HSSM Tutorial Winterbrain 2025¶

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', '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	10.496365	1.0	2.812863
1	2.163771	1.0	1.250355
2	2.788312	1.0	2.861103
3	3.030407	1.0	2.970084
4	3.987627	1.0	0.999557
...	...	...	...
995	1.251335	1.0	0.083161
996	6.996762	1.0	2.011492
997	6.273996	1.0	0.308007
998	0.823745	1.0	-1.294699
999	2.683449	1.0	1.990916

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

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

warmup:   5%|▌         | 50/1000 [00:00<00:13, 68.96it/s, 31 steps of size 1.43e-02. acc. prob=0.74]

warmup:  10%|█         | 100/1000 [00:00<00:06, 132.15it/s, 7 steps of size 2.22e-02. acc. prob=0.77]

warmup:  15%|█▌        | 150/1000 [00:01<00:09, 86.22it/s, 123 steps of size 3.15e-02. acc. prob=0.77]

warmup:  20%|██        | 200/1000 [00:01<00:07, 113.44it/s, 15 steps of size 2.92e-01. acc. prob=0.78]

warmup:  25%|██▌       | 250/1000 [00:02<00:05, 145.69it/s, 7 steps of size 7.89e-02. acc. prob=0.78] 

warmup:  30%|███       | 300/1000 [00:02<00:03, 180.82it/s, 15 steps of size 3.92e-01. acc. prob=0.78]

warmup:  35%|███▌      | 350/1000 [00:02<00:02, 226.48it/s, 3 steps of size 8.19e-01. acc. prob=0.79] 

warmup:  40%|████      | 400/1000 [00:02<00:02, 260.14it/s, 15 steps of size 4.12e-01. acc. prob=0.79]

warmup:  45%|████▌     | 451/1000 [00:02<00:01, 306.60it/s, 7 steps of size 4.77e+00. acc. prob=0.79] 

sample:  50%|█████     | 501/1000 [00:02<00:01, 329.87it/s, 15 steps of size 3.63e-01. acc. prob=0.97]

sample:  55%|█████▌    | 551/1000 [00:02<00:01, 366.88it/s, 15 steps of size 3.63e-01. acc. prob=0.85]

sample:  60%|██████    | 601/1000 [00:02<00:01, 376.87it/s, 7 steps of size 3.63e-01. acc. prob=0.87] 

sample:  65%|██████▌   | 651/1000 [00:03<00:00, 379.27it/s, 7 steps of size 3.63e-01. acc. prob=0.89]

sample:  70%|███████   | 702/1000 [00:03<00:00, 408.36it/s, 15 steps of size 3.63e-01. acc. prob=0.90]

sample:  75%|███████▌  | 752/1000 [00:03<00:00, 423.81it/s, 7 steps of size 3.63e-01. acc. prob=0.90] 

sample:  80%|████████  | 802/1000 [00:03<00:00, 420.67it/s, 15 steps of size 3.63e-01. acc. prob=0.90]

sample:  85%|████████▌ | 852/1000 [00:03<00:00, 423.73it/s, 15 steps of size 3.63e-01. acc. prob=0.90]

sample:  90%|█████████ | 902/1000 [00:03<00:00, 427.78it/s, 15 steps of size 3.63e-01. acc. prob=0.90]

sample:  95%|█████████▌| 952/1000 [00:03<00:00, 423.26it/s, 15 steps of size 3.63e-01. acc. prob=0.90]

sample: 100%|██████████| 1000/1000 [00:03<00:00, 260.95it/s, 3 steps of size 3.63e-01. acc. prob=0.90]

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

warmup:  29%|██▉       | 288/1000 [00:00<00:00, 2876.55it/s, 1 steps of size 1.18e-38. acc. prob=0.00]

sample:  58%|█████▊    | 583/1000 [00:00<00:00, 2916.77it/s, 1 steps of size 1.18e-38. acc. prob=0.00]

sample:  88%|████████▊ | 875/1000 [00:00<00:00, 2916.83it/s, 1 steps of size 1.18e-38. acc. prob=0.00]

sample: 100%|██████████| 1000/1000 [00:00<00:00, 2912.21it/s, 1 steps of size 1.18e-38. acc. prob=0.00]

There were 504 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

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_zSTN	-0.1	0.64	-0.73	0.58	2	2	2.25	0.45	0.011
v	0.5	0.08	0.44	0.63	2	2	2.24	0.054	0.014
z	0.6	0.1	0.47	0.69	2	22	2.24	0.07	0.0072
t	0.3	0.24	0.014	0.51	2	2	2.23	0.17	0.0096
a_Intercept	1.8	0.24	1.5	2	2	23	2.24	0.16	0.016

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 0x1277a4830>

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	0.747980	1.0	-0.158168	1.413627	1
1	1.233290	1.0	0.365957	2.550413	1
2	1.078334	1.0	2.792268	1.212261	1
3	0.988373	1.0	2.902137	-1.633718	1
4	1.382792	1.0	2.014373	-2.116652	1
...	...	...	...	...	...
2995	0.784602	1.0	2.236697	2.210372	10
2996	0.775152	1.0	3.069456	6.258231	10
2997	0.725300	1.0	4.501823	0.557991	10
2998	2.103001	1.0	0.337555	-2.215638	10
2999	0.904685	1.0	1.697219	0.776832	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]: [a, z, 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: 100%|██████████| 1000/1000 [00:22<00:00, 43.63it/s, 63 steps of size 1.67e-01. acc. prob=0.81]

