Chapter 5 Applications of TAMGP
5.1 Applying TAMGP to Educational Information and Measurement
5.1.4 Using TAM in GM(0,n)
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Experimental Data
The study has performed an experiment to evaluate for the academic achievement of 24 students in Taichung, Taiwan. Data are the academic achievement of junior high school students in three subjects (English, Native Language, and National Language). The data are shown in the Table 5-8.
Table 5-8
The academic achievement of 24 students in three subjects
ID English Native Language
National
Language ID English Native Language
National Language
S1 99 99 96 S13 96 97 95
S2 91 97 92 S14 94 92 92
S3 98 95 95 S15 99 99 93
S4 86 95 87 S16 91 92 97
S5 80 85 93 S17 98 99 95
S6 97 97 92 S18 78 71 78
S7 91 84 95 S19 94 82 93
S8 92 92 93 S20 85 92 97
S9 87 90 88 S21 99 99 97
S10 80 92 86 S22 88 97 95
S11 78 87 85 S23 85 82 92
S12 78 75 75 S24 97 97 93
Results
The study have used T-GM(0,n) based on the relationship between the academic achievement of junior high school students in three subjects. The results of T-GM(0,n) are shown in the Table 5-9. The details of the calculation using T-GM(0,n) to evaluate for the academic achievement of 24 students are described as follows:
Establishing the original sequence for the academic achievement of junior high school students in English, Native Language, and National Language, respectively:
) 0 (
x1 = (99, 91, 98, 86, 80,
, 85, 99, 88, 85, 97).) 0 (
x2 = (99, 97, 95, 95, 85,
, 92, 99, 97, 82, 97).) 0 (
x3 = (96, 92, 95, 87, 93,
, 97, 97, 95, 92, 93).Using GM(0,3) to calculate the parameters b2, b3 and a, the result obtained b2 = 0.2272, b3 = 0.7588,and a = 0.2633; the fitted values for the academic achievement of
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students in English: ˆx1(0)= (96, 92, 94, 88, 90,
, 95, 96, 94, 88, 93); and the fitted error of the GM(0,3) model: Q = 523.0, MAPE = 4.5%, and RMSPE = 5.4%.Using T-GM(0,3) to calculate the parameters b2, b3 and a, the result obtained b2 = 0.4017, b3 = 0.5828,and a = 3.2795; the fitted values for the academic achievement of students in English: ˆx1(0)= (99, 93, 94, 89, 88,
, 93, 96, 94, 87, 93); and the fitted error of the T-GM(0,3) model: Q = 487.8, MAPE = 4.3%, and RMSPE = 5.0%.Table 5-9
The results of GM(0,n) and T-GM(0,n)
ID English GM(0,n) T-GM(0,n) ID English GM(0,n) T-GM(0,n)
S1 99 96 99 S13 96 94 94
S2 91 92 93 S14 94 91 91
S3 98 94 94 S15 99 93 94
S4 86 88 89 S16 91 95 93
S5 80 90 88 S17 98 95 95
S6 97 92 93 S18 78 75 74
S7 91 91 89 S19 94 89 87
S8 92 91 91 S20 85 95 93
S9 87 87 87 S21 99 96 96
S10 80 86 87 S22 88 94 94
S11 78 84 84 S23 85 88 87
S12 78 74 74 S24 97 93 93
Discussion
The results of T-GM(0,n) for the academic achievement of junior high school students in English indicated that Taylor approximation method in GM(0,n) was a good alternative for optimizing parameters and the accuracy of GM(0,n) in this study. The results also indicated that the influence of National Language for the academic achievement of students in English is higher than the influence of Native Language.
5.2 Using TAM in RaschGSP IRT
Experimental Data
In this study, the researchers used the results of two classes in the seventh grade in the same school on a Math paper test in Taichung, Taiwan. The number of students per class was 30 students and the number of problems was 25 problems.
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Results
Firstly, this study carried out a reliability test for the answers of class A and class B by using Cronbach’s coefficient. The results showed that Cronbach’s alpha value of the test for class A was α = 0.879, and class B was α = 0.833, which meant the reliabilities were quite good. The results from the test are shown in Table 5-10 and Table 5-12. Secondly, data are calculated by S-P chart. The original data are processed and arranged according to the rule of the S-P chart. Thirdly, Gamma values for students and problems are calculated based on GRA. Next, data are arranged in order of Gamma values for students and problems with GSP chart.
Fourthly, parameters α, β and γ are calculated by using RaschGSP IRT function.
