📌 Event snapshot — Winterbrain 2025 workshop, Talk 1 (Nadja R. Ging-Jehli, Brown University). This notebook is preserved as taught and is not maintained. For current material, start with the HSSM tutorial; for model comparison, see How to compare and interpret models.

HSSM Tutorial Winterbrain 2025 (event snapshot)¶

Nadja R. Ging-Jehli¶

Brown University¶

LNCC Lab (Michael J. Frank)¶

This is a slightly simplified version of the original workshop tutorial which you can find here.

Load Modules¶

In [1]:

# Import modules
import arviz as az
import jax
import matplotlib as plt
import pytensor

import hssm

pytensor.config.floatX = "float32"
jax.config.update("jax_enable_x64", False)

plt.use("Agg")
%matplotlib inline
%config InlineBackend.figure_format='retina'

# hssm.set_floatX("float32")

# Import modules import arviz as az import jax import matplotlib as plt import pytensor import hssm pytensor.config.floatX = "float32" jax.config.update("jax_enable_x64", False) plt.use("Agg") %matplotlib inline %config InlineBackend.figure_format='retina' # hssm.set_floatX("float32")

1. Overview of Sequential Sampling Models¶

In [2]:

from IPython.display import Image

# Display the image
Image(filename="hssm_tutorial_workshop_1/ssm_overview.png")

from IPython.display import Image # Display the image Image(filename="hssm_tutorial_workshop_1/ssm_overview.png")

Out[2]:

Some Useful Papers relevant for Neuroscience (a small selection)¶

• Ratcliff, R., & Frank, M. J. (2012). Reinforcement-based decision making in corticostriatal circuits: mutual constraints by neurocomputational and diffusion models. Neural computation, 24(5), 1186-1229.

• Turner, B. M., Palestro, J. J., Miletić, S., & Forstmann, B. U. (2019). Advances in techniques for imposing reciprocity in brain-behavior relations. Neuroscience & Biobehavioral Reviews, 102, 327-336.

• Forstmann, B. U., Ratcliff, R., & Wagenmakers, E. J. (2016). Sequential sampling models in cognitive neuroscience: Advantages, applications, and extensions. Annual review of psychology, 67(1), 641-666.

Blogs (self-plug :))¶

https://www.gingjehli.com/research-blog

In [3]:

### Check supported models
hssm.defaults.SupportedModels

### Check supported models hssm.defaults.SupportedModels

Out[3]:

typing.Literal['ddm', 'ddm_sdv', 'full_ddm', 'angle', 'levy', 'ornstein', 'weibull', 'race_no_bias_angle_4', 'ddm_seq2_no_bias', 'lba3', 'lba4', 'lba2', 'racing_diffusion_3', 'poisson_race', 'softmax_inv_temperature_2', 'softmax_inv_temperature_3']

In [4]:

### Details about a particular model
hssm.modelconfig.get_default_model_config("weibull")
##### the bounds within "likelihoods" provide reasonable parameter
##### values (take this with a grain of salt)

### Details about a particular model hssm.modelconfig.get_default_model_config("weibull") ##### the bounds within "likelihoods" provide reasonable parameter ##### values (take this with a grain of salt)

Out[4]:

{'response': ['rt', 'response'],
 'list_params': ['v', 'a', 'z', 't', 'alpha', 'beta'],
 'choices': [-1, 1],
 'description': None,
 'likelihoods': {'approx_differentiable': {'loglik': 'weibull.onnx',
   'backend': 'jax',
   'default_priors': {},
   'bounds': {'v': (-2.5, 2.5),
    'a': (0.3, 2.5),
    'z': (0.2, 0.8),
    't': (0.001, 2.0),
    'alpha': (0.31, 4.99),
    'beta': (0.31, 6.99)},
   'extra_fields': None}}}

In [5]:

hssm.modelconfig.get_default_model_config("ddm")

hssm.modelconfig.get_default_model_config("ddm")

Out[5]:

{'response': ['rt', 'response'],
 'list_params': ['v', 'a', 'z', 't'],
 'choices': [-1, 1],
 'description': 'The Drift Diffusion Model (DDM)',
 'likelihoods': {'analytical': {'loglik': <function hssm.likelihoods.analytical.logp_ddm(data: numpy.ndarray, v: float, a: float, z: float, t: float, err: float = 1e-15, k_terms: int = 20, epsilon: float = 1e-15) -> numpy.ndarray>,
   'backend': None,
   'bounds': {'v': (-inf, inf),
    'a': (0.0, inf),
    'z': (0.0, 1.0),
    't': (0.0, inf)},
   'default_priors': {'t': {'name': 'HalfNormal', 'sigma': 2.0}},
   'extra_fields': None},
  'approx_differentiable': {'loglik': 'ddm.onnx',
   'backend': 'jax',
   'default_priors': {'t': {'name': 'HalfNormal', 'sigma': 2.0}},
   'bounds': {'v': (-3.0, 3.0),
    'a': (0.3, 2.5),
    'z': (0.0, 1.0),
    't': (0.0, 2.0)},
   'extra_fields': None},
  'blackbox': {'loglik': <function hssm.likelihoods.blackbox.hddm_to_hssm.<locals>.outer(data: numpy.ndarray, *args, **kwargs)>,
   'backend': None,
   'bounds': {'v': (-inf, inf),
    'a': (0.0, inf),
    'z': (0.0, 1.0),
    't': (0.0, inf)},
   'default_priors': {'t': {'name': 'HalfNormal', 'sigma': 2.0}},
   'extra_fields': None}}}

Notes¶

• there is a difference between "ddm" and "full_ddm"
• the bounds within "likelihoods" provide reasonable param values (but take this with a grain of salt)
• the order of "list_params" is important

2. Regression-based modeling¶

Case Study 1: 1 hierarchical layer (trial-level) with 1 within-subject coefficient)¶

• you have intracranial recording in STN and want to understand the decision-relevant dynamics in the activity
• we assume that activity in STN varies on a trial-by-trial basis in a systematic way
• your Hypothesis: STN is implicated in response cautiousness
• you want to test this hypothesis using a classical DDM

Question to audience: which model parameter does STN modulate?¶

Step 1: Regression-based data simulation (one subjects)¶

In [6]:

import numpy as np
import pandas as pd
from ssms.basic_simulators.simulator import simulator

v_true = 0.5
a_true = 1.5
z_true = 0.5
t_true = 0.5
# a changes trial wise
# a_trialwise = np.random.normal(loc=2, scale=0.3, size=1000)
Intercept = 1.5
zSTN = np.random.normal(loc=1, scale=1, size=1000).astype(np.float32)
a_trialwise = Intercept + 0.5 * zSTN

# a changes trial wise
theta_mat = np.zeros((1000, 4))
theta_mat[:, 0] = v_true  # v
theta_mat[:, 1] = a_trialwise  # a
theta_mat[:, 2] = z_true  # z
theta_mat[:, 3] = t_true  # t

# simulate data
sim_out_trialwise = simulator(
    theta=theta_mat,  # parameter_matrix
    model="ddm",  # specify model (many are included in ssms)
    n_samples=1,  # number of samples for each set of parameters
    # (plays the role of `size` parameter in `hssm.simulate_data`)
)

# Turn into nice dataset
df_ddm_reg_case1 = pd.DataFrame(
    np.column_stack(
        [sim_out_trialwise["rts"][:, 0], sim_out_trialwise["choices"][:, 0]]
    ),
    columns=["rt", "response"],
)

df_ddm_reg_case1["zSTN"] = zSTN
df_ddm_reg_case1

import numpy as np import pandas as pd from ssms.basic_simulators.simulator import simulator v_true = 0.5 a_true = 1.5 z_true = 0.5 t_true = 0.5 # a changes trial wise # a_trialwise = np.random.normal(loc=2, scale=0.3, size=1000) Intercept = 1.5 zSTN = np.random.normal(loc=1, scale=1, size=1000).astype(np.float32) a_trialwise = Intercept + 0.5 * zSTN # a changes trial wise theta_mat = np.zeros((1000, 4)) theta_mat[:, 0] = v_true # v theta_mat[:, 1] = a_trialwise # a theta_mat[:, 2] = z_true # z theta_mat[:, 3] = t_true # t # simulate data sim_out_trialwise = simulator( theta=theta_mat, # parameter_matrix model="ddm", # specify model (many are included in ssms) n_samples=1, # number of samples for each set of parameters # (plays the role of `size` parameter in `hssm.simulate_data`) ) # Turn into nice dataset df_ddm_reg_case1 = pd.DataFrame( np.column_stack( [sim_out_trialwise["rts"][:, 0], sim_out_trialwise["choices"][:, 0]] ), columns=["rt", "response"], ) df_ddm_reg_case1["zSTN"] = zSTN df_ddm_reg_case1

Out[6]:

	rt	response	zSTN
0	3.398903	1.0	2.815744
1	1.995214	1.0	1.548521
2	2.570446	1.0	2.068199
3	2.912847	1.0	1.467202
4	2.117052	1.0	1.230379
...	...	...	...
995	4.793263	1.0	0.683175
996	2.021674	1.0	-0.021068
997	5.266840	1.0	2.851962
998	3.453858	1.0	2.043670
999	2.457493	1.0	0.874597

1000 rows × 3 columns

Looking at the data, what do you notice?¶

Required Data Structure for HSSM¶

• we need variables labeled as "rt" and "response"
• make sure rt are in seconds
• responses are (1,-1) because you are in a classical DDM setting
• make sure regressors are z-scored (particularly important when you have more than two regressors)

Step 2: Model Setup & Priors¶

In [7]:

model_ddm_reg_case1 = hssm.HSSM(
    data=df_ddm_reg_case1,
    model="ddm",
    include=[
        {
            "name": "a",
            "formula": "a ~ 1 + zSTN",
            "prior": {
                # All ways to specify priors in the non-regression case
                # work the same way here.
                "Intercept": {"name": "Normal", "mu": 1.5, "sigma": 1.0},
                "zSTN": {"name": "Normal", "mu": 0, "sigma": 1.0},
            },
            "link": "identity",
        }
    ],
)

model_ddm_reg_case1 = hssm.HSSM( data=df_ddm_reg_case1, model="ddm", include=[ { "name": "a", "formula": "a ~ 1 + zSTN", "prior": { # All ways to specify priors in the non-regression case # work the same way here. "Intercept": {"name": "Normal", "mu": 1.5, "sigma": 1.0}, "zSTN": {"name": "Normal", "mu": 0, "sigma": 1.0}, }, "link": "identity", } ], )

Model initialized successfully.

Notes to Step 2¶

• make sure your priors are reasonable (always double-check)
• see my blog for useful notes for defining priors on each of these params https://www.gingjehli.com/single-post/choosing-effective-samplers-and-setting-priors
• priors are particularly important in these more complex regression-based models
• it can also be helpful to set "init_val" for the sampler so it starts in a reasonable region.

Step 3: Double-check model specification¶

In [8]:

# graphical illustration of model
model_ddm_reg_case1.graph()

# graphical illustration of model model_ddm_reg_case1.graph()

Out[8]:

Step 4: Sampling from the Posterior Distribution¶

In [9]:

# suppress warnings
import warnings

warnings.filterwarnings("ignore")
samples_model_ddm_reg_case1 = model_ddm_reg_case1.sample(
    sampler="numpyro",
    cores=2,
    chains=2,
    draws=500,
    tune=500,
    idata_kwargs=dict(log_likelihood=True),
)

# suppress warnings import warnings warnings.filterwarnings("ignore") samples_model_ddm_reg_case1 = model_ddm_reg_case1.sample( sampler="numpyro", cores=2, chains=2, draws=500, tune=500, idata_kwargs=dict(log_likelihood=True), )

Using default initvals. 

NUTS[numpyro]: [z, v, t, a_Intercept, a_zSTN]

  0%|          | 0/1000 [00:00<?, ?it/s]

warmup:   5%|▌         | 50/1000 [00:00<00:12, 73.56it/s, 3 steps of size 3.17e-02. acc. prob=0.75]

warmup:  10%|█         | 100/1000 [00:00<00:06, 133.70it/s, 3 steps of size 1.47e-02. acc. prob=0.76]

warmup:  15%|█▌        | 150/1000 [00:01<00:05, 151.20it/s, 63 steps of size 8.54e-02. acc. prob=0.77]

warmup:  20%|██        | 200/1000 [00:01<00:04, 189.55it/s, 3 steps of size 1.59e-01. acc. prob=0.78] 

warmup:  25%|██▌       | 250/1000 [00:01<00:03, 217.89it/s, 7 steps of size 3.13e-01. acc. prob=0.78]

warmup:  30%|███       | 300/1000 [00:01<00:02, 239.16it/s, 7 steps of size 8.03e-01. acc. prob=0.78]

warmup:  35%|███▌      | 350/1000 [00:01<00:02, 261.18it/s, 7 steps of size 6.75e-01. acc. prob=0.79]

warmup:  40%|████      | 400/1000 [00:01<00:02, 273.65it/s, 7 steps of size 7.45e-01. acc. prob=0.79]

warmup:  45%|████▌     | 450/1000 [00:02<00:01, 293.61it/s, 15 steps of size 2.44e-01. acc. prob=0.79]

warmup:  50%|█████     | 500/1000 [00:02<00:01, 262.55it/s, 6 steps of size 2.65e-01. acc. prob=0.78] 

sample:  55%|█████▌    | 550/1000 [00:02<00:01, 267.79it/s, 7 steps of size 2.65e-01. acc. prob=0.89]

sample:  60%|██████    | 600/1000 [00:02<00:01, 280.29it/s, 7 steps of size 2.65e-01. acc. prob=0.92]

sample:  65%|██████▌   | 650/1000 [00:02<00:01, 285.93it/s, 15 steps of size 2.65e-01. acc. prob=0.92]

sample:  70%|███████   | 700/1000 [00:02<00:01, 292.61it/s, 15 steps of size 2.65e-01. acc. prob=0.91]

sample:  75%|███████▌  | 750/1000 [00:03<00:00, 299.26it/s, 15 steps of size 2.65e-01. acc. prob=0.92]

sample:  80%|████████  | 800/1000 [00:03<00:00, 309.39it/s, 15 steps of size 2.65e-01. acc. prob=0.90]

sample:  85%|████████▌ | 850/1000 [00:03<00:00, 310.58it/s, 7 steps of size 2.65e-01. acc. prob=0.90] 

sample:  90%|█████████ | 900/1000 [00:03<00:00, 295.84it/s, 15 steps of size 2.65e-01. acc. prob=0.90]

sample:  95%|█████████▌| 950/1000 [00:03<00:00, 289.40it/s, 15 steps of size 2.65e-01. acc. prob=0.90]

sample: 100%|██████████| 1000/1000 [00:03<00:00, 285.00it/s, 7 steps of size 2.65e-01. acc. prob=0.90]

sample: 100%|██████████| 1000/1000 [00:04<00:00, 250.00it/s, 7 steps of size 2.65e-01. acc. prob=0.90]

