Statistics covariance

The neurobehavioral test battery used in the 66-month Seychelles study was designed to assess multiple developmental domains (Davidson et al. 1998). The tests were considered to be sufficiently sensitive and accurate to detect neurotoxicity in the presence of a number of statistical covariates. On-site test administration reliability was assessed by an independent scorer, and mean interclass correlations for interscorer reliability were 0.96-0.97 (Davidson et al. 1998). The sample size was determined to be sufficient to detect a 5.7-point difference on any test with a mean (SD) of 100 (16) between low (0-3 ppm) and high (>12 ppm) hair mercury concentration groups for a 2-sided test (A = 0.05 at 80% power). 

In this section the heat and mass transport coefficients for turbulent boundary layers are examined. In this case the model derivation is based on the governing Reynolds averaged equations. In these equations statistical covariances appear which involve fluctuating velocities, temperatures and concentrations. The nature of these terms is not known a priori and their effects must by estimated by semi-empirical turbulence modeling. The resulting parameterizations allow us to express the unknown turbulent fluctuations in terms of the mean flow field variables. It is emphasized that the Reynolds equations are not actually solved, merely semi-empirical relations are derived for the wall fluxes through the inner boundary layer. 

Pearson s correlation coefficient, Spearman s rho, Kendall s tau-b, univariate statistics, covariances and cross-products, outlier screening prior to analysis... 

The primary purpose for expressing experimental data through model equations is to obtain a representation that can be used confidently for systematic interpolations and extrapolations, especially to multicomponent systems. The confidence placed in the calculations depends on the confidence placed in the data and in the model. Therefore, the method of parameter estimation should also provide measures of reliability for the calculated results. This reliability depends on the uncertainties in the parameters, which, with the statistical method of data reduction used here, are estimated from the parameter variance-covariance matrix. This matrix is obtained as a last step in the iterative calculation of the parameters. 

Rollins, D.K. and J.F. Davis, Gross Error Detection when Variance-Covariance Matrices are Unknown, AlChE Journal, 39(8), 1993, 13.35-1341. (Unknown statistics)... 

There are instances where it is important to know if a given regression line is linear. For example, simple competitive antagonism should yield a linear Schild regression (see Chapter 6). A statistical method used to assess whether or not a regression is linear utilizes analysis of covariance. A prerequisite to this approach is that there... 

Finally, an infinite set of random vectors is defined to be statistically independent if all finite subfamilies are statistically independent. Given an infinite family of identically distributed, statistically independent random vectors having finite means and covariances, we define their normalized sum to be the vector sfn, , sj where... 

The fact that linear independence is a necessary condition for statistical independence is obvious. The sufficiency of the condition can be established by noting that the covariance matrix... 

Polynomial regression with indicator variables is another recommended statistical method for analysis of fish-mercury data. This procedure, described by Tremblay et al. (1998), allows rigorous statistical comparison of mercury-to-length relations among years and is considered superior to simple hnear regression and analysis of covariance for analysis of data on mercury-length relations in fish. 

Therefore, on statistical grounds, if the error terms (e,) are normally distributed with zero mean and with a known covariance matrix, then Q( should be the inverse of this covariance matrix, i.e.,... 

The above expressions for the CO l (k ) and of are valid, if the statistically correct choice of the weighting matrix Q, (i=1,...,N) is used in the formulation of the problem. Namely, if the errors in the response variables (e, i=l,...,N) are normally distributed with zero mean and covariance matrix,... 

Let us now consider models that have only more than one measured variable (w>l). The previously described model adequacy tests have multivariate extensions that can be found in several advanced statistics textbooks. For example, the book Introduction to Applied Multivariate Statistics by Srivastava and Carter (1983) presents several tests on covariance matrices. 

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. 

Note that with PROC PHREG all covariates need to be numeric, so treatment and gender need to be numeric. The p-values and hazard ratios that are useful for your statistical tables can be found in the ProbChiSq and HazardRatio variables, respectively, in the pvalue data set. 

