29 | J o u r n a l f o r E c o n o m i c E d u c a t o r s , 1 7 ( 2 ) , 2 0 1 7 29 Economics and Finance in the MBA Core: Sequence of Core Course Completion and Student Performance E. Anthon Eff and Christopher C. Klein 1 Abstract Does the order in which MBA students complete the core courses for the degree affect their performance in the program? There is little literature on this topic. To gain insight into this question, we examine 5,822 student/course records from a large public university for the years 2008-2012, encompassing part of the academic history of 1,384 MBA students. Standardized grades are our measure of performance in each of nine core courses. For each core course, both t-tests of differences in mean grade across different combinations of prior courses and fixed- effects OLS regressions indicate that the sequence of completing courses affects student performance. The Economics core course informs Accounting and Marketing, for example, while the core Finance course interacts with Management courses. While the data limit the strength of any conclusions, we find good indications for placing information systems and quantitative methods courses earlier in the core sequence. Key words: MBA core, course sequencing, student performance JEL: A12, A23, M10 Introduction To determine the effect of MBA core course sequencing on MBA student performance, we examine 5,822 records of student performance in each of nine MBA core courses at Middle Tennessee State University, a large AACSB accredited regional university, during the Spring 2008 to Spring 2012 period. This includes partial performance records of over 1,384 individual MBA students. Student performance is defined as the standardized grade a student earns in a course, where the raw grade is standardized by subtracting from the mean grade given by that student’s instructor in all sections of that particular core course. Two different methods are employed to test for the effect of course sequence. The first uses t-tests to compare the mean grade earned in course B when course A is taken first against the mean grade earned in course B when course B is taken first. The second runs OLS regressions of course B grades against dummy variables indicating the other core courses taken prior to course B. The second method allows for control of other student characteristics, so that the effect on course B’s grade is more likely due to course sequence than to differences in the kinds of students that choose a particular sequence. The composite results of the two methods reveal some surprises. Information Systems Management is a valuable precursor to five other core courses directly, and indirectly to two more, while there are no core courses that positively affect it. The courses that benefit the most from other core courses are Managerial Economics (five) and Financial Analysis (four), followed 1 Professors of Economics, Economics and Finance Department, Middle Tennessee State University, Murfreesboro, TN. Anthon.Eff@mtsu.edu, Chris.Klein@mtsu.edu. This research was supported by a summer research grant from the Jones College of Business at Middle Tennessee State University. mailto:Anthon.Eff@mtsu.edu mailto:Chris.Klein@mtsu.edu 30 | J o u r n a l f o r E c o n o m i c E d u c a t o r s , 1 7 ( 2 ) , 2 0 1 7 30 by Marketing Management and Accounting for Business Decisions (three each). Oddly, the capstone Strategic Management course benefits from only two other core courses, one course directly and another indirectly. We present both our recommendations for sequencing as well as a comparison of the optimal sequence to the actual sequences in which students took courses during the period. The paper is organized as follows. A literature review follows the introduction, then the data collection is discussed. A detailed discussion of the methodology and