# Visualising the predictive distribution of a log-transformed linear model

Last week I presented visualisations of theoretical distributions that predict ice cream sales statistics based on linear and generalised linear models, which I introduced in an earlier post.Theoretical distributions |

The posterior predictive distribution is what I am most interested in. From the simulations I can get the 95% prediction interval, which will be slightly wider than the theoretical 95% interval, as it takes into account the parameter uncertainty as well.

Ok, first I take my log-transformed linear model of my earlier post and turn it into a Stan model, including a section to generate output from the posterior predictive distribution.

After I have complied and run the model, I can extract the simulations and calculate various summary statistics. Furthermore, I use my parameters also to predict the median and mean, so that I can compare them against the sample statistics. Note again, that for the mean calculation of the log-normal distribution I have to take into account the variance as well.

Ok, that looks pretty reasonable, and also quite similar to my earlier output with

`glm`

. Using my plotting function of last week I can also create a nice 3D plot again.Posterior predictive distributions |

Just as expected, I note a slightly wider 95% interval range in the posterior predictive distributions compared to the theoretical distributions at the top.

### Session Info

```
R version 3.2.2 (2015-08-14)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: OS X 10.10.5 (Yosemite)
locale:
[1] en_GB.UTF-8/en_GB.UTF-8/en_GB.UTF-8/C/en_GB.UTF-8/en_GB.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets
[6] methods base
other attached packages:
[1] rstan_2.7.0-1 inline_0.3.14 Rcpp_0.12.0
loaded via a namespace (and not attached):
[1] tools_3.2.2 codetools_0.2-14 stats4_3.2.2
```

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