Variance F test

Source of variation Sum of squares Degrees of freedom Variance F-test... 

Hi. Test of Significance for Variances (F test). The need for this might arise as follows A small-scale pilot plant experiment has shown that a satisfactory mixture can be produced for further processing. All of the work was done by a research group where considerable attention was... 

The statistical treatment involves tests, e.g. to assess the conformity of the distributions of individual results and of laboratory means to normal distributions (Kolmogorov-Smimov-Lilliefors tests), to detect outlying values in the population of individual results and in the population of laboratory means (Nalimov test), to assess the overall consistency of the variance values obtained in the participating laboratories (Bartlett test), and to detect outlying values in the laboratory variances (s ) (Cochran test). One-way analysis of variance (F-test) may be used to compare and estimate... 

It is possible to compare the means of two relatively small sets of observations when the variances within the sets can be regarded as the same, as indicated by the F test. One can consider the distribution involving estimates of the true variance. With sj determined from a group of observations and S2 from a second group of N2 observations, the distribution of the ratio of the sample variances is given by the F statistic ... 

The larger variance is placed in the numerator. For example, the F test allows judgment regarding the existence of a significant difference in the precision between two sets of data or between two analysts. The hypothesis assumed is that both variances are indeed alike and a measure of the same a. 

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. 

F-test statistical test for comparing two variances to see if their difference is too large to be explained by indeterminate error, (p. 87)... 

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

This lack of sharpness of the 1-way F-test on REV s is sometimes seen when there is information spanned by some eigenvectors that is at or below the level of the noise spanned by those eigenvectors. Our data sets are a good example of such data. Here we have a 4 component system that contains some nonlinearities. This means that, to span the information in our data, we should expect to need at least 4 eigenvectors — one for each of the components, plus at least one additional eigenvector to span the additional variance in the data caused by the non-linearity. But the F-test on the reduced eigenvalues only... 

A central concept of statistical analysis is variance,105 which is simply the average squared difference of deviations from the mean, or the square of the standard deviation. Since the analyst can only take a limited number n of samples, the variance is estimated as the squared difference of deviations from the mean, divided by n - 1. Analysis of variance asks the question whether groups of samples are drawn from the same overall population or from different populations.105 The simplest example of analysis of variance is the F-test (and the closely related t-test) in which one takes the ratio of two variances and compares the result with tabular values to decide whether it is probable that the two samples came from the same population. Linear regression is also a form of analysis of variance, since one is asking the question whether the variance around the mean is equivalent to the variance around the least squares fit. 

This PROC TTEST runs a two-sample f-test to compare the LDL change-from-baseline means for active drug and placebo. ODS OUTPUT is used to send the p-values to a data set called pvalue and to send the test of equal mean variances to a data set called variance test. The final pvalue DATA step checks the test for unequal variances. If the test for unequal variances is significant at the alpha =. 05 level, then the mean variances are unequal and the unequal variances p-value is kept. If the test for unequal variances is insignificant, then the equal variances p-value is kept. The final pvalue data set contains the Probt variable, which is the p-value you want. 

The f-test in this form can only be applied under the condition that the variances of the two sample subsets, sf and sf, do not differ significantly. This has to be checked by the F-test beforehand. The test statistic f has to be compared to the related quantile of the (-distribution h-a,v where v = mx + n2 — 2. 

The variance between the labs is 0.1284 and the variance within the labs is 0.03295 therefore the F-test gives ... 

This Worksheet demonstrates using Mathcad s F distribution function and programming operators to conduct an analysis of variance (ANOVA) test. 

Figure 65-1 shows a schematic representation of the F-test for linearity. Note that there are some similarities to the Durbin-Watson test. The key difference between this test and the Durbin-Watson test is that in order to use the F-test as a test for (non) linearity, you must have measured many repeat samples at each value of the analyte. The variabilities of the readings for each sample are pooled, providing an estimate of the within-sample variance. This is indicated by the label Operative difference for denominator . By Analysis of Variance, we know that the total variation of residuals around the calibration line is the sum of the within-sample variance (52within) plus the variance of the means around the calibration line. Now, if the residuals are truly random, unbiased, and in particular the model is linear, then we know that the means for each sample will cluster... 

A detailed treatment of linearity evaluation is beyond the scope of this present book but a few general points are made below. It is important to establish the homogeneity of the variance ( homoscedasticity ) of the method across the working range. This can be done by carrying out ten replicate measurements at the extreme ends of the range. The variance of each set is calculated and a statistical test (F test) carried out to check if these two variances are statistically significantly different [9]. 

When a comparison of two separate replicate sets of data is required, the first stage is normally to compare their respective precisions by means of the F-test. This test uses the ratio of the variances of the two sets to establish any statistically significant difference in precision. F is calculated from... 

The Cochran test should be used to compare two groups of continuous data when the variances (as indicated by the F test) are heterogeneous and the numbers of data within the groups are not equal (N N2). This is the situation, for example, when the data, though expected to be randomly distributed, were found not to be (Cochran and Cox, 1975, pp. 100-102). 

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 ... 

---