results is followed by a summary and conclusion. Literature Little literature relates MBA course sequencing to student performance in MBA programs. Several articles address the effect of prerequisites on performance in individual courses (MacMillan-Capehart and Adeyemi-Bello, 2008), the determinants of student success in individual courses (Krausz, et al. 2002), and the need for individual courses and their prerequisites in MBA programs (Kolluri and Singamsett, 2007). Others examine MBA admission elements and find (Christensen, Nance and White, 2012; Malik, 2011; Sulaiman and Mohezar, 2006; Truell, Zhao, Alexander and Hill, 2006) that undergraduate grade point average (GPA) is the best predictor of a student’s MBA GPA, while GMAT quantitative score, performance in undergraduate composition courses, and certain undergraduate majors (especially economics and business statistics) positively influence MBA performance. The issue of overall sequencing of the MBA core courses and any possible effects on student performance due to differences in sequencing has not been addressed previously. Nevertheless, Dailey (2011) points out that flexibility in MBA course sequencing – interpreted as lack of prerequisites or other sequencing requirements in the core - can be a marketing advantage for MBA programs, especially part-time programs. An examination of websites for competing MBA programs in Tennessee reveals a range of sequencing requirements from minimal prerequisites (MTSU, University of Memphis) to limited prerequisites (ETSU, Belmont and Lipscomb Universities) to strict sequencing (UT-Knoxville, Vanderbilt University). This is mirrored in MBA programs at institutions in neighboring states such as the University of Alabama – Tuscaloosa (minimal) and the University of Kentucky (strict sequencing). Thus, the degree to which core courses are required to be completed in sequence appears to be one component of an MBA program’s product niche. All of this taken together suggests that MBA core sequencing requirements should not be adopted solely on the basis of student performance as measured below. Sequencing requirements are part of an MBA program’s product niche and should be adopted to serve consumers to whom the chosen product niche appeals. Any choice of sequencing requirements is likely to involve trade-offs among attraction of students to the program, student performance in specific courses, and the ability of students to complete the program. Data We obtained data comprising all students taking courses in the Jones College of Business at Middle Tennessee State University from Spring 2008 to Spring 2012. Each record contains an instructor ID, days and times at which the course met, basic demographic information on the student (birthdate, ethnicity, sex), information on previous college and high schools attended (including GPA), and test scores (ACT, SAT, GMAT, GRE). 31 | J o u r n a l f o r E c o n o m i c E d u c a t o r s , 1 7 ( 2 ) , 2 0 1 7 31 Not all of the data were usable. Among the problems were students with several different assignments of ethnicity or sex, birthdates after 2011, or extreme outliers for high school GPAs or test scores. We deleted all records for students with obviously bad data. We concluded that the most reliable variables were those that appear on student transcripts: semester, course, and grade. Our final dataset contains 124,840 records, with each record containing the grade for a specific course taken by a specific student in a specific semester. We extracted all records in which the course taken was one of nine core MBA courses. Since our main interest is in course sequencing, we dropped all students who took a particular course more than once. The result was 5,506 records that represent a portion of the academic history of 1,350 students. This dataset contains relatively few