Quality & Quantity 32: 433–439, 1998.
© 1998 Kluwer Academic Publishers. Printed in the Netherlands.
Graphical Analysis of the Two Parameters Logistic
Function (A Methodological Proposal)
CONCEPCIÓN SAN-LUIS and ALFONSO SÁNCHEZ-BRUNO
Área de Metodología de las Ciencias del Comportamiento, Facultad de Psicología, Universidad de
Abstract. It is proposed to study the graphical representation of the parametric space of maximum
likelihood function of two parameters logistic functions, as is used in Item Response Theory. This
proposal is made more from a point of view of understanding rather than of discovery.
Key words: graphical analysis, IRT, parametric space, maximum likelihood.
In the present paper we propose to study the graphical representation of the para-
metric space of maximum likelihood function of two parameters logistic function,
as is used in Item Response Theory, as a tool to aid understanding of the behavior
of the algorithms used to determine those parameters.
Research based on graphic analysis techniques acquires more importance every
day, as it is used and trusted by many authors (see, e.g., Hartwig & Dearing, 1984).
One of the reasons for its popularity is the development of the graphical capa-
bilities of modern computers, as well as their increasing computing power, which
makes graphical analysis and representation of extremely complex functions very
By ‘parametric space’ we mean the representation of the values of a function
related to the values of its parameters, assuming constant values of the variables
(both dependent and independent) (Draper & Smith, 1981). In case the function
under study has only two parameters, we shall get a three-dimensional representa-
tioninwhich,ontheX and Y axes, we shall represent a series of values of each
parameter,andontheZ axis the function values.
In the present case, the X and Y axes match respectively the a (discrimination)
and b (difﬁculty) parameters, while the Z axis, symbolized by means of contour
lines, matches the values that assumes the logarithmic likelihood function for every
combination of values of a and b.