🚩 Keywords: Mixed Effects Model, BIC Model Selection, Bayesian vs. Frequentist, StreetRx, Geographic Variation

This project was completed for the Fall 2025 section of STA 610: Multilevel and Hierarchical Models at Duke University.

Overview

Street drug pricing is shaped by a complex mix of pharmacological, economic, and geographic factors. Using crowdsourced data from the StreetRx platform, this study applies hierarchical linear mixed models to analyze what drives the price per milligram (ppm) of diazepam (a benzodiazepine) and whether significant pricing variation exists across U.S. states.

Research Questions

Which variables are associated with the price per mg of diazepam on the street market?
Is there significant geographic heterogeneity in pricing across U.S. states?

Data

Source: StreetRx — a crowdsourced database of self-reported street drug transaction prices
Outcome: Log-transformed price per milligram — log(ppm)
Key variables: Dosage strength (mgstr), bulk purchase indicator, information source (word-of-mouth, internet, personal), state, year
Preprocessing: Removed primary_reason (>50% missing); excluded erroneous/outlier entries

Modeling Approach

Model Structure

\[\log(\text{ppm}_{ij}) = \beta_0 + \beta_1 \cdot \text{mgstr}_{ij} + \beta_2 \cdot \text{bulk}_{ij} + \beta_3 \cdot \text{source}_{ij} + \beta_4 \cdot \text{year}_{ij} + u_j + \epsilon_{ij}\]

where $u_j \sim \mathcal{N}(0, \sigma_u^2)$ captures state-level random intercepts — the primary quantity of interest for geographic heterogeneity.

Model Selection

An exhaustive search over 1,024 candidate models (all subsets of fixed effects) was performed using BIC as the selection criterion. The optimal model includes dosage strength, bulk purchase, source, and year as fixed effects, with state random intercepts.

Frequentist vs. Bayesian Comparison

Both approaches were implemented and compared:

Framework	Method	Package
Frequentist	Restricted Maximum Likelihood (REML)	`lme4`
Bayesian	MCMC with weakly informative priors	`brms` / `rstan`

Posterior distributions and REML estimates were nearly identical, providing strong cross-validation of the findings.

Key Findings

Factor	Effect on log(ppm)	Interpretation
Bulk purchase	−0.124	~12.4% price reduction per mg for bulk transactions
Internet source	−0.330	~33% lower prices vs. word-of-mouth
Personal report	−0.103	~10% lower prices vs. word-of-mouth
State random effect (σ_u)	Significant	Substantial geographic variation in baseline pricing

Street Pricing of Diazepam: Hierarchical Modeling of Geographic Heterogeneity

Tags