# Estimating trial-level effects

## Introduction

By default, the `hmetad` package uses aggregated data (i.e., counts of the number of trials with the same stimulus, type 1 response, and type 2 response.) This is because data aggregation makes model fitting and simulation much more efficient. But sometimes researchers will be interested in trial-level effects.

One common example would be what is often called “crossed random effects”. For example, in a design where all participants make responses to the same set of items, a researcher might want to estimate both participant-level and item-level effects on their model parameters.

We can simulate data from such a design like so:

    library(tidyverse)
    library(tidybayes)
    library(hmetad)

    ## average model parameters
    K <- 3 ## number of confidence levels
    mu_log_M <- -0.5
    mu_dprime <- 1.5
    mu_c <- 0
    mu_c2_0 <- rep(-1, K - 1)
    mu_c2_1 <- rep(-1, K - 1)

    ## participant-level standard deviations
    sd_log_M_participant <- 0.25
    sd_dprime_participant <- 0.5
    sd_c_participant <- 0.33
    sd_c2_0_participant <- cov_matrix(rep(0.25, K - 1), diag(K - 1))
    sd_c2_1_participant <- cov_matrix(rep(0.25, K - 1), diag(K - 1))

    ## item-level standard deviations
    sd_log_M_item <- 0.1
    sd_dprime_item <- 0.5
    sd_c_item <- 0.75
    sd_c2_0_item <- cov_matrix(rep(0.1, K - 1), diag(K - 1))
    sd_c2_1_item <- cov_matrix(rep(0.1, K - 1), diag(K - 1))

    ## simulate data
    d <- expand_grid(
      participant = 1:50,
      item = 1:25
    ) |>
      ## simulate participant-level differences
      group_by(participant) |>
      mutate(
        z_log_M_participant = rnorm(1, sd = sd_log_M_participant),
        z_dprime_participant = rnorm(1, sd = sd_dprime_participant),
        z_c_participant = rnorm(1, sd = sd_c_participant),
        z_c2_0_participant = list(rmulti_normal(1, mu = rep(0, K - 1), Sigma = sd_c2_0_participant)),
        z_c2_1_participant = list(rmulti_normal(1, mu = rep(0, K - 1), Sigma = sd_c2_1_participant))
      ) |>
      ## simulate item-level differences
      group_by(item) |>
      mutate(
        z_log_M_item = rnorm(1, sd = sd_log_M_item),
        z_dprime_item = rnorm(1, sd = sd_dprime_item),
        z_c_item = rnorm(1, sd = sd_c_item),
        z_c2_0_item = list(rmulti_normal(1, mu = rep(0, K - 1), Sigma = sd_c2_0_item)),
        z_c2_1_item = list(rmulti_normal(1, mu = rep(0, K - 1), Sigma = sd_c2_1_item))
      ) |>
      ungroup() |>
      ## compute model parameters
      mutate(
        log_M = mu_log_M + z_log_M_participant + z_log_M_item,
        dprime = mu_dprime + z_dprime_participant + z_dprime_item,
        c = mu_c + z_c_participant + z_c_item,
        c2_0_diff = map2(
          z_c2_0_participant, z_c2_0_item,
          ~ exp(mu_c2_0 + .x + .y)
        ),
        c2_1_diff = map2(
          z_c2_1_participant, z_c2_1_item,
          ~ exp(mu_c2_1 + .x + .y)
        )
      ) |>
      ## simulate two trials per participant/item (stimulus = 0 and stimulus = 1)
      mutate(trial = pmap(list(dprime, c, log_M, c2_0_diff, c2_1_diff), sim_metad, N_trials = 2)) |>
      select(participant, item, trial) |>
      unnest(trial)

    #> # A tibble: 2,500 × 16
    #>    participant  item trial stimulus response correct confidence dprime
    #>          <int> <int> <int>    <int>    <int>   <int>      <int>  <dbl>
    #>  1           1     1     1        0        0       1          3   1.67
    #>  2           1     1     1        1        1       1          2   1.67
    #>  3           1     2     1        0        0       1          1   1.78
    #>  4           1     2     1        1        1       1          3   1.78
    #>  5           1     3     1        0        0       1          3   1.21
    #>  6           1     3     1        1        0       0          2   1.21
    #>  7           1     4     1        0        0       1          3   1.27
    #>  8           1     4     1        1        1       1          3   1.27
    #>  9           1     5     1        0        0       1          2   1.90
    #> 10           1     5     1        1        0       0          1   1.90
    #> # ℹ 2,490 more rows
    #> # ℹ 8 more variables: c <dbl>, meta_dprime <dbl>, M <dbl>,
    #> #   meta_c2_0 <list>, meta_c2_1 <list>, theta <dbl>, theta_1 <dbl>,
    #> #   theta_2 <dbl>

