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 c meta_dprime M meta_c2_0
#> <int> <int> <int> <int> <int> <int> <int> <dbl> <dbl> <dbl> <dbl> <list>
#> 1 1 1 1 0 1 0 3 0.725 -0.843 0.614 0.847 <dbl [2]>
#> 2 1 1 1 1 1 1 3 0.725 -0.843 0.614 0.847 <dbl [2]>
#> 3 1 2 1 0 0 1 2 1.74 -1.24 1.12 0.642 <dbl [2]>
#> 4 1 2 1 1 1 1 3 1.74 -1.24 1.12 0.642 <dbl [2]>
#> 5 1 3 1 0 0 1 3 0.746 0.614 0.565 0.758 <dbl [2]>
#> 6 1 3 1 1 0 0 2 0.746 0.614 0.565 0.758 <dbl [2]>
#> 7 1 4 1 0 0 1 2 1.50 -0.0110 1.21 0.809 <dbl [2]>
#> 8 1 4 1 1 1 1 3 1.50 -0.0110 1.21 0.809 <dbl [2]>
#> 9 1 5 1 0 0 1 1 1.21 -0.231 1.13 0.931 <dbl [2]>
#> 10 1 5 1 1 1 1 1 1.21 -0.231 1.13 0.931 <dbl [2]>
#> # ℹ 2,490 more rows
#> # ℹ 4 more variables: 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 rowsAnd 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 rowsStandard 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)
#> `hmetad` has inferred that there are K=3 confidence levels in the data. If this is incorrect, please set this manually using the argument `K=<K>`
#> # A tibble: 1,250 × 5
#> participant item N_0 N_1 N[,"N_0_1"] [,"N_0_2"] [,"N_0_3"] [,"N_0_4"] [,"N_0_5"] [,"N_0_6"] [,"N_1_1"]
#> <int> <int> <int> <int> <int> <int> <int> <int> <int> <int> <int>
#> 1 1 1 1 1 0 0 0 0 0 1 0
#> 2 1 2 1 1 0 1 0 0 0 0 0
#> 3 1 3 1 1 1 0 0 0 0 0 0
#> 4 1 4 1 1 0 1 0 0 0 0 0
#> 5 1 5 1 1 0 0 1 0 0 0 0
#> 6 1 6 1 1 1 0 0 0 0 0 0
#> 7 1 7 1 1 0 0 0 0 1 0 0
#> 8 1 8 1 1 1 0 0 0 0 0 0
#> 9 1 9 1 1 1 0 0 0 0 0 0
#> 10 1 10 1 1 1 0 0 0 0 0 0
#> # ℹ 1,240 more rows
#> # ℹ 1 more variable: N[8:12] <int>As you can see, the aggregated data set has 1250 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
# Priors are chosen arbitrarily for this example.
# Please choose your own wisely!
priors <- prior(normal(0, .25), class = Intercept) +
set_prior(
"normal(0, .25)",
class = "Intercept",
dpar = c("dprime", "c")
) +
set_prior("normal(-0.5, .1)", class = "Intercept", dpar = metac2_parameters(K = 3)) +
prior(normal(0, 1), class = sd) +
set_prior("normal(0, 1)", class = "sd", dpar = c("dprime", "c", metac2_parameters(K = 3)))
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, prior = priors
)
#> `hmetad` has inferred that there are K=3 confidence levels in the data. If this is incorrect, please set this manually using the argument `K=<K>`
#> Compiling Stan program...
