Fisher F-ratio

Assume the model = 0 + r, is used to describe the nine data points in Section 3.1. Calculate directly the sum of squares of residuals, the sum of squares due to purely experimental uncertainty, and the sum of squares due to lack of fit. How many degrees of freedom are associated with each sum of squares Do and SS add up to give SS l Calculate and What is the value of the Fisher F-ratio for lack of fit (Equation 6.27)7 Is the lack of fit significant at or above the 95% level of confidence ... 

In Section 6.4, it was shown for replicate experiments at one factor level that the sum of squares of residuals, SS can be partitioned into a sum of squares due to purely experimental uncertainty, SS, and a sum of squares due to lack of fit, SSi f. Each sum of squares divided by its associated degrees of freedom gives an estimated variance. Two of these variances, and were used to calculate a Fisher F-ratio from which the significance of the lack of fit could be estimated. 

Calculate the Fisher F-ratio for the significance of the factor effects (Equation 9.40) for the model and data of Problem 9.6. At approximately what level of confidence is the factor x, significant ... 

Note that the Fisher F-ratio for the significance of lack of fit cannot be tested because there are no degrees of freedom for purely experimental uncertainty. This lack of degrees of freedom for replication is a usual feature of observational data. Any information about lack of fit must be obtained from patterns in the residuals. 

The sum of squares and degrees of freedom tree for the fitted model is given in Figure 11.16. The R2 value is 0.9989. The Fisher F-ratio for the significance of the factor effects is F56 = 1096.70 which is significant at the 100.0000% level of confidence. The F-ratio for the lack of fit is F3 3 = 0.19 which is not very significant. As expected, the residuals are small ... 

Another commonly used measure of equation significance is the Fisher F ratio. This is the regression mean square divided by the error mean square,... 

The test of this hypothesis makes use of the calculated Fisher variance ratio, F. 

The test of this hypothesis makes use of the calculated Fisher variance ratio, F. DF ,DFd = S lof/Spe (6-27)... 

Here TV is the number of measurements, P is the number of parameters of the model, and Cj meas and cFcalc are measured and calculated solute concentrations for the /th observation, respectively. The presence of the number of parameters in the denominator makes the mean square lack-of-fit to be an unbiased estimator of the model s standard error (Whitmore, 1991). To test the null hypothesis, one has to compare the / -ratio of the mean of lack-of fit squares F=st ade2 st,fade2 to the critical value of the Fisher s statistic Fn pADE n pfade> where PADE = 2, and PFADE = 3. The null hypothesis can be rejected if F > Fn pADE> n-pfade- Data in Table 2-3 show that the F ratio exceeds the critical value taken at the 0.05 significance level, so that the FADE performs better. 

Fisher statistic, Fisher value ratio of variances for two models to be compared. It can be overall or partial F value. The overall Fisher statistic tests the entire equation, whether all coefficients are significant in the model. The partial F value is used to test whether the incriminated variable is significant in the model. 

Instead of the imivariate Fisher ratio, SLDA considers the ratio between the generalized within-category dispersion (the determinant of the pooled within-category covariance matrix) and total dispersion (the determinant of the generalized covariance matrix). This ratio is called Wilks lambda, and the smaller it is, the better the separation between categories. The selected variable is that that produces the maximum decrease of Wilks lambda, tested by a suitable F statistic for the input of a new variable or for the deletion of a previously selected one. 

As noted in Section 7.5, the test statistic in ANOVAs is called F, and the test is sometimes called the F-test. The name pays respect to Sir Ronald Fisher, the statistician who developed this approach. Similarly to the calculation of the test statistic t in a f-test, F is calculated as a ratio, as follows ... 

Flgurt 3.6 Optimum reflux ratio. (a) Capital and operating cost curves (6) effect of using expensive materials of construction on the optimum (c) effect of high energy costs on the optimum (optimum reflux, toluene-benzene separation, show a flat total cost curve near the optimum. (Part d reprinted with permission from W. R. Fisher, M. F. Doherty, and J. M. Douglas, bid. Eng. and Chem. Proc. Dee. and DeveL, Vol. 24. p. 955, Copyright (1985) American Chemical Society). 

Hgw U Continued) Optimum reflux ratio, id) optimum reflux, tolueoe-beniene separation.showing a flat total cost curve near the optimum. (Part d re-fainted with permission from W. R. Fisher, M. F. 

Lowe, D.J., Fisher, K., and Thomeley, R.N.F. (1993) Pre-steady-state absorbance changes show that redox changes occur in the Klebsiella pneumonia MoFe-protein that depend on substrate and components ratio a role for P-centers in reducing dinitrogen, Biochem. J. 292, 93-. 

The test of significance for this type of problem is due to Fisher (Fisher actually dealt with the natural logarithm of the ratio of the square roots of the variances, which he called z, but here we will use the simple variance ratio which is denoted by F). 

A more powerful criterion of goodness of fit is the F-test, pioneered by Fisher (1925), of the variance ratio... 

The F test In contrast to the t test, which is a comparison of means, the F test is a comparison of variances. The ratio between the two variances to be compared is the variance ratio F (for R. A. Fisher), defined by... 

The null hypothesis (statistical terminology), states that if there are no significant differences in the variances, then the ratio must be close to 1. Reference should therefore be made to the Fisher-Snedecor values of F, established for a variable number of observations (Table 22.3). If the calculated value for F exceeds that found in the table, the means are considered to be significantly different. Since the variability is greater than si, then the second series of measurements is therefore the more precise one. 

Fisher s test (F = MSuop IMS Kg) allows the two estimates of the variance, s/ and to be compared. A ratio much larger than 1 would indicate to us that the estimation j/ is too high and that therefore the model is inadequate, certain necessary terms having been omitted. In Fisher s tables, a value F, = 6.60 corresponds to a significance level of 0.05 (5%). Two cases may be envisaged ... 

Z. (1) The symbol for a standardized value of a Normal variable. (2) The symbol for the test statistic in Whitehead s boundary approach to sequential clinical trials (see Chapter 19). (3) The symbol which R.A. Fisher used to designate half the difference of the natural logarithms of two independent estimates of variances when comparing them. (Nowadays we tend to use the ratio instead, which we compare to the F-distribution.) (4) The last entry in this glossary. 

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