Fit a Drift-Diffusion Model

Estimate the parameters of a drift-diffusion model (DDM) from reaction time and accuracy data, evaluate model fit against observed quantiles, compare candidate model variants, and validate estimation quality through parameter recovery simulation.

When to Use

• Modeling binary decision-making with reaction time data
• Estimating cognitive parameters (drift rate, boundary separation, non-decision time) from experimental data
• Comparing sequential sampling model variants for a decision task
• Validating that a DDM fitting pipeline recovers known parameter values
• Decomposing speed-accuracy tradeoff effects into latent cognitive components

Inputs

• Required: Reaction time data with accuracy labels (correct/error) per trial
• Required: Subject and condition identifiers for each trial
• Required: Choice of DDM variant (basic 3-parameter, full 7-parameter, or hierarchical)
• Optional: Prior distributions for Bayesian estimation (default: weakly informative)
• Optional: Number of simulated datasets for parameter recovery (default: 100)
• Optional: RT filtering bounds in seconds (default: 0.1 to 5.0)

Procedure

Step 1: Prepare Reaction Time Data

Clean and format the raw behavioral data for DDM fitting.

1. Load the dataset and inspect columns for subject ID, condition, RT, and accuracy:

import pandas as pd

data = pd.read_csv("behavioral_data.csv")
required_columns = ["subject_id", "condition", "rt", "accuracy"]
assert all(col in data.columns for col in required_columns), \
    f"Missing columns: {set(required_columns) - set(data.columns)}"

2. Filter outlier RTs using configurable bounds:

rt_lower = 0.1  # seconds
rt_upper = 5.0  # seconds

n_before = len(data)
data = data[(data["rt"] >= rt_lower) & (data["rt"] <= rt_upper)]
n_removed = n_before - len(data)
print(f"Removed {n_removed} trials ({100*n_removed/n_before:.1f}%) outside [{rt_lower}, {rt_upper}]s")

3. Compute summary statistics per subject and condition:

summary = data.groupby(["subject_id", "condition"]).agg(
    n_trials=("rt", "count"),
    mean_rt=("rt", "mean"),
    accuracy=("accuracy", "mean")
).reset_index()
print(summary.describe())

4. Verify minimum trial counts (DDM needs sufficient data per cell):

min_trials = summary["n_trials"].min()
assert min_trials >= 40, f"Minimum trials per cell is {min_trials}; need at least 40 for stable estimation"

Expected: Cleaned dataframe with no RT outliers, at least 40 trials per subject-condition cell, and accuracy rates between 0.50 and 0.99.

On failure: If trial counts are too low, consider collapsing conditions or removing subjects with excessive missing data. If accuracy is at ceiling (>0.99) or floor (<0.55), the DDM may not be identifiable -- check task difficulty.

Step 2: Select DDM Variant

Choose the appropriate model complexity based on the research question.

1. Define the candidate model variants:

model_variants = {
    "basic": {
        "params": ["v", "a", "t"],
        "description": "Drift rate, boundary separation, non-decision time",
        "free_params": 3
    },
    "full": {
        "params": ["v", "a", "t", "z", "sv", "sz", "st"],
        "description": "Basic + starting point bias, cross-trial variability",
        "free_params": 7
    },
    "hddm": {
        "params": ["v", "a", "t", "z"],
        "description": "Hierarchical with group-level and subject-level parameters",
        "free_params": "4 per subject + 8 group-level"
    }
}

2. Select based on data characteristics:

Criterion	Basic (3-param)	Full (7-param)	Hierarchical
Trials per cell	40-100	200+	40+ (pooled)
Subjects	Any	Any	10+
Research goal	Group effects	Individual fits	Both levels
Error RT shape	Symmetric	Asymmetric	Either

3. Configure the selected variant:

selected_variant = "basic"  # adjust based on criteria above
model_config = model_variants[selected_variant]
print(f"Selected: {selected_variant} ({model_config['free_params']} free parameters)")
print(f"Parameters: {', '.join(model_config['params'])}")

Expected: A model variant selected with justification based on trial counts, subject count, and research question.

On failure: If unsure between variants, start with the basic model and add complexity only if residual diagnostics indicate systematic misfit (e.g., error RT distribution mismatch).

Step 3: Estimate Parameters

Fit the DDM to data using maximum likelihood or Bayesian estimation.

