Add exercise and clean up code for example

exercises/jsp.md

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+Exercise: Junior School Project
+================
+Nora Wickelmaier
+2025-06-20
+
+Load the Junior School Project collected from primary (U.S. term is
+elementary) schools in inner London in R. You might need to install the
+faraway package first with `install.packages("faraway")`.
+
+The data frame contains the following variables:
+
+|           |                                                                                                                                                              |
+|-----------|--------------------------------------------------------------------------------------------------------------------------------------------------------------|
+| `school`  | 50 schools code 1–50                                                                                                                                         |
+| `class`   | a factor with levels `1`, `2`, `3`, and `4`                                                                                                                  |
+| `gender`  | a factor with levels `boy` and `girl`                                                                                                                        |
+| `social`  | class of the father I = 1; II = 2; III nonmanual = 3; III manual = 4; IV = 5; V = 6; Long-term unemployed = 7; Not currently employed = 8; Father absent = 9 |
+| `raven`   | test score                                                                                                                                                   |
+| `id`      | student id coded 1–1402                                                                                                                                      |
+| `english` | score on English                                                                                                                                             |
+| `math`    | score on Maths                                                                                                                                               |
+| `year`    | year of school                                                                                                                                               |
+
+We want to investigate how math achievement is influenced by raven score
+and social class of the father. If you need a refresher on Raven’s
+Progressive Matrices, check here:
+<https://en.wikipedia.org/wiki/Raven%27s_Progressive_Matrices>.
+Basically, it is an intelligent test.
+
+We will take a subset of the data, so that each student provides only
+one data point, for simplicity:
+
+``` r
+data("jsp", package = "faraway")
+dat <- jsp |> subset(year == 0)
+```
+
+<img src="jsp_files/figure-gfm/unnamed-chunk-2-1.png" style="display: block; margin: auto;" />
+
+1.  Create a new variable `craven` where `raven` is centered over all
+    students
+
+2.  Create another variable `gcraven` where `craven` is centered over
+    all schools. Create a variable `mraven` containing the centered
+    average school means first, so that you can calculate
+
+``` r
+dat$gcraven <- dat$craven - dat$mraven
+
+# Check your results
+aggregate(craven ~ school, dat, mean) |> head()
+```
+
+    ##   school     craven
+    ## 1      1 -2.6886533
+    ## 2      2  0.1250722
+    ## 3      3  2.0584055
+    ## 4      4  0.9167389
+    ## 5      5  2.3512627
+    ## 6      6  0.4584055
+
+``` r
+aggregate(gcraven ~ school, dat, mean) |> head()
+```
+
+    ##   school       gcraven
+    ## 1      1  1.253746e-15
+    ## 2      2 -1.184075e-15
+    ## 3      3 -1.421172e-15
+    ## 4      4  1.184283e-15
+    ## 5      5  5.075722e-16
+    ## 6      6  0.000000e+00
+
+3.  Create a plot with `lattice::xyplot()` with `gcraven` on the
+    $x$-axis and `math` on the $y$-axis and one panel for each school.
+    Use `type = c("p",    "g", "r")`. You can also use `ggplot2` if you
+    want to. What would be your conclusion about the need for
+    school-specific slopes based on this plot?
+
+<img src="jsp_files/figure-gfm/unnamed-chunk-7-1.png" style="display: block; margin: auto;" />
+
+4.  We will consider the following levels of the data:
+
+    - Level 1: students
+    - Level 2: schools
+
+    And the variables associated with the levels:
+
+    | Level | Variable  | Description                                  |
+    |-------|-----------|----------------------------------------------|
+    | 2     | `school`  | 50 schools code 1–50                         |
+    | 2     | `mraven`  | mean raven score of school (overall mean 0)  |
+    | 1     | `social`  | class of the father (categorical)            |
+    | 1     | `gcraven` | centered test score (mean for each school 0) |
+    | 1     | `math`    | score on Maths                               |
+
+    Fit the following model containing school-specific intercepts and
+    slopes with `lme4::lmer()`
+
+    $$
+    \begin{align*}
+    \text{(Level 1)} \quad y_{ij} &= b_{0i} + b_{1i}\,gcraven_{ij} + b_{2i}\,social_{ij} + b_{3i}\,(gcraven_{ij}\times social_{ij}) + \varepsilon_{ij}\\
+    \text{(Level 2)} \quad b_{0i} &= \beta_0 + \beta_4\,mraven_i + \upsilon_{0i} \\
+                     \quad b_{1i} &= \beta_1 + \beta_5\,mraven_i + \upsilon_{1i}\\
+                     \quad b_{2i} &= \beta_2\\
+                     \quad b_{3i} &= \beta_3\\
+    \text{(2) in (1)} \quad y_{ij} &= \beta_{0} + \beta_{1}\,gcraven_{ij} + \beta_{2}\,social_{ij} + \beta_{3}(gcraven_{ij}\times social_{ij})\\
+                                &~~~ + \beta_{4}\,mraven_i + \beta_{5}\,(gcraven_{ij} \times mraven_{i})\\
+                                &~~~ + \upsilon_{0i} + \upsilon_{1i}\,gcraven_{ij} + \varepsilon_{ij}
+    \end{align*}
+    $$ with
+    $\boldsymbol\upsilon \sim N(\boldsymbol 0, \boldsymbol{\Sigma}_\upsilon)$
+    i.i.d., $\varepsilon_{ij} \sim N(0, \sigma^2)$ i.i.d.
+
+5.  Interpret the parameters of the model:
+
+    - How much does math score increases if the raven score for a
+      student increases by one point for the reference social class of
+      the father?
+    - How much does math score increases when the raven score per school
+      increases by one point for the reference social class of the
+      father?
+    - What is your conclusion about the interactions in the model. Are
+      they needed?
+    - Does the inclusion of `social` improve the model fit? How can we
+      test this?