Estimate covariance

Accordingly, we have for the estimate of the variables, the measurement errors, and the error estimate covariance... 

Sg covariance matrix of the error estimates covariance matrix of variable estimates X Lagrangian multipliers... 

Now, since the purpose of an experimental program is to gain information, in attempting to design the experiment we will try to plan the measurements in such a way that the final error estimate covariance is minimal. In our case, this can be achieved in a sequential manner by applying the matrix inversion lemma to Eq. (9.17), as we have shown in previous chapters. 

Number of variables is larger than the number of objects 2.3.2 Estimating Covariance and Correlation... 

Each of the upper left to lower right diagonal elements of V is an estimated variance of a parameter estimate, si, these elements correspond to the parameters as they appear in the model from left to right. Each of the off-diagonal elements is an estimated covariance between two of the parameter estimates [Dunn and Clark (1987)]. 

Let sIq be the estimated variance associated with the parameter estimate bo let be the estimated variance associated with b and let slf (or sl ) represent the estimated covariance between and b,. Then... 

Figure 7.4 Value of the element of (X X) for the estimated covariance between b and />, as a function of the location of a second experiment, one experiment fixed at x, = 1.

Figure 8.12 Value of the element of (X X) for the estimated covariance between b and...

The least squares regression results arc Q 20.7691 -. 840583. The estimated covariance matrix... 

The estimated covariance matrix for the least squares estimates is... 

The covariance between two variables is a method for determining how closely they follow similar trends. It will never exceed in magnitude the geometric mean of the variance of the two variables the lower is the value, the less close are the trends. Both variables must be measured for an identical number of samples, I in this case. The sample or estimated covariance between variables x and y is defined by... 

Set X = 0 and compute the estimated covariance matrix C = a . This gives the standard errors in the fitted parameters a. 

The advantages of the one-stage analysis are that interindividual variability of the parameters can be estimated, random residual error can be estimated, covariates can be included in the model, parameters for individuals can be estimated, and pharmacokinetic-pharmacodynamic models can be... 

Keywords Data Reconciliation, State Estimation, Covariance Estimation. 

CEM) filter. The output of the linear matched filter for a test input r, estimated covariance matrix is given by... 

A potential limit at this point in development is that many covariates are stiU being identified and explored, so the true effect is not yet known. Consequently, the precision of the estimated covariate effect may be relatively low. Although it would be advisable to limit the number of covariates included at this stage to those of direct clinical relevance, it is recommended that a full model approach (16) be employed. The full model would include all covariates of interest with associated mean estimates and precision (e.g., confidence interval (Cl) calculated using asymptotic standard errors or bootstrap replication procedures). Collinear covariates should be used with caution as they may affect the precision of the estimates (17). 

The results are presented in Table 6.8. Again, different between-subject covariance structured resulted in different results. The AIC and AICc selected the unstructured covariance (which has seven estimable covariance parameters) as the best model, whereas the BIC selected the simple covariance (which has four estimable covariance parameters) as the best model. 

Legend , model number OFV, NONMEM objective function value LRT, likelihood ratio test SBC, Schwarz s Bayesian criterion based on NONMEM OFV (larger is better) LRT, likelihood ratio test based on Wald s approximation SBC, Schwarz s Bayesian criterion based on Wald s approximation (larger is better) p, total number of estimable parameter values plus all estimable covariance terms in reference model q, number of estimable parameter values and estimable covariance terms in reduced model. SBC and SBC ranking based on the largest value. 

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