Equal variance requirement

As stated earlier, LDA requires that the variance-covariance matrices of the classes being considered can be pooled. This is only so when these matrices can be considered to be equal, in the same way that variances can only be pooled, when they are considered equal (see Section 2.1.4.4). Equal variance-covariance means that the 95% confidence ellipsoids have an equal volume (variance) and orientation in space (covariance). Figure 33.10 illustrates situations of unequal variance or covariance. Clearly, Fig. 33.1 displays unequal variance-covariance, so that one must expect that QDA gives better classification, as is indeed the case (Fig. 33.2). When the number of objects is smaller than the number of variables m, the variance-covariance matrix is singular. Clearly, this problem is more severe for QDA (which requires m < n ) than for LDA, where the variance-covariance matrix is pooled and therefore the number of objects N is the sum of all objects... 

To cany out a Lagrange multiplier test of the hypothesis of equal variances, we require the separate and common variance estimators based on the restricted slope estimator. This, in turn, is the pooled least squares estimator. For the combined sample, we obtain... 

Microarray data may require transformation to stabilize the variance by applying a global logarithmic transformation prior to normalization and further analysis. This may unpredictably skew the measurement, making further analysis difficult and unreliable (39). As many of the techniques used for the analysis of microarray data are not robust to the equal variance assumption, a family of variance stabilizing transformation (VST) methods for microarray data have been proposed (4CM14). 

The Wilcoxon s Rank-Sum Test (WRST) is a non-parametric alternative. The WRST is robust to the normal distribution assumption, but not to the assumption of equal variance. Furthermore, this test requires that the two groups of data under comparison have similarly shaped distributions. Non-parametric tests typically suffer from having less statistical power than their parametric counterparts. Similar to the /-test, the WRST will exhibit false positive rate inflation across a microarray dataset. It is possible to use the Wilcoxon test statistic as the single filtering mechanism however calculation of the false positive rate is challenging (48). 

The biological activity satisfies the second and third requirements, since each measurement is done independently (see, however, below). Upon repeated measurements the values will be normally distributed, and will have an equal variance. The independent variables in the Hansch approach are obtained from experimental measurements and contain therefore an experimental error. They cannot be considered as fixed variates. The experimental error in these parameters, however, is usually much smaller than the error in the dependent variable they can be treated therefore as fixed variates. The complications arising from correlations between observations that include experimental errors have been thoroughly analyzed (224). 

The contributory variance required for model matching should be less than or equal to this expected tolerance, implying the p conditions ... 

The statistical model also requires that the random errors (xi,X2) be independently distributed, with zero mean and equal variance, at all level combinations of the factorial design. This is exactly what we assumed when we combined all observations to obtain a pooled variance estimate. If we wish to perform a or an F-test on our results, we must also assume that the errors are normally distributed (Section 2.6). 

If one fits the data in the form of one of the reciprocal graphs, a linear plot is always obtained. Under no circumstances should one attempt to fit data in reciprocal form for statistical evaluations of kinetic constants, since this requires weighted fits using weights (when equal variance in velocities is assumed) or Uo weights (if the error is proportional). 

The standard requirements for the behavior of the errors are met, that is, the errors associated with the various measurements are random, independent, normally (i. e Gaussian) distributed, and are a random sample from a (hypothetical, perhaps) population of similar errors that have a mean of zero and a variance equal to some finite value of sigma-squared. 

Bartlett test. H0 the variances of the data distributions are equal. Requirements normal distribution of all data sets, independent samples. 

Generalized Covariance Models. When l x) is an intrinsic random function of order k, an alternative to the semi-variogram is the generalized covariance (GC) function of order k. Like the semi-variogram model, the GC model must be a conditionally positive definite function so that the variance of the linear functional of ZU) is greater than or equal to zero. The family of polynomial GC functions satisfy this requirement. The polynomial GC of order k is... 

Equally important is the use of suitable quantitative statistical analyses, including more complicated models, because they can hold certain variables constant, control for artifacts, and provide supplementary information. Whatever statistics are used, they should be explicitly described in sufficient detail so the reader knows exactly what was done and can make a judgment about their appropriateness. For example, there are many different types of analyses of variance (ANOVA), analyses of covariance (ANCOVA), or multivariate analyses of variance (MANOVA), and some are not appropriate to the task at hand. If only the results of an ANOVA with a p < 0.001 are provided, the reader should be justifiably dubious, because this model may not be proper ( p is an estimate of the probability that the results occurred by chance). Sufficient details are required to clarify which model was used, because the p value may be invalid with an inappropriate model. 

Rotatability guarantees equality of variances in estimating response values when moving in any direction from the center of the experiment. All these requirements are met by a 24 full factorial experiment. Considering the requirement for a minimal... 

In contrast to the four tetrahedrally oriented elliptic orbits of the Sommer-feld model, the new theory leads to only three, mutually orthogonal orbitals, at variance with the known structure of methane. A further new theory that developed to overcome this problem is known as the theory of orbital hybridization. In order to simulate the carbon atom s basicity of four an additional orbital is clearly required. The only possible candidate is the 2s orbital, but because it lies at a much lower energy and has no angular momentum to match, it cannot possibly mix with the eigenfunctions on an equal footing. The precise manoeuvre to overcome this dilemma is never fully disclosed and appears to rely on the process of chemical resonance, invented by Pauling to address this, and other, problems. With resonance, it is assumed that, linear combinations of an s and three p eigenfunctions produce a set of hybrid orbitals with the required tetrahedral properties. 

We can go further in improving the quality of the model. It was mentioned that the D-optimal design requires equal number of measurements at -1 and+1 however, we have made no mention about the total number of points. In fact, it is possible to reduce the number of measurement to four or even to two without loss of prediction ability. Because it is always useful to have extra degrees of freedom to avoid overfitting and for estimating residual variance, it is apparent that four measurements, two at -1.0 and two at +1, will give the best solution. 

In statistics, the reproducibility variance is a random variable having a number of degrees of freedom equal to u = N(m — 1). Without the reproducibility variances or any other equivalent variance, we cannot estimate the significance of the regression coefficients. It is important to remember that, for the calculation of this variance, we need to have new statistical data or, more precisely, statistical data not used in the procedures of the identification of the coefficients. This requirement explains the division of the statistical data of Fig. 5.3 into two parts one sigmficant part for the identification of the coefficients and one small part for the reproducibility variance calculation. 

Table 5.39 also contains the indications and calculations required to verify the zero hypothesis. This hypothesis considers the equality of the variance containing the effect of the factor on the process response (Sj) with the variance that shows the experimental reproducibility (s ). 

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