  0%|          | 0/1000 [00:00<?, ?it/s]

warmup:   5%|▌         | 50/1000 [00:00<00:03, 309.43it/s, 7 steps of size 1.23e-02. acc. prob=0.74]

warmup:  10%|█         | 100/1000 [00:00<00:03, 233.85it/s, 7 steps of size 1.45e-02. acc. prob=0.76]

warmup:  15%|█▌        | 150/1000 [00:00<00:05, 164.59it/s, 15 steps of size 2.65e-01. acc. prob=0.78]

warmup:  20%|██        | 200/1000 [00:00<00:03, 203.70it/s, 15 steps of size 2.14e-01. acc. prob=0.78]

warmup:  25%|██▌       | 250/1000 [00:01<00:03, 210.43it/s, 15 steps of size 3.88e-01. acc. prob=0.78]

warmup:  30%|███       | 300/1000 [00:01<00:03, 228.62it/s, 31 steps of size 1.97e-01. acc. prob=0.78]

warmup:  35%|███▌      | 350/1000 [00:01<00:02, 245.62it/s, 7 steps of size 2.59e-01. acc. prob=0.78] 

warmup:  40%|████      | 400/1000 [00:01<00:02, 271.88it/s, 5 steps of size 3.25e-01. acc. prob=0.78]

warmup:  45%|████▌     | 450/1000 [00:01<00:01, 297.88it/s, 3 steps of size 5.38e-01. acc. prob=0.79]

warmup:  50%|█████     | 500/1000 [00:02<00:01, 269.27it/s, 7 steps of size 3.33e-01. acc. prob=0.79]

sample:  55%|█████▌    | 550/1000 [00:02<00:01, 279.63it/s, 11 steps of size 3.33e-01. acc. prob=0.92]

sample:  60%|██████    | 600/1000 [00:02<00:01, 291.72it/s, 3 steps of size 3.33e-01. acc. prob=0.88] 

sample:  65%|██████▌   | 650/1000 [00:02<00:01, 302.45it/s, 15 steps of size 3.33e-01. acc. prob=0.86]

sample:  70%|███████   | 700/1000 [00:02<00:00, 313.88it/s, 3 steps of size 3.33e-01. acc. prob=0.87] 

sample:  75%|███████▌  | 750/1000 [00:02<00:00, 317.20it/s, 15 steps of size 3.33e-01. acc. prob=0.88]

sample:  80%|████████  | 800/1000 [00:02<00:00, 324.30it/s, 7 steps of size 3.33e-01. acc. prob=0.88] 

sample:  85%|████████▌ | 850/1000 [00:03<00:00, 312.82it/s, 15 steps of size 3.33e-01. acc. prob=0.88]

sample:  90%|█████████ | 900/1000 [00:03<00:00, 317.36it/s, 15 steps of size 3.33e-01. acc. prob=0.87]

sample:  95%|█████████▌| 950/1000 [00:03<00:00, 326.43it/s, 15 steps of size 3.33e-01. acc. prob=0.88]

sample: 100%|██████████| 1000/1000 [00:03<00:00, 315.33it/s, 11 steps of size 3.33e-01. acc. prob=0.88]

sample: 100%|██████████| 1000/1000 [00:03<00:00, 276.92it/s, 11 steps of size 3.33e-01. acc. prob=0.88]

There were 27 divergences after tuning. Increase `target_accept` or reparameterize.

We recommend running at least 4 chains for robust computation of convergence diagnostics

The rhat statistic is larger than 1.01 for some parameters. This indicates problems during sampling. See https://arxiv.org/abs/1903.08008 for details

Notes to Step 4¶

• in a real analysis, you would want to run more chains (usually >=4)
• if models don't converge, as a first step, it is typically a good idea to increase the chain length
• depending on task structure, some model params are more/less likely to trade-off (e.g., simple task design, t & a are often anti-correlated)
• if there are parameter trade-offs or depending on your regression equation, it can be helpful to play around with the sampler
• though: numpyro sampler does not work with blackbox likelihoods.
• sometime variational inference (VI) can be a helpful alternative to MCMC (check out the dedicated tutorials in the HSSM documentation)

Step 5: Model Validation¶

What do you think about these results below (also given Jensen's part)?¶

In [10]:

#### quick check on posterior statistics
az.summary(model_ddm_reg_case1.traces)

#### quick check on posterior statistics az.summary(model_ddm_reg_case1.traces)

Out[10]:

	mean	sd	eti89_lb	eti89_ub	ess_bulk	ess_tail	r_hat	mcse_mean	mcse_sd
a_Intercept	1.509	0.029	1.5	1.6	618	628	1.01	0.0012	0.00082
v	0.546	0.027	0.5	0.59	696	619	1.01	0.001	0.00072
a_zSTN	0.501	0.02	0.47	0.53	454	520	1.00	0.00094	0.00074
z	0.495	0.0125	0.47	0.51	536	539	1.00	0.00054	0.00041
t	0.512	0.014	0.49	0.53	343	313	1.01	0.00083	0.00073

In [11]:

#### trace plots
az.plot_trace_dist(model_ddm_reg_case1.traces)

#### trace plots az.plot_trace_dist(model_ddm_reg_case1.traces)

Out[11]:

<arviz_plots.plot_collection.PlotCollection at 0x130b37e00>

Case Study 2: Participant hierarchy on two within-subject coefficients¶

• besides from STN, we also have an indirect nogo pathway via GPe
• this time, we want to assume that activity in these components modulates drift rate
• even though there is evidence that STN modulates the boundary, we always want to check different model specifications

Step 1: Regression-based data simulation (multiple subjects)¶

In [12]:

# Function to simulate data for one participant
def simulate_participant(participant_id, size=300):
    """Simulate DDM trial data for one participant."""
    intercept = 1.5
    zSTN = np.random.normal(loc=1, scale=2, size=size)
    zGPe = np.random.normal(loc=1, scale=2, size=size)
    v = intercept + 0.8 * zSTN + 0.3 * zGPe

    # Assume `hssm.simulate_data` returns a DataFrame
    true_values = np.column_stack(
        [v, np.repeat([[1.5, 0.5, 0.5]], axis=0, repeats=300)]
    )

    dataset_reg_v = hssm.simulate_data(
        model="ddm",
        theta=true_values,
        size=1,  # Generate one data point for each of the 1000 set of true values
    )

    # Adding additional variables to the dataset
    dataset_reg_v["zSTN"] = zSTN
    dataset_reg_v["zGPe"] = zGPe
    dataset_reg_v["participant_id"] = str(participant_id)
    return dataset_reg_v


# Simulate data for 10 participants

# Combine datasets into one DataFrame
combined_dataset = pd.concat(
    [simulate_participant(i, size=300) for i in range(1, 11)], ignore_index=True
)
combined_dataset

# Function to simulate data for one participant def simulate_participant(participant_id, size=300): """Simulate DDM trial data for one participant.""" intercept = 1.5 zSTN = np.random.normal(loc=1, scale=2, size=size) zGPe = np.random.normal(loc=1, scale=2, size=size) v = intercept + 0.8 * zSTN + 0.3 * zGPe # Assume `hssm.simulate_data` returns a DataFrame true_values = np.column_stack( [v, np.repeat([[1.5, 0.5, 0.5]], axis=0, repeats=300)] ) dataset_reg_v = hssm.simulate_data( model="ddm", theta=true_values, size=1, # Generate one data point for each of the 1000 set of true values ) # Adding additional variables to the dataset dataset_reg_v["zSTN"] = zSTN dataset_reg_v["zGPe"] = zGPe dataset_reg_v["participant_id"] = str(participant_id) return dataset_reg_v # Simulate data for 10 participants # Combine datasets into one DataFrame combined_dataset = pd.concat( [simulate_participant(i, size=300) for i in range(1, 11)], ignore_index=True ) combined_dataset

Out[12]:

	rt	response	zSTN	zGPe	participant_id
0	2.279359	-1.0	-3.816594	1.321567	1
1	0.728926	1.0	0.974609	-0.437938	1
2	1.842637	1.0	0.013669	0.552476	1
3	1.498359	1.0	-1.236362	3.151876	1
4	0.749893	1.0	3.172414	1.738854	1
...	...	...	...	...	...
2995	0.779592	1.0	2.433109	4.903835	10
2996	0.747541	1.0	1.212487	4.565028	10
2997	1.093232	1.0	0.191995	0.527976	10
2998	1.158536	1.0	-0.378472	0.432769	10
2999	1.036517	1.0	3.097844	0.259007	10

3000 rows × 5 columns

Step 2: Model Setup & Priors¶

Alternative 1: A centered model¶

• specification where the subject-specific coefficients are coming from one hierarchical prior (mother distribution with mu and sigma)
• we can also call it as "0+" models
• this is the specification used in HDDM
• in this case, it's important to set "noncentered=False" (by default it's set to "True")
• The group distributions here will be recovered as v_1|subject_mu and v_x|subject_mu

In [13]:

model_reg_v_ex2_A1 = hssm.HSSM(
    data=combined_dataset,
    model="ddm",
    include=[
        {
            "name": "v",
            "formula": "v ~ 0 + (1 + zSTN + zGPe | participant_id)",
            "prior": {
                "1|participant_id": {
                    "name": "Normal",
                    "mu": {"name": "Normal", "mu": 1, "sigma": 1.0, "initval": 0},
                    "sigma": {"name": "HalfNormal", "sigma": 1, "initval": 0.25},
                },
                "zSTN|participant_id": {
                    "name": "Normal",
                    "mu": {"name": "Normal", "mu": 0, "sigma": 1.0, "initval": 0},
                    "sigma": {"name": "HalfNormal", "sigma": 1, "initval": 0.25},
                },
                "zGPe|participant_id": {
                    "name": "Normal",
                    "mu": {"name": "Normal", "mu": 0, "sigma": 1.0, "initval": 0},
                    "sigma": {"name": "HalfNormal", "sigma": 1, "initval": 0.25},
                },
            },
            "link": "identity",
        }
    ],
    noncentered=False,
    p_outlier=0.05,
)

model_reg_v_ex2_A1 = hssm.HSSM( data=combined_dataset, model="ddm", include=[ { "name": "v", "formula": "v ~ 0 + (1 + zSTN + zGPe | participant_id)", "prior": { "1|participant_id": { "name": "Normal", "mu": {"name": "Normal", "mu": 1, "sigma": 1.0, "initval": 0}, "sigma": {"name": "HalfNormal", "sigma": 1, "initval": 0.25}, }, "zSTN|participant_id": { "name": "Normal", "mu": {"name": "Normal", "mu": 0, "sigma": 1.0, "initval": 0}, "sigma": {"name": "HalfNormal", "sigma": 1, "initval": 0.25}, }, "zGPe|participant_id": { "name": "Normal", "mu": {"name": "Normal", "mu": 0, "sigma": 1.0, "initval": 0}, "sigma": {"name": "HalfNormal", "sigma": 1, "initval": 0.25}, }, }, "link": "identity", } ], noncentered=False, p_outlier=0.05, )

Model initialized successfully.

In [14]:

# graphical illustration of model
model_reg_v_ex2_A1.graph()

# graphical illustration of model model_reg_v_ex2_A1.graph()

Out[14]:

Alternative 2: A non-centered model¶

• Non-centered means that each subject is estimated as offset from the group mean
• in this case the group means will be zSTN and zGPe.
• The priors are different here because now we want the mu for the random subject effects to be 0
• the mu should not be estimated from a hyperprior simultaneously with the fixed effects, as those will tradeoff and could produce convergence issues - we still estimate the sigmas with a hyperprior to allow us to get indiv params
• now we keep noncentered=true.

In [15]:

model_reg_v_ex2_A2 = hssm.HSSM(
    data=combined_dataset,
    include=[
        {
            "name": "v",
            "formula": "v ~ 1 + zSTN + zGPe + (1 + zSTN + zGPe | participant_id)",
            "prior": {
                # All ways to specify priors in the non-regression case
                # work the same way here.
                "Intercept": {"name": "Normal", "mu": 1.0, "sigma": 1.0},
                "zSTN": {"name": "Normal", "mu": 0, "sigma": 1.0},
                "zGPe": {"name": "Normal", "mu": 0, "sigma": 1.0},
                "1|participant_id": {
                    "name": "Normal",
                    "mu": 0,
                    "sigma": {"name": "HalfNormal", "sigma": 1.0},
                },
                "zSTN|participant_id": {
                    "name": "Normal",
                    "mu": 0,
                    "sigma": {"name": "HalfNormal", "sigma": 1.0},
                },
                "zGPe|participant_id": {
                    "name": "Normal",
                    "mu": 0,
                    "sigma": {"name": "HalfNormal", "sigma": 1.0},
                },
            },
            "link": "identity",
        }
    ],
    noncentered=True,
    p_outlier=0.05,
)

model_reg_v_ex2_A2 = hssm.HSSM( data=combined_dataset, include=[ { "name": "v", "formula": "v ~ 1 + zSTN + zGPe + (1 + zSTN + zGPe | participant_id)", "prior": { # All ways to specify priors in the non-regression case # work the same way here. "Intercept": {"name": "Normal", "mu": 1.0, "sigma": 1.0}, "zSTN": {"name": "Normal", "mu": 0, "sigma": 1.0}, "zGPe": {"name": "Normal", "mu": 0, "sigma": 1.0}, "1|participant_id": { "name": "Normal", "mu": 0, "sigma": {"name": "HalfNormal", "sigma": 1.0}, }, "zSTN|participant_id": { "name": "Normal", "mu": 0, "sigma": {"name": "HalfNormal", "sigma": 1.0}, }, "zGPe|participant_id": { "name": "Normal", "mu": 0, "sigma": {"name": "HalfNormal", "sigma": 1.0}, }, }, "link": "identity", } ], noncentered=True, p_outlier=0.05, )

Model initialized successfully.

