the smaller the value of beta is ,the more difficult the item is ?

[rudy-letote]

hi ,there is a problem which confuse me a few of days .i run code as follows:
res = irt(data)
i check many items in the data. when the beta is smaller such as -1.9,-2 , but the liked rate of this item is very low which i think it should be more large.when the beta is bigger such as 1.9 , 2 ,but the liked rate of this item is very high which i think it should be smaller. could anyone explain sth to me when the beta is large ,the liked rate should be large or smaller. note: the liked rate equals the number of "1" divide all numbers of items .
the format of the data is as follows :

customer_id	product_id	liked
1086992	14	1
1112695	14	0
1112695	17	1
803222	18	1
1110363	19	0
273880	19	1
445253	19	0

[sadrasabouri]

I think there is a mistake in the library (maybe the real b is -b).
I ran this toy example to check. Clearly, the first question was the hardest since everyone correctly evaluated it, and the second one was the hardest since they all missed it.

from pyirt import irt
from pprint import pprint

item_param, user_param = irt([
    (1, 1, 0), (1, 2, 1), (1, 3, 1),
    (2, 1, 0), (2, 2, 1), (2, 3, 0),
    (3, 1, 0), (2, 2, 1), (2, 3, 1),
])

pprint(item_param)
# {1: {'alpha': 0.25, 'beta': -2.0, 'c': 0.0},
# 2: {'alpha': 0.5720948359800555, 'beta': 2.0, 'c': 0.0},
# 3: {'alpha': 0.25, 'beta': 0.634102890032698, 'c': 0.0}}
pprint(user_param)
# {1: 0.13484234439295528, 2: 0.07287218993120267, 3: 0.019347314772223796}

Based on the original paper in equation (4), the probability of successfully answering questions is as follows:

$$ P( y_{ij}=1 | a_i, b_i, c_i=0, \theta_j) = \frac{1}{1 + e^{-a_i(\theta_j - b_i)}} $$

Since $a_i>0$ given a fixed value for subject ability, $\theta_j$, higher values of $b_i$ should result in a lower probability of people getting the answer right. But that's not the case here.