Fisher significance 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,... 

Accordingly, we wish to test whether the Between Column Mean Square is significantly greater than the Within Column Mean Square. This can be done with the Fisher variance ratio test, discussed earlier in Chapter IV (a). 

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. 

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). 

The various mean squares are tested for significance with Fisher s variance ratio test. Here we are principally interested in the Row and Column effects,... 

And, taking into account the ratio Di/Dj s, criterion of statistical significance of the Fisher dispersion can be applied for further calculation. In (7.3), Dq is the total dispersion of the results relative to the total average value, Dj is the dispersion of the average results value by all r levels of the first factor, and D is the dispersion of the average results value by all r levels of the factor. More detailed explanation of the dispersion analysis theory can be found in [6, 7]. 

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. 

The flow coefficient C is determined by calibration with water, and it is not entirely satisfactory for predicting the flow rate of compressible fluids under choked flow conditions. This has to do with the fact that different valves exhibit different pressure recovery characteristics with gases and hence will choke at different pressure ratios, which is not significant for liquid flows. For this reason, another flow coefficient, C, is often determined by calibration with air under critical flow conditions (Fisher Controls, 1977). The corresponding flow equation for gas flow is... 

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. 

Finally in this section, we note that a number of tests exist for estimating the significance of the difference between ratios. These include the and Fisher tests, both of which are described in standard texts. They have occasional use in structure correlation studies. For example, x test was used to show that the distribution of non-bonded contacts around nitro groups in organic crystal structures is significantly anisotropic [34]. 

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 ... 

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