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Table 1 Characteristics of four different base models (no predictors). Lower deviance information criterion (DIC) represents a better trade off between model fit and complexity. Models 1 and 3 have a random intercept; models 2 and 4 follow a BYM2 structure. \(D\left (\overline \theta \right)\), deviance of mean model parameters θ; pD, effective number of parameters

From: An ecological study of socioeconomic predictors in detection of COVID-19 cases across neighborhoods in New York City

Model Distribution Parameters Hyperparameters \(D\left (\overline \theta \right)\) pD DIC
Model 1* Poisson β0, νi τν 1346.53 149.6 1645.73
Model 2** Poisson β0, \(\upsilon _{i}^{*}\), \(\nu _{i}^{*}\) τγ, φ 1362.37 124.68 1611.73
Model 3 Negative binomial β0, νi n, τν 1855.47 3.30 1862.07
Model 4 Negative binomial β0, \(\upsilon _{i}^{*}\), \(\nu _{i}^{*}\) n, τγ, φ 1455.71 103.58 1662.87
  1. *Model 1: yi|λiPois(λi), log(λi)=ηi+log(Ei)=β0+νi+log(Ei)
  2. **Model 2: yi|λiPois(λi), \(\log \left (\lambda _{i}\right)=\eta _{i}+\log \left (E_{i}\right)=\beta _{0}+\frac {1}{\sqrt {\tau _{\gamma }}}\left ({\sqrt {\varphi }\upsilon _{i}^{*}}+\sqrt {1-\varphi }\nu _{i}^{*}\right)+\log \left (E_{i}\right)\)
  3. Model 3: yi|λiNegBin(n,λi), log(λi)=ηi+log(Ei)=β0+νi+log(Ei)
  4. Model 4: yi|λiNegBin(n,λi), \(\log \left (\lambda _{i}\right)=\eta _{i}+\log \left (E_{i}\right)=\beta _{0}+\frac {1}{\sqrt {\tau _{\gamma }}}\left ({\sqrt {\varphi }\upsilon _{i}^{*}}+\sqrt {1-\varphi }\nu _{i}^{*}\right)+\log \left (E_{i}\right)\)
  5. Symbols: yi, count of cases in Zip Code Tabulation Area (ZCTA) i; λi, expected cases in ZCTA i; Ei, number of total COVID-19 tests in ZCTA i; ηi, linear predictor for ZCTA i; β0, intercept; νi, nonspatial random-effect; \(\nu _{i}^{*}\), scaled nonspatial random-effect; \(\upsilon _{i}^{*}\), scaled spatial random-effect with intrinsic conditional autoregressive structure; τν, precision for nonspatial random effect, log-gamma prior; τγ, overall precision, penalized complexity (PC) prior; φ, mixing parameter, PC prior; n, overdispersion parameter, PC gamma prior