There were 57 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]: [a, z, 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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warmup:  48%|████▊     | 485/1000 [00:41<00:09, 52.90it/s, 31 steps of size 1.78e-01. acc. prob=0.78]

warmup:  49%|████▉     | 491/1000 [00:41<00:09, 53.78it/s, 31 steps of size 2.68e-01. acc. prob=0.78]

warmup:  50%|████▉     | 498/1000 [00:41<00:08, 57.10it/s, 15 steps of size 2.17e-01. acc. prob=0.78]

sample:  50%|█████     | 505/1000 [00:41<00:08, 60.51it/s, 15 steps of size 1.93e-01. acc. prob=0.95]

sample:  51%|█████     | 512/1000 [00:41<00:07, 63.15it/s, 15 steps of size 1.93e-01. acc. prob=0.86]

sample:  52%|█████▏    | 520/1000 [00:41<00:07, 67.86it/s, 31 steps of size 1.93e-01. acc. prob=0.85]

sample:  53%|█████▎    | 528/1000 [00:41<00:06, 69.76it/s, 15 steps of size 1.93e-01. acc. prob=0.87]

sample:  54%|█████▎    | 535/1000 [00:41<00:07, 66.42it/s, 31 steps of size 1.93e-01. acc. prob=0.88]

sample:  54%|█████▍    | 542/1000 [00:42<00:06, 65.91it/s, 31 steps of size 1.93e-01. acc. prob=0.90]

sample:  55%|█████▌    | 550/1000 [00:42<00:06, 69.79it/s, 15 steps of size 1.93e-01. acc. prob=0.89]

sample:  56%|█████▌    | 555/1000 [00:42<00:06, 63.84it/s, 31 steps of size 1.93e-01. acc. prob=0.89]

sample:  56%|█████▌    | 561/1000 [00:42<00:07, 62.63it/s, 31 steps of size 1.93e-01. acc. prob=0.89]

sample:  57%|█████▋    | 568/1000 [00:42<00:06, 61.99it/s, 31 steps of size 1.93e-01. acc. prob=0.90]

sample:  57%|█████▊    | 575/1000 [00:42<00:06, 61.85it/s, 31 steps of size 1.93e-01. acc. prob=0.90]

sample:  58%|█████▊    | 583/1000 [00:42<00:06, 65.49it/s, 31 steps of size 1.93e-01. acc. prob=0.91]

sample:  59%|█████▉    | 589/1000 [00:42<00:06, 61.92it/s, 31 steps of size 1.93e-01. acc. prob=0.91]

sample:  60%|█████▉    | 596/1000 [00:42<00:06, 62.60it/s, 15 steps of size 1.93e-01. acc. prob=0.91]

sample:  60%|██████    | 605/1000 [00:42<00:05, 66.86it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  61%|██████    | 612/1000 [00:43<00:05, 66.53it/s, 15 steps of size 1.93e-01. acc. prob=0.92]

sample:  62%|██████▏   | 618/1000 [00:43<00:05, 64.53it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  63%|██████▎   | 626/1000 [00:43<00:05, 65.63it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  63%|██████▎   | 632/1000 [00:43<00:05, 62.50it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  64%|██████▍   | 640/1000 [00:43<00:05, 64.20it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  65%|██████▍   | 647/1000 [00:43<00:05, 65.63it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  65%|██████▌   | 653/1000 [00:43<00:05, 62.57it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  66%|██████▌   | 661/1000 [00:43<00:05, 64.61it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  67%|██████▋   | 668/1000 [00:43<00:05, 64.43it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  67%|██████▋   | 674/1000 [00:44<00:05, 61.15it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  68%|██████▊   | 681/1000 [00:44<00:05, 61.59it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  69%|██████▊   | 687/1000 [00:44<00:05, 58.34it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  69%|██████▉   | 694/1000 [00:44<00:05, 60.47it/s, 15 steps of size 1.93e-01. acc. prob=0.92]

sample:  70%|███████   | 701/1000 [00:44<00:04, 61.22it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  71%|███████   | 707/1000 [00:44<00:04, 60.70it/s, 15 steps of size 1.93e-01. acc. prob=0.92]

sample:  71%|███████   | 712/1000 [00:44<00:05, 56.66it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  72%|███████▏  | 720/1000 [00:44<00:04, 60.33it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  73%|███████▎  | 728/1000 [00:44<00:04, 64.14it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  73%|███████▎  | 734/1000 [00:45<00:04, 62.35it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  74%|███████▍  | 740/1000 [00:45<00:04, 61.58it/s, 15 steps of size 1.93e-01. acc. prob=0.92]

sample:  75%|███████▍  | 747/1000 [00:45<00:04, 62.50it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  76%|███████▌  | 755/1000 [00:45<00:03, 64.61it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  76%|███████▌  | 762/1000 [00:45<00:03, 65.53it/s, 15 steps of size 1.93e-01. acc. prob=0.92]

sample:  77%|███████▋  | 769/1000 [00:45<00:03, 66.77it/s, 15 steps of size 1.93e-01. acc. prob=0.92]

sample:  78%|███████▊  | 776/1000 [00:45<00:03, 67.67it/s, 15 steps of size 1.93e-01. acc. prob=0.92]

sample:  78%|███████▊  | 784/1000 [00:45<00:03, 67.87it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  79%|███████▉  | 793/1000 [00:45<00:02, 72.50it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  80%|████████  | 800/1000 [00:46<00:02, 70.01it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  81%|████████  | 807/1000 [00:46<00:02, 68.40it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  82%|████████▏ | 815/1000 [00:46<00:02, 70.46it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  82%|████████▏ | 822/1000 [00:46<00:02, 68.50it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  83%|████████▎ | 829/1000 [00:46<00:02, 66.95it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  84%|████████▎ | 835/1000 [00:46<00:02, 64.92it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  84%|████████▍ | 843/1000 [00:46<00:02, 67.48it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  85%|████████▌ | 850/1000 [00:46<00:02, 67.80it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  86%|████████▌ | 856/1000 [00:46<00:02, 63.99it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  86%|████████▋ | 863/1000 [00:47<00:02, 63.01it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  87%|████████▋ | 870/1000 [00:47<00:02, 62.00it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  88%|████████▊ | 877/1000 [00:47<00:01, 63.74it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  88%|████████▊ | 883/1000 [00:47<00:01, 62.44it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  89%|████████▉ | 890/1000 [00:47<00:01, 62.84it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  90%|████████▉ | 897/1000 [00:47<00:01, 61.77it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  90%|█████████ | 904/1000 [00:47<00:01, 61.00it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  91%|█████████ | 910/1000 [00:47<00:01, 60.39it/s, 31 steps of size 1.93e-01. acc. prob=0.92]

sample:  92%|█████████▏| 917/1000 [00:47<00:01, 61.54it/s, 31 steps of size 1.93e-01. acc. prob=0.91]