complete MBA student histories: only 40 students took all nine core courses during the sample period. Another 235 students took eight core classes, but many of these may be students with Accounting or Information Systems undergraduate degrees who are required to take only eight of the nine core classes. Most of the students either began taking courses before our sample period, or were still unfinished at the end. The short span of the sample period limits our analysis to semester-to-semester comparisons. Table 1: Frequency of core courses in dataset Course Description incidents ACTG.6910 Accounting and Business Decisions 207 ACTG.6920 Financial Statement Analysis 366 BUAD.6980 Strategic Management 550 ECON.6000 Managerial Economics 706 FIN.6710 Financial Analysis 594 INFS.6610 Information Systems Management and Applications 660 MGMT.6600 Study of Organizations 765 MGMT.6650 Operations Management 672 MKT.6800 Marketing Management 763 QM.6770 Computer-Based Decision Modeling 223 Total 5506 Table 1 shows the frequency with which each core course occurs in the dataset. There are ten courses listed, because students can take either of the two accounting courses. The low frequency for the QM 6770 course is due to its relatively recent addition in the core. Methods Our objective is to determine whether taking course A before course B will improve student performance in course B. We define student performance as the grade a student earns in course B, where that grade is standardized by subtracting from it the mean grade given by that student’s instructor in all sections of that particular core course in that particular semester. This controls for differences in grades across instructors. We use two different methods to investigate this question. The first employs t-tests to compare the mean grade earned in course B when course A is taken first with the mean grade earned in course B when course B is taken first or when A and B are taken at the same time. The second method uses OLS to estimate a model with course B grades as the dependent variable and dummy variables indicating which other core courses were taken prior to course B. The advantage of the second method is that it allows us to control for other characteristics of students, so that we can be more certain that the effect on course B’s grade is due to course 32 | J o u r n a l f o r E c o n o m i c E d u c a t o r s , 1 7 ( 2 ) , 2 0 1 7 32 sequence, and not simply to differences in the kinds of students that choose a particular course sequence. Table 2 reports the information used to calculate the t-statistics that underlie Figure 1. The top matrix gives the number of incidents in which a student takes the row course before the column course. The second matrix reports the mean grade in the column course for those taking the row course first, minus the mean grade in the column course for those not taking the row course first. The third matrix presents a p-value for a t-test applied to each group, with the null hypothesis that taking the row course before the column course will result in the same grade in the column course as not taking the row course first. We use the Welch t-statistic, for comparing two means with different variances (Welch 1947), which requires degrees of freedom produced using the Welch-Satterthwaite equation (Satterthwaite 1946). The last matrix is an adjacency matrix, where the non-zero entries represent cells for which the p-values are less than 0.1: “1” indicates that taking the row course first improves the column course grade; “-1” indicates that taking the row course first lowers the column course grade. 