Shrinkage Calculation

The Omega output worksheet contains h shrinkage data. It is based on the standard deviation definition:

where SD(h_j) is the empirical standard deviation of the j^th h over all Nsub subjects, and w_j,j is the estimate of the population variance of the j^th random effect, j = 1,2,…, NEta.

For all population engines other than QRPEM, the numerical h_j value used in the shrinkage computation is the mode (maximum) of the empirical Bayesian posterior distribution of the random effects h_j, evaluated at the final parameter estimates of the fixed and random effects. For QRPEM, the h_j value is the mean of the empirical Bayesian distribution.

Note that another common way to define the h-shrinkage is through the variance, and is given by

where Var(h_j) is the empirical variance of the j^th h over all Nsub subjects. By the two previous equations, one can see that standard deviation-based h-shrinkage can be computed from variance-based h-shrinkage, and vice versa. For example, if one has Shrinkage_Var_j, then Shrinkage_SD_j can be calculated as:

The Eta output worksheet contains the individual shrinkage, which is calculated as follows:

where i = 1, 2, …, Nsub and j = 1, 2, …, NEta. Here, h_SE_i,j denotes the standard error of the j^th individual parameter estimator for the i^th subject. For all population engines other than QRPEM, h_SE_i,j is calculated as the square root of the (j, j)^th element of the inverse of the negative Hessian (second derivative matrix) of the empirical Bayesian posterior distribution for the i^th subject. While, for the QRPEM engine, they are computed via the importance sampling of the empirical Bayesian posterior distribution.

The formula used to calculate Shrink_Sub_Var (the previous equation) is extended from the 1-1 relation between the population shrinkage and standard error of individual parameter estimator conjectured in Xu, et al, AAPS J., (2012) pp. 927-936. This relationship can be intuitively observed from the following important theoretical relationship obtained for the EM algorithm.

From which, one can see that the commonly used variance-based population shrinkage becomes

Note that population shrinkage calculated using the above formula is also reported in bluptable.dat (after the individual shrinkages for each h) and is denoted as shrinkageebd_var (see the highlighted text in yellow in the image below). From these results, we can see that the value of shrinkageebd_var is similar to the one for Shrinkage_Var.

In bluptable.dat, Shrink_Sub_SD is calculated by using the following formula

This relationship between Shrink_Sub_SD and Shrink_Sub_Var is an analog of the relationship shown in the earlier Shrinkage_SD equation.

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