In [16]:

model_reg_v_ex2_A2.graph()

model_reg_v_ex2_A2.graph()

Out[16]:

Important note¶

Note that in this non-centered model, you will need to specify each individual coefficient's prior (at least for your regression equation). See example below

Step 4: Sampling from the Posterior Distribution (Bayesian Model Fitting based on MCMC Procedure)¶

In [17]:

## centered model version
samples_model_reg_v_ex2_A1 = model_reg_v_ex2_A1.sample(
    sampler="numpyro",
    cores=3,
    chains=3,
    draws=500,
    tune=500,
    idata_kwargs=dict(log_likelihood=True),
)

## centered model version samples_model_reg_v_ex2_A1 = model_reg_v_ex2_A1.sample( sampler="numpyro", cores=3, chains=3, draws=500, tune=500, idata_kwargs=dict(log_likelihood=True), )

Using default initvals. 

NUTS[numpyro]: [z, a, t, v_1|participant_id_mu, v_1|participant_id_sigma, v_1|participant_id, v_zSTN|participant_id_mu, v_zSTN|participant_id_sigma, v_zSTN|participant_id, v_zGPe|participant_id_mu, v_zGPe|participant_id_sigma, v_zGPe|participant_id]

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sample:  55%|█████▌    | 550/1000 [00:28<00:26, 17.03it/s, 127 steps of size 4.82e-02. acc. prob=0.99]

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sample:  55%|█████▌    | 550/1000 [00:28<00:17, 25.64it/s, 15 steps of size 8.29e-02. acc. prob=0.73]

sample:  60%|██████    | 600/1000 [00:28<00:11, 34.48it/s, 5 steps of size 8.29e-02. acc. prob=0.41] 

sample:  65%|██████▌   | 650/1000 [00:28<00:07, 45.85it/s, 3 steps of size 8.29e-02. acc. prob=0.29]

sample:  70%|███████   | 700/1000 [00:30<00:07, 38.68it/s, 63 steps of size 8.29e-02. acc. prob=0.37]

sample:  75%|███████▌  | 750/1000 [00:32<00:07, 31.80it/s, 63 steps of size 8.29e-02. acc. prob=0.47]

sample:  80%|████████  | 800/1000 [00:35<00:07, 27.03it/s, 63 steps of size 8.29e-02. acc. prob=0.54]

sample:  85%|████████▌ | 850/1000 [00:35<00:04, 34.49it/s, 5 steps of size 8.29e-02. acc. prob=0.48] 

sample:  90%|█████████ | 900/1000 [00:37<00:02, 35.41it/s, 31 steps of size 8.29e-02. acc. prob=0.49]

sample:  95%|█████████▌| 950/1000 [00:39<00:01, 30.55it/s, 63 steps of size 8.29e-02. acc. prob=0.54]

sample: 100%|██████████| 1000/1000 [00:41<00:00, 28.01it/s, 63 steps of size 8.29e-02. acc. prob=0.58]

sample: 100%|██████████| 1000/1000 [00:41<00:00, 24.12it/s, 63 steps of size 8.29e-02. acc. prob=0.58]

There were 218 divergences after tuning. Increase `target_accept` or reparameterize.

We recommend running at least 4 chains for robust computation of convergence diagnostics

The rhat statistic is larger than 1.01 for some parameters. This indicates problems during sampling. See https://arxiv.org/abs/1903.08008 for details

The effective sample size per chain is smaller than 100 for some parameters.  A higher number is needed for reliable rhat and ess computation. See https://arxiv.org/abs/1903.08008 for details

In [18]:

## centered model version
samples_model_reg_v_ex2_A2 = model_reg_v_ex2_A2.sample(
    sampler="numpyro",
    cores=3,
    chains=3,
    draws=500,
    tune=500,
    idata_kwargs=dict(log_likelihood=True),
)

## centered model version samples_model_reg_v_ex2_A2 = model_reg_v_ex2_A2.sample( sampler="numpyro", cores=3, chains=3, draws=500, tune=500, idata_kwargs=dict(log_likelihood=True), )

Using default initvals. 

NUTS[numpyro]: [z, a, t, v_Intercept, v_zSTN, v_zGPe, v_1|participant_id_sigma, v_1|participant_id_offset, v_zSTN|participant_id_sigma, v_zSTN|participant_id_offset, v_zGPe|participant_id_sigma, v_zGPe|participant_id_offset]

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sample:  75%|███████▍  | 749/1000 [00:56<00:05, 46.99it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  75%|███████▌  | 754/1000 [00:56<00:05, 46.53it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  76%|███████▌  | 759/1000 [00:56<00:05, 46.17it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  76%|███████▋  | 764/1000 [00:56<00:05, 45.97it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  77%|███████▋  | 769/1000 [00:56<00:04, 46.94it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  77%|███████▋  | 774/1000 [00:56<00:04, 46.44it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  78%|███████▊  | 779/1000 [00:56<00:04, 46.13it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  78%|███████▊  | 784/1000 [00:56<00:04, 45.65it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  79%|███████▉  | 789/1000 [00:56<00:04, 45.14it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  79%|███████▉  | 794/1000 [00:57<00:04, 44.15it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  80%|███████▉  | 799/1000 [00:57<00:04, 42.83it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  80%|████████  | 804/1000 [00:57<00:04, 44.36it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  81%|████████  | 809/1000 [00:57<00:04, 43.66it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  81%|████████▏ | 814/1000 [00:57<00:04, 43.45it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  82%|████████▏ | 819/1000 [00:57<00:04, 43.21it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  82%|████████▏ | 824/1000 [00:57<00:04, 42.89it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  83%|████████▎ | 829/1000 [00:57<00:03, 43.69it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  84%|████████▎ | 835/1000 [00:58<00:03, 47.42it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  84%|████████▍ | 840/1000 [00:58<00:03, 47.17it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  84%|████████▍ | 845/1000 [00:58<00:03, 44.57it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  85%|████████▌ | 850/1000 [00:58<00:03, 45.67it/s, 15 steps of size 1.72e-01. acc. prob=0.93]

sample:  86%|████████▌ | 855/1000 [00:58<00:03, 45.37it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  86%|████████▌ | 860/1000 [00:58<00:03, 45.67it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  86%|████████▋ | 865/1000 [00:58<00:02, 45.38it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  87%|████████▋ | 870/1000 [00:58<00:02, 46.64it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  88%|████████▊ | 875/1000 [00:58<00:02, 45.94it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  88%|████████▊ | 880/1000 [00:59<00:02, 45.02it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  88%|████████▊ | 885/1000 [00:59<00:02, 45.94it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  89%|████████▉ | 892/1000 [00:59<00:02, 49.69it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  90%|████████▉ | 898/1000 [00:59<00:02, 50.51it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  90%|█████████ | 903/1000 [00:59<00:02, 47.79it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  91%|█████████ | 908/1000 [00:59<00:01, 46.72it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  91%|█████████▏| 913/1000 [00:59<00:01, 46.08it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  92%|█████████▏| 918/1000 [00:59<00:01, 46.21it/s, 15 steps of size 1.72e-01. acc. prob=0.93]

sample:  92%|█████████▏| 923/1000 [00:59<00:01, 46.46it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  93%|█████████▎| 927/1000 [01:00<00:01, 43.99it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  93%|█████████▎| 932/1000 [01:00<00:01, 44.83it/s, 31 steps of size 1.72e-01. acc. prob=0.94]

sample:  94%|█████████▎| 937/1000 [01:00<00:01, 44.82it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  94%|█████████▍| 942/1000 [01:00<00:01, 44.00it/s, 31 steps of size 1.72e-01. acc. prob=0.94]

sample:  95%|█████████▍| 948/1000 [01:00<00:01, 47.60it/s, 15 steps of size 1.72e-01. acc. prob=0.94]

sample:  95%|█████████▌| 953/1000 [01:00<00:01, 46.41it/s, 31 steps of size 1.72e-01. acc. prob=0.94]

sample:  96%|█████████▌| 958/1000 [01:00<00:00, 45.26it/s, 31 steps of size 1.72e-01. acc. prob=0.93]

sample:  96%|█████████▋| 963/1000 [01:00<00:00, 43.91it/s, 31 steps of size 1.72e-01. acc. prob=0.94]

sample:  97%|█████████▋| 968/1000 [01:00<00:00, 44.42it/s, 31 steps of size 1.72e-01. acc. prob=0.94]

sample:  97%|█████████▋| 973/1000 [01:01<00:00, 44.88it/s, 15 steps of size 1.72e-01. acc. prob=0.94]

sample:  98%|█████████▊| 978/1000 [01:01<00:00, 44.16it/s, 31 steps of size 1.72e-01. acc. prob=0.94]

sample:  98%|█████████▊| 983/1000 [01:01<00:00, 44.66it/s, 31 steps of size 1.72e-01. acc. prob=0.94]

sample:  99%|█████████▉| 989/1000 [01:01<00:00, 46.57it/s, 31 steps of size 1.72e-01. acc. prob=0.94]

sample:  99%|█████████▉| 994/1000 [01:01<00:00, 47.26it/s, 31 steps of size 1.72e-01. acc. prob=0.94]

sample: 100%|█████████▉| 999/1000 [01:01<00:00, 47.74it/s, 31 steps of size 1.72e-01. acc. prob=0.94]

sample: 100%|██████████| 1000/1000 [01:01<00:00, 16.23it/s, 31 steps of size 1.72e-01. acc. prob=0.94]

We recommend running at least 4 chains for robust computation of convergence diagnostics

Final notes:¶

• whether centered is better depends on the structure of your dataset
• the two models are mathematically equivalent, however the posterior geometry is affected by the parameterization
• in practice, you can try both and see which one works better
• here is a great (albeit non-trivial to digest) long blog post which touches on the topic in some detail`

Step 5: Model Validation¶

In [19]:

#### posterior statistics of centered model
az.summary(samples_model_reg_v_ex2_A2, var_names=["~_id"], filter_vars="like")

#### posterior statistics of centered model az.summary(samples_model_reg_v_ex2_A2, var_names=["~_id"], filter_vars="like")

Out[19]:

	mean	sd	eti89_lb	eti89_ub	ess_bulk	ess_tail	r_hat	mcse_mean	mcse_sd
a	1.474	0.0263	1.4	1.5	1477	907	1.00	0.00069	0.00046
v_zGPe	0.3099	0.0126	0.29	0.33	1339	1021	1.00	0.00035	0.00027
v_Intercept	1.538	0.043	1.5	1.6	1210	1144	1.00	0.0012	0.00091
z	0.474	0.0114	0.46	0.49	1256	1076	1.00	0.00033	0.00024
t	0.4978	0.0061	0.49	0.51	1296	1186	1.00	0.00017	0.00011
v_zSTN	0.807	0.0178	0.78	0.84	1156	1105	1.00	0.00052	0.0004

In [20]:

# Plotting the posterior predictive
hssm.plotting.plot_predictive(model_reg_v_ex2_A1, col="participant_id", col_wrap=5)

# Plotting the posterior predictive hssm.plotting.plot_predictive(model_reg_v_ex2_A1, col="participant_id", col_wrap=5)

No posterior_predictive samples found. Generating posterior_predictive samples using the provided DataTree object and the original data. This will modify the provided DataTree object, or if not provided, the traces object stored inside the model.