Next, TAM in RaschGSP IRT is used to optimize parameters α, β and γ, and improving the accuracy of RaschGSP IRT curves. After that, the slope of RaschGSP IRT curve (a) at the point (b, 0.5) and the test difficulty are determined. The slope of RaschGSP IRT curve is 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. The results and discussion of this study are described as follows: For class A, the results from Fig. 5-4 and Fig. 5-5 showed that the slope of RaschGSP curve for students a = 0.55, b = 65.8%, and the slope of RaschGSP IRT curve for problems a = 0.52, b = 74.1%. According to the results from Table 5-11 indicated that using TAM in RaschGSP IRT could optimize parameters α, β and γ, to improve the accuracy of RaschGSP IRT curves. The results from Table 5-11 showed that the MAPE values of T-RGSP are smaller than the MAPE values of RaschGSP IRT. For class B, the results from Fig. 5-6 and Fig. 5-7 showed that the slope of RaschGSP IRT curve for students a = 0.48, b = 76.8%, and the slope of RaschGSP IRT curve for problems a = 0.40, b = 85.4%. The results from Table 5-13 showed that parameters α, β and γ are optimized, and the accuracy of RaschGSP IRT curves is improved. The values of T-RGSP curves and RaschGSP IRT curves for students and problems compared with Gamma values showed that the MAPE values of T-RGSP are smaller than the MAPE values of RaschGSP IRT.
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Table 5-10
GSP chart for class A
S \ P 13 11 7 1 10 18 2 14 19 9 20 6 21 12 4 22 3 17 8 24 16 23 25 5 15 Total Ratio CS GS 10 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 24 0.96 0.16 0.80 23 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 24 0.96 0.72 0.80 14 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 24 0.96 0.88 0.80 22 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 23 0.92 0.92 0.72 28 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 21 0.84 0.00 0.60 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 21 0.84 0.15 0.60 9 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 0 1 1 0 21 0.84 0.49 0.60 11 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 0 1 0 0 20 0.80 0.21 0.55 16 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 0 1 1 0 1 0 20 0.80 0.40 0.55 20 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 19 0.76 0.16 0.51 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 0 1 1 0 1 0 0 19 0.76 0.22 0.51 4 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 0 0 1 1 1 1 0 0 0 0 17 0.68 0.21 0.43 15 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 0 17 0.68 0.31 0.43 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 0 0 0 0 0 0 16 0.64 0.04 0.40 25 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 0 0 0 1 0 0 0 16 0.64 0.24 0.40 13 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 16 0.64 0.44 0.40 29 1 0 1 0 1 1 0 1 1 1 1 1 1 0 0 0 1 1 1 0 1 0 0 0 1 15 0.60 0.76 0.37 27 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 14 0.56 0.35 0.34 17 1 1 1 1 1 1 0 1 1 0 1 0 0 0 0 0 1 0 0 1 0 1 1 1 0 14 0.56 0.69 0.34 26 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 0 0 0 0 1 0 12 0.48 0.54 0.28 6 1 1 1 1 1 1 0 0 1 0 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 11 0.44 0.18 0.25 8 1 1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 11 0.44 0.31 0.25 19 1 1 1 0 1 1 1 1 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 11 0.44 0.39 0.25 21 0 1 0 1 1 1 1 1 0 0 1 0 1 1 1 0 0 0 0 0 0 1 0 0 0 11 0.44 0.53 0.25 30 1 1 1 1 1 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 10 0.40 0.32 0.23 3 1 1 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 10 0.40 0.38 0.23 7 1 1 1 1 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 8 0.32 0.44 0.18 12 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 1 8 0.32 1.07 0.18 24 0 1 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 0 6 0.24 0.94 0.13 18 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 2 0.08 0.58 0.04 Total 27 27 26 24 24 23 23 21 21 19 19 19 19 18 18 17 17 17 16 16 15 11 10 8 6 461
Ratio 0.90 0.900.87 0.80 0.80 0.77 0.77 0.70 0.70 0.63 0.630.630.630.600.600.57 0.57 0.57 0.53 0.53 0.50 0.37 0.33 0.27 0.20
CP 0.10 0.430.32 0.29 0.60 0.15 0.17 0.11 0.13 0.14 0.260.360.450.210.710.15 0.38 0.63 0.14 0.39 0.29 0.79 0.36 0.62 0.42 S = 30 GP 0.68 0.680.63 0.55 0.55 0.52 0.52 0.45 0.45 0.39 0.390.390.390.370.370.34 0.34 0.34 0.32 0.32 0.29 0.20 0.18 0.14 0.11 P = 25
CS: Caution indices for students; CP: Caution indices for problems; GS: Gamma values for students; GP: Gamma values for problems
Fig. 5-4 T-RGSP graph for 30 students (class A)
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Fig. 5-5 T-RGSP graph for 25 problems (class A) Table 5-11
Parameters and error analysis (class A)
α β γ MAPE (%) RMSPE (%)
RaschGSP IRT (Students) 0.0328 -0.1622 -0.9360 7.6185 9.0986 T-RGSP (Students) 0.0266 -12.5661 -1.6952 3.9164 5.1465 RaschGSP IRT (Problems) 0.0305 0.3783 -0.7386 5.8627 6.9916
T-RGSP (Problems) 0.0337 2.1587 -0.8008 3.1886 3.8234
In this study, the study used Taylor approximation method in RaschGSP IRT curves for students and problems at four points to six points near Gamma value 0.5 to determine the slope of RaschGSP IRT curves at the point (b, 0.5) and the test difficulty. The experimental results showed that the discrimination for students of class A is higher than the discrimination for students of class B, and the problem's difficulty of class A is lower than the problem's difficulty of class B. When a is small, the discrimination for students and problems will be low. In contrast, when a is large, the discrimination for students and problems will be high. Besides, when b is small, the academic achievement of students will be high and the test difficulty will be low. In contrast, when b is large, the academic achievement of students will be low, and will the test difficulty will be high.
