# Modeling human response time data with an exponential model: model comparison issues

Hi Community,
The graph below displays mean reaction time human data as a function of age in a simple two-choice task with two experimental conditions (compatible and incompatible).

I would like to model these data with N(μ, σ), with either
μ = α * pm.math.exp(-β * Age) + γ [exponential model]
or μ = α * Age**-β + γ [power model]
For each model variant (exponential vs power), I would further like to constrain β to be identical between compatible and incompatible conditions.

For the power model, I computed:

with pm.Model() as model_pow:
α_c = pm.HalfCauchy('α_c', 10)
α_i = pm.HalfCauchy('α_i', 10)
β = pm.Normal('β', 1, 0.5)
γ_c = pm.Normal('γ_c', Mean_RT_comp_Indiv.mean(), .15)
γ_i = pm.Normal('γ_i', Mean_RT_incomp_Indiv.mean(), .15)
σ = pm.HalfNormal('σ', .1)
μ_c = α_c*Years_indiv**-β + γ_c
μ_i = α_i*Years_indiv**-β + γ_i
y_obs_comp = pm.Normal('y_obs_comp', μ_c, σ, observed=Mean_RT_comp_Indiv)
y_obs_incomp = pm.Normal('y_obs_incomp', μ_i, σ, observed=Mean_RT_incomp_Indiv)

if __name__ == "__main__":
with  model_pow:
trace_power = pm.sample (4000, chains = 4, core = 4, tune = 2000, target_accept = .95)
az.plot_trace(trace_power)


For the exponential model, I computed:

with pm.Model() as model_exp:
α_c = pm.HalfCauchy('α_c', 10)
α_i = pm.HalfCauchy('α_i', 10)
β = pm.Normal('β', 1, 0.5)
γ_c = pm.Normal('γ_c', Mean_RT_comp_Indiv.mean(), .15)
γ_i = pm.Normal('γ_i', Mean_RT_incomp_Indiv.mean(), .15)
σ = pm.HalfNormal('σ', .1)
μ_c = α_c*pm.math.exp(-β*Years_indiv) + γ_c
μ_i = α_i*pm.math.exp(-β*Years_indiv) + γ_i
y_obs_comp_exp = pm.Normal('y_obs_comp_exp', μ_c, σ, observed=Mean_RT_comp_Indiv)
y_obs_incomp_exp = pm.Normal('y_obs_incomp_exp', μ_c, σ, observed=Mean_RT_incomp_Indiv)
if __name__ == "__main__":
with  model_exp:
trace_exp = pm.sample (4000, chains = 4, core = 4, tune = 2000, target_accept = .95)
az.plot_trace(trace_exp)


Unfortunately, because I am new on this forum, I can’t incorporate additional files showing trace plots. I see sampling issues and get warnings such as:

There were 13 divergences after tuning. Increase target_accept or reparameterize.
There were 3 divergences after tuning. Increase target_accept or reparameterize.
The acceptance probability does not match the target. It is 0.879040376552122, but should be close to 0.8. Try to increase the number of tuning steps.
There were 18 divergences after tuning. Increase target_accept or reparameterize.
The number of effective samples is smaller than 25% for some parameters.

How can I solve or attenuate these sampling issues?

In addition, I would like to compare the performance of the power vs the exponential model using WAIC. I thus computed:

WAIC = az.compare({‘pow’:trace_power, ‘exp’:trace_exp})

returning the following error:

TypeError Traceback (most recent call last)
in
----> 1 WAIC = az.compare({‘pow’:trace_power, ‘exp’:trace_exp})
2 WAIC

~\Anaconda3\envs\bayes\lib\site-packages\arviz\stats\stats.py in compare(dataset_dict, ic, method, b_samples, alpha, seed, scale)
153 for name, dataset in dataset_dict.items():
154 names.append(name)
–> 155 ics = ics.append([ic_func(dataset, pointwise=True, scale=scale)])
156 ics.index = names
157 ics.sort_values(by=ic, inplace=True, ascending=ascending)

~\Anaconda3\envs\bayes\lib\site-packages\arviz\stats\stats.py in waic(data, pointwise, scale)
987 )
988 if “log_likelihood” not in inference_data.sample_stats:
–> 989 raise TypeError(“Data must include log_likelihood in sample_stats”)
990 log_likelihood = inference_data.sample_stats.log_likelihood
991

TypeError: Data must include log_likelihood in sample_stats

It looks like I need to compute the log likelihood but don’t know how to proceed. Any idea?

