The College Completion Puzzle: A Hidden Markov Model Approach

Abstract

Higher education in America is characterized by widespread access to college but low rates of completion, especially among undergraduates at less selective institutions. We analyze longitudinal transcript data to examine processes leading to graduation, using Hidden Markov modeling. We identify several latent states that are associated with patterns of course taking, and show that a trained Hidden Markov model can predict graduation or nongraduation based on only a few semesters of transcript data. We compare this approach to more conventional methods and conclude that certain college-specific processes, associated with graduation, should be analyzed in addition to socio-economic factors. The results from the Hidden Markov trajectories indicate that both graduating and nongraduating students take the more difficult mathematical and technical courses at an equal rate. However, undergraduates who complete their bachelor’s degree within 6 years are more likely to alternate between these semesters with a heavy course load and the less course-intense semesters. The course-taking patterns found among college students also indicate that nongraduates withdraw more often from coursework than average, yet when graduates withdraw, they tend do so in exactly those semesters of the college career in which more difficult courses are taken. These findings, as well as the sequence methodology itself, emphasize the importance of careful course selection and counseling early on in student’s college career.

Appendix

Appendix 1: Trellis diagram of a Hidden Markov Chain

Appendix 2: Building an HMM in Matlab

Notes O = number of categories of all variables together, Q = number of expected states, s_prior0 = random initial distributions, s_transmat0 = random transition probabilities, s_obsmat0 = random observation probabilities, s_prior1 = expected initial distributions, s_transmat1 = expected transition probabilities, s_obsmat1 = expected observation probabilities, LL_S = log-likelihood (of the Graduation-HMM).

Appendix 3: Assessing the HMM

Notes The ‘Start’ and ‘End’ indicate test of nongraduating students (N = 1045). The dmm_logprob function in the HMM toolbox was used to produce two log-likelihoods. One that matches each individual test case with the prior, transition, and observation matrices of the Graduation-HMM ( LL_S ) and one that matches each individual test case with the prior, transition, and observation matrices of the Non-Completion-HMM ( LL_F ). The :,i can be replaced with any selection of length of specific semester transcripts (e.g., semester 1 through 4). Finally, the algorithm classifies by comparing the log-likelihoods LL_F > LL_S

Appendix 4: Encoding and Decoding Transcripts

The input for each student-semester observation ($m_{1 \ldots n}$) is based on a vector ($v_{1 \ldots n}$) that has all feature values encoded using the following algorithm:

The student-semester observations (m…) include categorical and continuous values. Some categorical features include the binary features “did the student take a STEM class?” and “did the student drop any courses?” Continuous features include “number of attempted credits” and “cumulative GPA.” To simplify the modeling process, we represent all features as independent categorical features. The first step in this process is the discretization of continuous features. Each continuous value is represented as a categorical feature as described in Table 1. At this point, each vector (vi) is vector of k categorical features each of which can take one of m(k) values. The second step in the simplification process converts the vectors vi to v’i where v’i is a single categorical variable which can take one of m= \prod_{i=1}^k m^(k) values. This transformation is accomplished by a bijection, f(vi) = v’i. Since the HMM assumes that all elements in the original v vector are independent, no information is lost in via this transformation.

Subsequently, in the analysis phase, the encoded student-semester observation can be decoded through the inverse function f^-1(v’i) = v_i. Since f(v) is bijective, no information is lost in this inversion.