Using the Bernoulli model to analyze the distribution of course withdrawals at UPR-Bayamón

Uso del modelo de Bernoulli para analizar la distribución de bajas parciales por curso en la UPR-Bayamón

One of the most significant challenges that institutions of higher education face is the establishment of a selective admission policy which allows them to identify and admit, from a total pool of applicants, the candidates most able and likely to academically succeed: the greater the institutional economic shortage, the greater the urgency. Suppose that, to reach such an objective, the university administrators consider designing an ideal standardized test. According to Rothstein (2004), the ideal test should be able to predict as accurately as possible which applicants would be most successful if admitted. That is, all applicants whose performance in the test exceeds a determinate threshold after admission would be likely to (a) succeed academically and (b) fulfill all academic requisites during the allotted time. Should this be the case, the institutional admission policy would be quite simple and efficient. However, designing and implementing such a test is extremely difficult, if not impossible. Many diverse factors influence student academic achievement, which are difficult to identify and measure, and most are beyond student and institutional control.

Since the academic year 1979-80, the University of Puerto Rico (UPR) —the country’s public university system composed of eleven campuses across the Island— has adopted as its official admission policy a standardized test administered by the Puerto Rico (PR) and Latin America Office of the College Entrance Examination Board (CEEB), named the General Admission Index (GAI).[1] Every year, each of the UPR’s eleven campuses establishes the minimum GAI required by its different academic programs in response to trends in enrollment demand and the program's capacity.[2] The fact that the GAI required for each program is made public every year has led from its inception to a self-inclusion/exclusion process by which students themselves decide whether to apply to the UPR (and to a particular program), based on their GAI and the minimum established by the program. Students’ admissions to each academic program follow a strict order entirely defined by their GAI. It is expected that these students will be able to sort themselves in such a way that more (less) academically able are admitted to the highest (lowest) selective programs with more (less) inherent difficult content.

Thus, the role of the ideal test described earlier has been ascribed to the GAI. The issue to be settled is whether the GAI satisfies conditions (a) and (b)previously mentioned. Of course, the answer is no. Because of the inaccuracies of GAI, there are several endemic academic problems whose incidence varies among and within the eleven UPR campuses. For instance, to be admitted to the UPR at Bayamón (UPR-Bayamón), many students whose GAI is below the minimum apply to non-desired programs for which they qualify, looking for an eventual possible transfer to their desired program. The strategy consists of enrolling in courses in their desired program even though they are officially admitted to a different one. Because the academic requirements and contents of the programs could differ significantly, the likelihood of course failures and withdrawals increases. Moreover, such a strategy lengthens time until graduation, increasing the opportunity cost of schooling. Eventually, many such students withdraw from the institution because of failure to be admitted to their desired programs. Furthermore, it should be mentioned that these endemic problems are also prevalent even among students who were admitted to their desired programs from the beginning. Therefore, the official admission policy generates undesirable by-products like academic failures, too many applications for program transfers, as well as total and partial withdrawals. Among these problems, this study seeks to analyze the distribution of withdrawals (W) observed in the 39,337 courses offered during 41 consecutive terms (including summer sessions) at UPR-Bayamón from fall 1995-96 to fall 2015-16. To the best of my knowledge, the extant literature lacks studies devoted to analyzing the implications and consequences of the proportion of course withdrawals on the education process; this paper seeks to fill such a gap.

To this end, this paper adopts the Bernoulli probability model and derives the moment-generating functions around its origin and mean. For each course, the objective is to calculate the following four parameters: first, the proportion of withdrawals, which equals the first moment around the origin; second, the variance of its distribution, which is equal to the second moment around the mean; third, the coefficient of skewness, using the third moment around the mean; and fourth, the coefficient of kurtosis, using the fourth moment around the mean. Using different econometric specifications, including random- and fixed-effects models, allows the modeling of these four parameters. This paper uses a rich and detailed panel data containing time-varying variables describing faculty, student, and course characteristics to fulfill this objective.

This study contributes to the extant literature by (a) being the first to analyze in detail the distribution of withdrawals and its key moments at the course level, (b) using a rich panel data comprising all courses offered at UPR-Bayamón during 41 consecutive terms, (c) including time-varying variables describing in detail the faculty characteristics, and showing that courses, faculty and students characteristics exert strong and significant effects on the estimated models, and (d) presenting empirical evidence pointing to the conclusion that faculty and students engage in a symbiotic relationship where both parties improve their well-being at the expense of diminishing academic standards.

The remainder of the paper is organized as follows. Section 2 justifies adopting the Bernoulli model based on its simplicity and statistical properties. Section 3 describes the nature of the data and the specification of the statistical models to be estimated. Section 4 discusses the empirical results, as well as their policy implications. Finally, Section 5 closes the study with a summary and conclusions.