sample:  92%|█████████▏| 924/1000 [00:47<00:01, 63.82it/s, 15 steps of size 1.93e-01. acc. prob=0.91]

sample:  93%|█████████▎| 930/1000 [00:48<00:01, 62.58it/s, 31 steps of size 1.93e-01. acc. prob=0.91]

sample:  94%|█████████▎| 936/1000 [00:48<00:01, 60.38it/s, 31 steps of size 1.93e-01. acc. prob=0.91]

sample:  94%|█████████▍| 943/1000 [00:48<00:00, 61.73it/s, 31 steps of size 1.93e-01. acc. prob=0.91]

sample:  95%|█████████▌| 950/1000 [00:48<00:00, 62.94it/s, 31 steps of size 1.93e-01. acc. prob=0.91]

sample:  96%|█████████▌| 957/1000 [00:48<00:00, 63.34it/s, 47 steps of size 1.93e-01. acc. prob=0.91]

sample:  97%|█████████▋| 967/1000 [00:48<00:00, 70.06it/s, 31 steps of size 1.93e-01. acc. prob=0.91]

sample:  97%|█████████▋| 974/1000 [00:48<00:00, 68.11it/s, 31 steps of size 1.93e-01. acc. prob=0.91]

sample:  98%|█████████▊| 981/1000 [00:48<00:00, 65.38it/s, 31 steps of size 1.93e-01. acc. prob=0.91]

sample:  99%|█████████▉| 989/1000 [00:48<00:00, 66.51it/s, 31 steps of size 1.93e-01. acc. prob=0.91]

sample: 100%|█████████▉| 996/1000 [00:49<00:00, 65.14it/s, 31 steps of size 1.93e-01. acc. prob=0.91]

sample: 100%|██████████| 1000/1000 [00:49<00:00, 20.35it/s, 15 steps of size 1.93e-01. acc. prob=0.91]

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
v_zGPe	0.336	0.0145	0.31	0.36	1348	826	1.00	0.00044	0.0004
z	0.49	0.0113	0.47	0.51	1718	1166	1.00	0.00027	0.00019
v_zSTN	0.812	0.0188	0.78	0.84	1071	874	1.00	0.00058	0.00043
v_Intercept	1.56	0.053	1.5	1.6	968	812	1.00	0.0017	0.0014
t	0.5041	0.0057	0.49	0.51	1654	1217	1.01	0.00014	9.6e-05
a	1.48	0.0262	1.4	1.5	1715	1250	1.00	0.00063	0.00049

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 0x142d882f0>

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_1|participant_id_sigma	0.096	0.045	0.034	0.17	494	626	1.00	0.002	0.0024
v_zGPe	0.336	0.0145	0.31	0.36	1348	826	1.00	0.00044	0.0004
z	0.49	0.0113	0.47	0.51	1718	1166	1.00	0.00027	0.00019
v_zSTN|participant_id[1]	-0.022	0.027	-0.074	0.0099	936	1269	1.00	0.0009	0.00073
v_zSTN|participant_id[10]	-0.007	0.024	-0.048	0.028	1506	1134	1.00	0.00063	0.00063
v_zSTN|participant_id[2]	-0.006	0.025	-0.048	0.032	1554	1210	1.00	0.00064	0.00058
v_zSTN|participant_id[3]	-0.006	0.026	-0.051	0.033	1443	1062	1.00	0.0007	0.00069
v_zSTN|participant_id[4]	0.025	0.03	-0.0085	0.084	695	880	1.00	0.0012	0.00097
v_zSTN|participant_id[5]	-0.003	0.024	-0.041	0.035	1093	978	1.01	0.00073	0.00067
v_zSTN|participant_id[6]	-0.004	0.023	-0.042	0.031	1602	1173	1.00	0.00057	0.00054
v_zSTN|participant_id[7]	-0.011	0.026	-0.057	0.027	1486	1217	1.00	0.00067	0.00058
v_zSTN|participant_id[8]	0.022	0.029	-0.013	0.075	941	978	1.00	0.00097	0.00083
v_zSTN|participant_id[9]	0.011	0.025	-0.022	0.055	1421	958	1.00	0.00071	0.00076
v_zSTN	0.812	0.0188	0.78	0.84	1071	874	1.00	0.00058	0.00043
v_zGPe|participant_id[1]	-0.005	0.02	-0.039	0.022	1486	1231	1.00	0.00054	0.00073
v_zGPe|participant_id[10]	0.016	0.023	-0.0088	0.062	973	1232	1.00	0.0008	0.00086
v_zGPe|participant_id[2]	0.0029	0.019	-0.026	0.036	1764	1183	1.00	0.00048	0.00051
v_zGPe|participant_id[3]	-0.009	0.021	-0.048	0.016	1129	1077	1.00	0.00068	0.00078
v_zGPe|participant_id[4]	0.004	0.019	-0.025	0.036	1485	1111	1.00	0.00052	0.00056
v_zGPe|participant_id[5]	-0	0.02	-0.033	0.033	1294	1207	1.00	0.00057	0.00068
v_zGPe|participant_id[6]	-0.005	0.019	-0.036	0.02	1622	1165	1.01	0.00052	0.00069
v_zGPe|participant_id[7]	0.01	0.021	-0.015	0.047	1459	1267	1.00	0.00054	0.00056
v_zGPe|participant_id[8]	-0.006	0.019	-0.041	0.02	1366	1309	1.00	0.00053	0.00058
v_zGPe|participant_id[9]	-0.009	0.02	-0.046	0.017	1315	1230	1.01	0.00059	0.00062
v_zSTN|participant_id_sigma	0.029	0.019	0.0032	0.061	441	650	1.00	0.00092	0.00089
v_zGPe|participant_id_sigma	0.021	0.017	0.0017	0.05	446	594	1.01	0.00083	0.0012
v_Intercept	1.56	0.053	1.5	1.6	968	812	1.00	0.0017	0.0014
t	0.5041	0.0057	0.49	0.51	1654	1217	1.01	0.00014	9.6e-05
v_1|participant_id[1]	0.023	0.062	-0.069	0.13	1098	973	1.00	0.0019	0.0016
v_1|participant_id[10]	-0.031	0.061	-0.13	0.064	1331	844	1.00	0.0017	0.0013
v_1|participant_id[2]	0.039	0.06	-0.051	0.14	1261	1140	1.00	0.0017	0.0013
v_1|participant_id[3]	0.081	0.073	-0.022	0.2	1075	1182	1.00	0.0022	0.002
v_1|participant_id[4]	0.025	0.063	-0.074	0.13	1449	1129	1.00	0.0017	0.0016
v_1|participant_id[5]	0.099	0.072	-0.0038	0.22	976	972	1.00	0.0023	0.002
v_1|participant_id[6]	0.006	0.065	-0.091	0.11	1328	1172	1.01	0.0018	0.0019
v_1|participant_id[7]	-0.087	0.069	-0.21	0.011	1045	1069	1.00	0.0021	0.0016
v_1|participant_id[8]	-0.069	0.065	-0.18	0.024	1358	1371	1.00	0.0017	0.0014
v_1|participant_id[9]	-0.055	0.064	-0.16	0.037	1114	1119	1.00	0.0019	0.0015
a	1.48	0.0262	1.4	1.5	1715	1250	1.00	0.00063	0.00049

