Grey Relational Analysis and Grey Student-Problem Chart

The grey system theory was proposed by Deng in 1982, and the grey system theory includes internal information system model which is either insufficient or incomplete, and the grey system theory can be used for relational analysis (Liang et al, 2011a~2011c; Lin and Wen, 2009; Wang et al, 2011a~2011b;

Wang, Sheu and Nagai, 2011; Wang et al., 2012a~2012c; Yamaguchi, Li and Nagai, 2005). The grey system theory is to use discrete irregular data, generated by the accumulation of new data, and it has a form of regularity index and the establishment of differential equations, according to a new model to fit the data (Lee, Liang and Nagai, 2012; Liang et al, 2011a~2011c; Nagai et al, 2005; Wang et al, 2011a~2011b; Wang et al, 2012a~2012c; Wen, You, Nagai, Chang and Liang, 2010).

The GRA is an important approach of the grey system theory because GRA not only applies to cluster the data which have same features, but also measures their relationships (Hwu, Liang, Chiang, Chu and Nagai, 2012; Lai and Chu, 2011; Li, Masuda and Nagai, 2011; Lin and Wu, 2010). The procedure of GRA generation is summarized as follows (Chen, Chen and Cho, 2011; Hsu, Ken and Lein, 2008; Hwu et al, 2012; Lai, Chen, Chen, Yeh and Cheng, 2009;

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Nagai et al, 2005; Wang et al, 2011a~2011b; Wang et al, 2012a~2012c; Wen et al, 2010):

If we only use x₀ as reference sequence, and the rest are inspected sequences, it is called “Localization GRA.” If any sequence in x_i can be inspected sequence only, then is called “Globalization GRA.”

Step 2: Grey relational calculation.

min

In this paper, GRA calculation is used to reach the gamma value, which is

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between 0 to 1. Moreover, it is used to do the clustering of students’

performances and find the objective solution of educational decision-making fields.

To sum up, there are some publications which have indicated the GRA calculations is reliable for educational assessment (please see Wang et al, 2011a~2011b; Wang et al, 2012a~2012c).

Next, the Student-Problem chart analysis (S-P chart analysis) was first invented by Takahiro Sato, who cared about the differences of response data obtained from student answers; moreover, the students’ responses are shown in graphs (Sato, 1975, 1980, 1985). The abnormal performances of students or problems can be diagnosed through the S-P chart and teachers can benefit from the results for diagnosing the learning effects of learners (Harnisch, 1984;

Harnisch and Linn, 1981). Four numbered indices, such as disparity index, student caution index (CS), problem caution index (CP), and homogeneity index can be found in the S-P chart, and these indices help teachers diagnose student learning conditions, instructive achievement, and problem quality (Sato, 1980; Wu, 1999; Yih and Lin, 2010). Moreover, a performance profile curve of individual student can be drawn using the analyzed S-P chart data (Yu and Yu, 2006). At the end, teachers can provide remedial instructions and clear guidance for students based on the information of the data.

Table 2.3 shows the matrix of the original data where there are 10 students and 10 problems. In Table 2.3, when students get the correct answer on the problem, the cell will be marked as 1, and when they get the wrong answer, the cell will be marked as 0.

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Table 2.3 Example of original data S-P chart Problems

1 2 3 4 5 6 7 8 9 10 correct

1 0 1 1 1 0 1 0 1 1 1 7

2 0 0 1 1 0 1 1 1 0 0 5

3 1 1 1 1 1 1 1 1 1 1 10

4 0 0 1 1 1 1 1 0 1 1 7

5 0 1 1 1 0 1 1 0 0 1 6

Students

6 0 1 1 1 0 1 1 0 1 1 7

7 1 0 1 1 0 1 1 1 0 1 7

8 0 1 1 1 0 1 1 0 1 1 7

9 1 1 0 1 0 1 1 1 1 1 8

10 1 0 1 1 1 1 1 1 1 0 8

number 4 6 9 10 3 10 9 6 7 8

After all the test information is recorded, the students and problems are sorted from high to low (0 to 1), and the sorted results are shown in Table 2.4.