Out[20]:

<seaborn.axisgrid.FacetGrid at 0x131396ba0>

In [21]:

#### posterior statistics of non-centered model
az.summary(samples_model_reg_v_ex2_A2, var_names=["~_offset"], filter_vars="like")

#### posterior statistics of non-centered model az.summary(samples_model_reg_v_ex2_A2, var_names=["~_offset"], filter_vars="like")

Out[21]:

	mean	sd	eti89_lb	eti89_ub	ess_bulk	ess_tail	r_hat	mcse_mean	mcse_sd
v_zSTN|participant_id_sigma	0.028	0.019	0.0037	0.061	522	583	1.01	0.00077	0.00079
a	1.474	0.0263	1.4	1.5	1477	907	1.00	0.00069	0.00046
v_zGPe|participant_id_sigma	0.0142	0.0114	0.0012	0.036	766	579	1.00	0.00036	0.00036
v_zGPe|participant_id[1]	-0.0024	0.0146	-0.028	0.019	1645	1241	1.00	0.00037	0.00045
v_zGPe|participant_id[10]	-0.0026	0.0138	-0.025	0.017	2018	1430	1.00	0.00032	0.00039
v_zGPe|participant_id[2]	0.0073	0.016	-0.011	0.039	1669	1281	1.00	0.00041	0.00052
v_zGPe|participant_id[3]	0	0.0154	-0.023	0.024	1713	1122	1.00	0.00039	0.00048
v_zGPe|participant_id[4]	-0.0083	0.016	-0.038	0.0099	1272	1114	1.00	0.00046	0.00048
v_zGPe|participant_id[5]	0.0043	0.0163	-0.017	0.032	1878	1305	1.00	0.00038	0.00049
v_zGPe|participant_id[6]	-0.0007	0.0143	-0.024	0.022	1627	1226	1.00	0.00037	0.00044
v_zGPe|participant_id[7]	0.0001	0.0137	-0.022	0.02	1953	1311	1.00	0.00032	0.00036
v_zGPe|participant_id[8]	0.0013	0.0147	-0.02	0.025	1527	1192	1.01	0.00037	0.00049
v_zGPe|participant_id[9]	0.002	0.014	-0.018	0.025	2145	1314	1.00	0.00032	0.00037
v_1|participant_id[1]	0.009	0.046	-0.065	0.084	2030	1119	1.00	0.001	0.0011
v_1|participant_id[10]	-0.011	0.043	-0.082	0.053	1721	1234	1.00	0.001	0.00099
v_1|participant_id[2]	0.024	0.046	-0.037	0.11	1413	1180	1.00	0.0012	0.0012
v_1|participant_id[3]	-0.032	0.05	-0.12	0.031	1139	1085	1.00	0.0015	0.0013
v_1|participant_id[4]	-0.004	0.043	-0.073	0.063	1814	1325	1.00	0.001	0.00097
v_1|participant_id[5]	0.05	0.056	-0.014	0.15	750	1143	1.00	0.002	0.0017
v_1|participant_id[6]	-0.01	0.045	-0.088	0.058	1835	1384	1.00	0.001	0.0011
v_1|participant_id[7]	-0.029	0.048	-0.12	0.031	1210	1094	1.00	0.0014	0.0012
v_1|participant_id[8]	0.033	0.049	-0.026	0.12	928	1064	1.00	0.0016	0.0015
v_1|participant_id[9]	-0.016	0.044	-0.093	0.046	1384	1237	1.00	0.0012	0.0012
v_zGPe	0.3099	0.0126	0.29	0.33	1339	1021	1.00	0.00035	0.00027
v_Intercept	1.538	0.043	1.5	1.6	1210	1144	1.00	0.0012	0.00091
z	0.474	0.0114	0.46	0.49	1256	1076	1.00	0.00033	0.00024
t	0.4978	0.0061	0.49	0.51	1296	1186	1.00	0.00017	0.00011
v_zSTN	0.807	0.0178	0.78	0.84	1156	1105	1.00	0.00052	0.0004
v_zSTN|participant_id[1]	-0.008	0.023	-0.05	0.023	1500	1169	1.00	0.0006	0.00057
v_zSTN|participant_id[10]	-0.003	0.024	-0.041	0.035	1976	1299	1.00	0.00055	0.00054
v_zSTN|participant_id[2]	-0.023	0.028	-0.077	0.011	1150	1278	1.00	0.00082	0.00073
v_zSTN|participant_id[3]	0.024	0.03	-0.01	0.082	1313	1150	1.00	0.00081	0.00074
v_zSTN|participant_id[4]	0.011	0.023	-0.02	0.054	1531	1135	1.00	0.00061	0.00059
v_zSTN|participant_id[5]	0.016	0.026	-0.018	0.066	1205	1110	1.00	0.00075	0.00067
v_zSTN|participant_id[6]	0.001	0.024	-0.037	0.041	1556	1290	1.00	0.0006	0.00051
v_zSTN|participant_id[7]	-0.015	0.025	-0.061	0.017	1288	1181	1.00	0.00072	0.00065
v_zSTN|participant_id[8]	0.008	0.022	-0.025	0.048	1610	1328	1.00	0.00057	0.00051
v_zSTN|participant_id[9]	-0.008	0.023	-0.049	0.025	1407	1143	1.00	0.00062	0.00061
v_1|participant_id_sigma	0.052	0.035	0.0062	0.11	447	518	1.01	0.0015	0.0014

In [22]:

# Plotting the posterior predictive
hssm.plotting.plot_predictive(model_reg_v_ex2_A2, col="participant_id", col_wrap=5)

# Plotting the posterior predictive hssm.plotting.plot_predictive(model_reg_v_ex2_A2, col="participant_id", col_wrap=5)

No posterior_predictive samples found. Generating posterior_predictive samples using the provided DataTree object and the original data. This will modify the provided DataTree object, or if not provided, the traces object stored inside the model.

Out[22]:

<seaborn.axisgrid.FacetGrid at 0x14a4d2490>

Additional Case studies (between-subject & group variables)¶

Case Study 3: Participant-level hierarchy, two within-subject coefficients and one between-subject coefficient¶

• we still have the intracranial recordings
• but now you have different patients and they vary in symptom severity for Parkinson
• you wonder whether those with higher symptom severity have a lower drift rate because the modulation from STN is less efficient

Step 1: Regression-based data simulation (multiple subjects)¶

In [23]:

# Function to simulate data for one participant
def simulate_participant2(participant_id, sevScore, size=300):
    """Simulate DDM trial data for one participant with severity-dependent drift."""
    intercept = 0.5
    zSTN = np.random.normal(loc=1, scale=2, size=size)
    zGPe = np.random.normal(loc=1, scale=2, size=size)

    ## Strength of neural modulation depends on PD severity score

    #### Case 1: only the interaction btw sevScore & STN/GPe will turn out
    #### sign, the main effects of STN & GPe will disappear
    # v = intercept + 0.8 * zSTN * sevScore + 0.3 * zGPe * sevScore

    #### Case 2: main effects of STN/GPe will remain, plus a significant
    #### interaction, but no modulation on intercept
    if sevScore < 0.2:
        v = intercept + 0.8 * zSTN + 0.3 * zGPe
    elif sevScore >= 0.2 and sevScore < 0.4:
        v = intercept + 0.7 * zSTN + 0.25 * zGPe
    elif sevScore >= 0.4 and sevScore < 0.6:
        v = intercept + 0.6 * zSTN + 0.2 * zGPe
    elif sevScore >= 0.6 and sevScore < 0.8:
        v = intercept + 0.5 * zSTN + 0.15 * zGPe
    else:
        v = intercept + 0.4 * zSTN + 0.1 * zGPe

    # Assume `hssm.simulate_data` returns a DataFrame
    true_values = np.column_stack(
        [v, np.repeat([[1.5, 0.5, 0.5]], axis=0, repeats=size)]
    )
    dataset_reg_v = hssm.simulate_data(
        model="ddm",
        theta=true_values,
        size=1,  # Generate one data point for each of the 1000 set of true values
    )

    # Adding additional variables to the dataset
    dataset_reg_v["zSTN"] = zSTN
    dataset_reg_v["zGPe"] = zGPe
    dataset_reg_v["participant_id"] = str(participant_id)
    dataset_reg_v["sevScore"] = sevScore
    return dataset_reg_v


# Simulate data for four participants
### note that we assume that STN & GPe

# patients with severity scores
subj_list = [1, 1, 2, 3, 4, 5, 6, 7, 8, 8, 8]
sevscore_list = [0.00, 0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80, 0.90, 1.00]

# Combine datasets into one DataFrame
combined_dataset2 = pd.concat(
    [
        simulate_participant2(subj, sevscore)
        for subj, sevscore in zip(subj_list, sevscore_list)
    ],
    ignore_index=True,
)
combined_dataset2

# Function to simulate data for one participant def simulate_participant2(participant_id, sevScore, size=300): """Simulate DDM trial data for one participant with severity-dependent drift.""" intercept = 0.5 zSTN = np.random.normal(loc=1, scale=2, size=size) zGPe = np.random.normal(loc=1, scale=2, size=size) ## Strength of neural modulation depends on PD severity score #### Case 1: only the interaction btw sevScore & STN/GPe will turn out #### sign, the main effects of STN & GPe will disappear # v = intercept + 0.8 * zSTN * sevScore + 0.3 * zGPe * sevScore #### Case 2: main effects of STN/GPe will remain, plus a significant #### interaction, but no modulation on intercept if sevScore < 0.2: v = intercept + 0.8 * zSTN + 0.3 * zGPe elif sevScore >= 0.2 and sevScore < 0.4: v = intercept + 0.7 * zSTN + 0.25 * zGPe elif sevScore >= 0.4 and sevScore < 0.6: v = intercept + 0.6 * zSTN + 0.2 * zGPe elif sevScore >= 0.6 and sevScore < 0.8: v = intercept + 0.5 * zSTN + 0.15 * zGPe else: v = intercept + 0.4 * zSTN + 0.1 * zGPe # Assume `hssm.simulate_data` returns a DataFrame true_values = np.column_stack( [v, np.repeat([[1.5, 0.5, 0.5]], axis=0, repeats=size)] ) dataset_reg_v = hssm.simulate_data( model="ddm", theta=true_values, size=1, # Generate one data point for each of the 1000 set of true values ) # Adding additional variables to the dataset dataset_reg_v["zSTN"] = zSTN dataset_reg_v["zGPe"] = zGPe dataset_reg_v["participant_id"] = str(participant_id) dataset_reg_v["sevScore"] = sevScore return dataset_reg_v # Simulate data for four participants ### note that we assume that STN & GPe # patients with severity scores subj_list = [1, 1, 2, 3, 4, 5, 6, 7, 8, 8, 8] sevscore_list = [0.00, 0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80, 0.90, 1.00] # Combine datasets into one DataFrame combined_dataset2 = pd.concat( [ simulate_participant2(subj, sevscore) for subj, sevscore in zip(subj_list, sevscore_list) ], ignore_index=True, ) combined_dataset2

Out[23]:

	rt	response	zSTN	zGPe	participant_id	sevScore
0	1.899321	-1.0	-3.057895	0.802185	1	0.0
1	1.779316	1.0	1.111865	0.537509	1	0.0
2	1.962398	1.0	0.495962	-0.122734	1	0.0
3	2.923199	1.0	0.054011	1.787760	1	0.0
4	2.722710	1.0	-0.255625	1.421099	1	0.0
...	...	...	...	...	...	...
3295	2.610382	1.0	1.134913	0.561621	8	1.0
3296	2.092122	-1.0	-2.539877	0.449750	8	1.0
3297	1.699619	1.0	3.628208	0.140389	8	1.0
3298	1.112458	1.0	1.068393	-1.009358	8	1.0
3299	5.769248	1.0	1.490435	-0.354181	8	1.0

3300 rows × 6 columns

Step 2: Model Setup & Priors¶

In [24]:

model_reg_v_ex3_A1 = hssm.HSSM(
    data=combined_dataset2,
    include=[
        {
            "name": "v",
            "formula": (
                "v ~ 1 + (zSTN + zGPe)*sevScore + (1 + zSTN + zGPe | participant_id)"
            ),
            "prior": {
                # All ways to specify priors in the non-regression case
                # work the same way here.
                "Intercept": {"name": "Normal", "mu": 1.5, "sigma": 1.0},
                "zSTN": {"name": "Normal", "mu": 0, "sigma": 1.0},
                "zGPe": {"name": "Normal", "mu": 0, "sigma": 1.0},
                "sevScore": {"name": "Normal", "mu": 0, "sigma": 1.0},
                "zSTN:sevScore": {"name": "Normal", "mu": 0, "sigma": 1.0},
                "zGPe:sevScore": {"name": "Normal", "mu": 0, "sigma": 1.0},
                "1|participant_id": {
                    "name": "Normal",
                    "mu": 0,
                    "sigma": {"name": "HalfNormal", "sigma": 1.0},
                },
                "zSTN|participant_id": {
                    "name": "Normal",
                    "mu": 0,
                    "sigma": {"name": "HalfNormal", "sigma": 1.0},
                },
                "zGPe|participant_id": {
                    "name": "Normal",
                    "mu": 0,
                    "sigma": {"name": "HalfNormal", "sigma": 1.0},
                },
            },
            "link": "identity",
        }
    ],
    noncentered=True,
    p_outlier=0.05,
)

model_reg_v_ex3_A1 = hssm.HSSM( data=combined_dataset2, include=[ { "name": "v", "formula": ( "v ~ 1 + (zSTN + zGPe)*sevScore + (1 + zSTN + zGPe | participant_id)" ), "prior": { # All ways to specify priors in the non-regression case # work the same way here. "Intercept": {"name": "Normal", "mu": 1.5, "sigma": 1.0}, "zSTN": {"name": "Normal", "mu": 0, "sigma": 1.0}, "zGPe": {"name": "Normal", "mu": 0, "sigma": 1.0}, "sevScore": {"name": "Normal", "mu": 0, "sigma": 1.0}, "zSTN:sevScore": {"name": "Normal", "mu": 0, "sigma": 1.0}, "zGPe:sevScore": {"name": "Normal", "mu": 0, "sigma": 1.0}, "1|participant_id": { "name": "Normal", "mu": 0, "sigma": {"name": "HalfNormal", "sigma": 1.0}, }, "zSTN|participant_id": { "name": "Normal", "mu": 0, "sigma": {"name": "HalfNormal", "sigma": 1.0}, }, "zGPe|participant_id": { "name": "Normal", "mu": 0, "sigma": {"name": "HalfNormal", "sigma": 1.0}, }, }, "link": "identity", } ], noncentered=True, p_outlier=0.05, )

Model initialized successfully.

In [25]:

samples_model_reg_v_ex3_A1 = model_reg_v_ex3_A1.sample(
    sampler="numpyro",
    cores=3,
    chains=3,
    draws=500,
    tune=500,
    idata_kwargs=dict(log_likelihood=True),
)

samples_model_reg_v_ex3_A1 = model_reg_v_ex3_A1.sample( sampler="numpyro", cores=3, chains=3, draws=500, tune=500, idata_kwargs=dict(log_likelihood=True), )

Using default initvals. 

