Differing posterior predictive checks for logistic binomial model with and without addition-terms

Hi all,

Not so much an issue as double-checking expected behaviour:

I’m fitting a logistic binomial model where the response variable is the sum of how many times a target picture was looked in a certain time period (out of how many looks there were at all pictures during that period). This kind of response variable falls under “addition-terms” according to the brms documentation. The model estimates are plausible and the fit is good, but the posterior predictive check isn’t as good as if I fit a regular logistic binomial model to the same data but unaggregated. Below is a dummy example with models with and without addition-terms.

My questions are:

  • The posterior predictive check of the addition-terms model isn’t actually bad, but is it expected that it would not be as clean as the regular model? And if so, why?
  • Out of curiosity, is there any way to do a predictive check with the addition-terms model on the binomial scale rather than predicting the sum part of the addition-terms?
# fake data

## long format
(dat_long <- data.frame(
  subj = rep(1:10, each = 100),
  item = rep(1:10, each = 10),
  bin  = rep(1:10, times = 10),
  cond = c(-.5,.5),
  pTarget = rbinom(1000, 1, .6)
))

## aggregated over bins/items
(dat_aggregated <- dat_long %>%
  dplyr::group_by(subj, cond) %>%
  dplyr::summarise(sum = sum(pTarget), N = length(pTarget)))


# model using addition-terms
m_aggregated <- brm(formula = sum | trials(N) ~ cond,
                     family = binomial(),
                     iter = 5000,
                     prior = priors,
                     data =  dat_aggregated)

# regular model
m_long <- brm(formula = pTarget ~ cond,
                   iter = 1000,
                   family = binomial(),
                   prior = priors,
                   data =  dat_long)


# posterior predictive checks
pp_check(m_aggregated)
pp_check(m_long)

image
image

  • Operating System: Windows 10
  • brms Version: 2.14

Thanks!

2 Likes

Sorry for not getting to you earlier, your question is relevant and well written and I am glad you came up with a neat minimal example of the problem.

I would prefer to call the models “aggregated” and “disaggregated”, because you could (and maybe even should - for clarity) write the disaggregated model with addition terms (pTarget | trials(1) ~ cond) and addition terms are also used for other very different things.

I think what you observe is a combination of few things:

  1. the default “density” check is not very useful for binary outcomes (or generally discrete outcomes with few possible options). We know that only 0/1 outcomes are possible, so most of the density plot tells us basically nothing useful and only makes the scale weird. Using pp_check(m_long, type = "bars", nsamples = 1000) gives IMHO a better look:

  1. The aggregated model has much more possible outcomes for each row of the dataset - if we expect say 25 out of 40 trials to turn out succesful, seeing 24 or 26 is just not surprising at all. In the disaggregated model this gets distributed across many rows and averaged out, but looking at the aggregated model, we just expect a bit more variability. Once again, the density plot is a bit misleading as the actual data are not so smooth - when I run the bars check for the aggregated model, I get an ugly but somewhat revealing plot:

So we see that the actual data y are not really smooth and we expect a lot of variability in the middle portion of the plot (hope it is clear what the plot shows, if not, feel free to ask for clarification).

Yes, but you’ll need to do this manually - you can make predictions with posterior_predict, divide them by the number of trials for each row and do the same for the observed data, e.g.:

pred <- posterior_predict(m_aggregated, cores = 1)
# Divide the samples for each row by the corresponding number of trials
pred_divided <- sweep(pred, MARGIN = 1, STATS = dat_aggregated$N, FUN = "/") 
y_divided <- dat_aggregated$sum / dat_aggregated$N

samples_to_show <- sample(1:(dim(pred_divided)[1]), size = 30)
bayesplot::ppc_dens_overlay(y_divided, pred_divided[samples_to_show, ])

Hope that helps at least a little.

5 Likes

Thanks for the response Martin, it was extremely helpful!

Absolutely amazing!

Martin, I’ve been trying to adapt your approach to bayesplot::ppc_dens_overlay_grouped(), but can’t wrap my mind around it.

How would you plot this pp_check for each “cond” group?

This should be IMHO relatively straightforward - the format of both predicted and actual data is the same and you should be able to just add group = your_dataframe$cond and it should work.

Got it, thanks!

I have a couple of more questions. I re-wrote the code chunks showed before to better explain my rationale.

First, I will replicate the dataset in which all rows contain N = 50 and fit the model.

