Rasch*

Contents

import pymc as pm
import numpy as np
import arviz as az

%load_ext lab_black

4. Rasch*#

Adapted from Unit 10: rasch.odc.

Data can be found here.

True/False Questions
1 if answered correctly, 0 otherwise
n students
k questions
Assess (relative) ability of students
Assess (relative) difficulty of questions
Originally motivated by testing/education, applicable in different contexts

Notes:#

Model works well and matches BUGS results with simple broadcasting. Just need to figure out a better way to display the results.

y = np.loadtxt("../data/rasch.txt")
n, k = y.shape
n, k

(162, 33)

with pm.Model() as m:
    tau_alpha = pm.Gamma("tau_alpha", 0.01, 0.01)
    var_alpha = pm.Deterministic("var_alpha", 1 / tau_alpha)
    tau_delta = pm.Gamma("tau_delta", 0.01, 0.01)
    # there's a typo for mu in BUGS version
    mu_delta = pm.Normal("mu_delta", 0, tau=0.001)

    # the 1s in the shapes are for broadcasting
    delta = pm.Normal("delta", mu_delta, tau=tau_delta, shape=(1, 33))
    alpha = pm.Normal("alpha", 0, tau=tau_alpha, shape=(162, 1))

    p = alpha - delta

    pm.Bernoulli("likelihood", logit_p=p, observed=y)

    trace = pm.sample(3000)

az.summary(trace, var_names=["delta"])

	mean	sd	hdi_3%	hdi_97%	mcse_mean	mcse_sd	ess_bulk	ess_tail	r_hat
delta[0, 0]	0.951	0.231	0.509	1.375	0.005	0.003	2540.0	6221.0	1.0
delta[0, 1]	0.574	0.222	0.171	1.010	0.005	0.003	2302.0	4440.0	1.0
delta[0, 2]	0.255	0.219	-0.163	0.659	0.004	0.003	2399.0	5453.0	1.0
delta[0, 3]	1.451	0.245	0.982	1.901	0.005	0.003	2841.0	6140.0	1.0
delta[0, 4]	-1.018	0.225	-1.428	-0.584	0.005	0.003	2242.0	4786.0	1.0
delta[0, 5]	0.684	0.229	0.246	1.103	0.005	0.003	2355.0	5108.0	1.0
delta[0, 6]	0.953	0.231	0.511	1.378	0.005	0.003	2375.0	4748.0	1.0
delta[0, 7]	0.759	0.227	0.338	1.189	0.004	0.003	2814.0	5476.0	1.0
delta[0, 8]	0.649	0.227	0.219	1.070	0.005	0.003	2536.0	5835.0	1.0
delta[0, 9]	1.682	0.251	1.217	2.158	0.005	0.003	2722.0	5779.0	1.0
delta[0, 10]	1.152	0.235	0.713	1.595	0.004	0.003	2770.0	5242.0	1.0
delta[0, 11]	-1.201	0.227	-1.627	-0.779	0.005	0.003	2415.0	5362.0	1.0
delta[0, 12]	-0.187	0.223	-0.620	0.225	0.005	0.003	2194.0	5197.0	1.0
delta[0, 13]	-0.088	0.215	-0.500	0.301	0.005	0.003	1996.0	4825.0	1.0
delta[0, 14]	0.873	0.229	0.445	1.305	0.005	0.003	2496.0	5520.0	1.0
delta[0, 15]	2.271	0.274	1.743	2.776	0.005	0.003	3313.0	6834.0	1.0
delta[0, 16]	1.543	0.243	1.084	2.002	0.004	0.003	2953.0	6441.0	1.0
delta[0, 17]	-0.291	0.220	-0.724	0.091	0.005	0.003	2148.0	5395.0	1.0
delta[0, 18]	0.049	0.218	-0.360	0.461	0.004	0.003	2413.0	5464.0	1.0
delta[0, 19]	-1.315	0.233	-1.742	-0.873	0.005	0.003	2466.0	5697.0	1.0
delta[0, 20]	2.805	0.300	2.209	3.342	0.005	0.004	3432.0	6759.0	1.0
delta[0, 21]	-0.458	0.217	-0.854	-0.043	0.005	0.003	2296.0	4904.0	1.0
delta[0, 22]	0.573	0.224	0.134	0.984	0.004	0.003	2548.0	5782.0	1.0
delta[0, 23]	0.359	0.218	-0.053	0.769	0.004	0.003	2473.0	5795.0	1.0
delta[0, 24]	-1.767	0.243	-2.206	-1.291	0.005	0.003	2708.0	6129.0	1.0
delta[0, 25]	-1.474	0.232	-1.906	-1.038	0.005	0.003	2537.0	6302.0	1.0
delta[0, 26]	0.872	0.226	0.447	1.293	0.004	0.003	2635.0	5833.0	1.0
delta[0, 27]	0.084	0.219	-0.329	0.494	0.005	0.003	2220.0	5476.0	1.0
delta[0, 28]	1.319	0.239	0.881	1.774	0.005	0.003	2536.0	6135.0	1.0
delta[0, 29]	-0.394	0.219	-0.804	0.013	0.005	0.003	2209.0	4431.0	1.0
delta[0, 30]	0.398	0.224	-0.029	0.810	0.005	0.003	2232.0	5299.0	1.0
delta[0, 31]	-0.874	0.222	-1.279	-0.455	0.005	0.003	2324.0	5660.0	1.0
delta[0, 32]	0.723	0.227	0.308	1.159	0.005	0.003	2448.0	5629.0	1.0

%load_ext watermark
%watermark -n -u -v -iv -p pytensor

Last updated: Wed Mar 22 2023

Python implementation: CPython
Python version       : 3.11.0
IPython version      : 8.9.0

pytensor: 2.10.1

arviz: 0.15.1
numpy: 1.24.2
pymc : 5.1.2