An undergraduate student flow model: Semester system in university of Tabuk (KSA)

Introduction

A Markov chain is an important class of stochastic processes in which a future state of an experiment depends only on the present one, not on proceeding states (see Bharucha, 2012) [8] . There are various statistical techniques used for prediction such as time series models, cohort, regression, ratio, Markov chain and simulation. Among these techniques, Markov chain seems to be the most suitable model for this study, because it is a method that not only can estimate promotion and repetition rates, but it can also estimate the number of dropouts, graduates and death rates in the matrix (Johnstone, 1974; Borden & Dalphin, 1998; Armacost & Wilson 2004) [16,9,7] . Markov chain method can also measure detailed information on the students' progress such as the average time student spend in an education system whereby other techniques like regression and ratio are unable to measure this (Kinard & Krech, 1977; Healey & Brown, 1978; Grip & Young, 1999; Guo, 2002) [17,14,12,13] . The use of Markov chain to model and analyze the students flow in the higher education is not new. Reynolds & Porath (2008) [19] studied absorbing Markov chain with four transient and one absorbing states to model the academic progress of students attending the University of Wisconsin-Eau Claire over a specific length of time. Al-Awadhi & Konsowa (2007) [5] and Al-Awadhi & Konsowa (2010) [6] have modelled student flow in a high educational institution at Kuwait University by a finite Markov chain with eight states and with five transient and two absorbing states. Brezavšček & Baggia (2015) [10] and Brezavšček & Baggia (2017) [11] have built a model of student flow at Slovenian universities by a reducible discrete Markov chain with five transient and two absorbing states. Shah and Burke (1999) [20] used Markov chain to model the movement of undergraduates through the higher education system in Australia with 51 transient and two absorbing states. Rahel et al. (2013) [18] have developed an enrolment projection model based on the Markov chain for postgraduate students at University Utara Malaysia classified students by age, field of study, students' status whether they have graduated, dropped out, or deferred from their programs. Hlavatý & Dömeová (2014) [15] have used Markov chain to create a model of students' progress throughout the whole courses at the Czech university of life sciences (CULS) in Prague with four transient and four absorbing states. Very interesting and useful are the studies, which modelled the students' progression and their performance during higher education study using an absorbing Markov chain (see e.g., Adam, 2015; [2,3,4,1,21] . This paper provides a model, which can be used for analyzing the undergraduate students flow at University of Tabukfaculty of Science (KSA). The proposed model is built by a reducible discrete Markov chain with eight transient and three absorbing states. The eight transient states represent the eight stages (semesters) student should move around until graduate, while the three absorbing states are graduation, withdrawal from specific semester and apologized for the study program. The rest of the paper is outlined as follows. In Section 2, we describe states of the Markov chain. Mathematical model is demonstrated in Section 3. Numerical example is given in Section 4. Section 5 pertains to the results and section 6 provides the conclusions of the study.

Definition of states of the Markov Chain

The duration of bachelors' degree within University of Tabuk is four years divided into eight semesters. Therefore, to model the student academic progress we define the following states:

To develop the model, the following assumptions are considered:  Student who is currently enrolled into the first, second, third, fourth, fifth, sixth or seventh semester of the study program can next semester either progress to a higher level or repeat a semester (remaining at the same level).  Student who is currently enrolled into the eighth semester of the study program can be either remain into the current semester, or can graduate and finish the study program.  Student who has withdrawn from specific semester will never join this semester unless takes the acceptance of the academic manager.  Student who has apologized will leave this study program.

Mathematical model

The general form of the Markov chain model is given by Where 𝑛(𝑡) is the column vector whose 𝑖 th element represents the number of students in state 𝑖 at time 𝑡. 𝑍 (𝑡) is the square matrix whose 𝑖𝑗 th element represents the number of students moving from state 𝑖 at time 𝑡 to state 𝑗 at time 𝑡 + 1, 𝐼 is the column vector of ones, and 𝐻(𝑡) is the number of students moving from transient state 𝑖 at time 𝑡 to absorbing state 𝑗 at time 𝑡 + 1.

Eq. ( 1) refers that the number of students at the beginning of semesters consists of those who will survive to the next semester and those who will leave the program in that semester.

The general form of the probability transition matrix of an absorbing Markov chain with r absorbing and 𝑡 transient states is

Where 𝑄 is a square matrix expressing transitions between the transient states.

𝑅 is a matrix expressing transitions from the transient states to the absorbing states. 0 is a zero matrix 𝐼 is an identity matrix Base on the formulation in Shah & Burke (1999) [20] and Rahel et al. (2013) [18] , the transition matrix (𝑄) and the absorbing matrix (𝑅) are used to calculate the estimated average time student spend in the system and estimate the probability a student completing a course as follows, The matrix of transition probabilities given as where 𝑄(𝑡) is square matrix whose 𝑖𝑗 th element represents the probability of student moving from state 𝑖 at time 𝑡 to state 𝑗 at time 𝑡 + 1 and 𝑛 ̂(𝑡) is a diagonal matrix whose elements are the elements of 𝑛(𝑡).