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 0x139d9ad50>

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	1.167871	1.0	0.542555	3.558076	1	0.0
1	0.744359	1.0	2.945089	1.815673	1	0.0
2	1.561734	1.0	0.472066	-0.411510	1	0.0
3	1.042030	1.0	2.303357	0.950618	1	0.0
4	1.433222	-1.0	-0.543237	-1.771657	1	0.0
...	...	...	...	...	...	...
3295	1.487224	1.0	1.738488	2.102996	8	1.0
3296	1.201577	1.0	2.174505	-0.688230	8	1.0
3297	3.122706	1.0	0.943154	-1.270588	8	1.0
3298	1.008394	1.0	3.827790	2.551132	8	1.0
3299	1.413731	1.0	3.766832	4.183655	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]: [a, z, 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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warmup:  38%|███▊      | 380/1000 [00:37<00:13, 46.67it/s, 31 steps of size 1.68e-01. acc. prob=0.78]

warmup:  38%|███▊      | 385/1000 [00:37<00:13, 45.29it/s, 31 steps of size 1.48e-01. acc. prob=0.78]

warmup:  39%|███▉      | 390/1000 [00:37<00:13, 45.37it/s, 31 steps of size 1.77e-01. acc. prob=0.78]

warmup:  40%|███▉      | 396/1000 [00:37<00:13, 45.06it/s, 63 steps of size 1.10e-01. acc. prob=0.78]

warmup:  40%|████      | 402/1000 [00:37<00:12, 46.94it/s, 31 steps of size 1.29e-01. acc. prob=0.78]

warmup:  41%|████      | 406/1000 [00:37<00:13, 43.93it/s, 63 steps of size 9.92e-02. acc. prob=0.78]

warmup:  41%|████      | 411/1000 [00:37<00:13, 44.60it/s, 63 steps of size 9.75e-02. acc. prob=0.78]

warmup:  42%|████▏     | 416/1000 [00:37<00:12, 45.28it/s, 31 steps of size 1.00e-01. acc. prob=0.78]

warmup:  42%|████▏     | 422/1000 [00:37<00:12, 45.91it/s, 63 steps of size 9.42e-02. acc. prob=0.78]

warmup:  43%|████▎     | 426/1000 [00:38<00:13, 43.91it/s, 31 steps of size 1.14e-01. acc. prob=0.78]

warmup:  43%|████▎     | 430/1000 [00:38<00:13, 42.82it/s, 63 steps of size 1.54e-01. acc. prob=0.78]

warmup:  43%|████▎     | 433/1000 [00:38<00:16, 35.17it/s, 63 steps of size 8.67e-02. acc. prob=0.78]

warmup:  44%|████▍     | 438/1000 [00:38<00:14, 37.63it/s, 31 steps of size 1.73e-01. acc. prob=0.78]

warmup:  44%|████▍     | 444/1000 [00:38<00:13, 41.67it/s, 63 steps of size 1.01e-01. acc. prob=0.78]

warmup:  45%|████▍     | 449/1000 [00:38<00:13, 40.94it/s, 63 steps of size 6.86e-02. acc. prob=0.78]

warmup:  45%|████▌     | 454/1000 [00:38<00:12, 43.20it/s, 31 steps of size 1.49e-01. acc. prob=0.78]

warmup:  46%|████▌     | 459/1000 [00:38<00:13, 41.12it/s, 127 steps of size 6.34e-02. acc. prob=0.78]

warmup:  46%|████▋     | 463/1000 [00:39<00:15, 35.04it/s, 127 steps of size 8.27e-02. acc. prob=0.78]

warmup:  47%|████▋     | 468/1000 [00:39<00:13, 38.00it/s, 15 steps of size 1.44e-01. acc. prob=0.78] 

warmup:  47%|████▋     | 471/1000 [00:39<00:17, 30.57it/s, 255 steps of size 5.18e-02. acc. prob=0.78]

warmup:  48%|████▊     | 475/1000 [00:39<00:16, 32.53it/s, 15 steps of size 2.86e-01. acc. prob=0.78] 

warmup:  48%|████▊     | 482/1000 [00:39<00:13, 38.58it/s, 127 steps of size 8.48e-02. acc. prob=0.78]

warmup:  49%|████▉     | 488/1000 [00:39<00:12, 42.11it/s, 31 steps of size 2.71e-02. acc. prob=0.78] 

warmup:  49%|████▉     | 489/1000 [00:39<00:16, 30.87it/s, 255 steps of size 4.63e-02. acc. prob=0.78]

warmup:  49%|████▉     | 491/1000 [00:39<00:18, 27.66it/s, 63 steps of size 1.25e-01. acc. prob=0.78] 

warmup:  50%|████▉     | 499/1000 [00:40<00:12, 38.72it/s, 63 steps of size 1.04e-01. acc. prob=0.78]

sample:  51%|█████     | 506/1000 [00:40<00:10, 44.95it/s, 31 steps of size 1.27e-01. acc. prob=0.91]