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Table 5-12
GSP chart for class B
S \ P 11 1 13 19 2 10 7 18 9 4 6 14 12 20 17 24 22 8 16 23 15 21 3 25 5 Total Ratio CS GS 21 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 22 0.88 0.35 0.65
5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 0 0 1 21 0.84 0.34 0.60 15 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 0 0 0 20 0.80 0.11 0.55 24 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 1 0 0 20 0.80 0.20 0.55 28 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 0 0 1 1 1 1 0 1 20 0.80 0.79 0.55 13 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 0 0 1 1 0 0 19 0.76 0.29 0.51 6 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 19 0.76 0.48 0.51 10 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 1 1 1 0 1 0 19 0.76 0.55 0.51 26 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 0 0 1 0 0 0 18 0.72 0.16 0.47 25 1 1 1 1 1 1 0 0 1 0 0 1 1 0 1 1 1 1 1 0 1 0 1 1 1 18 0.72 0.87 0.47 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 0 0 0 0 0 0 17 0.68 0.09 0.43 2 1 1 1 1 1 0 1 0 1 1 1 1 1 1 0 1 1 1 1 0 0 0 0 1 0 17 0.68 0.35 0.43 14 1 1 1 0 1 1 1 1 0 1 1 0 1 1 1 1 1 1 0 0 0 0 1 0 0 16 0.64 0.39 0.40 30 1 1 1 1 1 1 0 1 0 1 0 1 0 1 1 1 0 1 1 0 1 0 1 0 0 16 0.64 0.50 0.40 20 1 1 1 1 1 0 1 1 1 0 0 1 1 1 0 1 1 1 0 0 0 0 0 1 1 16 0.64 0.52 0.40 11 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 0 0 0 0 1 1 1 0 1 0 16 0.64 0.82 0.40 29 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 0 0 0 1 0 0 0 0 0 1 15 0.60 0.26 0.37 17 1 1 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 0 1 0 0 1 0 15 0.60 0.52 0.37 22 1 1 0 1 1 1 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 1 13 0.52 0.41 0.31 19 1 1 1 1 1 0 1 0 0 1 0 1 1 0 0 0 0 1 0 0 1 0 0 0 0 11 0.44 0.35 0.25 3 1 1 1 1 1 1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 10 0.40 0.29 0.23 1 1 0 1 0 0 0 0 1 0 1 1 0 0 0 1 0 1 0 0 0 0 1 1 1 0 10 0.40 1.05 0.23 9 1 1 1 0 0 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 9 0.36 0.48 0.20 8 0 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 1 0 8 0.32 1.03 0.18 27 1 0 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 7 0.28 0.58 0.15 16 1 1 0 0 1 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 7 0.28 0.61 0.15 23 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 7 0.28 0.83 0.15 7 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 7 0.28 0.88 0.15 12 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 6 0.24 1.01 0.13 18 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 4 0.16 1.34 0.08 Total 26 24 23 23 22 20 20 19 19 18 18 17 17 17 17 17 16 15 15 11 11 10 10 10 8 423
Ratio 0.87 0.800.77 0.77 0.73 0.67 0.67 0.63 0.63 0.60 0.600.570.570.570.570.57 0.53 0.50 0.50 0.37 0.37 0.33 0.33 0.33 0.27
CP 0.31 0.060.27 0.27 0.05 0.27 0.79 0.26 0.43 0.30 0.560.260.330.380.560.68 0.29 0.38 0.48 0.66 0.78 0.65 0.71 1.11 0.66 S = 30 GP 0.63 0.550.52 0.52 0.48 0.42 0.42 0.39 0.39 0.37 0.370.340.340.340.340.34 0.32 0.29 0.29 0.20 0.20 0.18 0.18 0.18 0.14 P = 25
CS: Caution indices for students; CP: Caution indices for problems; GS: Gamma values for students; GP: Gamma values for problems.
