F-test for Equality of Variance

Our same clever engineer now observes that these results can be reached only if it can be assumed that the two population variances ate equal. At this point, the manager asks our statistician, Can we test this assumption The answer is yes, using an F test for equality of variances. 

Fig. 3 Scatter plot showing the mean levels of GAD67 mRNA-labehng in Golgi cells in seven control cases (circles) and eight subjects with autism (squares) sampled from 60 cells per subject (1.97 1.17 pixels/ttm control 2.21 2.36 pixels/ttm autism). Statistical analysis did not show a significant difference in GAD67 mRNA levels in the autistic group compared to age-, PMI-, and pH-matched controls (student t-test and Levene s test for equality of variances F = 2.981 p = 0.401 control, and p = 0.384 autism)...

A simple application of the F-test can be illustrated by examining the mineral water data in the previous examples for equality of variance. 

The analysis of variance technique for testing equality of means is a rather robust procedure. That is, when the assumption of normality and homogeneity of variances is slightly violated the F-test remains a good procedure to use. In the one-way model, for example, with an equal number of observations per column it has been exhibited that the F-test is not significantly effected. However, if the sample size varies across columns, then the validity of the F-test can be greatly affected. There are various techniques for testing the equality of k variances Oi, 02,..., crj,. We discuss... 

Assuming a normal multivariate distribution, with the same covariance matrices, in each of the populations, (X, X2,..., Xp) V(7t , 5), the multivariate analysis of variance MANOVA) for a single factor with k levels (extension of the single factor ANOVA to the case of p variables), permits the equality of the k mean vectors in p variables to be tested Hq = jl = 7 2 = = where ft. = fl, fif,..., fVp) is the mean vector of p variables in population Wi. The statistic used in the comparison is the A of Wilks, the value of which can be estimated by another statistic with F-distribution. If the calculated value is greater than the tabulated value, the null hypothesis for equality of the k mean vectors must be rejected. To establish whether the variables can distinguish each pair of groups a statistic is used with the F-distribution with p and n — p — k + i df, based on the square of Mahalanobis distance between the centroids, that permits the equality of the pairs of mean vectors to be compared Hq = jti = ft j) (Aflfl and Azen 1979 Marti n-Alvarez 2000). 

Table 3.4 F-Test values for comparison of variance of samples with equal degrees of freedom,...

The /-test is widely used in analytical laboratories for comparing samples and methods of analysis. Its application, however, relies on three basic assumptions. Firstly, it is assumed that the samples analysed are selected at random. This condition is met in most cases by careful design of the sampling procedure. The second assumption is that the parent populations from which the samples are taken are normally distributed. Fortunately, departure from normality rarely causes serious problems providing sufficient samples are analysed. Finally, the third assumption is that the population variances are equal. If this last criterion is not valid then errors may arise in applying the /-test and this assumption should be checked before other tests are applied. The equality of variances can be examined by application of the F-test. 

To test the equality of the variances, we can use the Goldfeld and Quandt test. Under the null hypothesis of equal variances, the ratio F = [ei ei/( i - 2)]/[e2 e2/( 2 - 2)] (or vice versa for the subscripts) is the ratio of two independent chi-squared variables each divided by then respective degrees of freedom. Although it might seem so fr om the discussion in the text (and the literature) there is nothing in the test which requhes that the coefficient vectors be assumed equal across gi oups. Since for ovn data, the second sample has the larger residual variance, we refer F 48,48] = s2 /si = 9.122 I iAll = 2.8 to the Stable. The critical value for 95% significance is 1.61, so the hypothesis of equal variances is rejected. 

F-distribution A statistical probability distribution used in the analysis of variance of two samples for statistical significance. It is calculated as the distribution of the ratio of two chi-square distributions and used, for the two samples, to compare and test the equality of the variances of the normally distributed variances. 

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 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). 

Up to now the technique of calculations in analysis of variance has been analyzed in more detail. Now let us briefly consider the analysis of variance theory. Let us consider the model for a one-way analysis of variance. Here it is assumed that the columns of data are J-random samples from J-independent normal populations with means i, i2,...,P, and common variance a2. The one-way analysis of variance technique will give us a procedure for testing the hypothesis H0 F.i=p.2=---=F-j against the alternative Hj at least two ij not equal. The statistical model gives us the structure of each observation in the IxJ matrix ... 

If we desire to study the effects of two independent variables (factors) on one dependent factor, we will have to use a two-way analysis of variance. For this case the columns represent various values or levels of one independent factor and the rows represent levels or values of the other independent factor. Each entry in the matrix of data points then represents one of the possible combinations of the two independent factors and how it affects the dependent factor. Here, we will consider the case of only one observation per data point. We now have two hypotheses to test. First, we wish to determine whether variation in the column variable affects the column means. Secondly, we want to know whether variation in the row variable has an effect on the row means. To test the first hypothesis, we calculate a between columns sum of squares and to test the second hypothesis, we calculate a between rows sum of squares. The between-rows mean square is an estimate of the population variance, providing that the row means are equal. If they are not equal, then the expected value of the between-rows mean square is higher than the population variance. Therefore, if we compare the between-rows mean square with another unbiased estimate of the population variance, we can construct an F test to determine whether the row variable has an effect. Definitional and calculational formulas for these quantities are given in Table 1.19. 

Only if the variances are found equal in the statistical sense it is allowed to proceed with the common /-test for the two corresponding means jq and x2. If the F-test signals significant differences one is forced to use special definitions of both the empirical /-value and the degrees of freedom. 

Table IV shows the overall analysis of variance (ANOVA) and lists some miscellaneous statistics. The ANOVA table breaks down the total sum of squares for the response variable into the portion attributable to the model, Equation 3, and the portion the model does not account for, which is attributed to error. The mean square for error is an estimate of the variance of the residuals — differences between observed values of suspensibility and those predicted by the empirical equation. The F-value provides a method for testing how well the model as a whole — after adjusting for the mean — accounts for the variation in suspensibility. A small value for the significance probability, labelled PR> F and 0.0006 in this case, indicates that the correlation is significant. The R2 (correlation coefficient) value of 0.90S5 indicates that Equation 3 accounts for 91% of the experimental variation in suspensibility. The coefficient of variation (C.V.) is a measure of the amount variation in suspensibility. It is equal to the standard deviation of the response variable (STD DEV) expressed as a percentage of the mean of the response response variable (SUSP MEAN). Since the coefficient of variation is unitless, it is often preferred for estimating the goodness of fit.

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... 

At times, there is a need to compare the variances (or standard deviations) of two populations. For example, the normal t test requires that the standard deviations of the data sets being compared are equal. A simple statistical test, called the F test, can be used to test this assumption under the provision that the populations follow the normal (Gaussian) distribution. The F test is also used in comparing more than two means (see Section 7C) and in linear regression analysis (see Section 8C-2). 

To apply ANOVA methods, we need to make a few assumptions concerning the populations under study. First, the usual ANOVA methods are based on an equal variance assumption. That is, the variances of the I populations are assumed to be identical. This assumption is sometimes tested (Hartley test) by comparing the maximum and minimum variances in the set with an F test (see Section 7B-4). However, the Hartley test is quite susceptible to departures from the normal distribution. As a rough rule of thumb, the largest s should not be much more than twice the smallest s for equal variances to be assumed. Transforming the data by working with a new variable such as or log x can also... 

The assumption for this so-called extended f-test is the comparability of the variances of the two random samples, and s. Comparability here means that the two variances are equal at a given statistical significance level. The significance of the differences between the two variances is tested by means of an f-test (see the following discussion). 

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