Don’t worry about the details of the simulation code- what matters is that we have a data set with repeated measures for participants:

    count(d, participant)
    #> # A tibble: 50 × 2
    #>    participant     n
    #>          <int> <int>
    #>  1           1    50
    #>  2           2    50
    #>  3           3    50
    #>  4           4    50
    #>  5           5    50
    #>  6           6    50
    #>  7           7    50
    #>  8           8    50
    #>  9           9    50
    #> 10          10    50
    #> # ℹ 40 more rows

And repeated measures for items:

    count(d, item)
    #> # A tibble: 25 × 2
    #>     item     n
    #>    <int> <int>
    #>  1     1   100
    #>  2     2   100
    #>  3     3   100
    #>  4     4   100
    #>  5     5   100
    #>  6     6   100
    #>  7     7   100
    #>  8     8   100
    #>  9     9   100
    #> 10    10   100
    #> # ℹ 15 more rows

## Standard model with data aggregation

If we would like, we can use the `fit_metad` function on this data with participant-level and item-level effects. However, if we aggregate the data ourselves, we can see that the aggregation doesn’t really help us here:

    aggregate_metad(d, participant, item)
    #> # A tibble: 1,250 × 5
    #>    participant item    N_0   N_1 N[,"N_0_1"] [,"N_0_2"] [,"N_0_3"]
    #>    <fct>       <fct> <int> <int>       <int>      <int>      <int>
    #>  1 1           1         1     1           1          0          0
    #>  2 1           2         1     1           0          0          1
    #>  3 1           3         1     1           1          0          0
    #>  4 1           4         1     1           1          0          0
    #>  5 1           5         1     1           0          1          0
    #>  6 1           6         1     1           0          0          0
    #>  7 1           7         1     1           1          0          0
    #>  8 1           8         1     1           1          0          0
    #>  9 1           9         1     1           0          0          1
    #> 10 1           10        1     1           1          0          0
    #> # ℹ 1,240 more rows
    #> # ℹ 1 more variable: N[4:12] <int>

As you can see, the aggregated data set has `500` rows (with two observations per row), which is not much smaller than the trial-level data that we started with! So, in this case, it will probably be easier not to aggregate our data. Nevertheless, there is nothing stopping us from fitting the model like normal:1

    m.multinomial <- fit_metad(
      bf(
        N ~ 1 + (1 | participant) + (1 | item),
        dprime + c +
          metac2zero1diff + metac2zero2diff +
          metac2one1diff + metac2one2diff ~
          1 + (1 | participant) + (1 | item)
      ),
      data = d, init = 0, cores = 4,
      file = "models/multinomial.rds",
      prior = prior(normal(0, .25), class = Intercept) +
        prior(normal(0, .25), class = Intercept, dpar = dprime) +
        prior(normal(0, .25), class = Intercept, dpar = c) +
        prior(normal(0, .1), class = Intercept, dpar = metac2zero1diff) +
        prior(normal(0, .1), class = Intercept, dpar = metac2zero2diff) +
        prior(normal(0, .1), class = Intercept, dpar = metac2one1diff) +
        prior(normal(0, .1), class = Intercept, dpar = metac2one2diff) +
        prior(normal(0, 1), class = sd) +
        prior(normal(0, 1), class = sd, dpar = dprime) +
        prior(normal(0, 1), class = sd, dpar = c) +
        prior(normal(0, 1), class = sd, dpar = metac2zero1diff) +
        prior(normal(0, 1), class = sd, dpar = metac2zero2diff) +
        prior(normal(0, 1), class = sd, dpar = metac2one1diff) +
        prior(normal(0, 1), class = sd, dpar = metac2one2diff)
    )

