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 y_ij, our assumptions are as follows:

where β_0i and β_1i are respectively the random intercept and slope parameters for the ith laboratory and x_ij is the true count in the jth measurement from the ith laboratory. Of course, we never know the true count (i.e., x_ij ), 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 y_ij and the corresponding x_ij 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 β_0i and β_1i, denoted by and .

Let y_il be a new observation from the ith laboratory. We do not know the value of the corresponding true observation x_il. Our goal is to estimate x_il and construct a confidence region for x_il using the previous estimates , , and of Σ, β_0i , and β_1i, respectively. We follow the likelihood-based procedure to estimate x_il (i.e., maximize the likelihood function of y_il with respect to x_il using , , and ). Denote this estimate of x_il by . The expression of is as follows:

(1)

This estimate is valid provided y_il > 0. In the case of y_ij = 0 we must set x_il = 0. Also note that the above estimate of x_il becomes nega-