Given is the following GLM: Y_{i} \sim \operatorname{Normal}\left(\alpha + \beta_{j} \cdot X_{ij}, \sigma\right)

where:
Y = quantitative response vector of size N
X^{N \times J} = design matrix with X_{ij} = 1 or X_{ij} =1;
with J \approx 10,000 and N \approx 50 
\beta_j \sim \operatorname{Normal}(\beta_{\mu}, \beta_{\sigma}) [actually noncentered]

and priors:
\alpha \sim \operatorname{Normal}(0, 10)
\beta_{\mu} \sim \operatorname{Normal}(0, 5)
\sigma, \beta_{\sigma} \sim \operatorname{Cauchy}^{+}(0, 5)
With this model I estimate the univariate (but partiallypooled) effects (\beta_j), and generate the loglikelihood matrix > \text{log_lik}^{P \times N \times J}, where P = Chains \cdot Iterations, N = number of observations, J = number of effects (columns in X)
My main question is how to estimate LOOIC (and to interpret p_loo) in univariate models?
Currently, I transform the original \text{log_lik} into \tilde{\text{log_lik}}^{O \times N} where O = Chain \cdot Iterations \cdot J and use this as input to loo. Any systematic error here?
Posterior predictive checks show that the model has high predictive accuracy (based on the observed data). I can include the model and the loo estimates if needed.