Homogeneity, variances

Both correlation and variance analysis results showed that the hypothesis on the linear correlation between inter-laboratory data and the homogeneity of the corresponding variances is true for all data sets, at the for 95% confidence level. Table 2 presents a typical example of such a comparison. Based on the detected property of homogeneous variances, root-mean-square standard deviation, S, for all melted snow samples was estimated S = 0.32 0.06 for 95% confidence level [3]. 

Box, M. J., and N. R. Draper, Estimation and design criteria for multiresponse non-linear models with non-homogeneous variance, J. Roy. Statist. Soc., Series C (Appl. Statist..) 21, 13-24 (1972). 

It is obvious that the lower end of the calibration curve does not provide an accurate representation of the calibrator concentrations, a problem that does not exist at the higher end of the curve. This problem illustrates what can happen when there is violation of the assumption of equal (homogeneous) variance for the y, values. In bioanalytical methods, the largest component of the random error can often be attributed to volume errors. One often finds that the standard deviation of the response (y,) is proportional to the concentration (x,), that is, e,- = kx, where k is a constant. This violates the assumption of homogeneity... 

The Lineweaver-Burk-transformation on papaverine for instance yields the calibration shown in Fig. 16. The scattering of the original data is distributed normally, according to the test designed by David 53,54a) and homogeneous variances exist, according to the test developed by Cochran 55,54 b) however, it is easy to see form Figs. 16 and 17 that this is not the case after transformation. 

Assumes a homoscedastic error structure (common or homogeneous variance regardless of response). The random error is the same for all observations. 

A multi-group criterion is the called Wilks X or McCabe U statistics (McCabe, 1975). This is a general statistic used as a measure for testing the difference among group centroids. All classes are assumed to be homogeneous variance-covarianoe matrices and the statistic is defined as... 

In this model, we assume homogeneous variance across all k treatments. For a three-arm trial, having two 5,2, we would further need to consider correlation between them. In this case, we would assume the 8 vector follows a mulfivari-ate normal distribution, typically with common correlation of 0.5 between two log odds ratios (a consequence of the usual assumption of consistency between direct and indirect evidence), as suggested by Lu and Ades (2006). 

The t-test is a parametric method for comparing two mean values, e.g., for the difference in means between two groups or one mean value with an expected value [33, 34], T-tests require a random sample, normally distributed and metric raw data, and homogeneous variances [33-35],... 

Consequently, the null hypothesis cannot be rejected at the 95% level, and one can as.sume homogeneous variances among the seven different measurement series. 

Evaluation Statistical tests can be used to evaluate relative homogeneity based on observed variations in spot sample composition. For a simple binaiy mixture such as that shown in Fig. 19-8, it can be shown (see Ref. 9) that the expected variance among samples containing n particles each is given by... 

As with the central moments in first-order statistics, we can first subtract the mean. We define the co-variance (for a statistically homogeneous process) as... 

Note that the Kolmogorov power spectrum is unphysical at low frequencies— the variance is infinite at k = 0. In fact the turbulence is only homogeneous within a finite range—the inertial subrange. The modified von Karman spectral model includes effects of finite inner and outer scales. 

Data Evaluation The Bartlett test (Section 1.7.3 cf. program MULTI using data file MOISTURE.dat) was first applied to determine whether the within-group variances were homogeneous, with the following intermediate results A = 0.1719, B = -424.16, C = 1.4286, D = 70, E = 3.50, F = 1.052, G = 3.32. 

The logarithmic transformation prior to column- or double-centered PCA (Section 31.3) can be considered as a special case of non-linear PCA. The procedure tends to make the row- and column-variances more homogeneous, and allows us to interpret the resulting biplots in terms of log ratios. 

A homogeneity index or significance coefficienf has been proposed to describe area or spatial homogeneity characteristics of solids based on data evaluation using chemometrical tools, such as analysis of variance, regression models, statistics of stochastic processes (time series analysis) and multivariate data analysis (Singer and... 

In order to extract the degree of homogeneity from the variance of repetitive determinations, it is mandatory to determine the variance of the method used for analysis as accurately as possible. On the other hand, it is obvious that the variance of the... 

The sampling variance of the material determined at a certain mass and the number of repetitive analyses can be used for the calculation of a sampling constant, K, a homogeneity factor, Hg or a statistical tolerance interval (m A) which will cover at least a 95 % probability at a probability level of r - a = 0.95 to obtain the expected result in the certified range (Pauwels et al. 1994). The value of A is computed as A = k 2R-s, a multiple of Rj, where is the standard deviation of the homogeneity determination,. The value of fe 2 depends on the number of measurements, n, the proportion, P, of the total population to be covered (95 %) and the probability level i - a (0.95). These factors for two-sided tolerance limits for normal distribution fe 2 can be found in various statistical textbooks (Owen 1962). The overall standard deviation S = (s/s/n) as determined from a series of replicate samples of approximately equal masses is composed of the analytical error, R , and an error due to sample inhomogeneity, Rj. As the variances are additive, one can write (Equation 4.2) ... 

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