Hierarchical GLMs
Case study: processing relative clauses
in most languages subject relative clauses
are easier than object relative clausesbut Chinese seems to be an exception
subject relative clause
The senator who interrogated the journalist …
object relative clause
The senator who the journalist interrogated …
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data: self-paced reading times
37 subjects read 15 sentences either with an SRC or an ORC in a self-paced reading task
# A tibble: 15 × 4
subj item so rt
<dbl> <dbl> <chr> <dbl>
1 1 13 1 1561
2 1 6 -1 959
3 1 5 1 582
4 1 9 1 294
5 1 14 -1 438
6 1 4 -1 286
7 1 8 -1 438
8 1 10 -1 278
9 1 2 -1 542
10 1 11 1 494
11 1 7 1 270
12 1 3 1 406
13 1 16 -1 374
14 1 15 1 286
15 1 1 1 246\(\Leftarrow\) contrast coding of categorical predictor so
data from Gibson & Wu (2013)
inspect data
# A tibble: 2 × 2
so mean_log_rt
<chr> <dbl>
1 -1 6.10
2 1 6.02
fixed effects model
predict log-reading times as affected by treatment
soassume improper priors for parameters
\[ \begin{align*} \log(\mathtt{rt}_i) & \sim \mathcal{N}(\eta_i, \sigma_{err}) & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \eta_{i} & = \beta_0 + \beta_1 \mathtt{so}_i \\ \sigma_{err} & \sim \mathcal{U}(0, \infty) & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \beta_0, \beta_1 & \sim \mathcal{U}(- \infty, \infty) \end{align*} \]
fixed effects model: results
Family: gaussian
Links: mu = identity; sigma = identity
Formula: log(rt) ~ so
Data: rt_data (Number of observations: 547)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 6.10 0.04 6.03 6.17 1.00 4102 3064
so1 -0.08 0.05 -0.17 0.02 1.00 4434 2854
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 0.60 0.02 0.57 0.64 1.00 3996 2748
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).fixed effects model: results
varying intercepts model
predict log-reading times as affected by treatment
soassume improper priors for parameters
assume that different subjects and items could be “slower” or “faster” throughout
\[ \begin{align*} \log(\mathtt{rt}_i) & \sim \mathcal{N}(\eta_i, \sigma_{err}) & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \eta_{i} & = \beta_0 + \underbrace{u_{0,\mathtt{subj}_i} + w_{0,\mathtt{item}_i}}_{\text{varying intercepts}} + \beta_1 \mathtt{so}_i \\ u_{0,\mathtt{subj}_i} & \sim \mathcal{N}(0, \sigma_{u_0}) & \ \ \ \ \ \ \ \ \ \ \ \ \ \ w_{0,\mathtt{subj}_i} & \sim \mathcal{N}(0, \sigma_{w_0}) \\ \sigma_{err}, \sigma_{u_0}, \sigma_{w_0} & \sim \mathcal{U}(0, \infty) & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \beta_0, \beta_1 & \sim \mathcal{U}(- \infty, \infty) \end{align*} \]
varying intercepts model: results
interc.+slopes model
predict log-reading times as affected by treatment
soassume improper priors for parameters
assume that different subjects and items could be “slower” or “faster” throughout
assume that different subjects and items react more or less to
somanipulation
\[ \begin{align*} \log(\mathtt{rt}_i) & \sim \mathcal{N}(\eta_i, \sigma_{err})\\ \eta_{i} & = \beta_0 + \underbrace{u_{0,\mathtt{subj}_i} + w_{0,\mathtt{item}_i}}_{\text{varying intercepts}} + (\beta_1 + \underbrace{u_{1,\mathtt{subj}_i} + w_{1,\mathtt{item}_i}}_{\text{varying slopes}} ) \mathtt{so}_i \end{align*} \] \[ \begin{align*} u_{0,\mathtt{subj}_i} & \sim \mathcal{N}(0, \sigma_{u_0}) & w_{0,\mathtt{subj}_i} & \sim \mathcal{N}(0, \sigma_{w_0}) \\ u_{1,\mathtt{subj}_i} & \sim \mathcal{N}(0, \sigma_{u_1}) & w_{1,\mathtt{subj}_i} & \sim \mathcal{N}(0, \sigma_{w_1}) \\ \sigma_{err}, \sigma_{u_{0|1}}, \sigma_{w_{0|1}} & \sim \mathcal{U}(0, \infty) & \beta_0, \beta_1 & \sim \mathcal{U}(- \infty, \infty) \end{align*} \]
interc.+slopes model: results
interc.+slopes model w/ correlation
predict log-reading times as affected by treatment
soassume improper priors for parameters
assume that different subjects and items could be “slower” or “faster” throughout
assume that different subjects and items react more or less to
somanipulationassume that random intercepts and slopes might be correlated
\[ \begin{align*} \log(\mathtt{rt}_i) & \sim \mathcal{N}(\eta_i, \sigma_{err}) \\ \eta_{i} & = \beta_0 + u_{0,\mathtt{subj}_i} + w_{0,\mathtt{item}_i} + \left (\beta_1 + u_{1,\mathtt{subj}_i} + w_{1,\mathtt{item}_i} \right ) \mathtt{so}_i \\ \begin{pmatrix}u_{0,\mathtt{subj}_i} \\ u_{1,\mathtt{subj}_i} \end{pmatrix} & \sim \mathcal{N} \left (\begin{pmatrix} 0 \\ 0 \end{pmatrix}, \Sigma_{u} \right ) \\ \Sigma_{w} & = \begin{pmatrix} \sigma_{u_{0}}^2 & \rho_u\sigma_{u{0}}\sigma_{u{1}} \\ \rho_u\sigma_{u{0}}\sigma_{u{1}} & \sigma_{u_{0}}^2 \end{pmatrix} \ \ \ \text{same for } \mathtt{item} \\ \beta_0, \beta_1 & \sim \mathcal{U}(- \infty, \infty) \ \ \ \ \ \ \ \ \rho_u, \rho_w \sim \mathcal{U}(-1,1) \ \ \ \ \ \ \ \ \sigma_{err}, \sigma_{u_{0|1}}, \sigma_{w_{0|1}} \sim \mathcal{U}(0, \infty) \end{align*} \]
interc.+slopes model w/ corr.: results
How to choose RE-structure
- two approaches:
- “keep it maximal”
- include the maximum RE structure that “makes sense”
- what makes sense can depend on a priori conceptual considerations
- data might not be sufficient to estimate some RE coefficients
- “let the data decide”
- fit models with varying RE-structures and compare
- the former is more careful / prudent in a science context (learning about the world from the model and the data), the latter may be more adequate for an engineering context (predicting well enough with efficient models)