Fisher’s F-test

The critical weight of samples can be derived from the general condition of representativeness of sampling expressed by the null hypothesis H0 °totai = °anai which is tested by means of Fisher s F-test... 

Influence on precision can be tested as usual by Fisher s F-test with the null hypothesis H0 ot0tai = aA and therefore a,j = 0 see Sect. 4.3.3 ... 

Thus, in order to apply the t-test, we must first test for differences between the two variances Fisher s F-test is used for this purpose. Again we shall omit all reference to the derivation of this test statistic and the rationale for statistical tests based on it. Instead we focus here on its application to testing two data sets to determine the confidence level at which we can assert that the variances i and Vx,2 (Equation [8.2b]) of the two are indistinguishable (the null hypothesis F(, for the test). The value of the experimental test statistic F is calculated simply as ... 

This homogeneity value will become positive when the null hypothesis is not rejected by Fisher s E-test (F < Ei a)Vl)V2) and will be closer to 1 the more homogeneous the material is. If inhomogeneity is statistically proved by the test statistic F > Fi a>VuV2, the homogeneity value becomes negative. In the limiting case F = hom(A) becomes zero. 

In statistics, ANalysis Of VAriance (ANOVA) is a collection of statistical models and their associated procedures in which the observed variance is partitioned into components because of different explanatory variables. The initial techniques of the analysis of variance were developed by the statistician and geneticist R.A. Fisher in the 1920s and 1930s, and are sometimes known as Fisher s ANOVA or Fisher s analysis of variance due to the use of Fisher s F-distribution as part of the test of statistical significance. 

Critical Value(s) The critical value for a hypothesis test is a threshold to which the value of the test statistic (e.g.. Student s t, or Fisher s F parameter) for a data set is compared to determine whether or not the null hypothesis is rejected. The critical value for any hypothesis test depends on the significance level at which the test is carried out, and whether the test is one sided or two sided. The critical value(s) determine the limits of the critical region. 

Test Statistic A test statistic is a quantity calculated from the experimental data set, that is used to decide whether or not the nidi hypothesis should be rejected in a hypothesis test. The choice of a test statistic depends on the assumed probability model and the hypotheses under question common choices are the Student s-t and Fisher s F parameters. 

Statistical comparisons were made using analysis of variance and Fisher s test, or Student s f-test. A statistically significant difference was defined as a p value <0.05. 

Once a significant difference has been demonstrated by an analysis of variance, a modified version of the f-test, known as Fisher s least significant difference, can be used to determine which analyst or analysts are responsible for the difference. The test statistic for comparing the mean values Xj and X2 is the f-test described in Chapter 4, except that Spool is replaced by the square root of the within-sample variance obtained from an analysis of variance. 

Fisher s least significant difference a modified form of the f-test for comparing several sets of data. (p. 696) flame ionization detector a nearly universal GC detector in which the solutes are combusted in an H2/air flame, producing a measurable current, (p. 570)... 

Statistical parameters, when available, indicating the significance of each of the descriptor s contribution to the final regression equation are listed under its corresponding term in the equation. These include the standard errors written as values, the Student t test values, and the VIF. The significance of the equation will be indicated by the sample size, n the variance explained, r the standard error of the estimate, s the Fisher index, F and the cross-validated correlation coefficient, q. When known, outliers will be mentioned. The equations are followed by a discussion of the physical significance of the descriptor terms. 

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 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 ... 

The two estimations, s and s, may be considered as significantly different. The mathematical model is thus rejected and it is therefore s, derived from the error sum of squares which is retained as an estimate of a. Fisher s test may be carried out a second time to compare s with our estimate of the experimental variance s- F = 161.8. With... 

The mathematical model may be accepted and a more precise estimate of obtained by combining our two estimations. In this case it is derived from the residual sum of squares, which is retained as an estimation of Fisher s test may be carried out again, this time to compare si with F = RESIP - 32.13/0.5 = 62.26. The significance level is still lower... 

Student s t-Test and Fisher F-Testfor Comparison of Means and Variances Applications to Two Data Sets... 

The universally accepted approach to statistical evaluation of small data sets is that originated by WUham Gosset (Gosset 1908) and developed and publicized by Richard Fisher (Fisher 1925, 1990). The so-caUed Student s t-distribution (see the accompanying text box) addresses the problem of estimating the uncertainty in the mean of an assumed normal distribution based on a small number of data points assumed to be random samples from that distribution. It forms the basis of f-tests commonly used for the statistical significance of the difference between the means of two small data sets believed to be drawn from the same normal distribution, and for determination of the confidence interval for the difference between two such data set means. 

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

Fisher E-test A statistical significance test which decides whether there is a significant difference between two variances (and therefore two sample standard deviations). This test is used in ANOVA. For two standard deviations X and s2, F=s /s where, V >,v2. (Sections 3.7, 4.4)... 

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. 

In a first step the chosen mathematical model, containing the functional model (27) or (34) and the stochastic model (33), has to be proved. By Baarda s test of the model the agreement of the a priori variance factor and the a posteriori variance factor will be tested. Supposing the modell is correct (hypothesis Ho), the test value Ti has a central Fisher-distribution with f degrees of freedoms (30) and will be provided to... 

The statistical quality of the QSAR models (15.43) and (15.44) is evaluated by means of r (correlation coefficient), s (standard deviation from the regression line), F (Fisher test), and r (cross-validation coefficient). The statistic represents the explained variance, and the numbers in brackets give the 95% confidence limits. In these equations we used only 11 data form data set because compound 12 from Table 15.10 is an outlier. The possible explanation of this exception may be connected with the peculiar nature of the substituent NHCOCH2NH2, which coti-tain a supplementary amino (NH2) group. 

Manufacturer s code A = Aldrich B = Burdick Jackson E = EM Science F = Fisher J = JT Baker M = MallinckrodL No ACS test exists for this solvent. 

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