However, there is a mathematical method for selecting those variables that best distinguish between formulations—those variables that change most drastically from one formulation to another and that should be the criteria on which one selects constraints. A multivariate statistical technique called principal component analysis (PCA) can effectively be used to answer these questions. PCA utilizes a variance-covariance matrix for the responses involved to determine their interrelationships. It has been applied successfully to this same tablet system by Bohidar et al. [18]. 

Just as in everyday life, in statistics a relation is a pair-wise interaction. Suppose we have two random variables, ga and gb (e.g., one can think of an axial S = 1/2 system with gN and g ). The g-value is a random variable and a function of two other random variables g = f(ga, gb). Each random variable is distributed according to its own, say, gaussian distribution with a mean and a standard deviation, for ga, for example, (g,) and oa. The standard deviation is a measure of how much a random variable can deviate from its mean, either in a positive or negative direction. The standard deviation itself is a positive number as it is defined as the square root of the variance ol. The extent to which two random variables are related, that is, how much their individual variation is intertwined, is then expressed in their covariance Cab ... 

All current taxometric procedures are based on a single statistical method termed Coherent Cut Kinetics (CCK). We decipher the meaning of this term in the next section in the example of the MAXCOV-HITMAX (MAXCOV stands for MAXimal COVariance the reason for this name will become clear in the next section) technique. However, we emphasize that it is not the shared statistical method that defines taxometrics. Adherence to a particular set of epistemological principles distinguishes taxometrics from other approaches. In other words, any analytic procedure that can identify taxa may... 

Zero-centered data means that each sensor is shifted across the zero value, so that the mean of the responses is zero. Zero-centered scaling may be important when the assumption of a known statistical distribution of the data is used. For instance, in case of a normal distribution, zero-centered data are completely described only by the covariance matrix. 

Statistical properties of a data set can be preserved only if the statistical distribution of the data is assumed. PCA assumes the multivariate data are described by a Gaussian distribution, and then PCA is calculated considering only the second moment of the probability distribution of the data (covariance matrix). Indeed, for normally distributed data the covariance matrix (XTX) completely describes the data, once they are zero-centered. From a geometric point of view, any covariance matrix, since it is a symmetric matrix, is associated with a hyper-ellipsoid in N dimensional space. PCA corresponds to a coordinate rotation from the natural sensor space axis to a novel axis basis formed by the principal... 

We assume a knowledge of the possible state covariances P generated by the tracking system. This knowledge is statistical and is represented by a probability distribution F(P) over the space of all positive definite matrices. 

The vector nk describes the unknown additive measurement noise, which is assumed in accordance with Kalman filter theory to be a Gaussian random variable with zero mean and covariance matrix R. Instead of the additive noise term nj( in equation (20), the errors of the different measurement values are assumed to be statistically independent and identically Gaussian distributed, so... 

Most techniques for process data reconciliation start with the assumption that the measurement errors are random variables obeying a known statistical distribution, and that the covariance matrix of measurement errors is given. In Chapter 10 direct and indirect approaches for estimating the variances of measurement errors are discussed, as well as a robust strategy for dealing with the presence of outliers in the data set. 

In the previous development it was assumed that only random, normally distributed measurement errors, with zero mean and known covariance, are present in the data. In practice, process data may also contain other types of errors, which are caused by nonrandom events. For instance, instruments may not be adequately compensated, measuring devices may malfunction, or process leaks may be present. These biases are usually referred as gross errors. The presence of gross errors invalidates the statistical basis of data reconciliation procedures. It is also impossible, for example, to prepare an adequate process model on the basis of erroneous measurements or to assess production accounting correctly. In order to avoid these shortcomings we need to check for the presence of gross systematic errors in the measurement data. 

Summarizing, the statistical characterisation of the random process (mean and covariance) can be projected through the interval tk < t < tk+1, and in this process there is an input noise that will increase the error, damaging the quality of the estimate. 

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