# Install/load packages
pacman::p_load(magrittr,
               dplyr,
               brms)

pacman::p_load_gh("stan/cmdstanr")

# fake data

## Version 1 (N = 50) ----

## long format
set.seed(123)
dat_long <- data.frame(
  subj = rep(1:10, each = 100),
  item = rep(1:10, each = 10),
  bin  = rep(1:10, times = 10),
  cond = c(-.5,.5),
  pTarget = rbinom(1000, 1, .6)
)

## aggregated over bins/items
dat_aggregated1 <- dat_long %>%
    dplyr::group_by(subj, cond) %>%
    dplyr::summarise(sum = sum(pTarget), N = length(pTarget))

# model using addition-terms
m_aggregated1 <- brm(formula = sum | trials(N) ~ cond,
                    family = binomial(),
                    iter = 5000,
                    backend = "cmdstanr",
                    data =  dat_aggregated1)

And here is the PPC code.

I generated figures using both brms::pp_check() and your approach to compare the shape of Y in each output. I noticed you used 1 for MARGIN in sweep(), which is a rowwise operation. Honestly I do not fully understand how MARGIN = 1 worked here, since length(dat_aggregated1$N) = 20 and dim(pred1) = 10000 20. I expected that we would need a vector with 10000 values in STATS when using MARGIN = 1 in this case. Nevertheless, it worked and both Y look identical.

# posterior predictive check

set.seed(123)
pred1 <- brms::posterior_predict(m_aggregated1, cores = 1)

# Divide the samples for each row by the corresponding number of trials
pred_divided1_row <- sweep(pred1,
                       MARGIN = 1, #rowwise
                       STATS = dat_aggregated1$N, # length = 20
                       FUN = "/") 

y_divided1 <- dat_aggregated1$sum / dat_aggregated1$N
samples_to_show1 <- sample(1:(dim(pred_divided1_row)[1]), size = 30)

# Same Y
brms::pp_check(m_aggregated1, ndraws = 30)
bayesplot::ppc_dens_overlay(y_divided1, pred_divided1_row[samples_to_show1, ])

I then generated grouped PPCs, and again both look perfect.

# Same Y
brms::pp_check(m_aggregated1,
               type = "dens_overlay_grouped",
               group = "cond",
               ndraws = 30)
bayesplot::ppc_dens_overlay_grouped(y_divided1,
                                    pred_divided1_row[samples_to_show1, ],
                                    group = dat_aggregated1$cond)

I now tried to generate the same figure with MARGIN = 2 (column wise operation) instead, which in this case looked identical to MARGIN = 1 output

# Above you ran sweep() with MARGIN = 1, thus rowwise, using
# values from dat_aggregated1$N, which has length = 20, for STATS

# pred1, however, has 20 columns, not 20 rows
dim(pred1)

# Now let's use MARGIN = 2, thus columnwise
pred_divided1_col <- sweep(pred1,
                           MARGIN = 2, #columnwise
                           STATS = dat_aggregated1$N, # length = 20
                           FUN = "/") 

# Same Y
bayesplot::ppc_dens_overlay(y_divided1,
                            # MARGIN = 1
                            pred_divided1_row[samples_to_show1, ])

bayesplot::ppc_dens_overlay(y_divided1, 
                            # MARGIN = 2
                            pred_divided1_col[samples_to_show1, ])

# Same Y
bayesplot::ppc_dens_overlay_grouped(y_divided1,
                                    # MARGIN = 1
                                    pred_divided1_row[samples_to_show1, ],
                                    group = dat_aggregated1$cond)

bayesplot::ppc_dens_overlay_grouped(y_divided1,
                                    # MARGIN = 2
                                    pred_divided1_col[samples_to_show1, ],
                                    group = dat_aggregated1$cond)

Later, I tried to replicate this workflow with a personal dataset and it looked very odd. I figured that it wasn’t working because my dataset does not contain identical Ns in each row, as it was the case above (N = 50).

Thus I now generated a fake dataset with different N and fit a corresponding model.

# Version 2 (varying N)

set.seed(123)
dat_aggregated2 <- dat_long %>%
  dplyr::group_by(subj, cond) %>%
  dplyr::summarise(sum = sum(pTarget), N = length(pTarget)) %>%
  # Add numbers to randomly change N rowwise
  dplyr::mutate(N = (N + round(runif(1, 0, 30))))

# model using addition-terms
m_aggregated2 <- brm(formula = sum | trials(N) ~ cond,
                     family = binomial(),
                     iter = 5000,
                     backend = "cmdstanr",
                     data =  dat_aggregated2)

And here is the PPC code.

