Skip to contents

Details on Model Output

Source: vignettes/articles/a01_model_output.Rmd
a01_model_output.Rmd

This article dives into the details of the output of the function stdglm().

Model Output

The output is an object of class stdglm which is a list with elements ave and out. The element ave contains the posterior means of the model parameters, while out contains the full MCMC output.

ave and out are lists as well, and their elements change depending on the model fitted.

ave list

Generally speaking, the list ave contains the following elements:

  • Yfitted_mean, Yfitted2_mean: p-by-T matrices with first two moments of draws from the posterior predictive distribution for the observed data points.
  • Ypred_mean, Ypred2_mean: p_{new}-by-T_{new} matrices with first two moments of draws from the posterior predictive distribution for the new data points (only if out-of-sample predictions are required).
  • B_postmean, B2_postmean: First two moments of the overall effect of varying coefficients.
  • Btime_postmean, Btime2_postmean: First two moments of the temporal effect of varying coefficients.
  • Bspace_postmean, Bspace2_postmean: First two moments of the spatial effect of varying coefficients.
  • Bspacetime_postmean, Bspacetime2_postmean: First two moments of the spatio-temporal effect of varying coefficients.
  • B2_c_t_s_st: 2nd moment of the varying coefficients, \beta_{it}.
  • Btime_pred_postmean, Btime_pred2_postmean: First two moments of the temporal effect of varying coefficients at the predicted time points (only if out-of-sample predictions are required).
  • Bspace_pred_postmean, Bspace_pred2_postmean: First two moments of the spatial effect of varying coefficients at the predicted spatial locations (only if out-of-sample predictions are required).
  • Bspacetime_pred_postmean, Bspacetime_pred2_postmean: First two moments of the spatio-temporal effect of varying coefficients at the predicted spatial locations and all time points (only if out-of-sample predictions are required).
  • B_pred2_c_t_s_st: 2nd moment of the varying coefficients, \beta_{it}, at the predicted spatial locations and all time points (only if out-of-sample predictions are required).
  • meanY1mean: Contribution of covariates with varying coefficients, i.e. \boldsymbol{x}_{it} multiplied by its effects.
  • meanZmean: Contribution of covariates with non-varying effects, i.e. \boldsymbol{z}_{it}' \boldsymbol{\gamma} (only if Z is specified).
  • thetay_mean: It is defined as meanY1mean + meanZmean + offset.
  • Eta_tilde_mean: Posterior mean of the linear predictor (for non-Gaussian outcomes). For Poisson outcomes, it is defined as Eta_tilde_mean = thetay_mean + epsilon, where epsilon is a Gaussian error term. For Bernoulli outcomes, it is obtained by drawing from a truncated normal distribution with mean thetay_mean.
  • DIC, Dbar, pD: Deviance Information Criterion, DIC = \bar{D} + pD.
  • WAIC, se_WAIC, pWAIC, se_pWAIC, elpd, se_elpd: Widely Applicable Information Criterion, the penalty term, and expected log pointwise predictive density. The prefix se_ denotes standard errors. See Gelman et al. (2014) for details.
  • CRPS: Continuous Ranked Probability Score. It is positively oriented, i.e. the model with the highest mean score is favoured (Gschlößl and Czado 2007): \mathrm{CRPS}\left(y_{it}\right)=\frac{1}{2} E\left|y_{r e p, {it}}-\tilde{y}_{r e p, {it}}\right|-E\left|y_{r e p, {it}}-y_{it}\right| where y_{r e p, {it}} and \tilde{y}_{r e p, {it}} are independent replicates from the posterior predictive distribution.
  • PMCC: Predictive model choice criterion. It is negatively oriented, i.e. the model with the lowest score is favoured (Gelfand and Ghosh 1998): \mathrm{PMCC}=\sum_{i=1}^p \sum_{t=1}^T \left\{y_{it}-E\left(y_{r e p, {it}} \mid \mathbf{y}\right)\right\}^2+\sum_{i=1}^p \sum_{t=1}^T \operatorname{Var}\left(y_{r e p, {it}} \mid \mathbf{y}\right) .
  • Bayesian p-values: following Gelman et al. (2014), the posterior predictive p-value is defined as: p_B=\operatorname{Pr}\left(T\left(y, \theta\right) \geq T(y_{rep}, \theta) \mid y\right), for some test quantity T(y, \theta) and some parameter vector \theta. The p-values returned by stdglm() are based on the following functions:
    • pvalue_YgrYhat: T(y_{it}, \theta) = y_{it}.
    • pvalue_ResgrReshat: T(y_{it}, \theta) = r, where r_{it}=\frac{y_{it} - E\left( y_{it} \mid \theta \right)}{\operatorname{Var}\left( y_{it} \mid \theta \right)} are the Pearson residuals.
    • pvalue_chisquare: T(y, \theta) = \sum_{i=1}^p \sum_{t=1}^T r_{it}^2
    • pvalue_perc95: T(y, \theta) is the 95-th percentile of the distribution of the outcome for each spatial location.
  • AccRate: Point-wise acceptance rate for the random-walk Metropolis-Hastings step (only for Poisson outcome).

Note that the criteria above (DIC, WAIC, p-values, etc.) are computed only using the non-missing values of the response variable.

out list

The list out contains the following elements:

  • please see the documentation of the function here.

Note that the function stdglm() does not return all the posterior draws for the varying coefficients, but only their posterior summaries (i.e., first two moments). This is done to save memory, as the storing matrices can be very large.

References

Gelfand, Alan E, and Sujit K Ghosh. 1998. “Model Choice: A Minimum Posterior Predictive Loss Approach.” Biometrika 85 (1): 1–11.
Gelman, Andrew, John B Carlin, Hal S Stern, and Donald B Rubin. 2014. Bayesian Data Analysis. 3rd ed. Chapman; Hall/CRC.
Gschlößl, Susanne, and Claudia Czado. 2007. “Spatial Modelling of Claim Frequency and Claim Size in Non-Life Insurance.” Scandinavian Actuarial Journal 2007 (3): 202–25.