NUTS[numpyro]: [z, a, t, v_Intercept, v_zSTN, v_zGPe, v_sevScore, v_zSTN:sevScore, v_zGPe:sevScore, v_1|participant_id_sigma, v_1|participant_id_offset, v_zSTN|participant_id_sigma, v_zSTN|participant_id_offset, v_zGPe|participant_id_sigma, v_zGPe|participant_id_offset]

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sample:  62%|██████▎   | 625/1000 [00:49<00:14, 25.35it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  63%|██████▎   | 628/1000 [00:49<00:15, 24.47it/s, 63 steps of size 9.14e-02. acc. prob=0.93]

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sample:  64%|██████▎   | 637/1000 [00:49<00:14, 25.06it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  64%|██████▍   | 640/1000 [00:49<00:14, 25.35it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  64%|██████▍   | 643/1000 [00:50<00:14, 24.46it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  65%|██████▍   | 647/1000 [00:50<00:13, 26.34it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  65%|██████▌   | 650/1000 [00:50<00:13, 26.49it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  65%|██████▌   | 654/1000 [00:50<00:11, 28.98it/s, 31 steps of size 9.14e-02. acc. prob=0.94]

sample:  66%|██████▌   | 657/1000 [00:50<00:12, 26.62it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  66%|██████▌   | 660/1000 [00:50<00:12, 26.67it/s, 31 steps of size 9.14e-02. acc. prob=0.94]

sample:  66%|██████▋   | 663/1000 [00:50<00:13, 25.21it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  67%|██████▋   | 666/1000 [00:50<00:13, 24.22it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

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sample:  69%|██████▉   | 688/1000 [00:51<00:12, 25.21it/s, 31 steps of size 9.14e-02. acc. prob=0.94]

sample:  69%|██████▉   | 691/1000 [00:51<00:12, 25.55it/s, 31 steps of size 9.14e-02. acc. prob=0.94]

sample:  70%|██████▉   | 695/1000 [00:52<00:10, 28.58it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

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sample:  76%|███████▋  | 763/1000 [00:54<00:08, 26.77it/s, 31 steps of size 9.14e-02. acc. prob=0.94]

sample:  77%|███████▋  | 768/1000 [00:54<00:07, 30.48it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  77%|███████▋  | 772/1000 [00:54<00:07, 30.33it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  78%|███████▊  | 775/1000 [00:55<00:07, 29.42it/s, 31 steps of size 9.14e-02. acc. prob=0.94]

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sample:  78%|███████▊  | 781/1000 [00:55<00:07, 28.23it/s, 31 steps of size 9.14e-02. acc. prob=0.94]

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sample:  79%|███████▉  | 789/1000 [00:55<00:06, 32.12it/s, 31 steps of size 9.14e-02. acc. prob=0.94]

sample:  79%|███████▉  | 794/1000 [00:55<00:05, 35.60it/s, 31 steps of size 9.14e-02. acc. prob=0.94]

sample:  80%|███████▉  | 797/1000 [00:55<00:06, 31.27it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  80%|████████  | 801/1000 [00:55<00:06, 30.89it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

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sample:  83%|████████▎ | 828/1000 [00:56<00:06, 27.59it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  83%|████████▎ | 831/1000 [00:57<00:06, 26.06it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  83%|████████▎ | 834/1000 [00:57<00:06, 26.38it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

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sample:  84%|████████▍ | 841/1000 [00:57<00:05, 27.08it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

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sample:  87%|████████▋ | 871/1000 [00:58<00:04, 26.61it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  87%|████████▋ | 874/1000 [00:58<00:04, 26.29it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

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sample:  88%|████████▊ | 880/1000 [00:58<00:05, 23.84it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  88%|████████▊ | 883/1000 [00:58<00:05, 23.12it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  89%|████████▊ | 886/1000 [00:59<00:04, 23.95it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  89%|████████▉ | 889/1000 [00:59<00:04, 24.69it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  89%|████████▉ | 892/1000 [00:59<00:04, 25.23it/s, 31 steps of size 9.14e-02. acc. prob=0.94]

sample:  90%|████████▉ | 895/1000 [00:59<00:04, 25.63it/s, 31 steps of size 9.14e-02. acc. prob=0.94]

sample:  90%|████████▉ | 898/1000 [00:59<00:04, 24.61it/s, 95 steps of size 9.14e-02. acc. prob=0.94]

sample:  90%|█████████ | 901/1000 [00:59<00:03, 25.21it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  90%|█████████ | 904/1000 [00:59<00:03, 25.83it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  91%|█████████ | 907/1000 [00:59<00:03, 26.17it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  91%|█████████ | 910/1000 [01:00<00:03, 26.47it/s, 31 steps of size 9.14e-02. acc. prob=0.94]

sample:  91%|█████████▏| 914/1000 [01:00<00:03, 27.71it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  92%|█████████▏| 917/1000 [01:00<00:03, 27.49it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

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sample:  93%|█████████▎| 926/1000 [01:00<00:02, 27.30it/s, 31 steps of size 9.14e-02. acc. prob=0.94]

sample:  93%|█████████▎| 930/1000 [01:00<00:02, 26.48it/s, 95 steps of size 9.14e-02. acc. prob=0.94]

sample:  93%|█████████▎| 933/1000 [01:00<00:02, 24.81it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  94%|█████████▎| 936/1000 [01:00<00:02, 25.14it/s, 31 steps of size 9.14e-02. acc. prob=0.94]

sample:  94%|█████████▍| 939/1000 [01:01<00:02, 25.38it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  94%|█████████▍| 943/1000 [01:01<00:02, 26.92it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  95%|█████████▍| 946/1000 [01:01<00:02, 25.19it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  95%|█████████▍| 949/1000 [01:01<00:01, 25.68it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  95%|█████████▌| 952/1000 [01:01<00:01, 24.41it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  96%|█████████▌| 955/1000 [01:01<00:01, 25.04it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  96%|█████████▌| 958/1000 [01:01<00:01, 24.23it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  96%|█████████▌| 961/1000 [01:01<00:01, 24.77it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  96%|█████████▋| 964/1000 [01:02<00:01, 23.94it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  97%|█████████▋| 967/1000 [01:02<00:01, 23.55it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  97%|█████████▋| 971/1000 [01:02<00:01, 25.51it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  97%|█████████▋| 974/1000 [01:02<00:01, 24.55it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  98%|█████████▊| 977/1000 [01:02<00:00, 23.82it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  98%|█████████▊| 979/1000 [01:02<00:00, 22.27it/s, 95 steps of size 9.14e-02. acc. prob=0.94]

sample:  98%|█████████▊| 982/1000 [01:02<00:00, 23.59it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  99%|█████████▊| 987/1000 [01:03<00:00, 28.04it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  99%|█████████▉| 990/1000 [01:03<00:00, 26.38it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample:  99%|█████████▉| 994/1000 [01:03<00:00, 27.56it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample: 100%|█████████▉| 997/1000 [01:03<00:00, 27.52it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample: 100%|██████████| 1000/1000 [01:03<00:00, 27.49it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

sample: 100%|██████████| 1000/1000 [01:03<00:00, 15.75it/s, 63 steps of size 9.14e-02. acc. prob=0.94]

We recommend running at least 4 chains for robust computation of convergence diagnostics

In [26]:

#### posterior statistics of non-centered model
az.summary(samples_model_reg_v_ex3_A1)

#### posterior statistics of non-centered model az.summary(samples_model_reg_v_ex3_A1)

Out[26]:

	mean	sd	eti89_lb	eti89_ub	ess_bulk	ess_tail	r_hat	mcse_mean	mcse_sd
v_zGPe|participant_id_sigma	0.025	0.019	0.003	0.058	445	572	1.00	0.00091	0.0012
v_zGPe|participant_id[1]	0.021	0.028	-0.009	0.072	638	551	1.00	0.0012	0.0016
v_zGPe|participant_id[2]	-0.001	0.022	-0.033	0.032	1306	906	1.00	0.00066	0.00096
v_zGPe|participant_id[3]	0.009	0.021	-0.019	0.044	1085	942	1.00	0.00072	0.00091
v_zGPe|participant_id[4]	-0.001	0.0197	-0.033	0.03	1789	1201	1.00	0.00049	0.00047
v_zGPe|participant_id[5]	-0.007	0.02	-0.038	0.022	1477	1169	1.00	0.00052	0.00055
v_zGPe|participant_id[6]	-0.0003	0.0183	-0.029	0.029	1540	1145	1.00	0.00046	0.00046
v_zGPe|participant_id[7]	-0.023	0.025	-0.072	0.0052	672	885	1.00	0.00098	0.00097
v_zGPe|participant_id[8]	0.003	0.023	-0.033	0.036	984	677	1.01	0.00082	0.0011
v_1|participant_id[1]	0	0.032	-0.047	0.047	1060	844	1.00	0.0012	0.0021
v_1|participant_id[2]	-0.012	0.033	-0.069	0.027	1385	1093	1.00	0.00099	0.0015
v_1|participant_id[3]	0.008	0.033	-0.037	0.061	1508	1232	1.00	0.001	0.0017
v_1|participant_id[4]	0.003	0.031	-0.04	0.055	1461	925	1.01	0.001	0.0015
v_1|participant_id[5]	0.014	0.034	-0.026	0.075	1144	950	1.01	0.0012	0.0017
v_1|participant_id[6]	-0.003	0.03	-0.052	0.041	1726	1146	1.00	0.00074	0.00087
v_1|participant_id[7]	0.004	0.03	-0.04	0.053	1709	1444	1.00	0.00073	0.00087
v_1|participant_id[8]	-0.014	0.034	-0.075	0.024	1287	960	1.00	0.001	0.0013
v_1|participant_id_offset[1]	-0.01	0.91	-1.5	1.4	1540	989	1.00	0.024	0.016
v_1|participant_id_offset[2]	-0.3	0.91	-1.7	1.1	1984	1008	1.01	0.021	0.016
v_1|participant_id_offset[3]	0.21	0.92	-1.3	1.7	2506	1194	1.00	0.018	0.013
v_1|participant_id_offset[4]	0.05	0.91	-1.4	1.4	2354	1116	1.00	0.019	0.015
v_1|participant_id_offset[5]	0.37	0.91	-1.1	1.8	1685	1176	1.00	0.022	0.015
v_1|participant_id_offset[6]	-0.06	0.92	-1.5	1.4	2548	1050	1.01	0.018	0.013
v_1|participant_id_offset[7]	0.1	0.91	-1.3	1.5	2001	1173	1.00	0.02	0.015
v_1|participant_id_offset[8]	-0.35	0.94	-1.8	1.1	1884	1295	1.00	0.022	0.016
v_zSTN	0.792	0.026	0.75	0.83	1022	843	1.00	0.00086	0.00073
v_zSTN|participant_id[1]	0.008	0.019	-0.013	0.043	1044	911	1.00	0.00065	0.0009
v_zSTN|participant_id[2]	0.0016	0.0149	-0.021	0.026	1657	1210	1.00	0.00039	0.00043
v_zSTN|participant_id[3]	-0.0015	0.0167	-0.028	0.023	2076	1123	1.00	0.00038	0.00049
v_zSTN|participant_id[4]	-0.0074	0.016	-0.037	0.012	1398	1110	1.00	0.00044	0.00052
v_zSTN|participant_id[5]	0.0052	0.0158	-0.016	0.033	2016	1381	1.00	0.00035	0.00044
v_zSTN|participant_id[6]	-0.0023	0.0148	-0.027	0.019	1915	1260	1.00	0.00038	0.00048
v_zSTN|participant_id[7]	-0.0011	0.015	-0.029	0.022	1779	1099	1.00	0.0004	0.0005
v_zSTN|participant_id[8]	-0.003	0.018	-0.031	0.023	1246	882	1.00	0.00057	0.00083
v_1|participant_id_sigma	0.031	0.028	0.0021	0.076	639	778	1.01	0.0012	0.003
v_zSTN:sevScore	-0.42	0.044	-0.49	-0.35	919	792	1.00	0.0015	0.0014
v_sevScore	0.043	0.077	-0.072	0.17	1024	1131	1.00	0.0025	0.0021
v_zSTN|participant_id_sigma	0.016	0.015	0.0014	0.04	707	794	1.01	0.00053	0.0012
v_zGPe|participant_id_offset[1]	0.7	0.87	-0.7	2	1118	1193	1.00	0.026	0.021
v_zGPe|participant_id_offset[2]	-0.12	0.83	-1.5	1.2	1879	1201	1.00	0.019	0.014
v_zGPe|participant_id_offset[3]	0.27	0.82	-1.1	1.5	2034	1203	1.00	0.018	0.014
v_zGPe|participant_id_offset[4]	-0.07	0.82	-1.3	1.2	2303	971	1.00	0.017	0.012
v_zGPe|participant_id_offset[5]	-0.25	0.83	-1.6	1.1	1953	1102	1.00	0.019	0.014
v_zGPe|participant_id_offset[6]	-0	0.77	-1.2	1.2	1652	960	1.00	0.019	0.014
v_zGPe|participant_id_offset[7]	-0.83	0.87	-2.1	0.56	1652	929	1.00	0.022	0.02
v_zGPe|participant_id_offset[8]	0.25	0.86	-1.2	1.6	1327	1094	1.00	0.024	0.016
v_zSTN|participant_id_offset[1]	0.37	0.96	-1.2	1.9	1417	964	1.00	0.025	0.019
v_zSTN|participant_id_offset[2]	0.06	0.89	-1.4	1.5	2003	953	1.01	0.02	0.014
v_zSTN|participant_id_offset[3]	-0.09	0.93	-1.6	1.4	2264	960	1.00	0.02	0.015
v_zSTN|participant_id_offset[4]	-0.36	0.91	-1.8	1.1	1684	1088	1.00	0.022	0.02
v_zSTN|participant_id_offset[5]	0.24	0.9	-1.3	1.7	2357	1165	1.00	0.019	0.013
v_zSTN|participant_id_offset[6]	-0.09	0.87	-1.4	1.3	2584	1230	1.01	0.017	0.013
v_zSTN|participant_id_offset[7]	-0.02	0.89	-1.4	1.4	2299	1040	1.01	0.019	0.015
v_zSTN|participant_id_offset[8]	-0.08	0.96	-1.6	1.5	1830	1049	1.00	0.022	0.016
v_zGPe	0.3	0.03	0.25	0.34	739	496	1.00	0.0013	0.0015
v_zGPe:sevScore	-0.223	0.054	-0.29	-0.13	667	563	1.01	0.0024	0.0027
a	1.4826	0.0193	1.5	1.5	1667	1247	1.00	0.00047	0.00034
v_Intercept	0.489	0.048	0.41	0.56	993	1077	1.00	0.0017	0.0019
z	0.4941	0.0073	0.48	0.51	1572	1027	1.00	0.00018	0.00012
t	0.503	0.0069	0.49	0.51	1530	1153	1.00	0.00018	0.00013