    #>  Family: metad__3__normal__absolute__multinomial
    #>   Links: mu = log; dprime = identity; c = identity; metac2zero1diff = log; metac2zero2diff = log; metac2one1diff = log; metac2one2diff = log
    #> Formula: N ~ 1 + (1 | participant) + (1 | item)
    #>          dprime ~ 1 + (1 | participant) + (1 | item)
    #>          c ~ 1 + (1 | participant) + (1 | item)
    #>          metac2zero1diff ~ 1 + (1 | participant) + (1 | item)
    #>          metac2zero2diff ~ 1 + (1 | participant) + (1 | item)
    #>          metac2one1diff ~ 1 + (1 | participant) + (1 | item)
    #>          metac2one2diff ~ 1 + (1 | participant) + (1 | item)
    #>    Data: data.aggregated (Number of observations: 1250)
    #>   Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
    #>          total post-warmup draws = 4000
    #>
    #> Multilevel Hyperparameters:
    #> ~item (Number of levels: 25)
    #>                               Estimate Est.Error l-95% CI u-95% CI Rhat
    #> sd(Intercept)                     0.14      0.11     0.01     0.40 1.00
    #> sd(dprime_Intercept)              0.52      0.13     0.32     0.83 1.00
    #> sd(c_Intercept)                   0.64      0.10     0.47     0.88 1.00
    #> sd(metac2zero1diff_Intercept)     0.80      0.24     0.12     1.24 1.03
    #> sd(metac2zero2diff_Intercept)     0.72      0.23     0.23     1.17 1.01
    #> sd(metac2one1diff_Intercept)      0.58      0.26     0.05     1.06 1.02
    #> sd(metac2one2diff_Intercept)      0.79      0.19     0.46     1.21 1.00
    #>                               Bulk_ESS Tail_ESS
    #> sd(Intercept)                     1614     2237
    #> sd(dprime_Intercept)              1384     2087
    #> sd(c_Intercept)                   1257     2043
    #> sd(metac2zero1diff_Intercept)      140       27
    #> sd(metac2zero2diff_Intercept)      405      368
    #> sd(metac2one1diff_Intercept)       204      227
    #> sd(metac2one2diff_Intercept)      1059     1874
    #>
    #> ~participant (Number of levels: 50)
    #>                               Estimate Est.Error l-95% CI u-95% CI Rhat
    #> sd(Intercept)                     0.32      0.15     0.04     0.62 1.00
    #> sd(dprime_Intercept)              0.50      0.09     0.35     0.69 1.00
    #> sd(c_Intercept)                   0.31      0.04     0.24     0.41 1.00
    #> sd(metac2zero1diff_Intercept)     0.13      0.11     0.00     0.36 1.02
    #> sd(metac2zero2diff_Intercept)     0.34      0.12     0.14     0.60 1.00
    #> sd(metac2one1diff_Intercept)      0.35      0.15     0.10     0.70 1.01
    #> sd(metac2one2diff_Intercept)      0.17      0.10     0.01     0.39 1.01
    #>                               Bulk_ESS Tail_ESS
    #> sd(Intercept)                      865     1116
    #> sd(dprime_Intercept)              1670     2770
    #> sd(c_Intercept)                   1486     2138
    #> sd(metac2zero1diff_Intercept)      196      110
    #> sd(metac2zero2diff_Intercept)      608      547
    #> sd(metac2one1diff_Intercept)       275      318
    #> sd(metac2one2diff_Intercept)       836     1817
    #>
    #> Regression Coefficients:
    #>                           Estimate Est.Error l-95% CI u-95% CI Rhat
    #> Intercept                    -0.32      0.12    -0.55    -0.10 1.00
    #> dprime_Intercept              1.04      0.14     0.74     1.29 1.00
    #> c_Intercept                   0.26      0.12     0.02     0.49 1.01
    #> metac2zero1diff_Intercept    -0.30      0.14    -0.65    -0.06 1.02
    #> metac2zero2diff_Intercept    -0.32      0.13    -0.58    -0.07 1.01
    #> metac2one1diff_Intercept     -0.37      0.14    -0.65    -0.09 1.01
    #> metac2one2diff_Intercept     -0.27      0.11    -0.49    -0.04 1.00
    #>                           Bulk_ESS Tail_ESS
    #> Intercept                     4558     2762
    #> dprime_Intercept              1544     2244
    #> c_Intercept                    665     1556
    #> metac2zero1diff_Intercept      201       51
    #> metac2zero2diff_Intercept      759     1241
    #> metac2one1diff_Intercept       377      679
    #> metac2one2diff_Intercept      1771     1925
    #>
    #> 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).

