Comparison of two variances

The extremly wide range is the consequence of the large sample variance. Comparison of two variances... 

Example 10 Comparison of two variances. A simplified analytical procedure is proposed for a routine laboratory test. It is necessary to determine not only whether the new procedure gives the same results as the old, i.e., whether the means of a duplicate set are the same, but also whether the precision of the new test is as good as the current test. The data for the two tests are as follows ... 

The statistical significance of apparent differences between two variances can be judged on the basis of a ratio of the two variances. The most frequently encountered situation is the comparison of two variance estimates, both estimates obtained from samples from two ostensibly different populations. The variance ratio, Sf/Ss, is called an F-ratio. and this ratio follows a sampling distribution called an F-distribution. The. shape of thc.se distributions depends on the number of degrees of freedom r// asstK iated with each of the variances (numerator, denominator). The statistical te.st to determine if the two variance estimates are equal is called an F-test this is conducted by comparing the calculated value of the ratio F(calc) to a critical F value called F(crit) that would be obtained on the basis of chance at some probability level w hen there is no real difference, i.c.. both estimates are draw n from the same population. There arc two potential situations or cases (1) from technical or operational conditions alone, one variance should have a larger value and (2) on a technical basis neither variance can be considered greater. 

A comparison of two or more means can be made with a one-way analysis of variance. This tool compares... 

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 Wilcoxon Rank-Sum test is commonly used for the comparison of two groups of nonparametric (inteval or not normally distributed) data, such as those which are not measured exactly but rather as falling within certain limits (for example, how many animals died during each hour of an acute study.) The test is also used when there is no variability (variance = 0) within one or more of the groups we wish to compare (Sokal and Rohlf, 1994). 

It is also necessary to decide how the primary endpoint variables will be analysed, what factors will be taken into account and how the result will be expressed. This most frequently involves analysis of variance or covariance. Predetermined comparisons of two or more treatments or doses can be made at specific time points, (e.g. each visit or selected visits) or may be assessed over time, giving an area imder the time curve analysis which will avoid multiple time-point analyses. 

CONTENTS 1. Chemometrics and the Analytical Process. 2. Precision and Accuracy. 3. Evaluation of Precision and Accuracy. Comparison of Two Procedures. 4. Evaluation of Sources of Variation in Data. Analysis of Variance. 5. Calibration. 6. Reliability and Drift. 7. Sensitivity and Limit of Detection. 8. Selectivity and Specificity. 9. Information. 10. Costs. 11. The Time Constant. 12. Signals and Data. 13. Regression Methods. 14. Correlation Methods. 15. Signal Processing. 16. Response Surfaces and Models. 17. Exploration of Response Surfaces. 18. Optimization of Analytical Chemical Methods. 19. Optimization of Chromatographic Methods. 20. The Multivariate Approach. 21. Principal Components and Factor Analysis. 22. Clustering Techniques. 23. Supervised Pattern Recognition. 24. Decisions in the Analytical Laboratory. 

The t-test provides a method for comparing two means while the F-test permits a comparison of two or more means. It can be shown that if a group of samples of size n from a total population has a sample variance s2(jc), then the means of the samples have a variance... 

The variance of the slope permits a comparison of two regression lines in the same way that two means were compared by the t-test. Thus,... 

The essential property of all these variances is that they can only be zero if the wave equation is satisfied at all (Rj). In practice, the valuable use for these is to assess two approximate wavefunctions such as Ca<1>a and CbOb- These could be results obtained either by different workers or in one calculation at different stages of the iteration. The comparison of two such results by an independent method would be very informative. If... 

Fritted-glass crucible A filtering crucible equipped with a porous glass bottom also called a sintered-glass crucible. Fronting Describes a nonideal chromatographic peak in which the early portions tend to be drawn out compare with tailing. F-test A statistical method that permits comparison of the variances of two sets of measurements. 

On average we would obtain an experimental variance value outside the range 0.00260 to 0.00957 only one time in twenty. These limiting variance values could be used to compare the mixture quality within a mixer at different mixing times or to compare the performance of two competing mixers. An F test would give a more refined statistical comparison of two mixture qualities as described by variance values. 

The Michaelis-Menten model was fitted to the experimental data using standard nonlinear regression techniques to obtain estimates of and K (Fig. 4.1). Best-fit values of and K of corresponding standard errors of the estimates plus the number of values used in the calculation of the standard error, and of the goodness-of-fit statistic are reported in Table 4.3. These results suggest that succinate is a competitive inhibitor of fumarase. This prediction is based on the observed apparent increase in Ks in the absence of changes in Vmax (see Table 4.1). At this point, however, the experimenter cannot state with any certainty whether the observed apparent increase in Ks is a tme effect of the inhibitor or merely an act of chance. A proper statistical analysis has to be carried out. For the comparison of two values, a two-tailed t-test is appropriate. When more than two values are compared, a one-way analysis of variance (ANOVA),... 

From a statistical viewpoint it is difficult to state a definite answer because several problems accumulate here. First, there is not an exact solution to the comparison of two population means, for which we have to estimate simultaneously (from the limited experimental data available) their average values and their associated variances, as is the case in laboratories. Second, the equations stated above were deduced for normal distributions but the slopes derived from a least-squares fit follow approximately a Student s distribution. We must bear in mind that although the theoretical slope and intercept of the population follow a normal distribution their estimators do not because the latter (along with the variance of the regression itself) must be estimated from (usually) a very reduced number of data and the number of degrees of freedom - dof- must be taken into account. In statistical terms an intermediate pivot statistic must be introduced to obtain an approximate Student s distribution. ... 

COMPARISON OF MORE THAN TWO MEANS (ANALYSIS OF VARIANCE)... 

The comparison of more than two means is a situation that often arises in analytical chemistry. It may be useful, for example, to compare (a) the mean results obtained from different spectrophotometers all using the same analytical sample (b) the performance of a number of analysts using the same titration method. In the latter example assume that three analysts, using the same solutions, each perform four replicate titrations. In this case there are two possible sources of error (a) the random error associated with replicate measurements and (b) the variation that may arise between the individual analysts. These variations may be calculated and their effects estimated by a statistical method known as the Analysis of Variance (ANOVA), where the... 

Because no direct way exists to test simultaneously more than two means for significant differences, the comparison of several means, J1,J2, >JV traced back to the comparison of variances. 

The variance approach may also be used to determine n. From Illustration 11.2 the variance of the response data based on dimensionless time is 30609/(374.4)2, or 0.218. From equation 11.1.76 it is evident that n is 4.59. Thus the results of the two approaches are consistent. However, a comparison of the F(t) curves for n = 4 and n = 5 with the experimental data indicates that these approaches do not provide very good representations of the data. For the reactor network in question it is difficult to model the residence time distribution function in terms of a single parameter. This is one of the potential difficulties inherent in using such simple models of reactor behavior. For more advanced methods of modeling residence time effects, consult the review article by Levenspiel and Bischoff (3) and textbooks written by these authors (2, 4). 

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