• there is no main effect of severity (makes sense because we didn't change the intercept by severity!)
• you can see that by looking at the credible interval (hdi's). They are interpret similar to confidence intervals
• the intercept itself is recovered well (input: 0.5)
• the effect of severity onto STN is significant. This is good because we did manipulate that (0.6 was around the mean effect)

Case Study 4: Two hierarchical layers (trial- and sbj-level) & Two within-sbj coefficients & one btw-sbj coefficient & group¶

• we still have the intracranial recordings
• but now you have different patients and they vary in symptom severity for Parkinson
• you wonder whether those with higher symptom severity have a lower drift rate because the modulation from STN is less efficient

BUT NOW: your friend is a clinician and tells you that some people might have Dystonia rather than Parkinson and they show different symptoms. So, you want to take this into account now

Audience Question: how would you do that? How would you change the data simulation?¶

Step 1: Regression-based data simulation (multiple subjects)¶

In [27]:

# Function to simulate data for one participant
def simulate_participant3(participant_id, sevScore, diagnosis, size=300):
    """Simulate DDM trial data for one participant with diagnosis-dependent drift."""
    # intercept = 0.5
    zSTN = np.random.normal(loc=1, scale=2, size=size)
    zGPe = np.random.normal(loc=1, scale=2, size=size)

    if diagnosis == "PD":
        intercept = 0.3
    else:
        intercept = 0.6

    ## Strength of neural modulation depends on PD severity score
    if sevScore <= 0.2:
        v = intercept + 0.8 * zSTN + 0.3 * zGPe
    elif sevScore > 0.2 and sevScore <= 0.4:
        v = intercept + 0.6 * zSTN + 0.2 * zGPe
    elif sevScore > 0.4 and sevScore < 0.6:
        v = intercept + 0.5 * zSTN + 0.1 * zGPe
    else:
        v = intercept + 0.4 * zSTN + 0.0 * zGPe

    # Assume `hssm.simulate_data` returns a DataFrame
    true_values = np.column_stack(
        [v, np.repeat([[1.5, 0.5, 0.5]], axis=0, repeats=size)]
    )
    dataset_reg_v = hssm.simulate_data(
        model="ddm",
        theta=true_values,
        size=1,  # Generate one data point for each of the 1000 set of true values
    )

    # Adding additional variables to the dataset
    dataset_reg_v["zSTN"] = zSTN
    dataset_reg_v["zGPe"] = zGPe
    dataset_reg_v["participant_id"] = str(participant_id)
    dataset_reg_v["sevScore"] = sevScore
    dataset_reg_v["diagnosis"] = diagnosis

    return dataset_reg_v


# Simulate data for four participants
### note that we assume that STN & GPe

## PD patients:
# patients with low severity scores
subj_list = [1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8]
sevscore_list = [
    0.10,
    0.20,
    0.30,
    0.40,
    0.50,
    0.60,
    0.70,
    0.80,
    0.10,
    0.20,
    0.30,
    0.40,
    0.50,
    0.60,
    0.70,
    0.80,
]
diagnosis_list = [
    "PD",
    "PD",
    "PD",
    "PD",
    "PD",
    "PD",
    "PD",
    "PD",
    "DD",
    "DD",
    "DD",
    "DD",
    "DD",
    "DD",
    "DD",
    "DD",
]

# [simulate_participant3(subj, sevscore, diagnosis) for subj, sevscore, diagnosis
# in zip(subj_list, sevscore_list, diagnosis_list)]
# dataset2_participant1 = simulate_participant3(1,0.10,"PD")
# dataset2_participant2 = simulate_participant3(2,0.20,"PD")
# dataset2_participant3 = simulate_participant3(3,0.30,"PD")
# dataset2_participant4 = simulate_participant3(4,0.40,"PD")
# # patients with high severity scores
# dataset2_participant5 = simulate_participant3(5,0.50,"PD")
# dataset2_participant6 = simulate_participant3(6,0.60,"PD")
# dataset2_participant7 = simulate_participant3(7,0.70,"PD")
# dataset2_participant8 = simulate_participant3(8,0.80,"PD")

# ## Dystonia patients:
# # patients with low severity scores
# dataset2_participant9 = simulate_participant3(1,0.10,"DD")
# dataset2_participant10 = simulate_participant3(2,0.20,"DD")
# dataset2_participant11 = simulate_participant3(3,0.30,"DD")
# dataset2_participant12 = simulate_participant3(4,0.40,"DD")
# # patients with high severity scores
# dataset2_participant13 = simulate_participant3(5,0.50,"DD")
# dataset2_participant14 = simulate_participant3(6,0.60,"DD")
# dataset2_participant15 = simulate_participant3(7,0.70,"DD")
# dataset2_participant16 = simulate_participant3(8,0.80,"DD")

# Combine datasets into one DataFrame
combined_dataset3 = pd.concat(
    [
        simulate_participant3(subj, sevscore, diagnosis)
        for subj, sevscore, diagnosis in zip(subj_list, sevscore_list, diagnosis_list)
    ],
    ignore_index=True,
)
combined_dataset3

# Function to simulate data for one participant def simulate_participant3(participant_id, sevScore, diagnosis, size=300): """Simulate DDM trial data for one participant with diagnosis-dependent drift.""" # intercept = 0.5 zSTN = np.random.normal(loc=1, scale=2, size=size) zGPe = np.random.normal(loc=1, scale=2, size=size) if diagnosis == "PD": intercept = 0.3 else: intercept = 0.6 ## Strength of neural modulation depends on PD severity score if sevScore <= 0.2: v = intercept + 0.8 * zSTN + 0.3 * zGPe elif sevScore > 0.2 and sevScore <= 0.4: v = intercept + 0.6 * zSTN + 0.2 * zGPe elif sevScore > 0.4 and sevScore < 0.6: v = intercept + 0.5 * zSTN + 0.1 * zGPe else: v = intercept + 0.4 * zSTN + 0.0 * zGPe # Assume `hssm.simulate_data` returns a DataFrame true_values = np.column_stack( [v, np.repeat([[1.5, 0.5, 0.5]], axis=0, repeats=size)] ) dataset_reg_v = hssm.simulate_data( model="ddm", theta=true_values, size=1, # Generate one data point for each of the 1000 set of true values ) # Adding additional variables to the dataset dataset_reg_v["zSTN"] = zSTN dataset_reg_v["zGPe"] = zGPe dataset_reg_v["participant_id"] = str(participant_id) dataset_reg_v["sevScore"] = sevScore dataset_reg_v["diagnosis"] = diagnosis return dataset_reg_v # Simulate data for four participants ### note that we assume that STN & GPe ## PD patients: # patients with low severity scores subj_list = [1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8] sevscore_list = [ 0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80, 0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80, ] diagnosis_list = [ "PD", "PD", "PD", "PD", "PD", "PD", "PD", "PD", "DD", "DD", "DD", "DD", "DD", "DD", "DD", "DD", ] # [simulate_participant3(subj, sevscore, diagnosis) for subj, sevscore, diagnosis # in zip(subj_list, sevscore_list, diagnosis_list)] # dataset2_participant1 = simulate_participant3(1,0.10,"PD") # dataset2_participant2 = simulate_participant3(2,0.20,"PD") # dataset2_participant3 = simulate_participant3(3,0.30,"PD") # dataset2_participant4 = simulate_participant3(4,0.40,"PD") # # patients with high severity scores # dataset2_participant5 = simulate_participant3(5,0.50,"PD") # dataset2_participant6 = simulate_participant3(6,0.60,"PD") # dataset2_participant7 = simulate_participant3(7,0.70,"PD") # dataset2_participant8 = simulate_participant3(8,0.80,"PD") # ## Dystonia patients: # # patients with low severity scores # dataset2_participant9 = simulate_participant3(1,0.10,"DD") # dataset2_participant10 = simulate_participant3(2,0.20,"DD") # dataset2_participant11 = simulate_participant3(3,0.30,"DD") # dataset2_participant12 = simulate_participant3(4,0.40,"DD") # # patients with high severity scores # dataset2_participant13 = simulate_participant3(5,0.50,"DD") # dataset2_participant14 = simulate_participant3(6,0.60,"DD") # dataset2_participant15 = simulate_participant3(7,0.70,"DD") # dataset2_participant16 = simulate_participant3(8,0.80,"DD") # Combine datasets into one DataFrame combined_dataset3 = pd.concat( [ simulate_participant3(subj, sevscore, diagnosis) for subj, sevscore, diagnosis in zip(subj_list, sevscore_list, diagnosis_list) ], ignore_index=True, ) combined_dataset3

Out[27]:

	rt	response	zSTN	zGPe	participant_id	sevScore	diagnosis
0	0.936833	-1.0	-2.754070	-1.719626	1	0.1	PD
1	1.049654	1.0	2.736417	2.045129	1	0.1	PD
2	1.191599	1.0	-0.021384	0.377893	1	0.1	PD
3	2.432412	1.0	-0.380539	4.469055	1	0.1	PD
4	1.587661	1.0	1.653595	-0.396669	1	0.1	PD
...	...	...	...	...	...	...	...
4795	1.000964	1.0	0.393771	1.020571	8	0.8	DD
4796	2.113130	1.0	-1.310476	2.243242	8	0.8	DD
4797	1.511142	1.0	3.044100	0.773762	8	0.8	DD
4798	3.110676	1.0	-0.831975	-2.348503	8	0.8	DD
4799	2.177231	1.0	-1.172113	0.847041	8	0.8	DD

4800 rows × 7 columns

In [28]:

model_reg_v_ex4_A1 = hssm.HSSM(
    data=combined_dataset3,
    include=[
        {
            "name": "v",
            "formula": (
                "v ~ 1 + (zSTN + zGPe)*sevScore*C(diagnosis) + "
                "((1 + zSTN + zGPe)|participant_id)"
            ),
            "prior": {
                # All ways to specify priors in the non-regression case
                # work the same way here.
                "Intercept": {"name": "Normal", "mu": 1.5, "sigma": 1.5},
                "zSTN": {"name": "Normal", "mu": 0, "sigma": 1.0},
                "zGPe": {"name": "Normal", "mu": 0, "sigma": 1.0},
                "sevScore": {"name": "Normal", "mu": 0, "sigma": 1.0},
                "zSTN:sevScore": {"name": "Normal", "mu": 0, "sigma": 1.0},
                "zGPe:sevScore": {"name": "Normal", "mu": 0, "sigma": 1.0},
                "1|participant_id": {
                    "name": "Normal",
                    "mu": 0,
                    "sigma": {"name": "HalfNormal", "sigma": 1},
                },
                "zSTN|participant_id": {
                    "name": "Normal",
                    "mu": 0,
                    "sigma": {"name": "HalfNormal", "sigma": 1},
                },
                "zGPe|participant_id": {
                    "name": "Normal",
                    "mu": 0,
                    "sigma": {"name": "HalfNormal", "sigma": 1},
                },
            },
            "link": "identity",
        }
    ],
    noncentered=True,
    p_outlier=0.05,
)
model_reg_v_ex4_A1

model_reg_v_ex4_A1 = hssm.HSSM( data=combined_dataset3, include=[ { "name": "v", "formula": ( "v ~ 1 + (zSTN + zGPe)*sevScore*C(diagnosis) + " "((1 + zSTN + zGPe)|participant_id)" ), "prior": { # All ways to specify priors in the non-regression case # work the same way here. "Intercept": {"name": "Normal", "mu": 1.5, "sigma": 1.5}, "zSTN": {"name": "Normal", "mu": 0, "sigma": 1.0}, "zGPe": {"name": "Normal", "mu": 0, "sigma": 1.0}, "sevScore": {"name": "Normal", "mu": 0, "sigma": 1.0}, "zSTN:sevScore": {"name": "Normal", "mu": 0, "sigma": 1.0}, "zGPe:sevScore": {"name": "Normal", "mu": 0, "sigma": 1.0}, "1|participant_id": { "name": "Normal", "mu": 0, "sigma": {"name": "HalfNormal", "sigma": 1}, }, "zSTN|participant_id": { "name": "Normal", "mu": 0, "sigma": {"name": "HalfNormal", "sigma": 1}, }, "zGPe|participant_id": { "name": "Normal", "mu": 0, "sigma": {"name": "HalfNormal", "sigma": 1}, }, }, "link": "identity", } ], noncentered=True, p_outlier=0.05, ) model_reg_v_ex4_A1

Model initialized successfully.