# posterior predictive check

set.seed(123)
pred2 <- brms::posterior_predict(m_aggregated2, cores = 1)

# Divide the samples for each row by the corresponding number of trials
pred_divided2_row <- sweep(pred2,
                       MARGIN = 1, #rowwise
                       STATS = dat_aggregated2$N,
                       FUN = "/") 
y_divided2 <- dat_aggregated2$sum / dat_aggregated2$N
samples_to_show2 <- sample(1:(dim(pred_divided2_row)[1]), size = 30)

# Different Y!
brms::pp_check(m_aggregated2, ndraws = 30)
bayesplot::ppc_dens_overlay(y_divided2, pred_divided2_row[samples_to_show2, ])

# Different Y!
brms::pp_check(m_aggregated2,
               type = "dens_overlay_grouped",
               group = "cond",
               ndraws = 30)
bayesplot::ppc_dens_overlay_grouped(y_divided2,
                                    pred_divided2_row[samples_to_show2, ],
                                    group = dat_aggregated2$cond)

# Now with MARGIN = 2
pred_divided2_col <- sweep(pred2,
                           MARGIN = 2, #columnwise
                           STATS = dat_aggregated2$N, # length = 20
                           FUN = "/") 

# Different Y
brms::pp_check(m_aggregated2, ndraws = 30)
bayesplot::ppc_dens_overlay(y_divided2, pred_divided2_col[samples_to_show2, ])

# Same Y
brms::pp_check(m_aggregated2,
               type = "dens_overlay_grouped",
               group = "cond",
               ndraws = 30)
bayesplot::ppc_dens_overlay_grouped(y_divided2,
                                    pred_divided2_col[samples_to_show2, ],
                                    group = dat_aggregated2$cond)


# MARGINs 1 vs. 2

# Different Y!
bayesplot::ppc_dens_overlay(y_divided2,
                            # MARGIN = 1
                            pred_divided2_row[samples_to_show2, ])

bayesplot::ppc_dens_overlay(y_divided2,
                            # MARGIN = 2
                            pred_divided2_col[samples_to_show2, ])

# Different Y!
bayesplot::ppc_dens_overlay_grouped(y_divided2,
                                    # MARGIN = 1
                                    pred_divided2_row[samples_to_show2, ],
                                    group = dat_aggregated2$cond)
bayesplot::ppc_dens_overlay_grouped(y_divided2,
                                    # MARGIN = 2
                                    pred_divided2_col[samples_to_show2, ],
                                    group = dat_aggregated2$cond)

Oddly, the shape of Y is different in each plot, ie with brms::pp_check(), MARGIN = 1, MARGIN = 2, grouped or not…

I do not understand why this is happening. Do you?

Lastly, shouldn’t MARGIN = 2 be the default here, given that brms::posterior_predict() generates a matrix where each column corresponds to each row in the original dataframe?

Thanks!

Full code below:

# Install/load packages
pacman::p_load(magrittr,
               dplyr,
               brms)

pacman::p_load_gh("stan/cmdstanr")

# fake data

## Version 1 (N = 50) ----

## long format
set.seed(123)
dat_long <- data.frame(
  subj = rep(1:10, each = 100),
  item = rep(1:10, each = 10),
  bin  = rep(1:10, times = 10),
  cond = c(-.5,.5),
  pTarget = rbinom(1000, 1, .6)
)

## aggregated over bins/items
dat_aggregated1 <- dat_long %>%
    dplyr::group_by(subj, cond) %>%
    dplyr::summarise(sum = sum(pTarget), N = length(pTarget))

# model using addition-terms
m_aggregated1 <- brm(formula = sum | trials(N) ~ cond,
                    family = binomial(),
                    iter = 5000,
                    backend = "cmdstanr",
                    data =  dat_aggregated1)


# posterior predictive check

set.seed(123)
pred1 <- brms::posterior_predict(m_aggregated1, cores = 1)

# Divide the samples for each row by the corresponding number of trials
pred_divided1_row <- sweep(pred1,
                       MARGIN = 1, #rowwise
                       STATS = dat_aggregated1$N, # length = 20
                       FUN = "/") 

y_divided1 <- dat_aggregated1$sum / dat_aggregated1$N
samples_to_show1 <- sample(1:(dim(pred_divided1_row)[1]), size = 30)

# Same Y
brms::pp_check(m_aggregated1, ndraws = 30)
bayesplot::ppc_dens_overlay(y_divided1, pred_divided1_row[samples_to_show1, ])