Fig. 5-6 T-RGSP graph for 30 students (class B)
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Fig. 5-7 T-RGSP graph for 25 problems (class B) Table 5-13
Parameters and error analysis (class B)
α β γ MAPE (%) RMSPE (%)
RaschGSP IRT (Students) 0.0262 0.2399 -0.8753 2.2408 2.6009 T-RGSP (Students) 0.0243 -5.6744 -1.1589 1.9587 2.2383 RaschGSP IRT (Problems) 0.0238 -1.3587 -0.7952 2.3543 3.0344 T-RGSP (Problems) 0.0269 4.1731 -0.6085 1.8243 1.9025
Discussion
The results showed that using the Taylor approximation method in RaschGSP IRT can obtain the optimal parameters and making the convergent error reduce to the minimum. This method will be useful for teachers as well as students to improve teaching and learning process, it not only can provide the important information for evaluating the academic achievement of classes and the test difficulty, but also can compare the academic achievement of classes and the test difficulty in the teaching and learning process.
5.3 Using the Combination of TAMGP and GRA
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Data
The study has performed an experiment to predict and evaluate for the academic achievement of 30 students in Mathematics in Taichung, Taiwan. Data are the academic achievement of students for three years corresponding to six semesters in junior high school (The data are shown in the Table 4-4).
Results Table 5-14
T Scores for the academic achievement of 30 students
ID The academic achievement of students
ID The academic achievement of students Sem1 Sem2 Sem3 Sem4 Sem5 Sem6 Sem1 Sem2 Sem3 Sem4 Sem5 Sem6 S1 48.0 54.6 56.3 58.5 57.7 57.8 S16 48.7 50.5 52.2 52.9 50.6 54.0 S2 48.0 49.1 50.2 50.6 54.2 53.4 S17 53.2 54.6 53.6 52.2 51.8 55.3 S3 59.1 59.5 59.7 60.9 60.7 57.8 S18 40.6 40.1 39.9 37.8 38.7 40.1 S4 57.6 53.2 57.0 57.7 56.5 57.2 S19 59.1 58.8 58.4 56.9 55.3 57.2 S5 57.6 55.3 58.4 57.7 57.7 57.2 S20 56.1 56.7 57.0 54.5 54.7 55.9 S6 48.7 48.4 50.8 49.8 54.2 54.6 S21 46.5 47.7 45.4 48.2 48.2 48.9 S7 54.6 48.4 48.8 50.6 52.4 52.1 S22 50.2 49.8 49.5 52.2 54.7 54.0 S8 19.0 22.1 27.6 25.9 24.5 21.7 S23 42.8 49.1 46.7 49.8 48.2 48.3 S9 59.1 58.8 57.7 58.5 58.9 58.4 S24 57.6 57.4 58.4 52.9 51.8 51.5 S10 59.1 58.1 54.9 56.9 57.7 57.2 S25 60.6 60.2 59.7 60.1 57.1 56.5 S11 37.6 31.1 25.6 24.3 24.5 25.5 S26 50.2 50.5 46.1 45.0 44.7 45.1 S12 59.8 59.5 61.8 58.5 57.7 55.9 S27 49.5 51.2 48.8 49.0 53.0 52.7 S13 27.2 28.4 28.3 27.5 28.6 28.0 S28 59.8 59.5 59.7 58.5 58.9 56.5 S14 47.2 49.8 52.2 49.0 48.2 48.9 S29 39.1 32.5 35.8 40.2 36.4 36.2 S15 45.0 45.6 42.6 45.0 45.8 47.0 S30 58.4 59.5 57.0 58.5 56.5 55.3 Note. “Sem” is the abbreviation of semester.
Firstly, the academic achievement of students in Mathematics transformed to T scores. Secondly, LGRA is used to calculate Gamma values based on T scores.
Thirdly, TAMGP is used to predict the next two semesters (Semester 7 and Semester 8), then error analysis is also performed. Results of error analysis showed that the MAPE and RMSPE are all relatively low and satisfy condition of less than 10%, so the predicted results are accepted. Fourthly, LGRA is used to calculate Gamma values based on the predicted academic achievement of students. Finally, drawing
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conclusions from the results of the combination of TAMGP and GRA for predicting and evaluating the academic achievement of students in Mathematics.