#> Start sampling
#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess#> 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 Bulk_ESS Tail_ESS
#> sd(Intercept) 0.14 0.10 0.01 0.37 1.00 1074 1538
#> sd(dprime_Intercept) 0.57 0.14 0.35 0.88 1.00 757 1379
#> sd(c_Intercept) 0.76 0.12 0.57 1.03 1.00 614 1061
#> sd(metac2zero1diff_Intercept) 0.07 0.05 0.00 0.20 1.00 1876 1525
#> sd(metac2zero2diff_Intercept) 0.13 0.09 0.01 0.32 1.00 927 1399
#> sd(metac2one1diff_Intercept) 0.11 0.08 0.00 0.31 1.00 1085 1964
#> sd(metac2one2diff_Intercept) 0.18 0.10 0.01 0.41 1.00 921 1546
#>
#> ~participant (Number of levels: 50)
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sd(Intercept) 0.12 0.09 0.01 0.32 1.00 1247 1931
#> sd(dprime_Intercept) 0.44 0.08 0.29 0.62 1.00 1200 2111
#> sd(c_Intercept) 0.35 0.05 0.27 0.45 1.00 1329 2091
#> sd(metac2zero1diff_Intercept) 0.19 0.10 0.02 0.38 1.01 717 955
#> sd(metac2zero2diff_Intercept) 0.30 0.10 0.11 0.49 1.01 1012 1093
#> sd(metac2one1diff_Intercept) 0.20 0.10 0.02 0.41 1.01 815 1241
#> sd(metac2one2diff_Intercept) 0.36 0.10 0.15 0.56 1.00 887 716
#>
#> Regression Coefficients:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> Intercept -0.23 0.09 -0.41 -0.07 1.00 3386 2946
#> dprime_Intercept 1.10 0.15 0.77 1.36 1.00 930 1255
#> c_Intercept 0.15 0.14 -0.12 0.42 1.02 275 496
#> metac2zero1diff_Intercept -0.83 0.06 -0.93 -0.72 1.00 3284 2931
#> metac2zero2diff_Intercept -0.79 0.06 -0.91 -0.67 1.00 2643 2965
#> metac2one1diff_Intercept -0.84 0.06 -0.95 -0.71 1.00 3121 2733
#> metac2one2diff_Intercept -0.79 0.07 -0.92 -0.65 1.00 2296 2603
#>
#> 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 (
0or1), and - a column containing the joint type 1/type 2 responses per trial
(between
1and2*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 confidence dprime c meta_dprime M
#> <int> <int> <int> <int> <int> <int> <int> <int> <dbl> <dbl> <dbl> <dbl>
#> 1 1 1 1 0 6 1 0 3 0.725 -0.843 0.614 0.847
#> 2 1 1 1 1 6 1 1 3 0.725 -0.843 0.614 0.847
#> 3 1 2 1 0 2 0 1 2 1.74 -1.24 1.12 0.642
#> 4 1 2 1 1 6 1 1 3 1.74 -1.24 1.12 0.642
#> 5 1 3 1 0 1 0 1 3 0.746 0.614 0.565 0.758
#> 6 1 3 1 1 2 0 0 2 0.746 0.614 0.565 0.758
#> 7 1 4 1 0 2 0 1 2 1.50 -0.0110 1.21 0.809
#> 8 1 4 1 1 6 1 1 3 1.50 -0.0110 1.21 0.809
#> 9 1 5 1 0 3 0 1 1 1.21 -0.231 1.13 0.931
#> 10 1 5 1 1 4 1 1 1 1.21 -0.231 1.13 0.931
#> # ℹ 2,490 more rows
#> # ℹ 5 more variables: 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, prior = priors
)
#> `hmetad` has inferred that there are K=3 confidence levels in the data. If this is incorrect, please set this manually using the argument `K=<K>`
#> Compiling Stan program...
