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

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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 |

Today I will take a closer look at the log-transformed linear model and use Stan/rstan, not only to model the sales statistics, but also to generate samples from the posterior predictive distribution.

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

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