When and why do students decide to withdraw from a course? Although the answer to this question has dramatic policy implications for students and universities since withdrawals entail significant cost consequences, the research published on this topic is limited. Wollman and Lawrence (1984), Adams and Becker (1990), Dunwoody and Frank (1995), and Miller (1997) constitute notable exceptions. However, inspired by the original work of Spady (1970, 1971) and Tinto (1975), there is an extensive and diverse body of published research related to the withdrawals of students from college. This is the first research to analyze the determinants of the distribution of withdrawals and their key moments at the course level.

The study of the distribution of course withdrawals among and within academic programs and across time is relevant for several reasons. Withdrawals can increase student time until graduation and the total cost of the degree. Moreover, they could predict or signal total withdrawals and attrition from college, decreasing college graduation rates. At one time, Dunwoody and Frank (1995) raised the issue that individual course withdrawal could have the highest impact on overall retention, attrition, and institutional success. For some researchers (e.g., Zwick & Sklar, 2005), the best criterion to measure an institution’s academic success is, precisely, the proportion of students who complete their degrees in the allotted time. In this context, low graduation rates negatively impact institutional rankings and, consequently, their ability to attract students with more significant academic potential. Moreover, student attrition represents a fiscal cost to institutions in terms of lost revenues from tuition, room and board, and alum donations (Raisman, 2013; Schuch, 2005). Attrition also constitutes a problem for society in general by reducing the availability of college-educated workers in the labor market (Bound et al., 2007). It also negatively impacts lower tax receipts for federal and state governments (Schneider & Yin, 2011). Although these considerations are beyond the scope of this research, they illustrate how important it is to model the determinants of the distribution of withdrawals at the course level.

From Adams and Becker (1990), it will be hypothesized that students want to maximize their utility function subject to the constraints imposed by their economic and academic environment. Students derive utility from their present and future stream of consumption of the goods and services they will be able to buy in the market as a product of their investment in human capital through education. Education is costly in terms of money and opportunity cost. Therefore, withdrawing from courses would entail a waste of money and increase the opportunity cost of schooling by lengthening the time until graduation. It seems reasonable to posit that the disutility derived by students directly varies with the intrinsic course difficulty level. However, such a concept is relative and unobservable. Thus, it will be assumed that a student will withdraw whenever the disutility (dissatisfaction) derived from the course is greater than the disutility induced by the cost of withdrawing it.

While student disutility or dissatisfaction is not observable, their actions of withdrawing or remaining in the course are. The indicator variable defined in (5) allows us to consider these actions.

Usually, the determination of the number of total courses offered by academic fields (AFs) and the number of students enrolled in each one occurs at the beginning of each term. Likewise, by the end of the term, each academic department head knows with certainty the number of students who withdraw from each course and those who remain in it. Suppose each academic unit adopts a coding system such that code “1” represents the students who withdraw and code “0” represents those who do not withdraw. Thus, expression (1) defines the random variable W.

Formulas

Formulas

Formula 6

.The first moment around the origin is equal to π. This calculation requires taking the first derivative with respect to t (evaluated at t = 0) from the corresponding moment-generating function.[6] The superior moments around the origin are all equal to πThe first moment around the origin is equal to π. This calculation requires taking the first derivative with respect to t (evaluated at t = 0) from the corresponding moment-generating function.6The superior moments around the origin are all equal to π. How-ever, moments around the mean are more interesting. Let μ1,μ2,μ3, and μ4 be the first four moments around the mean. To find them, it is necessary to take the first, second, third, and fourth derivatives with respect to t and to evaluate each one at t = 0; Table 1 reports their values, as well as the coefficients of skewness (CS) and kurtosis (CK). The units of measurement of μ3 and μ4 influence their respective size. Therefore, considered alone, they are poor measures of skewness and kurtosis, respectively. Such dimensionality disappears, defining each coefficient as a relative measure, as done in (

Formula 7

As expected, μ1 = 0. The second moment is equal to the variance. It is an open downward parabola, which reaches its absolute maximum (0.25) at π = 0.5. Obviously, it is zero in the extremes, at