Table 5-15
The predicted results, error analysis, Gamma values for 30 students
ID The predicted academic achievement of students
Model MAPE (%)
RMSPE (%)
Gamma Values Sem1 Sem2 Sem3 Sem4 Sem5 Sem6 Sem7 Sem8
S1 48.0 54.3 56.9 57.3 57.7 58.1 58.5 58.9 T-GM(2,1) 0.70 1.00 0.89 S2 48.0 49.0 50.2 51.4 52.7 54.0 55.3 56.7 T-GM(1,1) 0.98 1.40 0.79 S3 59.1 60.2 60.0 59.7 59.5 59.2 59.0 58.8 T-GM(1,1) 1.34 1.61 1.00 S4 57.6 54.9 55.6 56.3 57.1 57.8 58.5 59.3 T-GM(1,1) 1.66 1.98 0.93 S5 57.6 56.7 57.0 57.3 57.6 57.9 58.2 58.5 T-GM(1,1) 1.19 1.52 0.95 S6 48.7 48.5 50.0 51.5 53.1 54.8 56.5 58.2 T-GM(1,1) 1.25 1.79 0.80 S7 54.6 48.3 49.3 50.4 51.5 52.7 53.8 55.0 T-GM(1,1) 0.72 0.93 0.79 S8 19.0 21.9 27.6 25.7 23.7 21.8 20.0 18.4 TGM(2,1) 0.88 1.47 0.00 S9 59.1 58.4 58.4 58.5 58.5 58.6 58.6 58.7 T-GM(1,1) 0.50 0.66 0.98 S10 59.1 56.8 56.9 57.0 57.1 57.2 57.2 57.3 T-GM(1,1) 1.16 1.77 0.94 S11 37.6 29.3 26.4 24.5 24.0 25.5 30.0 39.0 T-GM(2,1) 1.99 2.82 0.19 S12 59.8 60.9 59.8 58.7 57.6 56.5 55.4 54.4 T-GM(1,1) 1.20 1.71 0.95 S13 27.2 28.2 28.2 28.2 28.1 28.1 28.0 28.0 T-GM(1,1) 0.92 1.31 0.15 S14 47.2 50.8 50.2 49.6 49.0 48.5 47.9 47.4 T-GM(1,1) 1.63 2.01 0.72 S15 45.0 44.5 43.8 44.6 46.0 48.0 50.2 52.6 T-GM(2,1) 1.44 1.77 0.65 S16 48.7 51.0 51.5 52.0 52.6 53.1 53.7 54.2 T-GM(1,1) 1.59 1.99 0.80 S17 53.2 53.6 53.5 53.5 53.4 53.4 53.3 53.3 T-GM(1,1) 1.86 2.31 0.84 S18 40.6 39.6 39.5 39.3 39.2 39.1 38.9 38.8 T-GM(1,1) 1.68 2.10 0.46 S19 59.1 58.6 57.9 57.3 56.7 56.1 55.4 54.8 T-GM(1,1) 1.01 1.33 0.93 S20 56.1 56.6 56.2 55.8 55.4 55.0 54.6 54.2 T-GM(1,1) 1.12 1.37 0.90 S21 46.5 46.6 47.2 47.7 48.2 48.7 49.3 49.8 T-GM(1,1) 1.27 1.90 0.69 S22 50.2 49.4 50.4 52.0 53.7 55.4 57.2 59.1 T-GM(2,1) 1.26 1.58 0.82 S23 42.8 48.5 48.4 48.4 48.4 48.4 48.4 48.4 T-GM(1,1) 1.37 1.93 0.68 S24 57.6 58.1 56.2 54.3 52.5 50.8 49.1 47.4 T-GM(1,1) 1.73 2.08 0.82 S25 60.6 60.7 59.7 58.7 57.7 56.8 55.8 54.9 T-GM(1,1) 0.80 1.13 0.95 S26 50.2 49.1 46.7 45.2 44.8 46.3 50.7 59.8 T-GM(2,1) 1.29 1.68 0.70 S27 49.5 50.0 49.2 50.2 52.1 54.7 57.8 61.1 T-GM(2,1) 1.85 2.21 0.80 S28 59.8 60.0 59.3 58.6 58.0 57.3 56.7 56.0 T-GM(1,1) 0.78 0.97 0.97 S29 39.1 34.7 35.5 36.2 37.0 37.8 38.6 39.4 T-GM(1,1) 3.95 5.26 0.40 S30 58.4 59.1 58.2 57.3 56.5 55.6 54.7 53.9 T-GM(1,1) 0.91 1.26 0.92 Note. “Sem” is the abbreviation of semester.
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In this study, Gamma values of the academic achievement of students in Mathematics transformed to T scores are compared with Gamma values of the predicted academic achievement of students based on the combination of Taylor approximation method in grey prediction and grey relational analysis. T Scores for the academic achievement of 30 students are described in the Table 5-14. The predicted results of the academic achievement of students, error analysis based on MAPE and RMSPE, and calculating Gamma values based on LGRA are described in the Table 5-15. Ranking the academic achievement of students based on Gamma values are described in the Table 5-16.