33 | J o u r n a l f o r E c o n o m i c E d u c a t o r s , 1 7 ( 2 ) , 2 0 1 7 33 Table 2: Data and results for t-tests course description ACTG.6910 ACTG.6920 ECON.6000 FIN.6710 INFS.6610 MGMT.6600 MGMT.6650 MKT.6800 QM.6770 BUAD.6980 Number of incidents of students taking row course before column course ACTG.6910 Accounting and Business Decisions 0 4 70 98 71 63 86 78 13 111 ACTG.6920 Financial Statement Analysis 1 0 112 160 97 100 137 97 49 229 ECON.6000 Managerial Economics 44 130 0 257 192 179 264 199 66 393 FIN.6710 Financial Analysis 30 76 135 0 122 128 177 127 52 382 INFS.6610 Information Systems Management and Applications 31 88 146 168 0 147 188 145 51 330 MGMT.6600 Study of Organizations 59 133 242 271 229 0 317 217 70 405 MGMT.6650 Operations Management 42 82 164 200 150 123 0 134 57 382 MKT.6800 Marketing Management 56 134 241 287 226 204 303 0 79 388 QM.6770 Computer-Based Decision Modeling 10 14 33 28 11 27 32 28 0 46 BUAD.6980 Strategic Management 3 2 4 3 1 1 2 0 1 0 Difference in mean standardized grades (those taking row course first minus all others) ACTG.6910 Accounting and Business Decisions - 0.0132 0.0614 0.0159 -0.0445 0.0445 0.0591 -0.0503 0.1252 0.3701 ACTG.6920 Financial Statement Analysis 1.7607 - -0.0149 0.0057 -0.1582 -0.1047 -0.0070 0.0352 -0.0721 -0.1003 ECON.6000 Managerial Economics 0.3232 0.0176 - 0.0061 -0.0229 -0.0089 -0.0326 0.0585 -0.0050 0.1730 FIN.6710 Financial Analysis 0.1442 0.0470 0.0587 - 0.0556 0.0777 0.0993 0.0016 0.1019 0.1073 INFS.6610 Information Systems Management and Applications 0.2985 0.0918 0.0983 0.0354 - 0.0525 -0.0284 0.0913 -0.1554 0.1344 MGMT.6600 Study of Organizations -0.0939 0.0312 -0.0827 -0.0348 -0.0005 - -0.0014 0.0755 -0.0461 0.0366 MGMT.6650 Operations Management 0.1827 0.0023 -0.0119 0.1467 -0.0711 0.0045 - 0.0631 -0.1800 -0.1083 MKT.6800 Marketing Management 0.1171 -0.0016 0.0021 0.0520 -0.0624 -0.0837 0.0249 - 0.1367 -0.1071 QM.6770 Computer-Based Decision Modeling -0.0323 0.0731 0.0716 0.0136 0.0487 -0.0892 0.1477 -0.0007 - 0.0585 BUAD.6980 Strategic Management -0.5989 -0.2013 -0.2740 -0.1465 -0.7048 -0.8730 -0.4352 - 0.0720 - P-values for Welch’s t-test (H0: Mean of row before column no different from all others) ACTG.6910 Accounting and Business Decisions - 0.4847 0.2445 0.4452 0.2828 0.3116 0.2081 0.2271 0.2792 0.0098 ACTG.6920 Financial Statement Analysis - - 0.4122 0.4681 0.0088 0.0716 0.4481 0.2823 0.3588 0.2317 ECON.6000 Managerial Economics 0.0089 0.3399 - 0.4492 0.3290 0.4206 0.1720 0.0449 0.4840 0.0402 FIN.6710 Financial Analysis 0.1370 0.1755 0.1273 - 0.1736 0.0498 0.0060 0.4851 0.2306 0.0610 INFS.6610 Information Systems Management and Applications 0.0160 0.0261 0.0430 0.2440 - 0.1531 0.2339 0.0146 0.1328 0.0955 MGMT.6600 Study of Organizations 0.2438 0.2826 0.0420 0.2540 0.4954 - 0.4880 0.0362 0.3630 0.3522 MGMT.6650 Operations Management 0.0863 0.4793 0.4120 0.0036 0.1037 0.4626 - 0.0741 0.0726 0.0530 MKT.6800 Marketing Management 0.1947 0.4873 0.4836 0.1749 0.1011 0.0560 0.2789 - 0.1514 0.1007 QM.6770 Computer-Based Decision Modeling 0.4709 0.2567 0.2984 0.4546 0.3440 0.2764 0.0141 0.4969 - 0.3241 BUAD.6980 Strategic Management 0.0356 0.3445 0.0464 0.3276 - - 0.2185 - - - Adjacency Matrix (1==first taking row course improves grade in column course; -1=first taking row course harms grade in column course) ACTG.6910 Accounting and Business Decisions 0 0 0 0 0 0 0 0 0 1 ACTG.6920 Financial Statement Analysis 0 0 0 0 -1 -1 0 0 0 0 ECON.6000 Managerial Economics 1 0 0 0 0 0 0 1 0 1 FIN.6710 Financial Analysis 0 0 0 0 0 1 1 0 0 1 INFS.6610 Information Systems Management and Applications 1 1 1 0 0 0 0 1 0 1 MGMT.6600 Study of Organizations 0 0 -1 0 0 0 0 1 0 0 MGMT.6650 Operations Management 1 0 0 1 0 0 0 1 -1 -1 MKT.6800 Marketing Management 0 0 0 0 0 -1 0 0 0 0 QM.6770 Computer-Based Decision Modeling 0 0 0 0 0 0 1 0 0 0 BUAD.6980 Strategic Management -1 0 -1 0 0 0 0 0 0 0 34 | J o u r n a l f o r E c o n o m i c E d u c a t o r s , 1 7 ( 2 ) , 2 0 1 7 34 Notes to Table 2: Standardized grades are grades received by students minus the mean