Table 1
Key Parameters of the Bernoulli Probability Model

or at π = 1 (→ –∞ ). It is a positive decreasing convex function (C”S> 0) on π∈(0; 0.5), reaches its inflection point at π = 0.5, and continues decreasing as a negative concave function (C”S<0) on π∈(0.5; 1). The coefficient of kurtosis (CK) is a positive U-shaped convex function symmetric with respect to the line π = 0.5. The coefficient, as well as its first and second derivatives are undefined at points π = 0 and π = 1. The coefficient decreases on π∈(0; 0.5), reaches its absolute minimum equal to 1 on π = 0.5, and increases unbounded on π∈(0.5; 1). As mentioned earlier, π determines, completely and uniquely, the four parameters of interest for this study. Three different points are of key interest in the range of π: π = 0, π = 0.5 and π = 1. There are 11,206 (28.49%) courses where π = 0, and none where π = 1 (see Section 4). For all courses where π = 0, σ2 (W) = 0; however, CS and CK are undefined. In all courses where π = 0.5, CS = 0 , implying a symmetric distribution of withdrawals, while σ2 (W) and CK reach their maximum and minimum values (0.25 and 1, respectively). On the other hand, the distribution of withdrawals is skewed to the left in all courses, where π > 0.5, since CS <0. Thus, once πj is known, it is straightforward to characterize the degrees of skewness and kurtosis prevailing in course j according to these analytical results. Moreover, the mean and variance of CS and CK distributed by specific courses or AFs can be computed and depicted over the 41 terms considered in this study for analytical comparison. Therefore, the simplicity of the Bernoulli model allows us to easily analyze the distribution of course withdrawals looking for policy measures that improve the academic process.

The UPR-Bayamón is an autonomous unit of the UPR system. Accredited by the Middle States Association of Colleges and Secondary Schools, it offers associate and bachelor’s degrees, as well as articulated transfer programs to the Río Piedras, Mayagüez, and Medical Sciences campuses. In the fall of 2024, total enrollment at UPR-Bayamón was 2,852, including 2,520 full-time students.

For each one of the 39,337 courses offered in UPR-Bayamón from 1995 to 2015, the following variables are available: enrollment; instructor who taught the course; letter grade distribution (As, Bs, Cs, Ds, Fs, and W); GPA; variance of the GPA, and AFs (21 dummies). As proxies to account for student quality at the course level, this research uses the mean and variance of the following variables: high school graduation GPA (HS-GPA), GAI, and the score on each of the five sections of the CEEB.[8] Furthermore, the proportions of students by gender and type of high school (public or private) are available for each offered course. Dummies control for academic schedules (weekdays and hours) and for summer terms. For each faculty member in the sample, the following time-varying variables are available: age, rank, degree, and tenure status. Dummies account for the instructor’s gender and the presence of courses subject to student evaluations of teaching (SET). The inclusion of a set of forty-one dummies, identifying term/year, allows for capturing time effects.[9] Table 2 describes the variables used.

The preceding discussion suggests that the model specified in (8) is appropriate to estimate the equations that predict the proportion of withdrawals and the distribution variance, as well as the coefficients of skewness and kurtosis observed in course j, taught by professor f. The matrices Xj,Zf, and Ms, consist of course (j), faculty (f), and student’s (s) characteristics, respective-ly. The vectors β, F,and c represent parameters to be estimated and ɛjf is the composite error term. The inclusion of random- and fixed-effects models allows to account for unobservable faculty heterogeneity (UFH = γf)

Table 2
Sample Statistics

Using a rich panel containing detailed information on the 39,337 courses offered during forty-one consecutive terms, this study analyzed the distribution of course withdrawals and its key moments at the UPR-Bayamón. Overall, the fit of the estimated models is very good. Evidence shows that courses, faculty, and students’ characteristics exert a strong and significant influence on π, σ2 (W), CS and CK. UFH, captured through random- and fixed-effects models, explains a significant proportion of the variation observed around the dependent variable of each estimated model. Empirical evidence does not allow rejection of the symbiotic relationship between faculty members and students, conjectured in the literature, since the estimated coefficients of the SET co-variate were highly significant and exhibited the correct pattern of signs in all models. That is, π and σ2 (W) tend to decrease, while CS and CK tend to increase for the simple reason that the SET was conducted in the course.

A similar result was observed in the case of summer covariate. Its estimated coefficients were highly significant in all estimated models. As discussed previously, π is expected to diminish by 8.8 points if the course is offered during the summer session. However, under the unlikely scenario that the institution would be able to recruit new entrant students with a GAI two standard deviations above the mean (GAI about 333 points), π would decrease by only 3.1 points. Hence, offering a Mathematics course during the summer session would have an expected effect on π equivalent to ad-mitting new entrant students with GAI 5.64 standard deviations above the mean, which is impossible. Therefore, the signs and significance of the coefficients of GAI, SET, summer covariates, and UFH have important implications for the institution’s academic policy mechanism design. Empirical evidence points to the conclusion that at UPR-Bayamón, there exists an environment where faculty members and students engage in a shopping-around process where both parties improve their well-being at the expense of re-ductions in academic standards and the quality of the education provided. Under such a scenario, it might be possible to explain the contradictions observed in the institution where, even though the indicators of student quality are consistently decreasing over time, the GPAs are increasing and π is decreasing simultaneously