Table 5-16
Ranking the academic achievement of students based on Gamma values
ID GRA
ID GRA
ID TAMGP-GRA
ID TAMGP-GRA Gamma Rank Gamma Rank Gamma Rank Gamma Rank
S3 1.00 1 S16 0.77 16 S3 1.00 1 S16 0.80 16
S28 0.98 2 S7 0.76 17 S9 0.98 2 S6 0.80 17 S25 0.98 3 S6 0.76 18 S28 0.97 3 S27 0.80 18 S12 0.98 4 S2 0.75 19 S25 0.95 4 S7 0.79 19
S9 0.97 5 S27 0.75 20 S5 0.95 5 S2 0.79 20
S30 0.94 6 S14 0.72 21 S12 0.95 6 S14 0.72 21 S19 0.94 7 S21 0.67 22 S10 0.94 7 S26 0.70 22
S5 0.94 8 S23 0.66 23 S4 0.93 8 S21 0.69 23
S10 0.93 9 S26 0.65 24 S19 0.93 9 S23 0.68 24 S4 0.91 10 S15 0.60 25 S30 0.92 10 S15 0.65 25 S20 0.90 11 S18 0.45 26 S20 0.90 11 S18 0.46 26 S24 0.86 12 S29 0.37 27 S1 0.89 12 S29 0.40 27 S1 0.86 13 S13 0.13 28 S17 0.84 13 S11 0.19 28 S17 0.83 14 S11 0.12 29 S22 0.82 14 S13 0.15 29 S22 0.78 15 S8 0.00 30 S24 0.82 15 S8 0.00 30
Discussion
Assessment of student learning outcomes provides feedback on the process of student learning and teaching process of teachers. This is the basis for determining the degree which students have achieved the goals of educational program and the success
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level of the teachers in teaching. Assessment of student learning outcomes has not only function that provided feedback, but also adjusts whole teaching process. Therefore, accurate assessment of learning outcomes is essential in the learning process, it will be an important foundation for planning and improving learning outcomes of students in the next time. This study improved the traditional method for selecting contestant and provided teachers an innovative method selecting contestants based on single subject and multiple subjects. The experimental results showed that using the combination of Taylor approximation method in grey prediction and grey relational analysis not only can predict and evaluate the academic achievement of students in Mathematics, but also can predict and evaluate the academic achievement of students in other subjects.
5.4 Using the Combination of TAMGP, GRA, and ROC
Experimental Data
The study has performed an experiment to set the standard for a 25-multiple choice question Mathematics test. There were 286 students who participated this mid-term test, they are the third grade of senior high school students in Taichung, Taiwan. The measurement data for applying the combination of TAMGP, GRA, and ROC are the Mathematics ability score of students in the previous five semesters and test scores (as the example shown in the Table 4-5).
Results
Firstly, for ease of comparisons, all raw data were transformed to T scores with a mean of 50 and a standard deviation of 10. Secondly, Taylor approximation method in grey prediction was used to calculate parameters and determine predicted values, and error analysis was also performed. The results of error analysis showed that MAPE and RMSPE were all relatively low and satisfy condition of less than 10%, so the predicted results were accepted.
Thirdly, LGRA was used to calculate Gamma values based on T scores and the predicted results. The prediction results were considered as the accurate and reliable basis for the
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application of GRA and ROC method. Next, ROC method is applied to provide information which can be used to calculate all possible cut scores according to relationships between the sensitivity (Se) and 1 – specificity (1 – Sp) of the test.