1 Like

Just from looking at the dataset that looks like a within subject design, is that correct? (ie you measured both comp and incomp for all subjects, so the measurements come in pairs). A linear model of the log(reation_times) or a glm with log link sound like the easiest models for a dataset like that (I don’t have any clue about reaction time modeling, so take that with a grain of salt). Or do you have specific reasons for your exponential and power model? The linear model would look something like that:

data = pd.DataFrame({
'subject_id': [0, 0, 1, 1, 2, 2, 3, 3, 4, 4],
'comp': [False, True] * 5,
'age': [12, 12, 13, 13, 14, 14, 25, 25, 40, 40],
'reaction_time': np.exp(np.random.randn(10)),
})
n_subjects = data.subject_id.nunique()

with pm.Model() as model:
intercept = pm.Flat('intercept')

sd = pm.HalfNormal('subject_sd', sd=2)
raw = pm.Normal('subject_raw', shape=n_subjects)
subject = pm.Deterministic('subject', sd * raw)

# Here we assume that the age has a linear relationship with the log(reaction_time)
age = pm.Normal('age', sd=2) # TODO what scale parameter is reasonable?

comp = pm.Normal('comp', sd=2)

mu = (
intercept
+ comp * data.comp
+ age * data.age
+ subject[data.subject_id])

# Only for convenience to make residual plots easier
pm.Deterministic('mu', mu)

sd = pm.HalfCauchy('sd', beta=2)
pm.Normal('log_reaction_time', mu=mu, sd=sd, observed=np.log(data.reaction_time))


This corresponds somewhat to the exponential model, as

\log(r) = \alpha + \beta \cdot \text{age} \\ r = \exp(\beta \cdot \text{age})\exp(\alpha).

You get the power model by replacing age = pm.Normal('age') by log_age = pm.Normal('log_age') and then also do + log_age * np.log(data.age) in the computation of mu. That is because

\log(r) = \alpha + \beta \cdot \log(\text{age}) \\ r = \exp(\alpha) \cdot \text{age}^\beta.

Instead of trusting formal model comparison (use LOO instead of WAIC) too much, I’d rather look at posterior predictive samples and plot the residuum. That allows you to figure out why one model is better.

1 Like

Oh, and you can get a quick idea about which of the two models might be better by looking at a log or a log-log plot. In the exponential model a plot of log(reation_time) vs age should be more or less a line. If it follows a power law you would expect log(reation_time) vs log(age) to be a straight line.

1 Like

Dear aseybolt,
Thank you so much for your help and interesting comments. I very much appreciate.
You are right, it is a within-subject design. Let me give you a little more information about that project. Previous research in developmental psychology has shown that age-related changes in mean RT in various tasks are well described by an exponential function of the form:

y = \gamma + \alpha \exp(-\beta x)

More specifically, it has been shown that (1) the rate of decay \beta remains the same whatever the task, and is equal to .334 (beautiful isn’t it?) (2) asymptotic performance levels are attained at the adult age.

We collected RT data in a decision-making task (within-subject design, two experimental conditions) from children of different ages. I then fit each individual dataset with a popular decision-making model to decompose the data into components of processing (this model has 5 parameters called v, a, ter, max_ampl, tau; each parameter quantifies a specific processing component) and determine whether (1) components of processing also follow an exponential-like function, and (2), whether all these components decay at the same rate as the mean RT data. To remove some noise, I averaged data of children from the same grade.

I thus fit the following model to data:

with pm.Model() as model_exp:
α_v = pm.HalfCauchy('α_v', 2)
α_a = pm.HalfCauchy('α_a', 2)
α_Ter = pm.HalfCauchy('α_Ter', 2)
α_max_ampl = pm.HalfCauchy('α_max_ampl', 2)
α_tau = pm.HalfCauchy('α_tau', 2)
α_c = pm.HalfCauchy('α_c',2)
α_i = pm.HalfCauchy('α_i', 2)

β = pm.Normal('β', 0.334, 0.1)
γ_v= pm.Normal('γ_v', mean_param_per_group_model[-1, 0], .1)
γ_a= pm.Normal('γ_a', mean_param_per_group_model[-1, 1], .1)
γ_Ter= pm.Normal('γ_Ter', mean_param_per_group_model[-1, 2], .1)
γ_max_ampl= pm.Normal('γ_max_ampl', mean_param_per_group_model[-1, 3], .1)
γ_tau= pm.Normal('γ_tau', mean_param_per_group_model[-1, 4], .1)
γ_c = pm.Normal('γ_c', Mean_RT_comp_General[-1], .1)
γ_i = pm.Normal('γ_i', Mean_RT_incomp_General[-1], .1)

σ_v = pm.HalfNormal('σ_v', .1)
σ_a = pm.HalfNormal('σ_a', .1)
σ_Ter = pm.HalfNormal('σ_Ter', .1)
σ_max_ampl = pm.HalfNormal('σ_max_ampl', .1)
σ_tau = pm.HalfNormal('σ_tau', .1)
σ_c = pm.HalfNormal('σ_c', .1)
σ_i = pm.HalfNormal('σ_i', .1)

