Multivariate models, random variables distributions

The unconditional model treats the sum of all tumors as a random variable. Then the exact unconditional null distribution is a multivariate binomial distribution. The distribution depends on the unknown probability. 

For the determination of the residual variability model it is assumed that s/j is a zero mean random variable with variance a2, that is multivariately symmetrically distributed. 

In most models developed for pharmacokinetic and pharmacodynamic data it is not possible to obtain a closed form solution of E(yi) and var(y ). The simplest algorithm available in NONMEM, the first-order estimation method (FO), overcomes this by providing an approximate solution through a first-order Taylor series expansion with respect to the random variables r i,Kiq, and Sij, where it is assumed that these random effect parameters are independently multivariately normally distributed with mean zero. During an iterative process the best estimates for the fixed and random effects are estimated. The individual parameters (conditional estimates) are calculated a posteriori based on the fixed effects, the random effects, and the individual observations using the maximum a posteriori Bayesian estimation method implemented as the post hoc option in NONMEM [10]. 

In this section we describe the six discrete probability distributions and five continuous probability distributions that occur most frequently in bioinformatics and computational biology. These are called univariate models. In the last three sections, we discuss probability models that involve more than one random variable called multivariate models. 

In our study we avoid this high dimensional optimization problem by applying the Nataf model (Nataf 1962), (Liu and Der Kiureghian 1986) to construct multivariate distributions. In this model a vector of standard normally distributed random variables... 

By applying the presented Nataf model the multivariate distribution function is obtained by solving the optimization problem with four parameters for each random variable independently. The successful application of the model requires a positive definite covari-ance matrix Czz and continuous and strictly increasing distribution functions Fxtixi). In our smdy Equation 21 is solved iteratively to obtain Py for each pair of marginal distributions from the known correlation coefficient pij. 

In CART models, no assumptions are necessary regarding the distribution of the input variables as made in many other multivariate methods. Another advantage is the treatment of missing values. In Section 5.1, we learned about column means or random numbers to deal with missing values. CART provides more sophisticated methods for this purpose, for example, by... 

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