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Inclusive Ranking of Indian States via Bayesian Bradley-Terry Model

Arshi Rizvi, Rahul Singh

Abstract

Evaluating the performance of different administrative regions within a country is crucial for its development and policy formulation. The performance evaluators are mostly based on health, education, per capita income, awareness, family planning and so on. Not only evaluating regions, but also ranking them is a crucial step, and various methods have been proposed to date. We aim to provide a ranking system for Indian states that uses a Bayesian approach via the famous Bradley-Terry model for paired comparisons. The ranking method uses indicators from the NFHS-5 dataset with the prior information of per-capita incomes of the states/UTs, thus leading to a holistic ranking, which not only includes human development factors but also take account the economic background of the states. We also carried out various Markov chain Monte Carlo diagnostics required for the reliability of the estimates of merits for these states. These merits thus provide a ranking for the states/UTs and can further be utilised to make informed policy decisions.

Inclusive Ranking of Indian States via Bayesian Bradley-Terry Model

Abstract

Evaluating the performance of different administrative regions within a country is crucial for its development and policy formulation. The performance evaluators are mostly based on health, education, per capita income, awareness, family planning and so on. Not only evaluating regions, but also ranking them is a crucial step, and various methods have been proposed to date. We aim to provide a ranking system for Indian states that uses a Bayesian approach via the famous Bradley-Terry model for paired comparisons. The ranking method uses indicators from the NFHS-5 dataset with the prior information of per-capita incomes of the states/UTs, thus leading to a holistic ranking, which not only includes human development factors but also take account the economic background of the states. We also carried out various Markov chain Monte Carlo diagnostics required for the reliability of the estimates of merits for these states. These merits thus provide a ranking for the states/UTs and can further be utilised to make informed policy decisions.
Paper Structure (14 sections, 2 theorems, 31 equations, 12 figures, 1 algorithm)

This paper contains 14 sections, 2 theorems, 31 equations, 12 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Datasets
  3. NFHS Dataset
  4. Per-Capita Income Dateset
  5. Modelling Framework
  6. The standard Bradley-Terry model
  7. The Bayesian framework
  8. Modelling covariance
  9. Model Fitting
  10. MCMC sampling
  11. Stopping rules/MCMC diagnostics
  12. Implementation and Results
  13. Simulation
  14. Discussion

Key Result

Proposition 1

Let $\bm{\mu} \in \mathbb{R}^M$ satisfy the linear constraint $\mathbf{1}^T \bm{\mu} = 0$. Assume a Gaussian prior distribution on $\bm{\mu}$, restricted to this subspace, and let the likelihood $\ell(\mathbf{x} \mid \bm{\mu})$ define the Then the preconditioned Crank--Nicolson (pCN) algorithm, combined with a Metropolis--Hastings accept--reject step, generates a Markov chain $(\bm{\mu}_t)_{t \ge

Figures (12)

  • Figure 1: Per-capita incomes of states in different income level zones.
  • Figure 2: Results when $116$ indicators are included.
  • Figure 3: Change in ranking shows the effect of likelihood on posterior based on NFHS data.
  • Figure 4: Histogram for variance hyperparameter $\alpha^2$ (when Chandigarh is removed).
  • Figure 5: Change in ranking shows the effect of likelihood on posterior based on NFHS data without Chandigarh.
  • ...and 7 more figures

Theorems & Definitions (3)

  • Proposition 1
  • Lemma 1
  • proof : Proof of \ref{['pcn-prop']}