grade given by that particular instructor in that particular course in that particular semester. The t-test is the Welch t-test for comparing two means with different variances (Welch 1947). The adjacency matrix takes on values of 1 when the row course, taken first, significantly improves the grade in the column course; it takes on values of -1 when it significantly lowers the column course grade. This adjacency matrix is presented graphically in Figure 1, considering only the positive values that represent grade improvement. Figure 1. Graph derived from t-test adjacency matrix (Table 2). Arrows point from the course occurring first to course receiving the grade improvement. Larger vertices are more “reachable”, while smaller vertices do more “reaching”. Table 3 reports descriptive statistics for variables used in regressions. We used the demographic controls age, sex, and ethnicity, despite our awareness that these data may not be accurate. The test score variable (“verbal” and its square) were constructed by standardizing (subtract mean, divide by standard deviation) each verbal test score across the entire sample of students (including undergraduates) and then taking the mean of the verbal and quantitative scores for each student. This procedure allowed us to avoid the problem of missing values in the GMAT scores. We also introduced a variable to capture whether students perform better in courses more aligned with their interests (“ownmaj”), as well as a dummy to indicate an online course. The variables of interest are the dummy variables indicating whether a specific core course was taken prior to the course represented in the observation. 35 | J o u r n a l f o r E c o n o m i c E d u c a t o r s , 1 7 ( 2 ) , 2 0 1 7 35 Table 3: Variables used in OLS variable description n mean sd min max grnorm Standardized course grade 5506 0.009 -3.846 1.567 0.589 eA Asian 5506 0.092 0 1 0.288 eB Black 5506 0.128 0 1 0.334 eW White 5506 0.679 0 1 0.467 male male 5506 0.609 0 1 0.488 age age 5506 27.477 20 60 5.984 age2 age squared 5506 790.809 400 3600 399.498 verbal composite verbal score 5197 -0.009 -2.084 2.721 0.959 verbal2 composite verbal score squared 5197 0.92 0 7.403 1.115 ownmaj course is in students major 5506 0.109 0 1 0.312 online online course 5506 0.173 0 1 0.378 ACTG.6910 Accounting and Business Decisions 5506 0.108 0 1 0.31 ECON.6000 Managerial Economics 5506 0.313 0 1 0.464 FIN.6710 Financial Analysis 5506 0.223 0 1 0.416 INFS.6610 Information Systems Management and Applications 5506 0.235 0 1 0.424 MGMT.6600 Study of Organizations 5506 0.353 0 1 0.478 MGMT.6650 Operations Management 5506 0.242 0 1 0.429 MKT.6800 Marketing Management 5506 0.348 0 1 0.476 QM.6770 Computer-Based Decision Modeling 5506 0.042 0 1 0.2 BUAD.6980 Strategic Management 5506 0.003 0 1 0.055 Table 4 reports the regression results underlying Figure 2. For each regression, the dependent variable is the grade earned in a particular course. In the top part of Table 4, columns give the names of the courses providing the dependent variable. The rows represent independent variables and cells show the standardized coefficients of independent variables that survived a stepwise procedure. Standardization allows comparison of effect sizes across variables, since the coefficient represents the number of standard deviations the dependent variable will change in response to a one standard deviation chance in the independent variable. Standard errors were obtained via bootstrapping (500 repetitions). The second panel in Table 4 partitions R2 (Chevan and Sutherland 1991; Gromping 2006) across four different groups of independent variables. The variables composing each group are identified in the first column of the first panel. Course sequence accounts for a respectable share of variation in the dependent variable, sometimes exceeding