Descriptive statistics covariance

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Each measure of an analysed variable, or variate, may be considered independent. By summing elements of each column vector the mean and standard deviation for each variate can be calculated (Table 7). Although these operations reduce the size of the data set to a smaller set of descriptive statistics, much relevant information can be lost. When performing any multivariate data analysis it is important that the variates are not considered in isolation but are combined to provide as complete a description of the total system as possible. Interaction between variables can be as important as the individual mean values and the distributions of the individual variates. Variables which exhibit no interaction are said to be statistically independent, as a change in the value in one variable cannot be predicted by a change in another measured variable. In many cases in analytical science the variates are not statistically independent, and some measure of their interaction is required in order to interpret the data and characterize the samples. The degree or extent of this interaction between variables can be estimated by calculating their covariances, the subject of the next section. 

The simplest use of statistical methods is to provide summary parameters characterising important statistical properties of input variables and of various measures of catalyst performance (such as yield or degree of conversion), or relationships between them. Such summary parameters are usually called descriptive statistics, their common representatives are mean, median, variance, standard deviation, covariance and correlation. 

It should be emphasized that for Markovian copolymers a knowledge of the values of structural parameters of such a kind will suffice to find the probability of any sequence Uk, i.e. for an exhaustive description of the microstructure of the chains of these copolymers with a given average composition. As for the composition distribution of Markovian copolymers, this obeys for any fraction of Z-mers the Gaussian formula whose covariance matrix elements are Dap/l where Dap depend solely on the values of structural parameters [2]. The calculation of their dependence on time, and the stoichiometric and kinetic parameters of the reaction system permits a complete statistical description of the chemical structure of Markovian copolymers to be accomplished. The above reasoning reveals to which extent the mathematical modeling of the processes of the copolymer synthesis is easier to perform provided the alternation of units in macromolecules is known to obey Markovian statistics. 

The variables XlfX2 are called uncorrelated when it is merely known that their covariance is zero, which is weaker than statistical independence. The reason why this property has a special name is that in many applications the first and second moments alone provide an adequate description. 

The statistical model is incorporated as described in Section 17.4. The covariate model allows a description of relationships between covariates and model parameters, explaining parts of the intersubject variability and identifying sub-populations at risk for concentrations below or above the therapeutic range. 

It should be emphasized that for the Markovian copolymers, the knowledge of these structure parameters will suffice for finding the probabilities of any sequences LZ, i.e., for a comprehensive description of the structure of the chains of such copolymers at their given average composition. As for the CD of the Markovian copolymers, for any fraction of Z-mers it is described at Z 1 by the normal Gaussian distribution with covariance matrix, which is controlled along with Z only by the values of structure parameters (Lowry, 1970). The calculation of their dependence on time and on the kinetic parameters of a reaction system enables a complete statistical description of the chemical structure of a Markovian copolymer. It is obvious therewith to which extent a mathematical modeling of the processes of the synthesis of linear copolymers becomes simpler when the sequence of units in their macromolecules is known to obey Markov statistics. 

Pot a description of the method of covariance see R. A. Fisher Statistical Methods for Research Workers, Section 49.1 (Oliver and Boyd). K. Mather Statistical Analysis in Biology, Section 34 (Methuen). G. W. Snedecor Statistical Methods, Chapter 12 (Iowa State College Press). 

Details of statistical analyses for potential toxicities that should be explicitly considered for all products and AEs of special interest Aiialyses for these events will in general be more comprehensive than for standard safety parameters. These analyses may include subject-year adjusted rates, Cox proportional hazards analysis of time to first event, and Kaplan-Meier curves. Detailed descriptions of the models would typically be provided. For example, if Cox proportional hazards analysis is specified, a detailed description of the model(s) that will be used should be provided. This would generally include study as a stratification factor, covariates, and model selection techniques. More advanced methods, such as multiple events models or competing risk analyses, should be described if used (as appropriate). It is recommended that graphical methods also be employed, for example, forest plot and risk-over-time plot (Xia et al., 2011). 

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