sample:  51%|█████     | 512/1000 [00:40<00:10, 48.29it/s, 31 steps of size 1.27e-01. acc. prob=0.93]

sample:  52%|█████▏    | 518/1000 [00:40<00:09, 50.96it/s, 31 steps of size 1.27e-01. acc. prob=0.94]

sample:  52%|█████▏    | 523/1000 [00:40<00:09, 49.76it/s, 31 steps of size 1.27e-01. acc. prob=0.95]

sample:  53%|█████▎    | 529/1000 [00:40<00:09, 51.60it/s, 31 steps of size 1.27e-01. acc. prob=0.95]

sample:  54%|█████▎    | 535/1000 [00:40<00:08, 52.42it/s, 31 steps of size 1.27e-01. acc. prob=0.95]

sample:  54%|█████▍    | 541/1000 [00:40<00:08, 53.44it/s, 31 steps of size 1.27e-01. acc. prob=0.96]

sample:  55%|█████▍    | 547/1000 [00:40<00:08, 53.96it/s, 31 steps of size 1.27e-01. acc. prob=0.95]

sample:  55%|█████▌    | 553/1000 [00:41<00:08, 54.53it/s, 31 steps of size 1.27e-01. acc. prob=0.95]

sample:  56%|█████▌    | 559/1000 [00:41<00:08, 55.08it/s, 31 steps of size 1.27e-01. acc. prob=0.95]

sample:  56%|█████▋    | 565/1000 [00:41<00:07, 56.05it/s, 31 steps of size 1.27e-01. acc. prob=0.94]

sample:  57%|█████▋    | 571/1000 [00:41<00:07, 56.16it/s, 31 steps of size 1.27e-01. acc. prob=0.94]

sample:  58%|█████▊    | 577/1000 [00:41<00:07, 56.18it/s, 31 steps of size 1.27e-01. acc. prob=0.92]

sample:  58%|█████▊    | 583/1000 [00:41<00:07, 56.35it/s, 31 steps of size 1.27e-01. acc. prob=0.92]

sample:  59%|█████▉    | 588/1000 [00:41<00:07, 53.33it/s, 63 steps of size 1.27e-01. acc. prob=0.92]

sample:  59%|█████▉    | 593/1000 [00:41<00:07, 51.32it/s, 31 steps of size 1.27e-01. acc. prob=0.92]

sample:  60%|█████▉    | 599/1000 [00:41<00:07, 52.68it/s, 31 steps of size 1.27e-01. acc. prob=0.92]

sample:  60%|██████    | 605/1000 [00:41<00:07, 53.69it/s, 31 steps of size 1.27e-01. acc. prob=0.92]

sample:  61%|██████    | 611/1000 [00:42<00:07, 54.21it/s, 31 steps of size 1.27e-01. acc. prob=0.92]

sample:  62%|██████▏   | 617/1000 [00:42<00:06, 54.74it/s, 31 steps of size 1.27e-01. acc. prob=0.93]

sample:  62%|██████▏   | 623/1000 [00:42<00:06, 55.65it/s, 31 steps of size 1.27e-01. acc. prob=0.93]

sample:  63%|██████▎   | 629/1000 [00:42<00:06, 56.11it/s, 31 steps of size 1.27e-01. acc. prob=0.93]

sample:  64%|██████▎   | 635/1000 [00:42<00:06, 56.21it/s, 31 steps of size 1.27e-01. acc. prob=0.93]

sample:  64%|██████▍   | 641/1000 [00:42<00:06, 56.03it/s, 31 steps of size 1.27e-01. acc. prob=0.93]

sample:  65%|██████▍   | 647/1000 [00:42<00:06, 55.81it/s, 31 steps of size 1.27e-01. acc. prob=0.93]

sample:  65%|██████▌   | 653/1000 [00:42<00:06, 55.71it/s, 31 steps of size 1.27e-01. acc. prob=0.93]

sample:  66%|██████▌   | 659/1000 [00:42<00:06, 55.31it/s, 31 steps of size 1.27e-01. acc. prob=0.94]

sample:  66%|██████▋   | 665/1000 [00:43<00:06, 55.07it/s, 31 steps of size 1.27e-01. acc. prob=0.94]

sample:  67%|██████▋   | 671/1000 [00:43<00:06, 54.70it/s, 31 steps of size 1.27e-01. acc. prob=0.94]

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sample:  69%|██████▊   | 687/1000 [00:43<00:06, 51.42it/s, 31 steps of size 1.27e-01. acc. prob=0.94]

sample:  69%|██████▉   | 693/1000 [00:43<00:05, 51.37it/s, 31 steps of size 1.27e-01. acc. prob=0.94]

sample:  70%|██████▉   | 699/1000 [00:43<00:05, 51.51it/s, 31 steps of size 1.27e-01. acc. prob=0.94]

sample:  70%|███████   | 705/1000 [00:43<00:05, 52.10it/s, 31 steps of size 1.27e-01. acc. prob=0.94]

sample:  71%|███████   | 709/1000 [00:43<00:06, 47.69it/s, 31 steps of size 1.27e-01. acc. prob=0.93]

sample:  72%|███████▏  | 721/1000 [00:44<00:04, 66.69it/s, 17 steps of size 1.27e-01. acc. prob=0.89]

sample:  74%|███████▍  | 738/1000 [00:44<00:02, 95.31it/s, 8 steps of size 1.27e-01. acc. prob=0.83] 

sample:  75%|███████▍  | 748/1000 [00:44<00:02, 96.12it/s, 3 steps of size 1.27e-01. acc. prob=0.80]

sample:  76%|███████▋  | 765/1000 [00:44<00:02, 117.39it/s, 9 steps of size 1.27e-01. acc. prob=0.75]

sample:  80%|████████  | 800/1000 [00:44<00:01, 184.65it/s, 9 steps of size 1.27e-01. acc. prob=0.67]

sample:  83%|████████▎ | 831/1000 [00:44<00:00, 220.93it/s, 7 steps of size 1.27e-01. acc. prob=0.61]

sample:  85%|████████▍ | 849/1000 [00:44<00:00, 204.58it/s, 15 steps of size 1.27e-01. acc. prob=0.59]

