Hi,

I am trying to reconstruct predicted values from a linear-mixed effects model with random-intercept. I use the data from Hox (see here and build a simple model with one predictor variable (originally *occas*, I called it *semester*). My goal is to understand how the model comes up with a final prediction based on the parameters (fixed and random) that have been estimated.

This is the output I get after I ran the following model: ‘gpa ~ 1 + semester + (1|student)’

```
Family: gaussian
Links: mu = identity; sigma = identity
Formula: gpa ~ 1 + semester + (1 | student)
Data: df (Number of observations: 1200)
Samples: 2 chains, each with iter = 3000; warmup = 1000; thin = 1;
total post-warmup samples = 4000
Group-Level Effects:
~student (Number of levels: 200)
Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
sd(Intercept) 0.25 0.01 0.23 0.28 1159 1.00
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
Intercept 2.60 0.02 2.55 2.64 969 1.01
semester 0.11 0.00 0.10 0.11 4000 1.00
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
sigma 0.24 0.01 0.23 0.25 4000 1.00
Samples were drawn using sampling(NUTS). For each parameter, Eff.Sample
is a crude measure of effective sample size, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).
```

In order to understand it better, I appended predictions + random effects to the original dataset. *preds1* is the prediction (calculated using *predict*), *u1* is the group-level effect estimated for that person (calculated using *ranef*), *sigma* is the residual error (*residuals*), *grand_gpa* is the grand mean. I wonder how to come up wih the prediction of, say student 1 at semester = 0 which is 2.5 and at semester = 1 which is 2.6.

student | semester | gpa | preds1 | u1 | sigma | grand_gpa |
---|---|---|---|---|---|---|

1 | 0 | 2.3 | 2.5 | -0.069 | -0.2284 | 2.9 |

1 | 1 | 2.1 | 2.6 | -0.069 | -0.5347 | 2.9 |

1 | 2 | 3.0 | 2.7 | -0.069 | 0.2590 | 2.9 |

1 | 3 | 3.0 | 2.8 | -0.069 | 0.1527 | 2.9 |

1 | 4 | 3.0 | 3.0 | -0.069 | 0.0463 | 2.9 |

1 | 5 | 3.3 | 3.1 | -0.069 | 0.2400 | 2.9 |

2 | 0 | 2.2 | 2.4 | -0.215 | -0.1827 | 2.9 |

2 | 1 | 2.5 | 2.5 | -0.215 | 0.0110 | 2.9 |

2 | 2 | 2.6 | 2.6 | -0.215 | 0.0047 | 2.9 |

2 | 3 | 2.6 | 2.7 | -0.215 | -0.1016 | 2.9 |

2 | 4 | 3.0 | 2.8 | -0.215 | 0.1920 | 2.9 |

2 | 5 | 2.8 | 2.9 | -0.215 | -0.1143 | 2.9 |

brms-version: 2.4.2

R version 3.4.2 (2017-09-28)

Platform: x86_64-apple-darwin15.6.0 (64-bit)

Running under: macOS High Sierra 10.13.6

Thanks in advance for your help.