#> Start sampling#> 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 Bulk_ESS Tail_ESS
#> sd(Intercept) 0.14 0.10 0.01 0.37 1.00 1467 1998
#> sd(dprime_Intercept) 0.57 0.14 0.36 0.89 1.00 854 1603
#> sd(c_Intercept) 0.76 0.12 0.57 1.02 1.00 792 1543
#> sd(metac2zero1diff_Intercept) 0.07 0.05 0.00 0.19 1.00 2130 2015
#> sd(metac2zero2diff_Intercept) 0.13 0.09 0.01 0.33 1.00 1264 1853
#> sd(metac2one1diff_Intercept) 0.12 0.09 0.00 0.31 1.00 1053 1268
#> sd(metac2one2diff_Intercept) 0.18 0.10 0.02 0.41 1.00 927 1443
#>
#> ~participant (Number of levels: 50)
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sd(Intercept) 0.12 0.09 0.00 0.31 1.00 1295 1610
#> sd(dprime_Intercept) 0.44 0.08 0.29 0.62 1.00 1513 2242
#> sd(c_Intercept) 0.35 0.05 0.27 0.45 1.00 1330 2340
#> sd(metac2zero1diff_Intercept) 0.18 0.10 0.01 0.38 1.00 739 1234
#> sd(metac2zero2diff_Intercept) 0.30 0.10 0.07 0.51 1.01 698 502
#> sd(metac2one1diff_Intercept) 0.21 0.10 0.02 0.41 1.00 702 973
#> sd(metac2one2diff_Intercept) 0.36 0.10 0.17 0.56 1.01 954 837
#>
#> Regression Coefficients:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> Intercept -0.23 0.09 -0.42 -0.06 1.00 3799 2616
#> dprime_Intercept 1.09 0.15 0.78 1.35 1.00 851 1703
#> c_Intercept 0.14 0.13 -0.12 0.40 1.01 501 1045
#> metac2zero1diff_Intercept -0.83 0.05 -0.93 -0.72 1.00 4309 2970
#> metac2zero2diff_Intercept -0.79 0.06 -0.91 -0.66 1.00 2891 2937
#> metac2one1diff_Intercept -0.83 0.06 -0.95 -0.71 1.00 2537 2565
#> metac2one2diff_Intercept -0.79 0.07 -0.92 -0.65 1.00 2479 2625
#>
#> 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 0.600 ± 0.058 0.89 ± 0.031
#> 2 1 2 0 2 0.413 ± 0.058 0.79 ± 0.047
#> 3 1 3 0 1 0.244 ± 0.048 0.65 ± 0.059
#> 4 1 4 1 1 0.136 ± 0.034 0.47 ± 0.062
#> 5 1 5 1 2 0.063 ± 0.020 0.29 ± 0.054The 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 0.605 ± 0.057 0.89 ± 0.028
#> 2 1 2 0 2 0.418 ± 0.057 0.79 ± 0.043
#> 3 1 3 0 1 0.248 ± 0.047 0.66 ± 0.055
#> 4 1 4 1 1 0.138 ± 0.033 0.47 ± 0.058
#> 5 1 5 1 2 0.064 ± 0.019 0.30 ± 0.051Next, 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 p_hit
#> <int> <int> <int> <int> <dbl> <rvar[1d]> <rvar[1d]>
#> 1 1 1 1 0 3 0.54 ± 0.103 0.93 ± 0.038
#> 2 2 2 1 0 3 0.75 ± 0.077 0.91 ± 0.042
#> 3 3 3 1 0 3 0.84 ± 0.066 0.96 ± 0.024
#> 4 4 4 1 0 3 0.45 ± 0.099 0.78 ± 0.073
#> 5 5 5 1 0 3 0.63 ± 0.093 0.93 ± 0.035
#> 6 6 6 1 0 3 0.44 ± 0.099 0.91 ± 0.043
#> 7 7 7 1 0 3 0.60 ± 0.092 0.88 ± 0.051
#> 8 8 8 1 0 3 0.40 ± 0.095 0.67 ± 0.088
#> 9 9 9 1 0 3 0.72 ± 0.083 0.88 ± 0.052
#> 10 10 10 1 0 3 0.50 ± 0.099 0.68 ± 0.087
#> # ℹ 240 more rowsWe 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 p_hit
#> <int> <int> <int> <int> <dbl> <rvar[1d]> <rvar[1d]>
#> 1 1 1 1 0 3 0.92 ± 0.026 0.98 ± 0.0115
#> 2 2 2 1 0 3 0.93 ± 0.023 1.00 ± 0.0023
#> 3 3 3 1 0 3 0.59 ± 0.064 0.74 ± 0.0521
#> 4 4 4 1 0 3 0.50 ± 0.066 0.91 ± 0.0292
#> 5 5 5 1 0 3 0.71 ± 0.055 0.93 ± 0.0252
#> 6 6 6 1 0 3 0.50 ± 0.067 0.92 ± 0.0249
#> 7 7 7 1 0 3 0.47 ± 0.067 0.84 ± 0.0413
#> 8 8 8 1 0 3 0.43 ± 0.064 0.89 ± 0.0324
#> 9 9 9 1 0 3 0.56 ± 0.065 0.88 ± 0.0334
#> 10 10 10 1 0 3 0.41 ± 0.065 0.87 ± 0.0349
#> # ℹ 115 more rowsOther 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!