Interpretations of Parameters with Generalized Logistic Function in a Hierarchical Model.

[JoeSu112]

Hi, all. I am running a DDM model with HSSM. I've got the estimation result of parameters.
So now I'm trying to do a data simulation with these parameters. However, I find it confusing to interpret these parameters.

Here's my model:

HDDM_0 = hssm.HSSM(
    data=data_0,
    model="ddm",
    loglik_kind="approx_differentiable",
    hierarchical=True,
    prior_settings="safe",
    link_settings="log_logit",
    include=[
        {
            "name": "v",
            "prior": {
                "Intercept": {"name": "Normal", "mu": 0, "sigma": 3},
                "month_delayed": {"name": "Normal", "mu": 0, "sigma": 3},
                "pay_diff": {"name": "Normal", "mu": 0, "sigma": 3},
                "time_advantage": {"name": "Normal", "mu": 0, "sigma": 3},
            },
            "formula": "v ~ 1 + month_delayed + pay_diff + time_advantage + (month_delayed|participant_id) + (pay_diff|participant_id) + (time_advantage|participant_id)",
            "link": "identity",
        },
    ],
)

As I have link_settings="log_logit", the link functions of a, z and t are all Generalized Logistic function:

v:
    Formula: v ~ 1 + month_delayed + pay_diff + time_advantage + (month_delayed|participant_id) + (pay_diff|participant_id) + (time_advantage|participant_id)
    Priors:
        v_Intercept ~ Normal(mu: 0.0, sigma: 3.0)
        v_month_delayed ~ Normal(mu: 0.0, sigma: 3.0)
        v_pay_diff ~ Normal(mu: 0.0, sigma: 3.0)
        v_time_advantage ~ Normal(mu: 0.0, sigma: 3.0)
        v_1|participant_id ~ Normal(mu: 0.0, sigma: Weibull(alpha: 1.5, beta: 0.30000001192092896))
        v_month_delayed|participant_id ~ Normal(mu: Normal(mu: 0.0, sigma: 0.25), sigma: Weibull(alpha: 1.5, beta: 0.30000001192092896))
        v_pay_diff|participant_id ~ Normal(mu: Normal(mu: 0.0, sigma: 0.25), sigma: Weibull(alpha: 1.5, beta: 0.30000001192092896))
        v_time_advantage|participant_id ~ Normal(mu: Normal(mu: 0.0, sigma: 0.25), sigma: Weibull(alpha: 1.5, beta: 0.30000001192092896))
    Link: identity
    Explicit bounds: (-3.0, 3.0)
 (ignored due to link function)
a:
    Formula: a ~ 1 + (1|participant_id)
    Priors:
        a_Intercept ~ Normal(mu: 0.0, sigma: 0.25)
        a_1|participant_id ~ Normal(mu: 0.0, sigma: Weibull(alpha: 1.5, beta: 0.30000001192092896))
    Link: Generalized logit link function with bounds (0.3, 2.5)
    Explicit bounds: (0.3, 2.5)
 (ignored due to link function)
z:
    Formula: z ~ 1 + (1|participant_id)
    Priors:
        z_Intercept ~ Normal(mu: 0.0, sigma: 0.25)
        z_1|participant_id ~ Normal(mu: 0.0, sigma: Weibull(alpha: 1.5, beta: 0.30000001192092896))
    Link: Generalized logit link function with bounds (0.0, 1.0)
    Explicit bounds: (0.0, 1.0)
 (ignored due to link function)
t:
    Formula: t ~ 1 + (1|participant_id)
    Priors:
        t_Intercept ~ Normal(mu: 0.0, sigma: 0.25)
        t_1|participant_id ~ Normal(mu: 0.0, sigma: Weibull(alpha: 1.5, beta: 0.30000001192092896))
    Link: Generalized logit link function with bounds (0.0, 2.0)
    Explicit bounds: (0.0, 2.0)
 (ignored due to link function)

The result table is as followed (take "a" as example):

	mean
a_Intercept	0.13032
a_1｜participant_id[1]	-0.44538
a_1｜participant_idt[2]	-0.26137
a_1｜participant_id[3]	-0.26907

As "a" is in the Generalized logistic link function with bounds (0.3, 2.5), is there any method or function that I can transformed the result (e.g., 0.13032) into its actual value so that I will be able to do the data simulation.
Also, because I want to do the simulation individually, I wonder what is the correct way to transform the above result into the
actual individual parameter. Should I add "a_Intercept" + "a_1｜participant_id[i]", then transform, or there is other way?

Thanks for any help!!!