the variation attributed to individual demographic or academic characteristics. Two of the models (ACTG.6920 and INFS.6610) had relatively poor fits with the existing data, but the others performed well on the diagnostics (shown in the third panel of Table 4). Note that in all cases except for course INFS 6610, we can reject the null hypothesis that taking preceding courses has no effect on course grade. At the bottom of Table 4 is an adjacency matrix giving the signs for significant (p-value ≤ 0.10) coefficients for preceding courses. Note that in six cases, taking course A before course B actually did damage, leading to a lower grade. Figure 2 shows a graphical representation of the adjacency matrix, omitting the negative entries. The two methods provide slightly different results, so we take the intersection of the adjacency matrices in Tables 2 and 4 to create a composite adjacency matrix, dropping the negative entries. This provides a relatively robust view of sequencing effects. Figure 3 shows a graphical representation of the composite adjacency matrix. 36 | J o u r n a l f o r E c o n o m i c E d u c a t o r s , 1 7 ( 2 ) , 2 0 1 7 36 Table 4: Results for regression on grades for each of the core courses Dependent variables: Standardized grades for the course group variable description ACTG.6910 ACTG.6920 ECON.6000 FIN.6710 INFS.6610 MGMT.6600 MGMT.6650 MKT.6800 QM.6770 BUAD.6980 IDC eA Asian 0.1394*** -0.0707 IDC eB Black -0.2380*** -0.0930** -0.0910** -0.1329*** -0.1656*** -0.1755*** -0.1637*** IDC eW White 0.1627** -0.1277** IDC male male 0.0907 0.0548 -0.0783* -0.0531 -0.0833* IDC age age 0.3404 -0.3692 0.0069 -0.2992 0.0410 -0.0517 -0.2149 0.1508 -0.0638 -0.0469 IDC age2 age squared -0.1423 0.4272 0.0180 0.3254 0.0586 0.0522 0.2219 -0.1427 -0.065 0.0598 IAC verbal composite verbal score 0.2202*** -0.0167 0.1666*** 0.0482 0.1213*** 0.0077 0.0037 0.1221*** 0.2072*** 0.1795*** IAC verbal2 composite verbal score squared 0.0213 -0.0683 0.1351*** 0.0861** -0.0445 0.0185 0.0197 -0.0546* 0.04 -0.0176 OCC ownmaj course is in students major 0.1169*** OCC online online course -0.1296 S ACTG.6910 Accounting and Business Decisions -0.0591* S ECON.6000 Magerial Economics 0.1278* 0.0796** S FIN.6710 Fincial Alysis 0.0674** 0.0890*** 0.1154*** S INFS.6610 Information Systems Magement and Applications 0.1825** 0.1101** 0.0571 0.0693** -0.1365* -0.0748 S MGMT.6600 Study of Organizations -0.1834** 0.0782* S MGMT.6650 Operations Magement 0.1066*** -0.0688 -0.1374* S MKT.6800 Marketing Magement -0.0947** 0.2371*** S QM.6770 Computer-Based Decision Modeling 0.1143 0.0635** 0.0767*** 0.0542* S BUAD.6980 Strategic Magement -0.0580*** Decomposition of R2 ACTG.6910 ACTG.6920 ECON.6000 FIN.6710 INFS.6610 MGMT.6600 MGMT.6650 MKT.6800 QM.6770 BUAD.6980 S sequencing 0.0717 0.0111 0.0077 0.0114 0.0044 0.0175 0.0190 0.0198 0.0463 0.0067 OCC other course characteristics 0.0000 0.0000 0.0121 0.0000 0.0000 0.0000 0.0000 0.0000 0.0139 0.0000 IAC individual academic characteristics 0.0615 0.0039 0.0511 0.0131 0.0184 0.0008 0.0008 0.0183 0.0404 0.0336 IDC individual demographic characteristics 0.0873 0.0122 0.0441 0.0358 0.0253 0.0193 0.0285 0.0323 0.0150 0.0311 Model diagnostics ACTG.6910 ACTG.6920 ECON.6000 FIN.6710 INFS.6610 MGMT.6600 MGMT.6650 MKT.6800 QM.6770 BUAD.6980 N Number of observations 197 345 657 560 633 725 634 720 207 519 R2 Model R2 0.2200 0.0270 0.1150 0.0600 0.0480 0.0380 0.0480 0.0700 0.1150 0.0710 RESET pval H0: model correct functional form 0.8812 0.2976 0.0649 0.2186 0.0287 0.1676 0.4900 0.5230 0.4067 0.1957 model F pval H0: none of the independent variables significant 0.0000 0.1537 0.0000 0.0000 0.0001 0.0021 0.0001 0.0000 0.0018 0.0000 restr F pval H0: dropped variables not significant 0.7912 0.9972 0.5235 0.6078 0.0003 0.5449 0.5606 0.6541 0.7231 0.9985 seqDummies F pval H0: course sequence not