Empirical Bayes estimates

As mentioned elsewhere in the chapter, EBEs are conditional estimates of an individual s model parameters given a model and the population parameter estimates. EBEs are useful for a variety of reasons, including  

A natural question that may arise is the quality of the estimate—how close do the EBEs compare to the true model parameters for a subject. That question is data-specific, but some general conclusions can be made which will be shown through example. 

Concentration time data were simulated from a 2-compartment model having a mean systemic clearance of 45 L/h, intercompartmental clearance of 15 L/h, central volume of 125 L, and peripheral volume of 125 L. 

FOCE-I, 250 subjects, 8 samples/subject FOCE-I, 75 subjects, 8 samples/subject 

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Parametric population methods also obtain estimates of the standard error of the coefficients, providing consistent significance tests for all proposed models. A hierarchy of successive joint runs, improving an objective criterion, leads to a final covariate model for the pharmacokinetic parameters. The latter step reduces the unexplained interindividual randomness in the parameters, achieving an extension of the deterministic component of the pharmacokinetic model at the expense of the random effects. Recently used individual empirical Bayes estimations exhibit more success in targeting a specific individual concentration after the same dose. 

The NONMEM program implements two alternative estimation methods, the first-order conditional estimation and the Laplacian methods. The first-order conditional estimation (FOCE) method uses a first-order expansion about conditional estimates (empirical Bayes estimates) of interindividual random effects, rather than about zero. In this respect, it is like the conditional first-order method of Lindstrom and Bates.f Unlike the latter, which is iterative, a single objective function is minimized, achieving a similar effect as with iteration. The Laplacian method uses second-order expansions about the conditional estimates of the random effects. ... 

Mixed models and empirical Bayes estimates of exposure... 

Friesen, M. C., Macnab, Y. C., Marion, S. A., Demers, P. A., Davies, H. W., and Teschke, K. (2006). Mixed models and empirical Bayes estimation for retrospective exposure assessment of dust exposures in Canadian sawmills. Ann Occup Hyg 50, 281-288. 

ESTIMATION OF THE RANDOM EFFECTS AND EMPIRICAL BAYES ESTIMATES (EBEs)... 

The linear mixed effect model assumes that the random effects are normally distributed and that the residuals are normally distributed. Butler and Louis (1992) showed that estimation of the fixed effects and covariance parameters, as well as residual variance terms, were very robust to deviations from normality. However, the standard errors of the estimates can be affected by deviations from normality, as much as five times too large or three times too small (Verbeke and Lesaffre, 1997). In contrast to the estimation of the mean model, the estimation of the random effects (and hence, variance components) are very sensitive to the normality assumption. Verbeke and Lesaffre (1996) studied the effects of deviation from normality on the empirical Bayes estimates of the random effects. Using computer simulation they simulated 1000 subjects with five measurements per subject, where each subject had a random intercept coming from a 50 50 mixture of normal distributions, which may arise if two subpopulations were examined each having equal variability and size. By assuming a unimodal normal distribution of the random effects, a histogram of the empirical Bayes estimates revealed a unimodal distribution, not a bimodal distribution as would be expected. They showed that the correct distributional shape of the random effects may not be observed if the error variability is large compared to the between-subject variability. 

Figure 9.8 Scatter plot of creatinine clearance against the empirical Bayes estimate for tobramycin systemic clearance under the 2-compartment model with reduced unstructured covariance matrix. Solid line is the LOESS smoother with 0.3 sampling proportion.

A third sort of measure is the empirical Bayes estimate. Here we assume that each Nij is drawn from Poisson distribution with mean i ij. We are really interested in... 

Examine distribution of empirical Bayes estimates and determine extremes/outliers. Determine if lOV or mixture model improves results. 

Davis CE, Leffingwell DP. Empirical Bayes estimates of subgroup effects in clinical trials. Controlled Clinical Trials, 11 37-42,1990. 

Liang, T. 2004. Empirical Bayes estimation with random right censoring. International J. Information and Management Sciences 15(4) 1-12. 

Susarla, V and Van Ryzin, J. 1978. Empirical Bayes estimation of a distribution (survival) function from right censored observations.Tnn. Statist. 6 740-754. 

The empirical Bayes estimate of the trigger rate is given by the expectation of the posterior distribution, which under the Gamma-Pois son model is Gamma (a -I- i, p + ti). The expectation of this distribution is ... 

Elvik, R. (2008). The predictive validity of empirical Bayes estimates of road safety. Accident Analysis and Prevention 40, 1964-1969. 

Sarhan, A.M. (2003). Empirical Bayes estimates in exponential reliability model. Applied Mathematics and Computation 135,319-332. 

The empirical Bayes estimator for a given asset of the expected number of failure events at time (1 + x) given the observations for that asset up to time t, is the mean of the posterior distribution of the scale parameter for the asset estimated using all observatiims for all assets up to time t multiplied by [t + xy, which reflects the non-linear degradation rate. Let n t) denote the vector representing the event history up to time t, where the ith element is the history of the fth asset denoted by , (r,), then the empirical Bayes estimate is given by... 

For the situation where we have the lowest level of the intensity at 0.0008 failures per annum, then we inevitably simulate many asset sample data sets with zero observed failures in a given year. For example, in some years we find that none of the assets experienced a failure event, but in other years only a single asset experienced a failure. Due to the properties of empirical Bayes estimation we find that in such circumstances an average weighted towards the pooled mean is allocated for zero event assets. In years when there is a failure event, even if only for one out of say the 20 assets in the pool, then this tends to increase the pool mean and hence the year ahead predicted number of failures. Equally in years where there are no observed failure events then the pooled mean decreases due to the additional exposure accumulated. It is also clear that the predicted number of failures for those particular assets which experience events are increased above the pool mean to account for their local history. 

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