Out[28]:

Hierarchical Sequential Sampling Model
Model: ddm

Response variable: rt,response
Likelihood: analytical
Observations: 4800

Parameters:

v:
    Formula: v ~ 1 + (zSTN + zGPe)*sevScore*C(diagnosis) + ((1 + zSTN + zGPe)|participant_id)
    Priors:
        v_Intercept ~ Normal(mu: 1.5, sigma: 1.5)
        v_zSTN ~ Normal(mu: 0.0, sigma: 1.0)
        v_zGPe ~ Normal(mu: 0.0, sigma: 1.0)
        v_sevScore ~ Normal(mu: 0.0, sigma: 1.0)
        v_zSTN:sevScore ~ Normal(mu: 0.0, sigma: 1.0)
        v_zGPe:sevScore ~ Normal(mu: 0.0, sigma: 1.0)
        v_C(diagnosis) ~ Normal(mu: 0.0, sigma: 0.25)
        v_zSTN:C(diagnosis) ~ Normal(mu: 0.0, sigma: 0.25)
        v_zGPe:C(diagnosis) ~ Normal(mu: 0.0, sigma: 0.25)
        v_sevScore:C(diagnosis) ~ Normal(mu: 0.0, sigma: 0.25)
        v_zSTN:sevScore:C(diagnosis) ~ Normal(mu: 0.0, sigma: 0.25)
        v_zGPe:sevScore:C(diagnosis) ~ Normal(mu: 0.0, sigma: 0.25)
        v_1|participant_id ~ Normal(mu: 0.0, sigma: HalfNormal(sigma: 1.0))
        v_zSTN|participant_id ~ Normal(mu: 0.0, sigma: HalfNormal(sigma: 1.0))
        v_zGPe|participant_id ~ Normal(mu: 0.0, sigma: HalfNormal(sigma: 1.0))
    Link: identity
    Explicit bounds: (-inf, inf)

a:
    Prior: HalfNormal(sigma: 2.0)
    Explicit bounds: (0.0, inf)

z:
    Prior: Uniform(lower: 0.0, upper: 1.0)
    Explicit bounds: (0.0, 1.0)

t:
    Prior: HalfNormal(sigma: 2.0)
    Explicit bounds: (0.0, inf)


Lapse probability: 0.05
Lapse distribution: Uniform(lower: 0.0, upper: 20.0)

In [29]:

samples_model_reg_v_ex4_A1 = model_reg_v_ex4_A1.sample(
    sampler="numpyro",
    cores=3,
    chains=3,
    draws=500,
    tune=500,
    idata_kwargs=dict(log_likelihood=True),
)

samples_model_reg_v_ex4_A1 = model_reg_v_ex4_A1.sample( sampler="numpyro", cores=3, chains=3, draws=500, tune=500, idata_kwargs=dict(log_likelihood=True), )

Using default initvals. 

NUTS[numpyro]: [z, a, t, v_Intercept, v_zSTN, v_zGPe, v_sevScore, v_zSTN:sevScore, v_zGPe:sevScore, v_C(diagnosis), v_zSTN:C(diagnosis), v_zGPe:C(diagnosis), v_sevScore:C(diagnosis), v_zSTN:sevScore:C(diagnosis), v_zGPe:sevScore:C(diagnosis), v_1|participant_id_sigma, v_1|participant_id_offset, v_zSTN|participant_id_sigma, v_zSTN|participant_id_offset, v_zGPe|participant_id_sigma, v_zGPe|participant_id_offset]

  0%|          | 0/1000 [00:00<?, ?it/s]

warmup:   0%|          | 0/1000 [00:12<?, ?it/s, 1023 steps of size 7.34e-03. acc. prob=0.73]

warmup:   4%|▍         | 41/1000 [00:12<04:50,  3.31it/s, 511 steps of size 5.41e-03. acc. prob=0.73]

warmup:   4%|▍         | 42/1000 [00:12<04:54,  3.26it/s, 511 steps of size 9.07e-03. acc. prob=0.73]

warmup:   4%|▍         | 44/1000 [00:13<05:07,  3.11it/s, 1023 steps of size 2.39e-03. acc. prob=0.72]

warmup:   4%|▍         | 45/1000 [00:14<05:36,  2.84it/s, 1023 steps of size 3.91e-03. acc. prob=0.73]

warmup:   5%|▍         | 46/1000 [00:15<06:13,  2.56it/s, 1023 steps of size 5.76e-03. acc. prob=0.73]

warmup:   5%|▍         | 47/1000 [00:15<06:15,  2.54it/s, 511 steps of size 9.19e-03. acc. prob=0.74] 

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sample:  60%|█████▉    | 595/1000 [01:07<00:22, 18.14it/s, 127 steps of size 6.65e-02. acc. prob=0.94]

sample:  60%|█████▉    | 598/1000 [01:07<00:21, 18.81it/s, 63 steps of size 6.65e-02. acc. prob=0.94] 

sample:  60%|██████    | 601/1000 [01:07<00:19, 20.69it/s, 31 steps of size 6.65e-02. acc. prob=0.94]

sample:  60%|██████    | 604/1000 [01:07<00:18, 21.06it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

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sample:  62%|██████▏   | 616/1000 [01:08<00:17, 21.74it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  62%|██████▏   | 619/1000 [01:08<00:16, 22.94it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

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sample:  66%|██████▌   | 661/1000 [01:10<00:15, 21.60it/s, 63 steps of size 6.65e-02. acc. prob=0.92]

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sample:  68%|██████▊   | 676/1000 [01:11<00:14, 21.80it/s, 63 steps of size 6.65e-02. acc. prob=0.92]

sample:  68%|██████▊   | 679/1000 [01:11<00:16, 19.74it/s, 127 steps of size 6.65e-02. acc. prob=0.92]

sample:  68%|██████▊   | 682/1000 [01:11<00:15, 20.32it/s, 63 steps of size 6.65e-02. acc. prob=0.92] 

sample:  68%|██████▊   | 685/1000 [01:11<00:15, 20.74it/s, 63 steps of size 6.65e-02. acc. prob=0.92]

sample:  69%|██████▉   | 688/1000 [01:11<00:14, 22.02it/s, 63 steps of size 6.65e-02. acc. prob=0.92]

sample:  69%|██████▉   | 691/1000 [01:11<00:14, 21.93it/s, 63 steps of size 6.65e-02. acc. prob=0.92]

sample:  69%|██████▉   | 694/1000 [01:11<00:13, 21.98it/s, 63 steps of size 6.65e-02. acc. prob=0.92]

sample:  70%|██████▉   | 697/1000 [01:12<00:13, 21.92it/s, 63 steps of size 6.65e-02. acc. prob=0.92]

sample:  70%|███████   | 700/1000 [01:12<00:13, 21.76it/s, 63 steps of size 6.65e-02. acc. prob=0.92]

sample:  70%|███████   | 703/1000 [01:12<00:13, 21.70it/s, 63 steps of size 6.65e-02. acc. prob=0.92]

sample:  70%|███████   | 705/1000 [01:12<00:15, 19.44it/s, 127 steps of size 6.65e-02. acc. prob=0.92]

sample:  71%|███████   | 708/1000 [01:12<00:14, 19.89it/s, 63 steps of size 6.65e-02. acc. prob=0.92] 

sample:  71%|███████   | 711/1000 [01:12<00:14, 20.49it/s, 63 steps of size 6.65e-02. acc. prob=0.92]

sample:  71%|███████▏  | 713/1000 [01:12<00:15, 18.64it/s, 63 steps of size 6.65e-02. acc. prob=0.92]

sample:  72%|███████▏  | 715/1000 [01:13<00:16, 17.51it/s, 63 steps of size 6.65e-02. acc. prob=0.92]

sample:  72%|███████▏  | 718/1000 [01:13<00:16, 17.19it/s, 127 steps of size 6.65e-02. acc. prob=0.92]

sample:  72%|███████▏  | 720/1000 [01:13<00:17, 16.42it/s, 63 steps of size 6.65e-02. acc. prob=0.92] 

sample:  72%|███████▏  | 723/1000 [01:13<00:15, 18.01it/s, 63 steps of size 6.65e-02. acc. prob=0.92]

sample:  73%|███████▎  | 726/1000 [01:13<00:15, 17.39it/s, 127 steps of size 6.65e-02. acc. prob=0.92]

sample:  73%|███████▎  | 729/1000 [01:13<00:14, 18.65it/s, 63 steps of size 6.65e-02. acc. prob=0.92] 

sample:  73%|███████▎  | 732/1000 [01:13<00:13, 19.58it/s, 63 steps of size 6.65e-02. acc. prob=0.92]

sample:  74%|███████▎  | 735/1000 [01:14<00:13, 20.17it/s, 63 steps of size 6.65e-02. acc. prob=0.92]

sample:  74%|███████▍  | 738/1000 [01:14<00:12, 20.67it/s, 63 steps of size 6.65e-02. acc. prob=0.92]

sample:  74%|███████▍  | 741/1000 [01:14<00:13, 18.63it/s, 127 steps of size 6.65e-02. acc. prob=0.92]

sample:  74%|███████▍  | 743/1000 [01:14<00:13, 18.87it/s, 63 steps of size 6.65e-02. acc. prob=0.92] 

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sample:  76%|███████▌  | 759/1000 [01:15<00:12, 19.16it/s, 63 steps of size 6.65e-02. acc. prob=0.93] 

sample:  76%|███████▌  | 762/1000 [01:15<00:11, 20.03it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  76%|███████▋  | 765/1000 [01:15<00:11, 20.69it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  77%|███████▋  | 768/1000 [01:15<00:11, 21.04it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  77%|███████▋  | 770/1000 [01:15<00:12, 19.06it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  77%|███████▋  | 773/1000 [01:16<00:11, 19.93it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  78%|███████▊  | 776/1000 [01:16<00:10, 20.49it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  78%|███████▊  | 779/1000 [01:16<00:10, 20.85it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  78%|███████▊  | 782/1000 [01:16<00:11, 19.27it/s, 127 steps of size 6.65e-02. acc. prob=0.93]

sample:  78%|███████▊  | 785/1000 [01:16<00:10, 20.08it/s, 63 steps of size 6.65e-02. acc. prob=0.93] 

sample:  79%|███████▊  | 787/1000 [01:16<00:11, 18.51it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  79%|███████▉  | 790/1000 [01:16<00:10, 19.40it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  79%|███████▉  | 792/1000 [01:17<00:11, 17.72it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  80%|███████▉  | 795/1000 [01:17<00:11, 18.58it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  80%|███████▉  | 798/1000 [01:17<00:10, 19.24it/s, 63 steps of size 6.65e-02. acc. prob=0.92]

sample:  80%|████████  | 800/1000 [01:17<00:12, 16.27it/s, 127 steps of size 6.65e-02. acc. prob=0.93]

sample:  80%|████████  | 802/1000 [01:17<00:12, 15.88it/s, 63 steps of size 6.65e-02. acc. prob=0.93] 

sample:  80%|████████  | 805/1000 [01:17<00:11, 17.54it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  81%|████████  | 808/1000 [01:17<00:10, 18.80it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  81%|████████  | 811/1000 [01:18<00:09, 19.66it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  81%|████████▏ | 814/1000 [01:18<00:09, 20.36it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  82%|████████▏ | 817/1000 [01:18<00:08, 20.92it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  82%|████████▏ | 819/1000 [01:18<00:09, 19.02it/s, 127 steps of size 6.65e-02. acc. prob=0.93]

sample:  82%|████████▏ | 821/1000 [01:18<00:10, 17.70it/s, 63 steps of size 6.65e-02. acc. prob=0.93] 

sample:  82%|████████▏ | 824/1000 [01:18<00:09, 18.95it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  83%|████████▎ | 826/1000 [01:18<00:09, 18.53it/s, 95 steps of size 6.65e-02. acc. prob=0.93]

sample:  83%|████████▎ | 829/1000 [01:18<00:08, 19.61it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  83%|████████▎ | 832/1000 [01:19<00:08, 20.33it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  84%|████████▎ | 835/1000 [01:19<00:07, 20.83it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  84%|████████▍ | 838/1000 [01:19<00:07, 21.14it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  84%|████████▍ | 841/1000 [01:19<00:07, 21.39it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  84%|████████▍ | 844/1000 [01:19<00:07, 21.52it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  85%|████████▍ | 846/1000 [01:19<00:07, 19.41it/s, 127 steps of size 6.65e-02. acc. prob=0.93]

sample:  85%|████████▍ | 848/1000 [01:19<00:08, 18.02it/s, 127 steps of size 6.65e-02. acc. prob=0.93]

sample:  85%|████████▌ | 851/1000 [01:20<00:07, 19.16it/s, 63 steps of size 6.65e-02. acc. prob=0.93] 

sample:  85%|████████▌ | 854/1000 [01:20<00:07, 19.99it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  86%|████████▌ | 857/1000 [01:20<00:06, 20.54it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  86%|████████▌ | 860/1000 [01:20<00:06, 20.82it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  86%|████████▋ | 863/1000 [01:20<00:06, 21.13it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  87%|████████▋ | 866/1000 [01:20<00:06, 21.29it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  87%|████████▋ | 869/1000 [01:20<00:06, 21.41it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  87%|████████▋ | 872/1000 [01:21<00:05, 21.49it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  88%|████████▊ | 875/1000 [01:21<00:05, 21.57it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  88%|████████▊ | 878/1000 [01:21<00:05, 22.44it/s, 31 steps of size 6.65e-02. acc. prob=0.93]

sample:  88%|████████▊ | 881/1000 [01:21<00:05, 22.28it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  88%|████████▊ | 883/1000 [01:21<00:05, 19.94it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  89%|████████▊ | 886/1000 [01:21<00:05, 20.59it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  89%|████████▉ | 889/1000 [01:21<00:05, 21.00it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  89%|████████▉ | 892/1000 [01:21<00:05, 21.33it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  90%|████████▉ | 895/1000 [01:22<00:04, 21.50it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  90%|████████▉ | 898/1000 [01:22<00:04, 21.53it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  90%|█████████ | 901/1000 [01:22<00:04, 21.49it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  90%|█████████ | 903/1000 [01:22<00:04, 19.43it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  90%|█████████ | 905/1000 [01:22<00:05, 17.30it/s, 127 steps of size 6.65e-02. acc. prob=0.93]

sample:  91%|█████████ | 908/1000 [01:22<00:04, 18.46it/s, 63 steps of size 6.65e-02. acc. prob=0.93] 