Table 2.4 Sorted S-P chart of Table 2.3 Problems

4 6 3 7 10 9 2 8 1 5 correct %

3 1 1 1 1 1 1 1 1 1 1 10 100

9 1 1 0 1 1 1 1 1 1 0 8 80

10 1 1 1 1 0 1 0 1 1 1 8 80

1 1 1 1 0 1 1 1 1 0 0 7 70

4 1 1 1 1 1 1 0 0 0 1 7 70

Students

6 1 1 1 1 1 1 1 0 0 0 7 70

7 1 1 1 1 1 0 0 1 1 0 7 70

8 1 1 1 1 1 1 1 0 0 0 7 70

5 1 1 1 1 1 0 1 0 0 0 6 60

2 1 1 1 1 0 0 0 1 0 0 5 50

number 10 10 9 9 8 7 6 6 4 3

% 100 100 90 90 80 70 60 60 40 30

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According to Table 2.4, students with higher scores are in the upper part of the S-P chart, and upper-left part shows the corrected problems answered by the students. Besides, it is able to draw the S-curve (solid line) and P-curve (dotted line).

The function of the S-P chart is to evaluate the learning progress and provide the remedial plan to improve the curriculum and teaching based on Bloom’s learning evaluation (Chacko, 1998; Chen, Lai and Liu, 2005): diagnostic evaluation, formative evaluation and summative evaluation. The S-P chart can not only be applied to the diagnostic evaluation during the learning process, but also make the progress in formative evaluation.

Then, Nagai first introduced the Grey Student-Problem chart in 2010, and it is the combination of GRA and S-P chart which not only makes the analysis more concrete and accurate, but can also be applied to analyze uncertain factors (Sheu et al, 2011; Wang et al, 2011a~2011b; Wang, Sheu and Nagai, 2011;

Wang et al, 2012c). Through using the equations, the GSP can make the readable chart effectively and find out the weighting or ordinal numbers between the discrete data. The GSP provides the educational assessment of English listening performances, product design and product professional courses to define curriculum assessment results which are based on grey relational analysis (Sheu, Liang, Wang, Tzeng and Nagai, 2010; Sheu, Tzeng, Liang, Wang and Nagai, 2010; Sheu, Wang, Liang, Tzeng and Nagai, 2010). To sum up, it is an effective way to deal with complicated factors and cause-effect analysis (Sheu et al, 2011; Wang et al, 2011a~2011b; Wang et al, 2012c).

For the algorithm of GSP chart formation, they are shown as follows:

Step 1: Construct decision matrix and grey relational construction.

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Step 2: Normalize the data of the decision matrix and follow three principles of establishing the sequences:

1. Non-dimension: the factor of the sequence does not have units;

2. Scaling: the factor of the sequence values should be less than 100;

3. Polarization: the factor of the sequence description should be in the same direction.

Fig. 2.2 Grey S-P chart framework

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shortcomings of the S-P chart, and they are summarized as follows (Wang et al, 2011a~2011b; Wang et al, 2012a~2012c):

1. Through the GRA calculation, the data are in the interval of 0 and 1 in the GSP chart.

2. There is no restriction on the number of students and problems in the GSP chart.

3. The traditional S-P chart fails to compare the students with the same level;

however, the proposed GSP  distribution figure can make up the defect.

The GSP method has also proven to be a reliable and innovative method in some educational publications (please see Wang et al, 2011a~2011b; Wang et al, 2012a~2012c).

For example, based on Table 2.3, the students’ GSP matrix can be established in Table 2.5.

Table 2.5 Example of students’ matrix of Table 2.3 Problems

Students P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

gamma value larger-the-better 1 1 1 1 1 1 1 1 1 1

S1 0 1 1 1 0 1 0 1 1 1 0.225

S2 0 0 1 1 0 1 1 1 0 0 0.000

S3 1 1 1 1 1 1 1 1 1 1 1.000

S4 0 0 1 1 1 1 1 0 1 1 0.225

S5 0 1 1 1 0 1 1 0 0 1 0.106

S6 0 1 1 1 0 1 1 0 1 1 0.225

S7 1 0 1 1 0 1 1 1 0 1 0.225

S8 0 1 1 1 0 1 1 0 1 1 0.225

S9 1 1 0 1 0 1 1 1 1 1 0.368

S10 1 0 1 1 1 1 1 1 1 0 0.368

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After transforming the data in Table 2.5, the matrix of problems can be established in Table 2.6.

Table 2.6 Example of problem’s matrix of Table 2.3 Students

Finally, the GSP chart can be established based on the information in Table 2.5 and Table 2.6, and the example of GSP figure is presented in Fig. 2.3.

Fig. 2.3 The example of GSP

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