we assume a mixed-effect Poisson regression model with random parameters β0 and β1. We further assume that the joint distribution of β0 and β1 is bivariate normal, with mean vector γ0 and covariance matrix Σ. Hence, regarding the complete distributions of yij, our assumptions are as follows:

where β0i and β1i are respectively the random intercept and slope parameters for the ith laboratory and xij is the true count in the jth measurement from the ith laboratory. Of course, we never know the true count (i.e., xij ), but a reasonable substitute is a consensus estimate based on a series of leading laboratories or analysts.

At this stage, we assume that all of the yij and the corresponding xij are known. We estimate the model parameter Σ by the method of marginal maximum likelihood (MML). The resulting estimate of Σ denoted by  is consistent and MML also provides the standard error of . Given MML estimates of the means and covariance matrix, we can obtain empirical Bayes estimates of β0i and β1i, denoted by and .

Let yil be a new observation from the ith laboratory. We do not know the value of the corresponding true observation xil. Our goal is to estimate xil and construct a confidence region for xil using the previous estimates , , and of Σ, β0i , and β1i, respectively. We follow the likelihood-based procedure to estimate xil (i.e., maximize the likelihood function of yil with respect to xil using , , and ). Denote this estimate of xil by . The expression of is as follows:

(1)

This estimate is valid provided yil > 0. In the case of yij = 0 we must set xil = 0. Also note that the above estimate of xil becomes nega-



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