126 lines
6.1 KiB
Markdown
126 lines
6.1 KiB
Markdown
# Exercise: Junior School Project
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Load the Junior School Project collected from primary (U.S. term is
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elementary) schools in inner London in R. You might need to install the
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faraway package first with `install.packages("faraway")`.
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The data frame contains the following variables:
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| | |
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|-----------|--------------------------------------------------------------------------------------------------------------------------------------------------------------|
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| `school` | 50 schools code 1–50 |
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| `class` | a factor with levels `1`, `2`, `3`, and `4` |
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| `gender` | a factor with levels `boy` and `girl` |
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| `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 |
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| `raven` | test score |
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| `id` | student id coded 1–1402 |
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| `english` | score on English |
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| `math` | score on Maths |
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| `year` | year of school |
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We want to investigate how math achievement is influenced by raven score
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and social class of the father. If you need a refresher on Raven’s
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Progressive Matrices, check here:
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<https://en.wikipedia.org/wiki/Raven%27s_Progressive_Matrices>.
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Basically, it is an intelligent test.
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We will take a subset of the data, so that each student provides only
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one data point, for simplicity:
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``` r
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data("jsp", package = "faraway")
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dat <- jsp |> subset(year == 0)
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```
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<img src="jsp_files/figure-gfm/unnamed-chunk-2-1.png" style="display: block; margin: auto;" />
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1. Create a new variable `craven` where `raven` is centered over all
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students
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2. Create another variable `gcraven` where `craven` is centered over
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all schools. Create a variable `mraven` containing the centered
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average school means first, so that you can calculate
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``` r
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dat$gcraven <- dat$craven - dat$mraven
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# Check your results
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aggregate(craven ~ school, dat, mean) |> head()
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```
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## school craven
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## 1 1 -2.6886533
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## 2 2 0.1250722
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## 3 3 2.0584055
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## 4 4 0.9167389
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## 5 5 2.3512627
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## 6 6 0.4584055
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``` r
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aggregate(gcraven ~ school, dat, mean) |> head()
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```
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## school gcraven
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## 1 1 1.253746e-15
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## 2 2 -1.184075e-15
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## 3 3 -1.421172e-15
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## 4 4 1.184283e-15
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## 5 5 5.075722e-16
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## 6 6 0.000000e+00
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3. Create a plot with `lattice::xyplot()` with `gcraven` on the x-axis
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and `math` on the y-axis and one panel for each school. Use
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`type = c("p", "g", "r")`. You can also use `ggplot2` if you want
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to. What would be your conclusion about the need for school-specific
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slopes based on this plot?
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<img src="jsp_files/figure-gfm/unnamed-chunk-7-1.png" style="display: block; margin: auto;" />
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4. We will consider the following levels of the data:
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- Level 1: students
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- Level 2: schools
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And the variables associated with the levels:
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| Level | Variable | Description |
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|-------|-----------|----------------------------------------------|
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| 2 | `school` | 50 schools code 1–50 |
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| 2 | `mraven` | mean raven score of school (overall mean 0) |
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| 1 | `social` | class of the father (categorical) |
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| 1 | `gcraven` | centered test score (mean for each school 0) |
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| 1 | `math` | score on Maths |
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Fit the following model containing school-specific intercepts and
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slopes with `lme4::lmer()`
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$$
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\begin{align*}
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\text{(Level 1)} \quad y_{ij} &= b_{0i} + b_{1i}\,gcraven_{ij} + b_{2i}\,social_{ij} + b_{3i}\,(gcraven_{ij}\times social_{ij}) + \varepsilon_{ij}\\
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\text{(Level 2)} \quad b_{0i} &= \beta_0 + \beta_4\,mraven_i + \upsilon_{0i} \\
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\quad b_{1i} &= \beta_1 + \beta_5\,mraven_i + \upsilon_{1i}\\
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\quad b_{2i} &= \beta_2\\
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\quad b_{3i} &= \beta_3\\
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\text{(2) in (1)} \quad y_{ij} &= \beta_{0} + \beta_{1}\,gcraven_{ij} + \beta_{2}\,social_{ij} + \beta_{3}(gcraven_{ij}\times social_{ij})\\
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&~~~ + \beta_{4}\,mraven_i + \beta_{5}\,(gcraven_{ij} \times mraven_{i})\\
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&~~~ + \upsilon_{0i} + \upsilon_{1i}\,gcraven_{ij} + \varepsilon_{ij}
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\end{align*}
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$$
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with
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$\boldsymbol\upsilon \sim N(\boldsymbol 0, \boldsymbol{\Sigma}_\upsilon)$
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i.i.d., $\varepsilon_{ij} \sim N(0, \sigma^2)$ i.i.d.
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5. Interpret the parameters of the model:
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- How much does math score increases if the raven score for a
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student increases by one point for the reference social class of
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the father?
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- How much does math score increases when the raven score per school
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increases by one point for the reference social class of the
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father?
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- What is your conclusion about the interactions in the model. Are
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they needed?
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- Does the inclusion of `social` improve the model fit? How can we
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test this?
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