Table 5-17
The predicted results, error analysis, Gamma values, test scores and state for 286 students (a part of the results)
ID The predicted values
Model MAPE (%)
RMSPE (%)
Gamma Values
Test scores State Sem1 Sem2 Sem3 Sem4 Sem5 Sem6
S1 54.9 56.2 57.2 58.2 59.3 60.4 T-GM(1,1) 2.28 2.86 0.84 94 1 S2 51.0 54.1 54.2 54.3 54.5 54.6 T-GM(1,1) 2.48 3.33 0.75 88 1 S3 41.5 47.2 48.8 50.5 52.2 54.0 T-GM(1,1) 1.15 1.53 0.64 91 1 S4 51.0 53.0 54.3 55.5 56.8 58.1 T-GM(1,1) 0.52 0.66 0.78 90 1 S5 62.9 65.2 65.2 65.1 65.1 65.0 T-GM(1,1) 0.33 0.45 0.98 97 1 S6 48.6 46.7 45.5 47.8 52.8 60.0 T-GM(2,1) 2.76 3.39 0.67 86 1 S7 56.5 52.3 53.2 54.2 55.1 56.1 T-GM(1,1) 2.24 2.80 0.77 91 1 S8 61.3 59.6 60.2 60.8 61.5 62.1 T-GM(1,1) 2.20 2.47 0.91 96 1 S9 49.4 38.1 49.0 48.1 36.2 22.1 T-GVM 2.57 3.08 0.38 54 0 S10 25.6 30.8 33.2 35.8 38.6 41.7 T-GM(1,1) 5.70 7.31 0.30 63 0
S277 57.3 59.3 58.1 56.8 55.6 54.4 T-GM(1,1) 2.24 2.83 0.81 91 1 S278 35.9 39.7 42.8 46.1 49.7 53.6 T-GM(1,1) 3.15 3.97 0.54 78 1 S279 25.6 29.3 30.8 32.3 34.0 35.7 T-GM(1,1) 1.55 1.76 0.23 63 0 S280 49.4 47.8 49.0 52.7 57.5 63.0 T-GM(2,1) 1.73 2.10 0.74 82 1 S281 56.5 55.0 56.7 58.4 60.1 62.0 T-GM(1,1) 3.05 3.83 0.85 92 1 S282 47.0 35.5 36.7 40.2 44.8 50.1 T-GM(2,1) 2.34 2.85 0.49 71 0 S283 37.5 39.4 36.5 33.9 31.5 29.2 T-GM(1,1) 4.25 5.47 0.30 61 0 S284 50.2 53.2 54.7 56.1 57.6 59.2 T-GM(1,1) 1.71 2.19 0.79 87 1 S285 44.6 44.8 42.3 43.1 47.8 57.4 T-GM(2,1) 3.33 4.15 0.58 74 1 S286 58.1 56.1 53.3 54.8 61.7 76.3 T-GM(2,1) 2.24 2.89 0.88 89 1 Note. “Sem” is the abbreviation of semester.
Finally, the optimal cut score that can differentiate between passing and failing students is found out satisfying the ultimate goal of the proposed method. Pass mark is determined by using the calculation of Youden index (J) for each cut score. Result of modeling predicted values in six semesters are presented in Table 5-17. According to predicted results for Math
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ability score of students in the sixth semester, setting the standard classified 286 students into two categories, 219 for positive and 67 for negative. ROC analysis and the Youden indexes corresponding to possible cut scores are calculated to pick the optimal cut score. The optimal cut score was 74 in this study (as shown in the Table 5-18). In addition, the AUC of the test is calculated 0.96 greater than 0.8 (as Fig. 5-8), this confirms that the proposed method is reliable in the mentioned condition.
Table 5-18
Hypothetical data for the sensitivity and specificity at various cut-off scores (a part of the results)
Score Sensitivity (Se) 1 – Specificity (1 – Sp) Youden index (J)
100 0.00 0.00 0.00
99 0.02 0.00 0.02
98 0.03 0.00 0.03
97 0.08 0.00 0.08
96 0.13 0.00 0.13
95 0.18 0.00 0.18
78 0.84 0.03 0.81
77 0.86 0.06 0.80
76 0.89 0.06 0.83
75 0.91 0.07 0.83
74 0.93 0.09 0.84
73 0.93 0.10 0.83
72 0.94 0.15 0.79
71 0.95 0.19 0.76
70 0.95 0.22 0.73
46 1.00 0.91 0.09
45 1.00 0.94 0.06
44 1.00 0.96 0.04
40 1.00 0.97 0.03
36 1.00 0.99 0.01
32 1.00 1.00 0.00
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Fig. 5-8 The ROC curve and the area under the ROC curve (AUC)
Discussion
With the hypothesis that in a specific condition, it is possible to design the standard method which is considered the best one and the most suitable. This study has proposed a combination of TAMGP, GRA and ROC method to set the standard method for mid-term test, belonging to the type of examinee-centered method, with advantage that students’ ability scores can be accurately predicted. It can be seen that the proposed method has made the task of setting the standard for tests and determining the pass mark reasonably and reliably.
The finding above indicates that the proposed method can perform its task very well in the case of the difficulty of the test is easier than or more difficult than the ability of students. The outstanding advantage is easy to apply and not time consuming, but the results obtained are reliable and objective. The limitation of the proposed method is that the accurate prediction of ability scores of students in semester is a hard work that requires measurement data to be accurate data.
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5.5 Conclusion of Chapter 5
The experimental results showed that Taylor approximation method in grey prediction is actually useful for prediction problems, evaluation, and setting the standard for tests of uncertainty problems under discrete data and incomplete information in educational information and measurement.
This study will be useful for teachers as well as students to improve the overall quality of teaching and learning process, it not only can provide the important information for evaluating and comparing the academic achievement of classes and the test difficulty but also can assist in improving the quality of education.