sample:  86%|████████▋ | 863/1000 [00:44<00:00, 185.45it/s, 5 steps of size 1.27e-01. acc. prob=0.57] 

sample:  88%|████████▊ | 877/1000 [00:44<00:00, 164.06it/s, 31 steps of size 1.27e-01. acc. prob=0.55]

sample:  89%|████████▉ | 889/1000 [00:44<00:00, 150.22it/s, 31 steps of size 1.27e-01. acc. prob=0.54]

sample:  90%|████████▉ | 896/1000 [00:45<00:00, 123.25it/s, 31 steps of size 1.27e-01. acc. prob=0.53]

sample:  90%|█████████ | 905/1000 [00:45<00:00, 98.14it/s, 127 steps of size 1.27e-01. acc. prob=0.52]

sample:  91%|█████████ | 912/1000 [00:45<00:00, 90.79it/s, 7 steps of size 1.27e-01. acc. prob=0.51]  

sample:  92%|█████████▏| 920/1000 [00:45<00:01, 79.92it/s, 127 steps of size 1.27e-01. acc. prob=0.50]

sample:  93%|█████████▎| 928/1000 [00:45<00:00, 76.58it/s, 31 steps of size 1.27e-01. acc. prob=0.50] 

sample:  93%|█████████▎| 934/1000 [00:45<00:00, 70.99it/s, 31 steps of size 1.27e-01. acc. prob=0.49]

sample:  94%|█████████▍| 940/1000 [00:45<00:00, 65.95it/s, 31 steps of size 1.27e-01. acc. prob=0.49]

sample:  95%|█████████▍| 947/1000 [00:45<00:00, 64.57it/s, 31 steps of size 1.27e-01. acc. prob=0.49]

sample:  96%|█████████▌| 955/1000 [00:46<00:00, 68.56it/s, 15 steps of size 1.27e-01. acc. prob=0.48]

sample:  96%|█████████▌| 962/1000 [00:46<00:00, 63.11it/s, 63 steps of size 1.27e-01. acc. prob=0.48]

sample:  97%|█████████▋| 968/1000 [00:46<00:00, 61.06it/s, 31 steps of size 1.27e-01. acc. prob=0.49]

sample:  97%|█████████▋| 974/1000 [00:46<00:00, 59.49it/s, 31 steps of size 1.27e-01. acc. prob=0.50]

sample:  98%|█████████▊| 980/1000 [00:46<00:00, 58.26it/s, 31 steps of size 1.27e-01. acc. prob=0.50]

sample:  99%|█████████▊| 986/1000 [00:46<00:00, 57.47it/s, 31 steps of size 1.27e-01. acc. prob=0.51]

sample:  99%|█████████▉| 992/1000 [00:46<00:00, 56.85it/s, 31 steps of size 1.27e-01. acc. prob=0.51]

sample: 100%|█████████▉| 998/1000 [00:46<00:00, 56.45it/s, 31 steps of size 1.27e-01. acc. prob=0.52]

sample: 100%|██████████| 1000/1000 [00:46<00:00, 21.34it/s, 31 steps of size 1.27e-01. acc. prob=0.52]