μ_v = -α_v*pm.math.exp(-β*age_categ) + γ_v
μ_a = α_a*pm.math.exp(-β*age_categ) + γ_a
μ_Ter = α_Ter*pm.math.exp(-β*age_categ) + γ_Ter
μ_max_ampl = α_max_ampl*pm.math.exp(-β*age_categ) + γ_max_ampl
μ_tau = α_tau*pm.math.exp(-β*age_categ) + γ_tau
μ_c = α_c*pm.math.exp(-β*Age_General) + γ_c
μ_i = α_i*pm.math.exp(-β*Age_General) + γ_i

y_v = pm.Normal('y_v', μ_v, σ_v, observed=mean_param[:, 0])
y_a = pm.Normal('y_a', μ_a, σ_a, observed=mean_param[:, 1])
y_Ter = pm.Normal('y_Ter', μ_Ter, σ_Ter, observed=mean_param[:, 2])
y_max_ampl = pm.Normal('y_max_ampl', μ_max_ampl, σ_max_ampl, observed=mean_param[:, 3])
y_tau = pm.Normal('y_tau', μ_tau, σ_tau, observed=mean_param[:, 4])
y_meanRT_comp = pm.Normal('y_meanRT_comp', μ_c, σ_c, observed=Mean_RT_comp)
y_meanRT_incomp = pm.Normal('y_meanRT_incomp', μ_i, σ_i, observed=Mean_RT_incomp)

if __name__ == "__main__":
with  model_exp:
trace_exp = pm.sample (4000, chains = 4, tune = 2000, target_accept = .95)
az.plot_trace(trace_exp)


which fits the data well and suggests that every processing component evolves at the same rate:

I wonder what I could do to provide more convincing evidence in favor of this conclusion. I am thinking of comparing this model with a variant in which the decay rate \beta is free to vary. However, I can’t compute a LOO statistic (I get the following error: “Data must include log_likelihood in sample_stats” when doing az.LOO). In addition, I would appreciate some advice about the best way to analyze residuum from my posterior predictive checks (right now I have just plotted an histogram of residuals, they are normally distributed and centered on 0 so everything looks ok).

Many thanks!

Hi all,

ArviZ is not saving the log_likelihood when more than one observed variable is present. We will try to fix this ASAP.

Is this still an issue?

I’m also getting this error ( Data must include log_likelihood in sample_stats). It was fixed?

Hi all,
This should be fixed on ArviZ most recent version: pointwise log likelihood values from multiple variables are now stored automatically.
The only case where no log likelihood is stored is when logp_elemwise cannot be evaluated (for example with models modified after sampling or when variables have been removed from trace). There should be a warning in these cases that log likelihood cannot be evaluated.

Note that IC functions do not work yet – some extra work has to be done. The workaround is to manually combine log likelihood data into a single array and store it in sample_stats as log_likelihood variable.

Hope this helps clarify things
cc @OriolAbril

1 Like

Hi everyone!

I wanted to build with some specific examples to @AlexAndorra’s answer.

I used this same model to perform some experiments on handling inference criteria calculations on models with multiple observations with ArviZ, I uploaded it on GitHub some days ago, I wanted to finish the work and add explanations to each of the steps but could never find the time. Today I dedicated some time to this other answer which I find is quite similar to this situation and also incorporates a hierarchical structure (which adds some extra possibilities to the mix).

I would recommend first reading the other discourse thread, however, as it is quite long, if you don’t have the time or the motivation, I copied below the main takeaway (from what I have gathered in other discourse threads and GitHub issues):

When working with several observed variables, there is not a single way of computing waic/loo nor or performing cross-validation.

As said above, from the ArviZ side, we’ll try to simplify the workflow, however, there is no unique way to do the computation, it will still require users to define explicitly what predictive task they are interested in.

Disclaimer: I have not checked the exchangeability criteria of the examples below. I use them to illustrate only ArviZ+xarray usage, not statistical correctness.

The GitHub link above goes to a folder with several files in it:

• pymc3_example shows 4 possibilities I came up with without any domain knowledge about the model, some of them may make no sense or may be incorrect (see disclaimer!). I hope the code itself is self explanatory enough.
• pystan_example is purely a translation of the pymc3_example but only with the exponential model (and therefore no model comparison either). If you are interested in information criteria, it can be interesting to read. I hope it can be followed even tough it is written in Stan, I tried to make the model as similar as possible as the PyMC3 one. I think it will be useful to people not familiar with Stan as a step before going into the next two notebooks.
• pystan_exact_loo_subject and pystan_exact_loo_observation are comparisons between the PSIS loo obtained by ArviZ after combining the pointwise log likelihood values and exact cross-validation. If you want to play with them, keep in mind that the model is refitted one time per excluded observation.

In this example, the results of leave one observation out and leave one subject out is extremely similar, which can happen, but do not rely on the two values being similar for any model.

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