## Data preparation

Fitting the trial-level model does not require data aggregation, however it still requires a small amount of data preparation. To fit the model, we will need two things:

- a column with the stimulus per trial (`0` or `1`), and
- a column containing the joint type 1/type 2 responses per trial (between `1` and `2*K`).

Our data already has a `stimulus` column but separate columns for the two responses. So, we can add in a joint response column now:

    d <- d |>
      mutate(joint_response = joint_response(response, confidence, K)) |>
      relocate(joint_response, .after = "stimulus")

    #> # A tibble: 2,500 × 17
    #>    participant  item trial stimulus joint_response response correct
    #>          <int> <int> <int>    <int>          <dbl>    <int>   <int>
    #>  1           1     1     1        0              1        0       1
    #>  2           1     1     1        1              5        1       1
    #>  3           1     2     1        0              3        0       1
    #>  4           1     2     1        1              6        1       1
    #>  5           1     3     1        0              1        0       1
    #>  6           1     3     1        1              2        0       0
    #>  7           1     4     1        0              1        0       1
    #>  8           1     4     1        1              6        1       1
    #>  9           1     5     1        0              2        0       1
    #> 10           1     5     1        1              3        0       0
    #> # ℹ 2,490 more rows
    #> # ℹ 10 more variables: confidence <int>, dprime <dbl>, c <dbl>,
    #> #   meta_dprime <dbl>, M <dbl>, meta_c2_0 <list>, meta_c2_1 <list>,
    #> #   theta <dbl>, theta_1 <dbl>, theta_2 <dbl>

## Model fitting

Now that we have our data, we can fit the trial-level model using `joint_response` as our response variable, `stimulus` as an extra variable passed to `brms`, and the argument `categorical=TRUE` to tell `fit_metad` not to aggregate the data:

    m.categorical <- fit_metad(
      bf(
        joint_response | vint(stimulus) ~ 1 + (1 | participant) + (1 | item),
        dprime + c +
          metac2zero1diff + metac2zero2diff +
          metac2one1diff + metac2one2diff ~
          1 + (1 | participant) + (1 | item)
      ),
      data = d, categorical = TRUE, init = 0, cores = 4,
      file = "models/categorical.rds",
      prior = prior(normal(0, .25), class = Intercept) +
        prior(normal(0, .25), class = Intercept, dpar = dprime) +
        prior(normal(0, .25), class = Intercept, dpar = c) +
        prior(normal(0, .1), class = Intercept, dpar = metac2zero1diff) +
        prior(normal(0, .1), class = Intercept, dpar = metac2zero2diff) +
        prior(normal(0, .1), class = Intercept, dpar = metac2one1diff) +
        prior(normal(0, .1), class = Intercept, dpar = metac2one2diff) +
        prior(normal(0, 1), class = sd) +
        prior(normal(0, 1), class = sd, dpar = dprime) +
        prior(normal(0, 1), class = sd, dpar = c) +
        prior(normal(0, 1), class = sd, dpar = metac2zero1diff) +
        prior(normal(0, 1), class = sd, dpar = metac2zero2diff) +
        prior(normal(0, 1), class = sd, dpar = metac2one1diff) +
        prior(normal(0, 1), class = sd, dpar = metac2one2diff)
    )