There were 133 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 [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_1|participant_id_sigma	0.039	0.033	0.0034	0.099	253	449	1.01	0.002	0.003
v_zGPe	0.284	0.029	0.24	0.32	75	157	1.04	0.0027	0.003
z	0.49	0.0077	0.48	0.5	553	691	1.01	0.00032	0.00023
v_zSTN|participant_id[1]	0.02	0.05	-0.023	0.13	8	5	1.27	0.022	0.019
v_zSTN|participant_id[2]	0.01	0.03	-0.022	0.076	9	8	1.22	0.012	0.014
v_zSTN|participant_id[3]	0.01	0.02	-0.022	0.054	16	18	1.14	0.0068	0.0063
v_zSTN|participant_id[4]	-0.004	0.023	-0.042	0.03	161	66	1.03	0.0026	0.0032
v_zSTN|participant_id[5]	0	0.03	-0.047	0.04	17	8	1.13	0.007	0.0069
v_zSTN|participant_id[6]	-0.02	0.04	-0.12	0.016	8	7	1.26	0.017	0.017
v_zSTN|participant_id[7]	-0.02	0.05	-0.15	0.025	9	8	1.24	0.023	0.024
v_zSTN|participant_id[8]	-0.05	0.09	-0.27	0.0094	8	7	1.26	0.042	0.038
v_sevScore	0.042	0.086	-0.07	0.19	326	457	1.01	0.0047	0.0046
v_1|participant_id_offset[1]	-0.07	0.83	-1.3	1.3	413	561	1.00	0.041	0.03
v_1|participant_id_offset[2]	-0.05	0.86	-1.4	1.3	318	221	1.01	0.048	0.034
v_1|participant_id_offset[3]	0.55	0.87	-0.77	1.9	587	708	1.02	0.036	0.027
v_1|participant_id_offset[4]	-0.26	0.81	-1.5	1.1	323	789	1.01	0.044	0.034
v_1|participant_id_offset[5]	0.1	0.78	-1.1	1.4	1487	798	1.03	0.019	0.016
v_1|participant_id_offset[6]	-0.5	0.9	-2	1.1	30	60	1.06	0.17	0.11
v_1|participant_id_offset[7]	0.15	0.81	-1.1	1.4	821	714	1.03	0.027	0.021
v_1|participant_id_offset[8]	-0.3	0.9	-1.6	1.2	74	212	1.04	0.1	0.074
v_zSTN|participant_id_sigma	0.04	0.05	0.002	0.15	8	8	1.25	0.021	0.021
v_zGPe|participant_id_sigma	0.025	0.02	0.0023	0.057	103	193	1.04	0.0021	0.0035
v_1|participant_id[1]	-0.002	0.037	-0.053	0.057	298	595	1.01	0.0018	0.0022
v_1|participant_id[2]	-0.004	0.035	-0.061	0.047	383	269	1.01	0.0017	0.0019
v_1|participant_id[3]	0.028	0.044	-0.018	0.11	363	594	1.02	0.0022	0.0025
v_1|participant_id[4]	-0.011	0.033	-0.065	0.033	729	584	1.01	0.0011	0.0013
v_1|participant_id[5]	0.005	0.032	-0.043	0.062	966	961	1.02	0.00091	0.0011
v_1|participant_id[6]	-0.02	0.04	-0.1	0.026	53	80	1.04	0.0052	0.0053
v_1|participant_id[7]	0.003	0.035	-0.052	0.06	899	705	1.02	0.0011	0.0015
v_1|participant_id[8]	-0.015	0.04	-0.083	0.035	87	127	1.03	0.0028	0.0036
v_zGPe|participant_id_offset[1]	0.3	0.88	-1.1	1.6	219	522	1.02	0.061	0.045
v_zGPe|participant_id_offset[2]	-0.23	0.81	-1.6	0.99	1243	733	1.05	0.023	0.018
v_zGPe|participant_id_offset[3]	0.6	0.9	-0.95	1.9	209	126	1.02	0.065	0.05
v_zGPe|participant_id_offset[4]	0.17	0.78	-1.2	1.3	955	480	1.01	0.025	0.02
v_zGPe|participant_id_offset[5]	-0.19	0.8	-1.5	1.1	1181	865	1.04	0.023	0.019
v_zGPe|participant_id_offset[6]	-0.38	0.85	-1.8	1	617	478	1.02	0.035	0.028
v_zGPe|participant_id_offset[7]	0.38	0.82	-1	1.6	453	492	1.01	0.04	0.031
v_zGPe|participant_id_offset[8]	-0.4	0.8	-1.8	0.84	119	303	1.03	0.076	0.058
v_zGPe:sevScore	-0.187	0.051	-0.25	-0.1	158	261	1.02	0.0037	0.0043
v_zGPe|participant_id[1]	0.012	0.026	-0.019	0.056	37	182	1.06	0.0037	0.0046
v_zGPe|participant_id[2]	-0.004	0.019	-0.036	0.022	686	550	1.03	0.00076	0.00088
v_zGPe|participant_id[3]	0.02	0.027	-0.0096	0.069	98	86	1.04	0.0025	0.0029
v_zGPe|participant_id[4]	0.005	0.018	-0.02	0.035	375	532	1.01	0.00086	0.00096
v_zGPe|participant_id[5]	-0.005	0.018	-0.036	0.019	449	835	1.01	0.00067	0.00071
v_zGPe|participant_id[6]	-0.012	0.02	-0.048	0.013	314	332	1.00	0.0011	0.0011
v_zGPe|participant_id[7]	0.01	0.02	-0.017	0.044	33	65	1.07	0.0039	0.0033
v_zGPe|participant_id[8]	-0.013	0.025	-0.06	0.016	134	106	1.03	0.0019	0.0022
v_zSTN	0.81	0.06	0.7	0.87	8	5	1.28	0.025	0.018
v_Intercept	0.53	0.053	0.45	0.61	92	185	1.02	0.0054	0.004
v_zSTN:sevScore	-0.4	0.2	-0.54	-0.061	8	6	1.27	0.073	0.06
t	0.5021	0.0069	0.49	0.51	324	503	1.01	0.00038	0.00029
v_zSTN|participant_id_offset[1]	0.2	0.9	-1.4	1.6	32	259	1.07	0.16	0.12
v_zSTN|participant_id_offset[2]	0.14	0.84	-1.3	1.4	470	731	1.05	0.042	0.03
v_zSTN|participant_id_offset[3]	0.17	0.81	-1.2	1.4	1118	947	1.08	0.024	0.019
v_zSTN|participant_id_offset[4]	-0.05	0.87	-1.4	1.4	1794	843	1.10	0.02	0.017
v_zSTN|participant_id_offset[5]	0.3	0.8	-1	1.6	138	721	1.02	0.069	0.052
v_zSTN|participant_id_offset[6]	-0.3	0.8	-1.5	1.1	138	508	1.03	0.073	0.057
v_zSTN|participant_id_offset[7]	-0.1	0.9	-1.4	1.4	25	129	1.08	0.19	0.12
v_zSTN|participant_id_offset[8]	-0.8	1	-2.2	0.89	17	57	1.13	0.24	0.16
a	1.474	0.02	1.4	1.5	23	72	1.09	0.004	0.0027

• 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	0.857299	1.0	4.174684	1.511771	1	0.1	PD
1	0.986127	1.0	3.633133	-0.442452	1	0.1	PD
2	1.410487	1.0	1.415882	1.524683	1	0.1	PD
3	1.979731	1.0	0.930716	-0.456628	1	0.1	PD
4	0.721911	1.0	4.502797	2.518944	1	0.1	PD
...	...	...	...	...	...	...	...
4795	1.022368	1.0	4.160187	5.785448	8	0.8	DD
4796	1.241501	-1.0	0.315886	-0.101750	8	0.8	DD
4797	1.106278	1.0	5.438028	-0.364264	8	0.8	DD
4798	1.351487	1.0	0.714511	2.704185	8	0.8	DD
4799	1.178348	1.0	5.107501	2.936340	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]: [a, z, 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]

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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_zGPe	0.376	0.053	0.3	0.45	650	723	1.00	0.0022	0.0025
v_C(diagnosis)[PD]	-0.237	0.061	-0.33	-0.14	1281	1178	1.00	0.0017	0.0012
z	0.4984	0.0062	0.49	0.51	2125	1195	1.00	0.00013	9.4e-05
v_sevScore	-0.096	0.105	-0.26	0.076	1197	1093	1.00	0.003	0.0023
v_zSTN:sevScore:C(diagnosis)[PD]	-0.014	0.067	-0.12	0.094	1222	1064	1.00	0.0019	0.0014
v_sevScore:C(diagnosis)[PD]	-0.054	0.117	-0.24	0.13	1096	1108	1.00	0.0035	0.0026
v_zSTN:C(diagnosis)[PD]	0.014	0.036	-0.043	0.073	1243	992	1.00	0.001	0.00072
v_zGPe:sevScore:C(diagnosis)[PD]	-0.003	0.058	-0.098	0.087	1122	937	1.00	0.0017	0.0013
v_zGPe:sevScore	-0.498	0.1	-0.65	-0.34	713	783	1.00	0.0038	0.0035
v_zSTN	0.856	0.066	0.75	0.95	567	640	1.00	0.0029	0.0028
v_Intercept	0.627	0.057	0.54	0.72	1288	1002	1.00	0.0016	0.0011
v_zGPe:C(diagnosis)[PD]	-0.006	0.031	-0.056	0.044	1189	1029	1.00	0.00091	0.00067
v_zSTN:sevScore	-0.64	0.13	-0.83	-0.43	545	582	1.00	0.0058	0.0058
t	0.5182	0.0062	0.51	0.53	1700	1248	1.00	0.00015	0.00011
a	1.4978	0.0154	1.5	1.5	2226	1258	1.00	0.00033	0.00024

