Effect of Correlated Outcome on Estimates in Bayesian Hierarchical Poisson Regression Models

Effect of Correlated Outcome on Estimates in Bayesian Hierarchical Poisson Regression Models

The app demonstrates the effect of correlated outcome on model estimates in Bayesian hierarchical Poisson regression models. The first model (NREM) assumes homogeneity in obesity counts across 117 health regions of Canada. The second model (REM) relaxes the homogeneity assumption, allowing the obesity counts to vary across the health regions. The third model (BYM) assumes that the random variability in the obesity counts can be partitioned into two components: unstructured variability as in the second model and structured variability attributable to the geographical proximity of health regions. This app generates synthetic data from the BYM model using chosen parameter values. The generated data is fitted using the three models. The model performances are then compared using estimated parameter values, observed vs estimated obesity counts, and observed vs predicted obesity counts. The predicted obesity counts are mapped to demonstrate the correlation effect in a spatial context. The user can choose their parameter values and investigate the correlation effects on estimates in real-time. Any comments, questions, or queries can be directed at masud.usask@gmail.com. Average run time ~ 5 minutes.

Masud Rana, PhD (Biostatistics)

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Acknowledgement

The app is designed based on the materials presented in the chapter Modeling Obesity Rate with Spatial Auto-correlation: A Case Study from the book Applied Statistics and Data Science. The detailed formulation of the statistical models can be found there.

Parameter	Description
$$\beta_{0}$$	Intercept
$$\beta_{1}$$	Effect of first covariate
$$\beta_{2}$$	Effect of second covariate
$$\beta_{3}$$	Effect of third covariate
Offset	Population size
$$\theta$$	Unstructured variation in obesity counts attributed to the random variation among the health regions.
$$\phi$$	Structured variation in obesity counts attributed to the geographical proximity of health regions. The app utilizes the geographical (neighborhood) structure of 117 health regions from Canada in simulation.
$$b$$	Combined structured and unstructured variation in obesity counts among the health regions
$$\rho$$	Proportion of variation coming form structured components in combined variation. A value of 0 would indicate a random effects model (REM).
$$\sigma$$	Overall variability of combined variation. A value of 0 would indicate a no random effect model (NREM).

Choose value for parameters

Parameter	Value	Parameter	Value
$$\beta_{0}$$		$$\beta_{1}$$
$$\beta_{2}$$		$$\beta_{3}$$
$$\rho$$		$$\sigma$$
Offset

Statistical Models

No Random Effect Model (NREM):

$$\text{Obesity Count}_{i} = \beta_{0} + \beta_{1} x_{1i} + \beta_{2} x_{2i} + \beta_{3} x_{3i} + \text{Offset}_{i}$$

Random Effect Model (REM):

$$\text{Obesity Count}_{i} = \beta_{0} + \beta_{1} x_{1i} + \beta_{2} x_{2i} + \beta_{3} x_{3i} + \text{Offset}_{i} + \theta_{i}$$

Besag-York-Mollie Model (BYM):

$$\text{Obesity Count}_{i} = \beta_{0} + \beta_{1} x_{1i} + \beta_{2} x_{2i} + \beta_{3} x_{3i} + \text{Offset}_{i} + b_{i}$$

$$b_{i} = (\sqrt(\rho/S)\phi_{i} + \sqrt(1 - \rho)\theta_{i}) \sigma$$

Parameter Estimate Comparison

Observed vs. Estimated Obesity Count Comparison

Observed Obesity Counts Across Health Regions

BYM Predicted Obesity Counts Across Health Regions

REM Predicted Obesity Counts Across Health Regions

NREM Predicted Obesity Counts Across Health Regions