Generalized Sample Variance

The determinant of S is often called the generalized sample variance. It is proportional to the square of the volume generated by the p deviation vectors, x - x.  

The generalized sample variance describes the scatter in the multivariate distribution. A large volume indicates a large generalized variance and a large amount of scatter in the multivariate distribution. A small volume indicates a small generalized 

The total sample variance is the sum of the diagonal elements of the sample variance-covariance matrix, S. Total variance =. S 2,, +. S 2 +...+ s2pp. Geometrically, the total sample variance is the sum of the squared lengths of the p deviation vectors, xx. 

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The criterion of mean-unbiasedness seems to be occasionally overemphasized. For example, the bias of an MLE may be mentioned in such a way as to suggest that it is an important drawback, without mention of other statistical performance criteria. Particularly for small samples, precision may be a more important consideration than bias, for purposes of an estimate that is likely to be close to the true value. It can happen that an attempt to correct bias results in lowered precision. An insistence that all estimators be UB would conflict with another valuable criterion, namely parameter invariance (Casella and Berger 1990). Consider the estimation of variance. As remarked in Sokal and Rohlf (1995), the familiar sample variance (usually denoted i ) is UB for the population variance (a ). However, the sample standard deviation (s = l is not UB for the corresponding parameter o. That unbiasedness cannot be eliminated for all transformations of a parameter simply results from the fact that the mean of a nonlinearly transformed variable does not generally equal the result of applying the transformation to the mean of the original variable. It seems that it would rarely be reasonable to argue that bias is important in one scale, and unimportant in any other scale. 

The sample of individuals is assumed to represent the patient population at large, sharing the same pathophysiological and pharmacokinetic-dynamic parameter distributions. The individual parameter 0 is assumed to arise from some multivariate probability distribution 0 / (T), where jk is the vector of so-called hyperparameters or population characteristics. In the mixed-effects formulation, the collection of jk is composed of population typical values (generally the mean vector) and of population variability values (generally the variance-covariance matrix). Mean and variance characterize the location and dispersion of the probability distribution of 0 in statistical terms. 

If a finite number n of observations is made with a sample mean x, there are n individual deviations x, — x. The sum of the n deviations, however, is zero, and only n — 1 of the deviations are necessary to define the nth. Hence, only n — 1 independently variable deviations or degrees of freedom are left. We can regard the sample variance as the average of the square of the independently variable deviations. For small numbers of observations we want s to approximate a as closely as possible, a condition that would be observed with an infinite number of observations. The sample mean x in general will not coincide with the population mean g. It can be shown that the use of n — 1 as a divisor, when averaged over all values of n, just compensates for the fact that the sample mean and the population mean are not identical. Because of the relative improbability of drawing large deviations in a small sample, the sample variance would otherwise underestimate the population variance. 

Additionally, many sources of variability, such as model misspecification, or dosing and sampling history, may lead to residual errors that are time dependent. For example, the residual variance may be larger in the absorption phase than in the elimination phase of a drug. Hence, it may be necessary to include time in the residual variance model. One can use a more general residual variance model where time is explicitly taken into account or one can use a threshold model where one residual variance model accounts for the residual variability up to time t, but another model applies thereafter. Such models have been shown to result in significant model improvements (Karlsson, Beal, and Sheiner, 1995). 

To continue, the variance (tr ) of the error term (written as for a population estimate or for the sample variance) needs to be estimated. As a general... 

In analytical chemistry, generally, we deal with a relatively limited number of replicate determinations and, therefore, use the sample variance and standard deviation. Spreadsheet programs have convenient (functions to calculate the most common statistical functions both Lotus 123 and QPro permit the calculation of V ( VAR) and s ( STD), but only the latter allow direct calculation of V(( VARS) and s( STDS). While s is larger than s, the difference depends on the number of values in the set increases. At what value of n does this difference become 10%, 5%, 1% (Show that the n values are 6, 11, and 51, respectively.) Obviously, as n increases, the differences in s and s (as do V and V) become negligible. 

To illustrate the general procedure, suppose we wish a 95% confidence interval for the mean of a random variable. Suppose the sample size is n. As described earlier, the sample mean x and the sample variance are unbiased estimators for the mean and variance of the underlying distribution. For a symmetric confidence interval the requirement is... 

The probabilistic nature of a confidence interval provides an opportunity to ask and answer questions comparing a sample s mean or variance to either the accepted values for its population or similar values obtained for other samples. For example, confidence intervals can be used to answer questions such as Does a newly developed method for the analysis of cholesterol in blood give results that are significantly different from those obtained when using a standard method or Is there a significant variation in the chemical composition of rainwater collected at different sites downwind from a coalburning utility plant In this section we introduce a general approach to the statistical analysis of data. Specific statistical methods of analysis are covered in Section 4F. 

In effect, the standard deviation quantifies the relative magnitude of the deviation numbers, i.e., a special type of average of the distance of points from their center. In statistical theory, it turns out that the corresponding variance quantities s have remarkable properties which make possible broad generalities for sample statistics and therefore also their counterparts, the standard deviations. 

F Distribution In reference to the tensile-strength table, the successive pairs of daily standard deviations could be ratioed and squared. These ratios of variance would represent a sample from a distribution called the F distribution or F ratio. In general, the F ratio is defined by the identity... 

The procedure can be readily extended to multicomponent systems by applying the test to each component in turn. In real systems, it is generally convenient to take samples of fixed volume or mass rather than fixed number of particles. In such cases, the expected variance can be computed using (see Refs. 19 and 20)... 

Equation (1) can be used in a general way to determine the variance resulting from the different dispersion processes that occur in an LC column. However, although the application of equation (1) to physical chemical processes may be simple, there is often a problem in identifying the average step and, sometimes, the total number of steps associated with the particular process being considered. To illustrate the use of the Random Walk model, equation (1) will be first applied to the problem of radial dispersion that occurs when a sample is placed on a packed LC column in the manner of Horne et al. [3]. 

The general principles of testing chemical homogeneity of solids are given e.g. by Malissa [1973], Cochran [1977], and Danzer et al. [1979]. The terms of variation o20tal and o2nal can be separated by analysis of variance (Sect. 5.1.1). According to Danzer and Kuchler [1977] there exists an exponential dependence between the total variance and the reciprocal sample mass... 

A statistical study of the conversion with tetralin of 68 coals (60) must now be regarded as superseded by a later, more comprehensive paper (61), but it did show very clearly that bivariate plots are of little value in interrelating liquefaction behavior with coal properties at least two or three coal properties must be taken into account in seeking to explain the variance of liquefaction behavior, and some of these properties are not related to the rank of the coal. The paper implies strongly that any interrelationships of coal characteristics must necessarily be multivariate. Hence in any study of coal a large sample and data base is essential if worthwhile generalizations are to be made. 

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