Chapter 3 TAM in RaschGSP IRT
3.2 RaschGSP IRT
GSP chart was first proposed by Nagai in 2010. Essentially, GSP chart is a
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combination of GRA and S-P chart. The GSP chart was developed in order to overcome the weaknesses of the S-P chart. GSP chart can make the analysis more concrete and accurate, and the uncertain factors in the studies can also be analyzed (Sheu et al., 2013).
Its description is shown in Table 3-1.
Table 3-1 GSP chart
Note. Adapted and modified from “A Matlab Toolbox for Problem Chart and Grey Student-Problem Chart and Its Application,” by Sheu et al., 2013, International Journal of Kansei Information 4(2), p. 77.
Definition 3-1: Gamma value
In GSP chart, GSi is the localized grey relational grade of the i-th student, and
GPj is the localized grey relational grade of the j-th problem. They are general called Gamma value, and in specific, S is called Gamma value for student and LGRG-P is called Gamma value for problem (Sheu et al., 2014).
m
52 n j
GPj j j , 1,2, ,
max min max 0
0
(3-3)
Nagai applied the view of Rasch model in GSP chart to propose the RaschGSP method that can analyze the relationship between two sets of data which were sets of the order value of students (or problems) and the localized grey relational grades. The purpose was to find a function that represented the characteristics of the entire data. This function is called RaschGSP function, and its graph is called the RaschGSP graph (Tzeng et al., 2012). According to the RaschGSP method, RaschGSP IRT is developed in 2014 (Sheu et al., 2014). Logistic regression for RaschGSP IRT is presented as follows:
Call N is the number of students taking the test, test results are reported in interval data by range of Gamma values
S (localized grey relational grade), the greater
S the higher the proficiency level on the test content.Suppose that the students are graded into two types: pass and fail, pass is denoted by 1 and fail is denoted by 0. Logistic regression is applied to determine the relationship between test scores and ability if a student passed the exam, as follows: In the OXY plane, X-axis describes the order of students sorted by ascending
S value, Y-axis shows only two states of pass and fail (Fig. 3-2).Fig. 3-2 Test results of the students plotted against the pass-fail categories 0
Г 1
Order of students’ abilities xxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxx
x
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0 Г 1
Order of students’ abilities x xxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxx
With Y is called the event of pass-fail of a student, the results are transformed into probabilities. The probabilities of event Y are plotted at each value of X, the curve is plotted to fit into cumulative probability curve, the graph is presented in Figure. 3-3.
According to that basis, logistic regression model is determined as follows:
Y: event of pass-fail of a student, Y=1 corresponding to S 0.5 : pass, Y=0 corresponding to S 0.5: fail
x: the order of students’ abilities corresponding to their GS value increasing.
Fig. 3-3 Test results plotted against probability of allocation to pass-fail categories
)
S( x
: the probability of event Y bound by the condition x, 0S(x)1 The form of the logistic regression equation is:
x x x
S S
1
) 0
( 1
)
ln (
(3-4)
where
0: intercept and
1: slope where 0S 1Continue for mathematical transformations, obtained:
) ( 1
) exp 0 1 (
x x x
S S
(3-5)
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Note. For the problems, the procedure is also performed the same with procedure above, the similar formula is obtained for P( x).
The graph is plotted again for clarity with some important points.
Fig. 3-4 Logistic regression curve of test results 0
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Referring to and selecting the advantages of nonparametric IRT model and kernel smoothing - nonparametric technique for estimating regression functions which was proposed by Ramsay (Lee, 2007; Ramsay, 1991), the results above lead to the following definitions and properties.
Definition 3-2: The one-parameter RaschGSP IRT function
Let 1 exp ( )
) 1
(
x D x
y be the one-parameter logistic regression function, where βR is regression coefficient; D = 1.702 is the scaling constant. When x is the order of student ability or the order of item difficulty and y is the localized grey relational grade, the above function y(x) is called RaschGSP IRT function. If x is the order of students’ abilities and y is LGRG-S then the above function y ( x)
is called the one-parameter RaschGSP IRT function for students, similarly, if x is the order of items’ difficulties and y is the LGRG-P then y( x) is called RaschGSP IRT function for problems (Sheu et al., 2014).
Definition 3-3: The two-parameter RaschGSP IRT function Let ( )1exp 1( )
x x D
y be the two-parameter logistic regression
function, where α and βR are regression coefficients; D = 1.702 is the scaling constant. When x is the order of student ability or the order of item difficulty and y is the localized grey relational grade, the above function y( x) is called the two-parameter RaschGSP IRT function. If x is the order of students’ abilities and y is LGRG-S then the above function y(x) is called RaschGSP IRT function for students, similarly, if x is the order of items’ difficulties and y is the LGRG-P then
) ( x
y is called RaschGSP IRT function for problems (Sheu et al., 2014).
Definition 3-4: The three-parameter RaschGSP IRT function
Let 1 exp ( )
) 1
(
x D x
y be the three-parameter logistic regression
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function, where α, β, and γ R are regression coefficients; D = 1.702 is the scaling constant. When x is the order of student ability or the order of item difficulty and y is the localized grey relational grade, the above function y( x) is called the three-parameter RaschGSP IRT function. If x is the order of students’ abilities and y is LGRG-S then the above function y(x) is called RaschGSP IRT function for students, similarly, if x is the order of items’ difficulties and y is the LGRG-P then
) (x
y is called RaschGSP IRT function for problems (Sheu et al., 2014).
Definition 3-5: The slope of the three-parameter RaschGSP IRT curve
Let
a D be the first derivative value of the RaschGSP IRT
function y’ at x = b, where a is the slope of the tangent of logistic regression line at the point (b, 0.5), and b is the abscissa of the intersection (Sheu et al., 2014).