Variance-ratio test

There are two common methods for comparing results (a) Student s -test and (b) the variance ratio test (F-test). 

F-tost (The Variance Ratio Test) a method for determining, with a given degree of probability, whether the variances of two populations differ significantly from one another ... 

Variance ratio test a test used to determine the difference in variability between two sets of deta. Used in the analysis of variance to compare variation due to a perticular factor with the experimental error. 

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 ... 

The Fisher variance ratio test will compare two variances, but not more than two. Suppose we have a group of ten machines turning out batches of Some product, and we measure some quality x on each batch. Suppose we suspect that some machines manufacture the product more regularly than others, i.e. that... 

It should be clear that if the largest and smallest variances of a set do not differ significantly as tested by the variance ratio test, then clearly those lying between cannot differ significantly either, and the whole set can be reasonably regarded as coming from a single population. In these circumstances there would be no need to employ the Bartlett Test. 

In evaluating a I and Ob we proceed by first calculating (n oi-i- oj) and o f, (in the generd case where there are n units per batch). The reason for this is that it makes the estimation of the significance of o practicable. The function (n a - - <4 ) has (m— 1) degrees of freedom, where m is the number of batches, arid 05 has m (n — 1) degrees of freedom. For a to exist, therefore, (n oH- <4) must be greater than o, and we can readily test this with the variance ratio test. 

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). 

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,... 

Table 9 shows the construction of the ANOVA table. If the variance estimate of a class variable MS variabie deviates significantly from that obtained by that for random error MSettot, then the null hypothesis that the means at the different levels for that variable are equal is rejected. In other words, the classification of data by that variable is explanatory of the variation observed in the data. We conduct the test by using the variance ratio test F = MSvariabie/AfSError, with... 

Outliers can be tested by three different methods, i.e. Bartlett s homogeneity of variance test, Cochran s one sided outlier test and Hartley s variance ratio test. The ISO (1986) recommends Cochran s test, because (1) the two other tests cannot be applied when one of the variances in a set is zero and (2) the two other tests are very sensitive to the value of the smallest variance. Although in our opinion a zero variance may be easily overcome by setting the variance to a small finite value, the other argument is certainly true for outliers. Cochran s criteria C is given by... 

Stegun (1964) to give equation 2.6-26, where F (equation 2.6-27) is the variance ratio distribution function and Q is the cumulative integral over F. This is similar to the classical result (equation 2 5 73) which means that pseudo-failures, a-1, are added to the failures, M, and pseudo-tests, p-a, are added to the tests, N. 

Mendal et al. (1993) compared eight tests of normality to detect a mixture consisting of two normally distributed components with different means but equal variances. Fisher s skewness statistic was preferable when one component comprised less than 15% of the total distribution. When the two components comprised more nearly equal proportions (35-65%) of the total distribution, the Engelman and Hartigan test (1969) was preferable. For other mixing proportions, the maximum likelihood ratio test was best. Thus, the maximum likelihood ratio test appears to perform very well, with only small loss from optimality, even when it is not the best procedure. 

Is there any significant difference between the precision of these two sets of results Applying the variance-ratio or F-Test from Eq. (zz) we have ... 

Procedure 1.2 Statistical comparison of the relative precision of two methods using the variance ratio or iFt test... 

The sums of squares of the individual items discussed above divided by its degrees of freedom are termed mean squares. Regardless of the validity of the model, a pure-error mean square is a measure of the experimental error variance. A test of whether a model is grossly adequate, then, can be made by acertaining the ratio of the lack-of-fit mean square to the pure-error mean square if this ratio is very large, it suggests that the model inadequately fits the data. Since an F statistic is defined as the ratio of sum of squares of independent normal deviates, the test of inadequacy can frequently be stated... 

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

If we are interested in comparing variance A with variance B, and would like to know whether A is larger than B, and also whethej A is smaller than B, we use a two-tailed test on the variance ratio. F% will be the ratio of variance A to variance B. Fz will be the ratio of variance B to variance A. Both these ratios are then... 

In evaluating two testraethods for repeatability or reproducibility, the variances (standard deviation squared) of the two can be com pared by means of the variance ratio or F test. This is a tried and true statistical technique which has many other uses, as well as being a critical part of analysis of variance. 

An F ratio test is hardly needed to decide whether the variances are different A t-test is still applicable but needs to be modified to take into account the variance differences. This is done by calculating the effective number of degrees of freedom using Satterthwaite s method.This is still an area which is controversial and a number of differing approaches and equations have been proposed. 

In order to compare the results between two laboratories for the same sample or, for example, two instruments for the same analysis method, it is essential to know whether the standard deviation, i of the first set of results is significantly different from that of the second set, s2. This is accomplished by using the variance equality test. In this test, an F factor is calculated, which is the ratio of the two variances such that F > 1 ... 

Now, to compute the likelihood ratio statistic for a likelihood ratio test of the hypothesis of equal variances, we refer %2 = 401n.58333 - 201n.847071 - 201n.320506 to the chi-squared table. (Under the null hypothesis, the pooled least squares estimator is maximum likelihood.) Thus, %2 = 4.5164, which is roughly equal to the LM statistic and leads once again to rejection of the null hypothesis. 

Suppose that the following sample is drawn from a nonnal distribution with mean u and standard deviation ct y = 3.1, -.1,. 3, 1.4, 2.9,. 3, 2.2, 1.5, 4.2,. 4. Test the hypothesis that the mean of the distribution which produced these data is the same as that which produced the data in Exercise 1. Test the hypothesis assuming that the variances are the same. Test the hypothesis that the variances are the same using an F test and using a likelihood ratio test. (Do not assume that the means are the same.)... 

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

The technique known as analysis of variance (ANOVA)2) uses tests based on variance ratios to determine whether or not significant differences exist among the means of several groups of observations, where each group follows a normal distribution. The analysis of variance technique extends the t-test used to determine whether or not two means differ to the case where there are three or more means. 

It gives a more precise estimate of the treatment effect. Some of the variation in the outcome can be ascribed to concomitant variation in the covariate. This allows this variation to be removed from the total error variation against which the effect variation is compared. This means that the denominator in the effect variance/error variance ratio is smaller, making the test statistic calculated of larger magnitude. 

Analytical methods should be precise, accurate, sensitive, and specific. The precision or reproducibility of a method is the extent to which a number of replicate measurements of a sample agree with one another and is expressed numerically in terms of the standard deviation of a large number of replicate determinations. Statistical comparison of the relative precision of two methods uses the variance ratio (F j or the F test. 

Liao (2000) derived a test statistic for single dispersion effects in 2" k designs. He applied the generalized likelihood ratio test for a normal model to the residuals after fitting a location model, which results in Bartlett s (1937) classical test for comparing variances in one-way layouts. The test is then applied, in turn, to compare the variances at the two levels of each of the k experimental factors. We caution that the test statistic (equation (3) in Liao) is written incorrectly. 

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