# Same Y
brms::pp_check(m_aggregated1,
               type = "dens_overlay_grouped",
               group = "cond",
               ndraws = 30)
bayesplot::ppc_dens_overlay_grouped(y_divided1,
                                    pred_divided1_row[samples_to_show1, ],
                                    group = dat_aggregated1$cond)

# Above you ran sweep() with MARGIN = 1, thus rowwise, using
# values from dat_aggregated1$N, which has length = 20, for STATS

# pred1, however, has 20 columns, not 20 rows
dim(pred1)

# Now let's use MARGIN = 2, thus columnwise
pred_divided1_col <- sweep(pred1,
                           MARGIN = 2, #columnwise
                           STATS = dat_aggregated1$N, # length = 20
                           FUN = "/") 

# Same Y
bayesplot::ppc_dens_overlay(y_divided1,
                            # MARGIN = 1
                            pred_divided1_row[samples_to_show1, ])

bayesplot::ppc_dens_overlay(y_divided1, 
                            # MARGIN = 2
                            pred_divided1_col[samples_to_show1, ])

# Same Y
bayesplot::ppc_dens_overlay_grouped(y_divided1,
                                    # MARGIN = 1
                                    pred_divided1_row[samples_to_show1, ],
                                    group = dat_aggregated1$cond)

bayesplot::ppc_dens_overlay_grouped(y_divided1,
                                    # MARGIN = 2
                                    pred_divided1_col[samples_to_show1, ],
                                    group = dat_aggregated1$cond)

# Version 2 (varying N)

set.seed(123)
dat_aggregated2 <- dat_long %>%
  dplyr::group_by(subj, cond) %>%
  dplyr::summarise(sum = sum(pTarget), N = length(pTarget)) %>%
  # Add numbers to randomly change N rowwise
  dplyr::mutate(N = (N + round(runif(1, 0, 30))))

# model using addition-terms
m_aggregated2 <- brm(formula = sum | trials(N) ~ cond,
                     family = binomial(),
                     iter = 5000,
                     backend = "cmdstanr",
                     data =  dat_aggregated2)


# posterior predictive check

set.seed(123)
pred2 <- brms::posterior_predict(m_aggregated2, cores = 1)

# Divide the samples for each row by the corresponding number of trials
pred_divided2_row <- sweep(pred2,
                       MARGIN = 1, #rowwise
                       STATS = dat_aggregated2$N,
                       FUN = "/") 
y_divided2 <- dat_aggregated2$sum / dat_aggregated2$N
samples_to_show2 <- sample(1:(dim(pred_divided2_row)[1]), size = 30)

# Different Y!
brms::pp_check(m_aggregated2, ndraws = 30)
bayesplot::ppc_dens_overlay(y_divided2, pred_divided2_row[samples_to_show2, ])

# Different Y!
brms::pp_check(m_aggregated2,
               type = "dens_overlay_grouped",
               group = "cond",
               ndraws = 30)
bayesplot::ppc_dens_overlay_grouped(y_divided2,
                                    pred_divided2_row[samples_to_show2, ],
                                    group = dat_aggregated2$cond)

# Now with MARGIN = 2
pred_divided2_col <- sweep(pred2,
                           MARGIN = 2, #columnwise
                           STATS = dat_aggregated2$N, # length = 20
                           FUN = "/") 

# Different Y
brms::pp_check(m_aggregated2, ndraws = 30)
bayesplot::ppc_dens_overlay(y_divided2, pred_divided2_col[samples_to_show2, ])

# Same Y
brms::pp_check(m_aggregated2,
               type = "dens_overlay_grouped",
               group = "cond",
               ndraws = 30)
bayesplot::ppc_dens_overlay_grouped(y_divided2,
                                    pred_divided2_col[samples_to_show2, ],
                                    group = dat_aggregated2$cond)


# MARGINs 1 vs. 2

# Different Y!
bayesplot::ppc_dens_overlay(y_divided2,
                            # MARGIN = 1
                            pred_divided2_row[samples_to_show2, ])

bayesplot::ppc_dens_overlay(y_divided2,
                            # MARGIN = 2
                            pred_divided2_col[samples_to_show2, ])

# Different Y!
bayesplot::ppc_dens_overlay_grouped(y_divided2,
                                    # MARGIN = 1
                                    pred_divided2_row[samples_to_show2, ],
                                    group = dat_aggregated2$cond)
bayesplot::ppc_dens_overlay_grouped(y_divided2,
                                    # MARGIN = 2
                                    pred_divided2_col[samples_to_show2, ],
                                    group = dat_aggregated2$cond)