1. For MLE fitting using the fast-dm or Python pyddm approach:

import pyddm

model = pyddm.Model(
    drift=pyddm.DriftConstant(drift=pyddm.Fittable(minval=0, maxval=5)),
    bound=pyddm.BoundConstant(B=pyddm.Fittable(minval=0.3, maxval=3.0)),
    nondecision=pyddm.NonDecisionConstant(t=pyddm.Fittable(minval=0.1, maxval=0.5)),
    overlay=pyddm.OverlayNonDecision(nondectime=pyddm.Fittable(minval=0.1, maxval=0.5)),
    T_dur=5.0,
    dt=0.001,
    dx=0.001
)

2. For Bayesian estimation using HDDM:

import hddm

hddm_model = hddm.HDDM(data, depends_on={"v": "condition"})
hddm_model.find_starting_values()
hddm_model.sample(5000, burn=1000, thin=2, dbname="traces.db", db="pickle")

3. Extract and store estimated parameters:

params = hddm_model.get_group_estimates()
print("Group-level parameter estimates:")
for param_name, stats in params.items():
    print(f"  {param_name}: {stats['mean']:.3f} [{stats['2.5q']:.3f}, {stats['97.5q']:.3f}]")

4. Check convergence (Bayesian only):

from kabuki.analyze import gelman_rubin

convergence = gelman_rubin(hddm_model)
max_rhat = max(convergence.values())
print(f"Max Gelman-Rubin R-hat: {max_rhat:.3f}")
assert max_rhat < 1.1, f"Chains have not converged (R-hat = {max_rhat:.3f})"

Expected: Parameter estimates with standard errors or credible intervals. For Bayesian fits, Gelman-Rubin R-hat < 1.1 for all parameters. Drift rate typically 0.5-4.0, boundary 0.5-2.5, non-decision time 0.15-0.50s.

On failure: If estimation fails to converge, try: (a) tighter parameter bounds, (b) better starting values via grid search, (c) longer chains with more burn-in. If MLE hits boundary values, the model may be misspecified.

Step 4: Evaluate Model Fit

Compare predicted and observed RT distributions using quantile-based diagnostics.

1. Generate predicted RT quantiles from the fitted model:

import numpy as np

quantiles = [0.1, 0.3, 0.5, 0.7, 0.9]

predicted_rts = model.simulate(n_trials=10000)
pred_quantiles = np.quantile(predicted_rts[predicted_rts > 0], quantiles)  # correct
pred_quantiles_err = np.quantile(np.abs(predicted_rts[predicted_rts < 0]), quantiles)  # error

2. Compute observed RT quantiles:

obs_correct = data[data["accuracy"] == 1]["rt"]
obs_error = data[data["accuracy"] == 0]["rt"]

obs_quantiles = np.quantile(obs_correct, quantiles)
obs_quantiles_err = np.quantile(obs_error, quantiles) if len(obs_error) > 10 else None

3. Create a quantile-probability plot (QP plot):

import matplotlib.pyplot as plt

fig, ax = plt.subplots(1, 1, figsize=(8, 6))
ax.scatter(obs_quantiles, quantiles, marker="o", label="Observed (correct)")
ax.scatter(pred_quantiles, quantiles, marker="x", label="Predicted (correct)")
if obs_quantiles_err is not None:
    ax.scatter(obs_quantiles_err, quantiles, marker="o", facecolors="none", label="Observed (error)")
    ax.scatter(pred_quantiles_err, quantiles, marker="x", label="Predicted (error)")
ax.set_xlabel("RT (s)")
ax.set_ylabel("Quantile")
ax.legend()
ax.set_title("Quantile-Probability Plot")
fig.savefig("qp_plot.png", dpi=150)

4. Compute fit statistic (chi-square on quantile bins):

from scipy.stats import chisquare

observed_proportions = np.diff(np.concatenate([[0], quantiles, [1]]))
predicted_proportions = np.diff(np.concatenate([[0], quantiles, [1]]))
chi2, p_value = chisquare(observed_proportions, predicted_proportions)
print(f"Chi-square fit: chi2={chi2:.3f}, p={p_value:.3f}")

Expected: QP plot shows predicted quantiles closely tracking observed quantiles for both correct and error RTs. Chi-square test is non-significant (p > 0.05), indicating adequate fit.

On failure: If the model systematically misses fast or slow quantiles, consider adding cross-trial variability parameters (sv, st). If error RT shape is wrong, add starting point variability (sz). Refit with the extended model.

Step 5: Compare Models

Use information criteria to select among candidate DDM variants.

1. Fit each candidate model and collect fit statistics:

model_results = {}
for variant_name in ["basic", "full"]:
    fitted_model = fit_ddm(data, variant=variant_name)
    model_results[variant_name] = {
        "log_likelihood": fitted_model.log_likelihood,
        "n_params": fitted_model.n_free_params,
        "bic": fitted_model.bic,
        "aic": fitted_model.aic
    }

2. Compute and compare BIC values:

print("Model Comparison (BIC):")
print(f"{'Model':<15} {'LL':>10} {'k':>5} {'BIC':>12} {'delta_BIC':>12}")
print("-" * 55)

best_bic = min(r["bic"] for r in model_results.values())
for name, result in sorted(model_results.items(), key=lambda x: x[1]["bic"]):
    delta = result["bic"] - best_bic
    print(f"{name:<15} {result['log_likelihood']:>10.1f} {result['n_params']:>5} "
          f"{result['bic']:>12.1f} {delta:>12.1f}")

3. Interpret BIC differences using standard guidelines:

BIC difference interpretation (Kass & Raftery, 1995):

• 0-2: Not worth mentioning
• 2-6: Positive evidence
• 6-10: Strong evidence
• >10: Very strong evidence

4. For Bayesian models, use DIC or WAIC:

dic = hddm_model.dic
print(f"DIC: {dic:.1f}")

Expected: A clear winner among models with BIC difference > 6, or a justified decision to retain the simpler model when the difference is < 2.