sample:  91%|█████████ | 911/1000 [01:22<00:04, 19.23it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  91%|█████████▏| 914/1000 [01:23<00:04, 20.02it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  92%|█████████▏| 917/1000 [01:23<00:04, 20.42it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  92%|█████████▏| 920/1000 [01:23<00:03, 20.78it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  92%|█████████▏| 923/1000 [01:23<00:03, 21.10it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  93%|█████████▎| 926/1000 [01:23<00:03, 21.24it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  93%|█████████▎| 929/1000 [01:23<00:03, 19.35it/s, 127 steps of size 6.65e-02. acc. prob=0.93]

sample:  93%|█████████▎| 932/1000 [01:24<00:03, 19.94it/s, 63 steps of size 6.65e-02. acc. prob=0.93] 

sample:  94%|█████████▎| 935/1000 [01:24<00:03, 20.36it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  94%|█████████▎| 937/1000 [01:24<00:03, 18.61it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  94%|█████████▍| 940/1000 [01:24<00:03, 19.53it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  94%|█████████▍| 943/1000 [01:24<00:02, 20.25it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  94%|█████████▍| 945/1000 [01:24<00:02, 18.52it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  95%|█████████▍| 948/1000 [01:24<00:02, 20.53it/s, 31 steps of size 6.65e-02. acc. prob=0.93]

sample:  95%|█████████▌| 950/1000 [01:24<00:02, 18.76it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  95%|█████████▌| 953/1000 [01:25<00:02, 19.65it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  96%|█████████▌| 955/1000 [01:25<00:02, 18.16it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  96%|█████████▌| 958/1000 [01:25<00:02, 19.24it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  96%|█████████▌| 960/1000 [01:25<00:02, 17.76it/s, 127 steps of size 6.65e-02. acc. prob=0.93]

sample:  96%|█████████▋| 963/1000 [01:25<00:01, 18.99it/s, 63 steps of size 6.65e-02. acc. prob=0.93] 

sample:  97%|█████████▋| 966/1000 [01:25<00:01, 19.69it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  97%|█████████▋| 969/1000 [01:25<00:01, 20.29it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  97%|█████████▋| 972/1000 [01:26<00:01, 20.72it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  98%|█████████▊| 975/1000 [01:26<00:01, 21.05it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  98%|█████████▊| 978/1000 [01:26<00:01, 21.31it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  98%|█████████▊| 981/1000 [01:26<00:00, 21.39it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  98%|█████████▊| 984/1000 [01:26<00:00, 21.57it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample:  99%|█████████▊| 987/1000 [01:26<00:00, 19.76it/s, 127 steps of size 6.65e-02. acc. prob=0.93]

sample:  99%|█████████▉| 989/1000 [01:26<00:00, 19.24it/s, 127 steps of size 6.65e-02. acc. prob=0.93]

sample:  99%|█████████▉| 991/1000 [01:27<00:00, 17.85it/s, 63 steps of size 6.65e-02. acc. prob=0.93] 

sample:  99%|█████████▉| 993/1000 [01:27<00:00, 16.95it/s, 127 steps of size 6.65e-02. acc. prob=0.93]

sample: 100%|█████████▉| 996/1000 [01:27<00:00, 18.43it/s, 63 steps of size 6.65e-02. acc. prob=0.93] 

sample: 100%|█████████▉| 999/1000 [01:27<00:00, 19.42it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

sample: 100%|██████████| 1000/1000 [01:27<00:00, 11.43it/s, 63 steps of size 6.65e-02. acc. prob=0.93]

We recommend running at least 4 chains for robust computation of convergence diagnostics

In [30]:

#### posterior statistics of non-centered model
az.summary(
    samples_model_reg_v_ex4_A1, var_names=["~_offset", "~_id"], filter_vars="like"
)  # var_names= ["~_offset"])

#### posterior statistics of non-centered model az.summary( samples_model_reg_v_ex4_A1, var_names=["~_offset", "~_id"], filter_vars="like" ) # var_names= ["~_offset"])

Out[30]:

	mean	sd	eti89_lb	eti89_ub	ess_bulk	ess_tail	r_hat	mcse_mean	mcse_sd
v_zSTN:sevScore:C(diagnosis)[PD]	0.03	0.066	-0.072	0.14	1191	992	1.00	0.0019	0.0013
v_zSTN	0.864	0.068	0.76	0.97	550	703	1.01	0.003	0.0028
v_zGPe	0.388	0.04	0.33	0.45	482	497	1.00	0.0019	0.0021
v_sevScore	0.172	0.101	0.012	0.33	896	910	1.00	0.0034	0.0028
z	0.501	0.0063	0.49	0.51	1761	1118	1.00	0.00015	0.0001
v_zGPe:C(diagnosis)[PD]	-0.042	0.03	-0.092	0.0042	732	899	1.00	0.0011	0.00079
v_sevScore:C(diagnosis)[PD]	-0.272	0.113	-0.45	-0.099	919	977	1.00	0.0037	0.0027
v_zGPe:sevScore	-0.583	0.075	-0.7	-0.47	531	615	1.00	0.0033	0.0032
v_zSTN:C(diagnosis)[PD]	-0.024	0.036	-0.083	0.032	1239	1177	1.00	0.001	0.0007
a	1.5001	0.0156	1.5	1.5	2215	1329	1.01	0.00033	0.00024
v_zGPe:sevScore:C(diagnosis)[PD]	0.121	0.057	0.032	0.21	753	993	1.00	0.0021	0.0014
v_Intercept	0.568	0.055	0.48	0.65	911	1021	1.00	0.0018	0.0014
v_zSTN:sevScore	-0.64	0.13	-0.83	-0.44	496	643	1.01	0.006	0.0063
t	0.5135	0.0061	0.5	0.52	1790	1243	1.00	0.00014	0.00011
v_C(diagnosis)[PD]	-0.203	0.06	-0.3	-0.1	911	1089	1.00	0.002	0.0014

Other Example 2¶

In [31]:

model_reg_v_ex4_A2 = hssm.HSSM(
    data=combined_dataset3,
    include=[
        {
            "name": "v",
            "formula": (
                "v ~ 1 + (zSTN + zGPe)*sevScore + "
                "((1 + zSTN + zGPe)|participant_id/C(diagnosis))"
            ),
            "prior": {
                # All ways to specify priors in the non-regression case
                # work the same way here.
                "Intercept": {"name": "Normal", "mu": 1.5, "sigma": 1.0},
                "zSTN": {"name": "Normal", "mu": 0, "sigma": 1.0},
                "zGPe": {"name": "Normal", "mu": 0, "sigma": 1.0},
                "sevScore": {"name": "Normal", "mu": 0, "sigma": 1.0},
                "zSTN:sevScore": {"name": "Normal", "mu": 0, "sigma": 1.0},
                "zGPe:sevScore": {"name": "Normal", "mu": 0, "sigma": 1.0},
                "1|participant_id": {
                    "name": "Normal",
                    "mu": 0,
                    "sigma": {"name": "HalfNormal", "sigma": 1},
                },
                "zSTN|participant_id": {
                    "name": "Normal",
                    "mu": 0,
                    "sigma": {"name": "HalfNormal", "sigma": 1},
                },
                "zGPe|participant_id": {
                    "name": "Normal",
                    "mu": 0,
                    "sigma": {"name": "HalfNormal", "sigma": 1},
                },
            },
            "link": "identity",
        }
    ],
    noncentered=True,
    p_outlier=0.05,
)

model_reg_v_ex4_A2 = hssm.HSSM( data=combined_dataset3, include=[ { "name": "v", "formula": ( "v ~ 1 + (zSTN + zGPe)*sevScore + " "((1 + zSTN + zGPe)|participant_id/C(diagnosis))" ), "prior": { # All ways to specify priors in the non-regression case # work the same way here. "Intercept": {"name": "Normal", "mu": 1.5, "sigma": 1.0}, "zSTN": {"name": "Normal", "mu": 0, "sigma": 1.0}, "zGPe": {"name": "Normal", "mu": 0, "sigma": 1.0}, "sevScore": {"name": "Normal", "mu": 0, "sigma": 1.0}, "zSTN:sevScore": {"name": "Normal", "mu": 0, "sigma": 1.0}, "zGPe:sevScore": {"name": "Normal", "mu": 0, "sigma": 1.0}, "1|participant_id": { "name": "Normal", "mu": 0, "sigma": {"name": "HalfNormal", "sigma": 1}, }, "zSTN|participant_id": { "name": "Normal", "mu": 0, "sigma": {"name": "HalfNormal", "sigma": 1}, }, "zGPe|participant_id": { "name": "Normal", "mu": 0, "sigma": {"name": "HalfNormal", "sigma": 1}, }, }, "link": "identity", } ], noncentered=True, p_outlier=0.05, )

The PyMC graph contains free random variables that do not influence the likelihood: 'v_zSTN|participant_id:C(diagnosis)_mu', 'v_zGPe|participant_id:C(diagnosis)_mu'. This typically happens when a hyperprior is supplied for a parameter that the chosen parameterization does not use (e.g. `mu` under `noncentered=True`). These nodes will be sampled but will not affect inference; consider switching the parameterization or adjusting the prior specification.

Model initialized successfully.

In [32]:

samples_model_reg_v_ex4_A2 = model_reg_v_ex4_A2.sample(
    sampler="numpyro",
    cores=3,
    chains=3,
    draws=500,
    tune=500,
    idata_kwargs=dict(log_likelihood=True),
)

samples_model_reg_v_ex4_A2 = model_reg_v_ex4_A2.sample( sampler="numpyro", cores=3, chains=3, draws=500, tune=500, idata_kwargs=dict(log_likelihood=True), )

Using default initvals. 

NUTS[numpyro]: [z, a, t, v_Intercept, v_zSTN, v_zGPe, v_sevScore, v_zSTN:sevScore, v_zGPe:sevScore, v_1|participant_id_sigma, v_1|participant_id_offset, v_1|participant_id:C(diagnosis)_sigma, v_1|participant_id:C(diagnosis)_offset, v_zSTN|participant_id_sigma, v_zSTN|participant_id_offset, v_zSTN|participant_id:C(diagnosis)_mu, v_zSTN|participant_id:C(diagnosis)_sigma, v_zSTN|participant_id:C(diagnosis)_offset, v_zGPe|participant_id_sigma, v_zGPe|participant_id_offset, v_zGPe|participant_id:C(diagnosis)_mu, v_zGPe|participant_id:C(diagnosis)_sigma, v_zGPe|participant_id:C(diagnosis)_offset]

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sample:  99%|█████████▊| 987/1000 [01:27<00:00, 18.02it/s, 63 steps of size 7.14e-02. acc. prob=0.92]

sample:  99%|█████████▉| 989/1000 [01:27<00:00, 17.73it/s, 63 steps of size 7.14e-02. acc. prob=0.92]

sample:  99%|█████████▉| 991/1000 [01:28<00:00, 17.76it/s, 63 steps of size 7.14e-02. acc. prob=0.92]

sample:  99%|█████████▉| 993/1000 [01:28<00:00, 17.81it/s, 63 steps of size 7.14e-02. acc. prob=0.92]

sample: 100%|█████████▉| 995/1000 [01:28<00:00, 17.89it/s, 63 steps of size 7.14e-02. acc. prob=0.92]

sample: 100%|█████████▉| 997/1000 [01:28<00:00, 17.87it/s, 63 steps of size 7.14e-02. acc. prob=0.92]

sample: 100%|█████████▉| 999/1000 [01:28<00:00, 17.87it/s, 63 steps of size 7.14e-02. acc. prob=0.92]

sample: 100%|██████████| 1000/1000 [01:28<00:00, 11.30it/s, 63 steps of size 7.14e-02. acc. prob=0.92]

There were 6 divergences after tuning. Increase `target_accept` or reparameterize.

We recommend running at least 4 chains for robust computation of convergence diagnostics

The rhat statistic is larger than 1.01 for some parameters. This indicates problems during sampling. See https://arxiv.org/abs/1903.08008 for details

The effective sample size per chain is smaller than 100 for some parameters.  A higher number is needed for reliable rhat and ess computation. See https://arxiv.org/abs/1903.08008 for details

In [33]:

#### posterior statistics of non-centered model
az.summary(
    samples_model_reg_v_ex4_A2, var_names=["~_offset", "~_id"], filter_vars="like"
)

#### posterior statistics of non-centered model az.summary( samples_model_reg_v_ex4_A2, var_names=["~_offset", "~_id"], filter_vars="like" )

Out[33]:

	mean	sd	eti89_lb	eti89_ub	ess_bulk	ess_tail	r_hat	mcse_mean	mcse_sd
v_zGPe	0.363	0.034	0.31	0.42	858	758	1.00	0.0012	0.0011
v_zSTN	0.853	0.067	0.75	0.96	481	645	1.01	0.0032	0.0032
v_sevScore	0.04	0.27	-0.38	0.46	598	786	1.00	0.012	0.011
z	0.5012	0.0064	0.49	0.51	2245	1071	1.00	0.00013	0.0001
a	1.501	0.0154	1.5	1.5	2943	1032	1.01	0.00028	0.0002
v_zGPe:sevScore	-0.516	0.066	-0.62	-0.41	852	805	1.00	0.0023	0.0021
v_Intercept	0.46	0.14	0.25	0.69	627	721	1.00	0.0057	0.005
v_zSTN:sevScore	-0.63	0.13	-0.81	-0.42	538	664	1.00	0.0059	0.0065
t	0.5133	0.0059	0.5	0.52	1862	958	1.01	0.00014	9.6e-05

Comparison¶

In [34]:

az.compare(
    {"Model 1": samples_model_reg_v_ex4_A1, "Model 2": samples_model_reg_v_ex4_A2}
)

az.compare( {"Model 1": samples_model_reg_v_ex4_A1, "Model 2": samples_model_reg_v_ex4_A2} )

Out[34]:

	rank	elpd_diff	dse	p_worse	diag_diff	diag_elpd	p	elpd	se	weight
Model 1	0	0.0	0.0	NaN			25.3	-5400.0	90.0	0.67
Model 2	1	-3.0	4.1	0.76	|elpd_diff| < 4		36.4	-5400.0	90.0	0.33