significant 0.0042 0.0308 0.0074 0.0059 0.1215 0.0000 0.0000 0.0000 0.0081 0.1366 Adjacency Matrix ACTG.6910 ACTG.6920 ECON.6000 FIN.6710 INFS.6610 MGMT.6600 MGMT.6650 MKT.6800 QM.6770 BUAD.6980 ACTG.6910 Accounting and Business Decisions 0 0 0 0 0 0 0 -1 0 0 ACTG.6920 Financial Statement Analysis 0 0 0 0 0 0 0 0 0 0 ECON.6000 Managerial Economics 1 0 0 0 0 0 0 1 0 0 FIN.6710 Financial Analysis 0 0 1 0 0 1 1 0 0 0 INFS.6610 Information Systems Management and Applications 1 1 0 0 0 0 0 1 -1 0 MGMT.6600 Study of Organizations -1 0 0 0 0 0 0 0 0 1 MGMT.6650 Operations Management 0 0 0 1 0 0 0 0 -1 0 MKT.6800 Marketing Management 0 0 0 0 0 -1 0 0 1 0 QM.6770 Computer-Based Decision Modeling 0 0 1 0 0 0 1 1 0 0 BUAD.6980 Strategic Management 0 0 0 0 0 -1 0 0 0 0 37 | J o u r n a l f o r E c o n o m i c E d u c a t o r s , 1 7 ( 2 ) , 2 0 1 7 37 Notes to Table 4: Stepwise regression, showing standardized coefficient for final restricted model. Bootstrapping used to produce standard errors (* p-value≤ 0.1; ** p-value≤ 0.05; *** p-value≤ 0.01). Adjacency matrix shows sign of all course dummy coefficients with p-value≤ 0.1. Figure 2. Adjacency matrix from dummies in regressions. Arrows point from the course occurring first to course receiving the grade improvement. Larger vertices are more “reachable”, while smaller vertices do more “reaching”. Figure 3: Composite optimal sequences, based on the intersection of graphs in Figure 1 and Figure 2. Larger vertices are more “reachable”, while smaller vertices do more “reaching”. Table 5 shows the shortest path lengths between vertices in Figure 3. Some courses are clearly more co nnected than others: MGMT 6600, for example, receives links directly from 1 course and indirectly from two others. 38 | J o u r n a l f o r E c o n o m i c E d u c a t o r s , 1 7 ( 2 ) , 2 0 1 7 38 Table 5: Shortest path length between vertices in composite graph shown in Figure 3. course description ACTG.6910 ACTG.6920 ECON.6000 FIN.6710 INFS.6610 MGMT.6600 MGMT.6650 MKT.6800 QM.6770 BUAD.6980 ACTG.6910 Accounting and Business Decisions 0 0 0 0 0 0 0 0 0 0 ACTG.6920 Financial Statement Analysis 0 0 0 0 0 0 0 0 0 0 ECON.6000 Managerial Economics 1 0 0 0 0 0 0 1 0 0 FIN.6710 Financial Analysis 0 0 0 0 0 1 1 0 0 0 INFS.6610 Information Systems Management and Applications 1 1 0 0 0 0 0 1 0 0 MGMT.6600 Study of Organizations 0 0 0 0 0 0 0 0 0 0 MGMT.6650 Operations Management 0 0 0 1 0 2 0 0 0 0 MKT.6800 Marketing Management 0 0 0 0 0 0 0 0 0 0 QM.6770 Computer-Based Decision Modeling 0 0 0 2 0 3 1 0 0 0 BUAD.6980 Strategic Management 0 0 0 0 0 0 0 0 0 0 In the jargon of graph theory, MGMT 6600 is reachable by 3 other courses. Since we interpret each directed edge as indicating knowledge from the origin course is important for learning in the destination course, then a vertex that is highly reachable is one which builds on much prerequisite knowledge. Other courses are not reachable, but themselves reach, as for example INFS 6610, which sends links directly to 3 courses. Reachability provides one way to think about optimal course sequencing: highly reachable courses, which don’t reach much or at all, should come last in the course sequence. One way to operationalize this is the following: reachratio = ∑j xij - ∑i xij (1) where X is a square matrix, where xij =1 indicates that a path connects vertex i to vertex j, and xij =0 indicates that vertex i cannot reach vertex j. In other words, reachratio is simply the column sums minus the row sums. Figure 3 shows that, in this most robust version, no courses reach the capstone course BUAD 6980. It is odd that material from earlier courses provides no advantage to students taking the capstone. There are two possible reasons for this missing connection: first, the course content of the capstone might fail to utilize material from earlier courses; second, the