Bayesian inference in measurement error models from objective priors for the bivariate normal distribution

Abstract

In regression analysis, when the covariates are not exactly observed, measurement error models extend the usual regression models toward a more realistic representation of the covariates. It is common in the literature to directly propose prior distributions for the parameters in normal measurement error models. Posterior inference requires Markov chain Monte Carlo (MCMC) computations. However, the regression model can be seen as a reparameterization of the bivariate normal distribution. In this paper, general results for objective Bayesian inference under the bivariate normal distribution were adapted to the regression framework. So, posterior inferences for the structural parameters of a measurement error model under a great variety of priors were obtained in a simple way. The methodology is illustrated by using five common prior distributions showing good performance for all prior distributions considered. MCMC methods are not necessary at all. Model assessment is also discussed. Results from a simulation study and applications to real data sets are reported.

Appendices

Appendix 1: Sampling under the right-Haar prior

For the right-Haar prior in Section 3, samples from the posterior distribution of the parameters in (3) are drawn by cycling the steps 1–6 below:

• (1) Draw independent $Z_1, Z_2, Z_3 \sim N(0,1)$, $C_1 \sim \chi ^2_{n-1}$ and $C_2 \sim \chi ^2_{n-2}$,

• (2) Compute $\sigma _1 = (s_{11} / C_1)^{1/2}$,

• (3) Compute

  $$\begin{aligned} \sigma _2 = \big [s_{22} (1 - r^2)\big ]^{1/2} \left[ \frac{1}{C_2} + \frac{1}{C_1} \bigg ( \frac{Z_3}{C_2^{1/2}} - \frac{r}{(1-r^2)^{1/2}} \bigg )^2\right] ^{1/2}, \end{aligned}$$

• (4) Compute $\rho = \psi (W)$ and

  $$\begin{aligned} W = -\frac{Z_3}{C_1^{1/2}} + \frac{C_2^{1/2}}{C_1^{1/2}} \frac{r}{\big (1-r^2\big )^{1/2}}, \quad \text {where} \ \psi (v) = \frac{v}{\big (1+v^2\big )^{1/2}}, \end{aligned}$$

• (5) Compute $\mu _1 = \bar{x} + Z_1 s_{11}^{1/2} / (n C_1)^{1/2}$ and

• (6) Compute

  $$\begin{aligned} \mu _2 = \bar{y} + \frac{Z_1 r s_{22}^{1/2}}{(n C_1)^{1/2}} + \bigg ( \frac{Z_2 }{C_2^{1/2}} - \frac{Z_1 Z_3}{(C_1 C_2)^{1/2}} \bigg ) \bigg (\frac{s_{22} \big (1-r^2\big )}{n} \bigg )^{1/2}. \end{aligned}$$

Appendix 2: Tables

In this appendix, we present tables summarizing some results from our simulation study. ‘Median’, ‘Length’ and ‘CP’ stand for the average of the posterior medians, the average of the length of the 95% HPD intervals and the coverage probability of the 95% HPD intervals, respectively (Tables 3, 4, 5, 6, 7 and 8).

Table 3 Case 1

Table 4 Case 2

Table 5 Case 3

Table 6 Case 4

Table 7 Case 5

Table 8 Case 6