared to the interindividual variability. The variance ratio, k, can be calculated from the Rgs values, since the standard deviation of each exposure distribution is equal to In R95/3.92, and the square of the standard deviation gives the variance. For the respiratory exposure, this results in a variance ratio k of 32.8, whereas for dermal exposure of the wrist the variance ratio is considerably lower, approximately 3.0. What are the implications of these variance ratios for the number of measurements per study subject For a bias of less than 10 % (or /P > 0.90), the number of repeated measurements per subject... 

Values of F are available in statistical tables at various levels of significance. The values depend on the number of degrees of freedom Vi and Vj for the variances and Fj. The tables are arranged with F values greater than unity therefore the two variances are compared in the order Fj > V2. Table 26-5 is a small section of a table of F values taken at the 95% confidence level. The entries correspond to a probability of 0.95 that the variance ratio will not exceed the value in the table. Extensive tables are available for various confidence levels and degrees of freedom. 

These observations may be summarized conveniently in an analysis-of-variance table-. Table 26-7 illustrates this type of table for the above case. The overall variance (total mean square) Sj(N — 1) contains contributions due to variances within as well as between classes. The variation between classes contains both variation within classes and a variation associated with the classes themselves and is given by the expected mean square aj + not. Whether not is significant can be determined by the F test. Under the null hypothesis, = 0. Whether the ratio... 

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