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, )

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]: [a, z, 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:  90%|█████████ | 901/1000 [01:02<00:03, 29.65it/s, 63 steps of size 9.98e-02. acc. prob=0.88]

sample:  90%|█████████ | 905/1000 [01:02<00:03, 30.00it/s, 63 steps of size 9.98e-02. acc. prob=0.88]

sample:  91%|█████████ | 910/1000 [01:02<00:02, 32.70it/s, 63 steps of size 9.98e-02. acc. prob=0.88]

sample:  91%|█████████▏| 913/1000 [01:02<00:02, 30.94it/s, 63 steps of size 9.98e-02. acc. prob=0.88]

sample:  92%|█████████▏| 916/1000 [01:02<00:02, 29.98it/s, 63 steps of size 9.98e-02. acc. prob=0.88]

sample:  92%|█████████▏| 920/1000 [01:03<00:02, 31.70it/s, 31 steps of size 9.98e-02. acc. prob=0.88]

sample:  92%|█████████▏| 923/1000 [01:03<00:02, 30.43it/s, 63 steps of size 9.98e-02. acc. prob=0.88]

sample:  93%|█████████▎| 927/1000 [01:03<00:02, 32.14it/s, 63 steps of size 9.98e-02. acc. prob=0.88]

sample:  93%|█████████▎| 931/1000 [01:03<00:02, 33.53it/s, 31 steps of size 9.98e-02. acc. prob=0.88]

sample:  94%|█████████▎| 936/1000 [01:03<00:01, 35.33it/s, 63 steps of size 9.98e-02. acc. prob=0.88]

sample:  94%|█████████▍| 939/1000 [01:03<00:01, 33.04it/s, 63 steps of size 9.98e-02. acc. prob=0.88]

sample:  94%|█████████▍| 943/1000 [01:03<00:01, 32.46it/s, 63 steps of size 9.98e-02. acc. prob=0.88]

sample:  95%|█████████▍| 947/1000 [01:03<00:01, 33.55it/s, 63 steps of size 9.98e-02. acc. prob=0.88]

sample:  95%|█████████▌| 952/1000 [01:04<00:01, 35.13it/s, 63 steps of size 9.98e-02. acc. prob=0.88]

sample:  96%|█████████▌| 956/1000 [01:04<00:01, 32.50it/s, 63 steps of size 9.98e-02. acc. prob=0.88]

sample:  96%|█████████▌| 959/1000 [01:04<00:01, 31.44it/s, 63 steps of size 9.98e-02. acc. prob=0.88]

sample:  96%|█████████▌| 962/1000 [01:04<00:01, 31.06it/s, 63 steps of size 9.98e-02. acc. prob=0.88]

sample:  96%|█████████▋| 965/1000 [01:04<00:01, 28.91it/s, 63 steps of size 9.98e-02. acc. prob=0.88]

sample:  97%|█████████▋| 969/1000 [01:04<00:01, 28.90it/s, 63 steps of size 9.98e-02. acc. prob=0.88]

sample:  97%|█████████▋| 972/1000 [01:04<00:00, 28.07it/s, 63 steps of size 9.98e-02. acc. prob=0.88]

sample:  98%|█████████▊| 975/1000 [01:04<00:00, 27.28it/s, 63 steps of size 9.98e-02. acc. prob=0.88]

sample:  98%|█████████▊| 978/1000 [01:04<00:00, 26.78it/s, 63 steps of size 9.98e-02. acc. prob=0.88]

sample:  98%|█████████▊| 982/1000 [01:05<00:00, 27.46it/s, 63 steps of size 9.98e-02. acc. prob=0.88]

sample:  99%|█████████▊| 986/1000 [01:05<00:00, 29.69it/s, 31 steps of size 9.98e-02. acc. prob=0.88]

sample:  99%|█████████▉| 989/1000 [01:05<00:00, 28.89it/s, 63 steps of size 9.98e-02. acc. prob=0.88]

sample:  99%|█████████▉| 993/1000 [01:05<00:00, 30.87it/s, 63 steps of size 9.98e-02. acc. prob=0.88]

sample: 100%|█████████▉| 996/1000 [01:05<00:00, 29.70it/s, 63 steps of size 9.98e-02. acc. prob=0.88]

sample: 100%|██████████| 1000/1000 [01:05<00:00, 15.23it/s, 31 steps of size 9.98e-02. acc. prob=0.88]

There was 1 divergence 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	0.375	0.047	0.31	0.45	435	478	1.01	0.0023	0.0021
z	0.4979	0.0062	0.49	0.51	2266	1120	1.00	0.00013	9.2e-05
v_sevScore	-0.12	0.21	-0.45	0.2	758	759	1.01	0.0075	0.0058
v_zGPe:sevScore	-0.502	0.095	-0.66	-0.36	491	608	1.00	0.0043	0.0038
v_zSTN	0.867	0.058	0.78	0.96	721	866	1.00	0.0023	0.0022
v_Intercept	0.507	0.109	0.34	0.68	625	679	1.00	0.0044	0.0038
v_zSTN:sevScore	-0.66	0.12	-0.84	-0.48	568	580	1.00	0.0053	0.0055
t	0.5177	0.0064	0.51	0.53	1959	1199	1.00	0.00014	0.00011
a	1.4981	0.0165	1.5	1.5	2272	1074	1.00	0.00035	0.00025

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	p	elpd_diff	weight	se	dse	warning
Model 1	0	-5500.0	24.9	0.0	1.0	90.0	0.0	False
Model 2	1	-5500.0	34.6	-6.0	0.0	90.0	2.6	False