On failure: If models are indistinguishable (BIC difference < 2), prefer the simpler model (parsimony). If the full model wins by a large margin, ensure the basic model was not misspecified due to data issues.

Step 6: Validate with Parameter Recovery Simulation

Verify the estimation pipeline recovers known parameter values from simulated data.

1. Define the ground-truth parameter grid:

true_params = {
    "v": [0.5, 1.0, 2.0, 3.0],
    "a": [0.6, 1.0, 1.5, 2.0],
    "t": [0.2, 0.3, 0.4]
}

2. Simulate datasets and re-estimate for each combination:

from itertools import product

recovery_results = []
n_simulated_trials = 500  # match empirical trial count

for v_true, a_true, t_true in product(true_params["v"], true_params["a"], true_params["t"]):
    simulated_data = simulate_ddm(v=v_true, a=a_true, t=t_true, n=n_simulated_trials)
    fitted = fit_ddm(simulated_data, variant="basic")
    recovery_results.append({
        "v_true": v_true, "v_est": fitted.params["v"],
        "a_true": a_true, "a_est": fitted.params["a"],
        "t_true": t_true, "t_est": fitted.params["t"]
    })

3. Compute recovery statistics:

recovery_df = pd.DataFrame(recovery_results)
for param in ["v", "a", "t"]:
    correlation = recovery_df[f"{param}_true"].corr(recovery_df[f"{param}_est"])
    bias = (recovery_df[f"{param}_est"] - recovery_df[f"{param}_true"]).mean()
    rmse = np.sqrt(((recovery_df[f"{param}_est"] - recovery_df[f"{param}_true"])**2).mean())
    print(f"{param}: r={correlation:.3f}, bias={bias:.4f}, RMSE={rmse:.4f}")

4. Generate recovery scatter plots:

fig, axes = plt.subplots(1, 3, figsize=(15, 5))
for idx, param in enumerate(["v", "a", "t"]):
    ax = axes[idx]
    ax.scatter(recovery_df[f"{param}_true"], recovery_df[f"{param}_est"], alpha=0.5)
    lims = [recovery_df[f"{param}_true"].min(), recovery_df[f"{param}_true"].max()]
    ax.plot(lims, lims, "k--", label="Identity")
    ax.set_xlabel(f"True {param}")
    ax.set_ylabel(f"Estimated {param}")
    ax.set_title(f"Recovery: {param} (r={recovery_df[f'{param}_true'].corr(recovery_df[f'{param}_est']):.3f})")
    ax.legend()
fig.tight_layout()
fig.savefig("parameter_recovery.png", dpi=150)

Expected: Recovery correlations r > 0.85 for all parameters, bias close to zero (< 5% of parameter range), and RMSE within acceptable bounds for the application.

On failure: Low recovery for a specific parameter usually means: (a) insufficient trials -- increase n_simulated_trials, (b) parameter tradeoffs -- drift rate and boundary can trade off; fix one to test recoverability, (c) flat likelihood surface -- consider reparameterization or Bayesian estimation with informative priors.

Validation

• Input data has RT and accuracy columns with correct types
• Outlier filtering removed fewer than 10% of trials
• Every subject-condition cell has at least 40 trials
• Parameter estimates are within plausible ranges (v: 0-5, a: 0.3-3.0, t: 0.1-0.6)
• Convergence diagnostics pass (R-hat < 1.1 for Bayesian, gradient near zero for MLE)
• QP plot shows predicted quantiles within 50ms of observed quantiles
• Model comparison yields a clear ranking or justified parsimony decision
• Parameter recovery correlations exceed r = 0.85 for all free parameters
• Recovery bias is less than 5% of the parameter range

Common Pitfalls

• Insufficient trial counts: DDM estimation is data-hungry. Fewer than 40 trials per cell leads to unstable estimates and poor recovery. Always verify trial counts before fitting.
• Ignoring error RTs: The DDM jointly models correct and error RT distributions. Discarding error trials throws away information about boundary separation and starting point bias.
• Not filtering fast guesses: RTs below 100ms are likely contaminants (anticipatory responses). Include them and they distort non-decision time estimates.
• Confusing DDM variants: The basic model assumes no cross-trial variability. If error RTs are systematically faster than correct RTs, you need the full model with sv and sz parameters.
• Overfitting with the full model: The 7-parameter DDM can overfit sparse data. Use BIC (which penalizes complexity) rather than AIC for model selection with DDMs.
• Skipping parameter recovery: Without recovery validation, you cannot distinguish estimation bias from true experimental effects. Always run recovery before interpreting condition differences.

Related Skills

• analyze-diffusion-dynamics - mathematical analysis of the diffusion process underlying the DDM
• implement-diffusion-network - generative diffusion models that share the forward-process framework
• design-experiment - experimental design considerations for collecting DDM-quality data
• write-testthat-tests - testing parameter estimation pipelines in R