    #>  Family: metad__3__normal__absolute__categorical
    #>   Links: mu = log; dprime = identity; c = identity; metac2zero1diff = log; metac2zero2diff = log; metac2one1diff = log; metac2one2diff = log
    #> Formula: joint_response | vint(stimulus) ~ 1 + (1 | participant) + (1 | item)
    #>          dprime ~ 1 + (1 | participant) + (1 | item)
    #>          c ~ 1 + (1 | participant) + (1 | item)
    #>          metac2zero1diff ~ 1 + (1 | participant) + (1 | item)
    #>          metac2zero2diff ~ 1 + (1 | participant) + (1 | item)
    #>          metac2one1diff ~ 1 + (1 | participant) + (1 | item)
    #>          metac2one2diff ~ 1 + (1 | participant) + (1 | item)
    #>    Data: data.aggregated (Number of observations: 2500)
    #>   Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
    #>          total post-warmup draws = 4000
    #>
    #> Multilevel Hyperparameters:
    #> ~item (Number of levels: 25)
    #>                               Estimate Est.Error l-95% CI u-95% CI Rhat
    #> sd(Intercept)                     0.15      0.11     0.01     0.40 1.00
    #> sd(dprime_Intercept)              0.51      0.12     0.32     0.80 1.00
    #> sd(c_Intercept)                   0.64      0.10     0.47     0.87 1.00
    #> sd(metac2zero1diff_Intercept)     0.85      0.20     0.50     1.26 1.00
    #> sd(metac2zero2diff_Intercept)     0.70      0.24     0.14     1.16 1.00
    #> sd(metac2one1diff_Intercept)      0.58      0.27     0.05     1.06 1.01
    #> sd(metac2one2diff_Intercept)      0.79      0.19     0.46     1.20 1.00
    #>                               Bulk_ESS Tail_ESS
    #> sd(Intercept)                     1675     2246
    #> sd(dprime_Intercept)              1244     2394
    #> sd(c_Intercept)                   1832     2722
    #> sd(metac2zero1diff_Intercept)     1372     1765
    #> sd(metac2zero2diff_Intercept)      402      239
    #> sd(metac2one1diff_Intercept)       257      379
    #> sd(metac2one2diff_Intercept)      1453     2325
    #>
    #> ~participant (Number of levels: 50)
    #>                               Estimate Est.Error l-95% CI u-95% CI Rhat
    #> sd(Intercept)                     0.32      0.14     0.04     0.61 1.01
    #> sd(dprime_Intercept)              0.51      0.09     0.35     0.68 1.00
    #> sd(c_Intercept)                   0.31      0.04     0.24     0.41 1.00
    #> sd(metac2zero1diff_Intercept)     0.12      0.08     0.00     0.29 1.00
    #> sd(metac2zero2diff_Intercept)     0.34      0.12     0.14     0.64 1.01
    #> sd(metac2one1diff_Intercept)      0.35      0.15     0.07     0.74 1.00
    #> sd(metac2one2diff_Intercept)      0.16      0.10     0.01     0.38 1.00
    #>                               Bulk_ESS Tail_ESS
    #> sd(Intercept)                      777     1074
    #> sd(dprime_Intercept)              1547     2407
    #> sd(c_Intercept)                   1663     2537
    #> sd(metac2zero1diff_Intercept)     1421     2084
    #> sd(metac2zero2diff_Intercept)      496      335
    #> sd(metac2one1diff_Intercept)       349      520
    #> sd(metac2one2diff_Intercept)      1197     1815
    #>
    #> Regression Coefficients:
    #>                           Estimate Est.Error l-95% CI u-95% CI Rhat
    #> Intercept                    -0.32      0.11    -0.55    -0.11 1.00
    #> dprime_Intercept              1.04      0.14     0.74     1.29 1.00
    #> c_Intercept                   0.25      0.12    -0.00     0.49 1.00
    #> metac2zero1diff_Intercept    -0.28      0.12    -0.53    -0.05 1.00
    #> metac2zero2diff_Intercept    -0.32      0.13    -0.60    -0.07 1.00
    #> metac2one1diff_Intercept     -0.37      0.14    -0.65    -0.10 1.00
    #> metac2one2diff_Intercept     -0.27      0.11    -0.50    -0.05 1.00
    #>                           Bulk_ESS Tail_ESS
    #> Intercept                     4309     3080
    #> dprime_Intercept              2057     2154
    #> c_Intercept                   1040     1931
    #> metac2zero1diff_Intercept     2206     1900
    #> metac2zero2diff_Intercept      675      604
    #> metac2one1diff_Intercept       446      944
    #> metac2one2diff_Intercept      2391     2490
    #>
    #> 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).

As you can see, aside from the way that the data is formatted, this model is exactly the same as the multinomial model above.

## Extracting model estimates

Obtaining posterior estimates over model parameters, predictions, and other estimates is very similar to the multinomial model (for details, see `vignette("hmetad")`). So, here we will focus on the type 1 ROC curves, this time using `roc1_rvars` instead of `roc1_draws` for increased efficiency.