The matrix of absorption probabilities is given by:

Where 𝑅(𝑡) is matrix whose 𝑖𝑗 th element represents the probability of a student in state 𝑖 at time 𝑡 departing into an absorbing state 𝑗 at time 𝑡 + 1.

The fundamental matrix 𝑁 of an absorbing Markov chain plays an important role in the assessment of the student completion attributes, and it defined as

Where 𝑁 is a square matrix whose 𝑖𝑗th element represents the average time (in 6 month (semester)) that a student who commenced in state 𝑖, remains in state 𝑗 before departing the program and 𝐼 is the identity matrix.

The sum of the entries in the diagonal of 𝑁 represents the expected time student enrolled in the first semester can expect to spend before graduating. That is,

The average time a student spends in the program is calculated by, Where 𝜇 is the mean time a student remains in the study program given that he commenced in state 𝑖.

The probability of a student moving into an absorbing state 𝑗, given that he commenced in transient state 𝑖, is given by the 𝑖𝑗 th element of the matrix Formula below can be used to predict the future enrolment of the student

Where, 𝑝 (0) represents the initial probability distribution, 𝑃 𝑛 is the transition matrix 𝑃 after 𝑛 academic years and 𝑝 (𝑛) is the probability distribution after 𝑛 academic years. The state transition diagram for the students' progression is illustrated in Figure 1.

Numerical example

To apply the model, data were obtained from the students' intake records at University of Tabuk-Faculty of Science (KSA). In our analysis, only the full time bachelor degree students were included. The frequency data during five consecutive academic years from 1428/29 to 1432/33 Hijri calender are presented in Table 1. The elements of each fundamental matrix 𝑁 𝑖 represent the expected number of semesters student will spend when he is enrolled in a particular stage of the study program. For example, let we assume student is enrolled in the first semester of academic year 1428/29, it is expected that he will spend 1.0085 semesters (6 months and 1 day) for the first level, 1 semester (6 months) for the second level, 0.9830 semesters (5 months and 27 days) for the third level, 1.0589 semesters (6 months and 10 days) for the fourth level, 0.9978 semesters (5 months and 29 days) for the fifth level, 0.8930 semesters (5 months and 10 days) for the sixth level, 0.8375 semesters (5 months) for the seventh level, and 0.7861 semesters (4 months and 21 days) for the eight level. The mean time 𝜇 i until absorption of Eq. ( 7) and the probability of absorption 𝐵 i of Eq.( 8) for each academic year are respectively given as:

Results

Moving between different semesters estimation

Table 2 gives the probabilities of students' progression between successive semesters of the study program for each academic year, which can be directly obtained from the probability transition matrices 𝑃 𝑖 𝑖 = 1,2, … ,5. and 93.89% of students graduated. The last row represents the probabilities averages of the students moved successfully between semesters until graduate for the five academic years.

Figure 1 shows probabilities of the students moved successfully between semesters until graduate for academic year 1428/29 and the averages of overall five academic years. Expected time spend before graduation Table 3 represents the expected time (semesters) student enrolled in the first semester can expect to spend before graduating for each academic year, which has been calculated according to Eq.( 6) From Table 3 we note that, regarding the academic year 1429/30, the expected time student spend until graduate is 8.4718 semesters (4 years, 2 months and 24 days) greater than other academic years, while student of the academic year 1431/32 spend 8.2166 semesters (4 years, 1 month and 8 days) less than student of the rest academic years. Moreover, student of the two last academic years 1431/32 and 1432/33 spend time less than the average time of overall five academic years (see Figure 2).

Probabilities of withdrawal and apology

Table 4 shows the probabilities of student withdraw from a particular semester and apologize of study program for each academic year. Note that the last two column give the averages of the five academic years. From Table 4 it can be seen that the probability of withdrawal and apology decrease as the students' progress to higher levels. This may be the result of the fact that they understand the system better as they pass from one level to another (see also Figure 3).

Predicting the future enrolment of students

To predict the future probability of student repeat specific semester, withdraw, apologize or graduate, let us assume that the initial state is in the last academic year 1432/33. So, form in Table 1 the total number of students who repeated each semester, graduated, withdrawn or apologized from study program of academic year 1432/33 are (1, 3, 9, 29, 13, 6, 14, 5, 173, 92, 15) which used to estimate the initial vector 𝑃 (0) , Using Eq.( 9) we can predict the future enrolment of the students in next 4 academic years 1434/35, 1435/36, 1436/37 and 1437/38. The results are given in Table 5.

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