This study will provide the important information for educational administrators to proactively propose the appropriate policy and to build the educational development strategy in accordance with the new conditions in the future. It is not only conducted to serve as a reference for the educational administrators, but also can assist the government in developing future policies regarding educational management.
This study will also provide the educational administrators with important information which are the basis for accelerating the construction, upgrading and rehabilitation of new and existing infrastructure in education, and enhancing the capacity to deliver infrastructure in education to continue to attract and efficient use of resources, bring international cooperation for training and scientific research. This is critical for improving the overall quality of teaching and learning and improving the quality of education across the country.
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Chapter 6 A MATLAB Toolbox for TAMGP
In this study, there is a difficult problem of the calculation for Taylor approximation method in grey prediction. Therefore, the design of a MATLAB toolbox for this method is necessary. The MATLAB toolbox not only helps to process data quickly and accurately, but also displays the results on a graphical user interface visually. MATLAB is a programming language that is very useful for numerical simulation and data analysis. It helps simplify solving computational techniques compared with traditional programming languages such as C, C ++, and Fortran. MATLAB is used in many fields, including signal and image processing, communications, control design automation, measurement, analysis, financial modeling, and computational science. With millions of engineers and scientists working in industrial environments as well as in academic environments, MATLAB is the language of scientific computing.
A graphical user interface (GUI) in MATLAB is a graphical display in one or more windows containing controls, called components that enable a user to perform interactive tasks. The user of the GUI does not have to create a script or type commands at the command line to accomplish the tasks. Unlike coding programs to accomplish tasks, the user of the GUI need not understand the details of how the tasks are performed. In recent years, the MATLAB software has been successfully employed in many fields. Many researchers have developed the MATLAB toolboxes for the purpose of the study and learning of grey system theory (Sheu et al., 2013; Sheu et al., 2014a; Sheu et al., 2013;
Sheu et al., 2014; Sheu et al., 2014; Wen, 2008; Wen & Chang, 2005; Wen & Huang, 2005). Therefore, this study is used the MATLAB software to design a MATLAB toolbox based on Taylor approximation method in grey prediction.
The GUI of the MATALAB toolbox starts up when file “MATLAB_Toolbox.m” is opened, the graphical user interface of the program is displayed on the screen of the computer. Firstly, the user needs to click a button of the panel (as Fig. 6-1).
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Fig. 6-1 Graphical user interface of the MATALAB toolbox
The figure 6-1 showed that the user can select a the button TAMGP1 (Taylor approximation method in GM(1,1), GVM, and GM(2,1)), or TAMGP2 (Taylor approximation method in GM(1,n) and GM(2,n)), or TAM in RaschGSP IRT (Taylor approximation method in RaschGSP IRT), or TAMGP – GRA (The combination of Taylor approximation method in grey prediction and grey relational analysis), or TAMGP – GRA – ROC (The combination of Taylor approximation method in grey prediction, grey relational analysis, and receiver operating characteristic), or EXIT (close program).
6.1 Graphical User Interface of MATLAB for Taylor Approximation Method in Grey Prediction
6.1.1 Using GUI of MATLAB for TAMGP1
The graphical user interface starts up when the button “TAMGP1” is clicked, the graphical user interface of the program is displayed on the screen of the computer (as
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Fig. 6-2). Firstly, the user needs to click the button “Input Data” to input the data.
Fig. 6-2 Using GUI of MATLAB for T-GM(1,1), T-GVM, and T-GM(2,1) to input data
According to the figure 6-2 showed that the user clicked the button “Input Data”, then the user selected “Data T-GM11.csv” file. Finally, the user clicked the button “Open”
to input the data. In this study, the data are the number of students of Asia studying in Taiwan from the 2005-2006 to the 2013-2014 school year which are numerical and written in “Data T-GM11.xlsx” file.
Secondly, the user clicked the button “OK” to test the data. In this study, the result of testing data showed that the data consistent with T-GM(1,1) model (as Fig. 5-3).
Thirdly, the user needs to select a prediction model (GM(1,1) or GVM or T-GM(2,1)) . In the panel, the user can type the number of Times K, the number of Delta h, and the number of Coefficient H.
The figure 6-3 showed that the number of Times K was 90, the number of Delta h was 500, and the number of Coefficient H was 20. When the T-GM(1,1) model was selected, the data were processed by MATLAB software, the results and the graphs were displayed in the panel (as Fig. 6-4). The figure 6-4 showed that the data of the number
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of students of Asia studying in Taiwan in nine years was used to predict for the next year.
Fig. 6-3 Using GUI of MATLAB for T-GM(1,1), T-GVM, and T-GM(2,1) to test data
Fig. 6-4 The results and the graphs for the T-GM(1,1) model