To get the posterior estimates, we need to specify a dataset to make predictions for, as well as a random effects formula to use in model predictions. For example, to estimate a ROC averaging over participants and items, we can use an empty data set with `re_formula=NA`:

    roc1_rvars(m.multinomial, tibble(.row = 1), re_formula = NA)
    #> # A tibble: 5 × 6
    #> # Groups:   .row, joint_response, response, confidence [5]
    #>    .row joint_response response confidence            p_fa         p_hit
    #>   <int>          <int>    <int>      <dbl>      <rvar[1d]>    <rvar[1d]>
    #> 1     1              1        0          3  0.785 ± 0.0568  0.95 ± 0.020
    #> 2     1              2        0          2  0.517 ± 0.0690  0.83 ± 0.043
    #> 3     1              3        0          1  0.221 ± 0.0421  0.60 ± 0.052
    #> 4     1              4        1          1  0.079 ± 0.0249  0.31 ± 0.060
    #> 5     1              5        1          2  0.016 ± 0.0077  0.10 ± 0.034

The process is exactly the same for the categorical model:

    roc1_rvars(m.categorical, tibble(.row = 1), re_formula = NA)
    #> # A tibble: 5 × 6
    #> # Groups:   .row, joint_response, response, confidence [5]
    #>    .row joint_response response confidence            p_fa         p_hit
    #>   <int>          <int>    <int>      <dbl>      <rvar[1d]>    <rvar[1d]>
    #> 1     1              1        0          3  0.790 ± 0.0553  0.95 ± 0.020
    #> 2     1              2        0          2  0.526 ± 0.0664  0.83 ± 0.041
    #> 3     1              3        0          1  0.224 ± 0.0433  0.60 ± 0.053
    #> 4     1              4        1          1  0.080 ± 0.0261  0.31 ± 0.061
    #> 5     1              5        1          2  0.017 ± 0.0082  0.10 ± 0.034

Next, to get participant-level ROCs (averaging over items), we can use a dataset with one row per participant and only the participant-level random effects:

    roc1_rvars(m.categorical, distinct(d, participant), re_formula = ~ (1 | participant))
    #> # A tibble: 250 × 7
    #> # Groups:   .row, participant, joint_response, response, confidence [250]
    #>     .row participant joint_response response confidence          p_fa
    #>    <int>       <int>          <int>    <int>      <dbl>    <rvar[1d]>
    #>  1     1           1              1        0          3  0.84 ± 0.075
    #>  2     2           2              1        0          3  0.89 ± 0.061
    #>  3     3           3              1        0          3  0.68 ± 0.106
    #>  4     4           4              1        0          3  0.73 ± 0.105
    #>  5     5           5              1        0          3  0.76 ± 0.096
    #>  6     6           6              1        0          3  0.86 ± 0.079
    #>  7     7           7              1        0          3  0.83 ± 0.082
    #>  8     8           8              1        0          3  0.68 ± 0.111
    #>  9     9           9              1        0          3  0.75 ± 0.099
    #> 10    10          10              1        0          3  0.87 ± 0.067
    #> # ℹ 240 more rows
    #> # ℹ 1 more variable: p_hit <rvar[1d]>

We can use a similar process to get item-level ROCs (averaging over participants):

    roc1_rvars(m.categorical, distinct(d, item), re_formula = ~ (1 | item))
    #> # A tibble: 125 × 7
    #> # Groups:   .row, item, joint_response, response, confidence [125]
    #>     .row  item joint_response response confidence          p_fa
    #>    <int> <int>          <int>    <int>      <dbl>    <rvar[1d]>
    #>  1     1     1              1        0          3  0.23 ± 0.057
    #>  2     2     2              1        0          3  0.26 ± 0.063
    #>  3     3     3              1        0          3  0.32 ± 0.065
    #>  4     4     4              1        0          3  0.62 ± 0.067
    #>  5     5     5              1        0          3  0.74 ± 0.062
    #>  6     6     6              1        0          3  0.36 ± 0.069
    #>  7     7     7              1        0          3  0.79 ± 0.057
    #>  8     8     8              1        0          3  0.31 ± 0.064
    #>  9     9     9              1        0          3  0.79 ± 0.057
    #> 10    10    10              1        0          3  0.57 ± 0.069
    #> # ℹ 115 more rows
    #> # ℹ 1 more variable: p_hit <rvar[1d]>

## Other benefits

Aside from representing the data in a more convenient format, the trial-level model should be more useful for things like model comparison using the `loo` package, multivariate models, and mediation models. These features should mostly work out of the box but they are still under active development, so stay tuned!

1. Note that in practice, fitting hierarchical models will usually require setting informed priors and adjusting the Stan sampler settings.↩︎