Simulation_with_parameter_uncertainty

Simulation with parameter uncertainty included

Incorporation of parameter uncertainty into simulation gives a more realistic representation of the system being studied. The vcvfixef statement can be used to do such simulation with uncertainty in parameter estimators for fixed effects and sigma (standard deviations of residual error variables) included. Specifically, the values of fixed effects and sigmas are sampled from a multivariate normal distribution once per replicate with mean being their initial estimates (i.e., provided through their associated fixef and error statements (see “Fixed effects” and “Observe statement for Gaussian Residual models”, respectively)) and variance-covariance matrix being specified inside of the vcvfixef statement.

The syntax of the vcvfixef statement is as follows. Select the link for a full description.


        vcvfixef(MatrixForm(ThetasSigmasList) = c(MatrixValues))

There can be more than one such statement in a model. The following example demonstrates the usage of vcvfixef statement to sample tvV, tvCl, and standard deviation of CEps from a multivariate normal distribution with mean being:

and variance-covariance matrix being:

        
        ##=========================================================
        ## Residual error model 
        ##=========================================================
        error(CEps = 0.12) 
        observe(CObs = C * (1 + CEps))
        
        ##=========================================================
        ## Model parameters
        ##=========================================================
        stparm(V = tvV * exp(nV))
        stparm(Cl = tvCl * exp(nCl))
        
        ## Fixed effects 
        fixef(tvV = c(,4.5,)) 
        fixef(tvCl = c(,1.2,)) 
        
        ## Random effects 
        ranef(diag(nV, nCl) = c(0.1, 0.1)) 
        
        ## Covariance matrix of parameter estimator for 
        ## Theta and Sigma
        vcvfixef(block(tvV, tvCl, CEps) = 
            c(0.01, 0.003, 0.02, -6.1E-05, -2.1E-05, 8.2E-06))

For the case where the associated fixef statement includes bounds, if the simulated value is not in the bound, then it will sample again. If this fails 1000 times, then it just gives up with an error outputted. The same rule applies to sigma whose lower bound is automatically set to 0.

The vcvfixef statement is only applicable for simulation tables, and it will be ignored in estimation modes.