1, 2, 3-Parameter Logistic Rasch and Birnbaum Models and Item Analysis

item difficulty

parametric coefficient

K

a

guessing parameter Ci

filling

none

axis

The item difficulty statistic is an appropriate choice for achievement or aptitude tests when the items are scored dichotomously (i.e., they are either correct or incorrect). Several methods were developed by G. Rasch and A. Birnbaum to estimate test reliability. The One-Parameter Logistic Model (1PLM) is

P

i

(

B

i

)=1/(1+exp[-(

B

i

-

D

i

)])

, or

P

i

(

B

i

)=exp[

B

i

-

D

i

]/(1+exp[

B

i

-

D

i

])

,

where

B

i

is the parameter describing the ability of the person being tested,

P

i

is the probability of getting a correct response, and

D

i

is the parameter describing the difficulty of item

i

.

The Two-Parameter Logistic Model (2PLM) is

P

i

(

B

i

)=1/(1+exp[-

K

a

(P-

D

i

)])

,

where

K

a

is the slope parameter; when

K

a

=1

, this is the same as 1PLM.

The Three-Parameter Logistic Model (3PLM, or Birnbaum's Model):

P

i

(

B

i

)=

C

i

+(1-

C

i

)/(1+exp[-

K

a

(

B

i

-

D

i

)])

.

The parameter

C

i

is used when item

i

is constructed so that guessing the correct answer is possible.

The Demonstration plots the 1PLM, 2PLM, and 3PLM item characteristic curves in red, blue, and green, respectively.