earlier courses might fail to provide students with the tools needed to suceed in the capstone. Both explanations suggest a failure of MBA faculty to interact each other in integrating their courses into a coherent cumulative learning process. Actual Course Sequencing There is no single course sequence adhered to by most students at this institution, but some course sequences are more common than others. The first complication is that students take multiple courses each semester, so that a precise sequence from one course to an other is never possible. Table 6 shows that about one out of three faces in an MBA class belongs to a student who is taking at least three core courses simultaneously in the same semester. 39 | J o u r n a l f o r E c o n o m i c E d u c a t o r s , 1 7 ( 2 ) , 2 0 1 7 39 Table 6: Number of core courses taken in same semester Number of core courses taken in semester Number of occurances Number of course seats Percent of course seats 1 1960 1960 36% 2 1105 2210 40% 3 359 1077 20% 4 61 244 4% 5 3 15 0% Total 3488 5506 100% Table 7 lists the courses most commonly taken together—all dyads and the most common triads. The lists are sorted, so that the most frequently associated courses are at the top. When at least two of the associated courses are linked in Figure 3, the columns labeled “opt” indicate that relationship with the numeral “1”. One can see that many of the courses taken together are identified as courses where improvement in one can be achieved by taking the other in an earlier semester. A rough view of actual course sequences can be produced using the top matrix in Table 2 (“number of incidents of students taking row course before column course”). A course with a high column sum is preceded by many other courses; if it has a high row sum it precedes many other courses. We use the same calculation expressed in Equation 1 above, subtracting the row sum from the column sum. Courses with high values come later in the sequence. Table 8 presents the sequences calculated from Equation 1, both actual (using the top matrix in Table 2), and three versions of an optimal sequence, based on Figures 1 through 3. Table 8 presents the ranks, rather than raw values, for each of these with some ties for the optimal sequences. If one were to make recommendations based on Table 8, the most defensible are to push INFS 6610 and QM 6770 earlier in the course sequence. Looking at Economics and Finance specifically, Managerial Economics has been taken at about the right place in the sequence, while Financial Analysis is shown to have been taken a little too late. The first three courses generally taken by students (ACTG 6910, MGMT 6600, and MKT 6800) appear to be better placed near the end of the sequence. 40 | J o u r n a l f o r E c o n o m i c E d u c a t o r s , 1 7 ( 2 ) , 2 0 1 7 40 Table 7: Courses taken in the same semester All dyads Most frequent triads course A course B occu rren ces opt course A course B course C occu rren ces opt MGMT.6600 MKT.6800 187 0 ECON.6000 MGMT.6600 MKT.6800 24 0 MGMT.6650 MKT.6800 137 0 MGMT.6600 MGMT.6650 MKT.6800 24 0 ECON.6000 MGMT.6600 120 0 FIN.6710 INFS.6610 MGMT.6650 23 0 MGMT.6600 MGMT.6650 117 0 ECON.6000 FIN.6710 INFS.6610 21 0 ECON.6000 MKT.6800 116 1 ACTG.6920 MGMT.6600 MKT.6800 20 0 FIN.6710 INFS.6610 107 0 ACTG.6920 INFS.6610 MGMT.6650 17 1 ECON.6000 FIN.6710 102 0 ACTG.6920 MGMT.6650 MKT.6800 16 0 ECON.6000 MGMT.6650 101 0 ECON.6000 FIN.6710 MGMT.6600 16 1 FIN.6710 MGMT.6650 97 1 ECON.6000 INFS.6610 MKT.6800 16 1 FIN.6710 MGMT.6650 97 1 ECON.6000 MGMT.6650 MKT.6800 16 0 INFS.6610 MGMT.6650 97 0 FIN.6710 INFS.6610 MGMT.6600 16 0 ACTG.6920 MGMT.6650 91 0 ECON.6000 FIN.6710 MKT.6800 15 0 FIN.6710 MGMT.6600 90 1 FIN.6710 MGMT.6600 MGMT.6650 15 1 ECON.6000 INFS.6610 87 0 FIN.6710 MGMT.6600 MKT.6800 15 1 INFS.6610 MGMT.6600 82 0 INFS.6610 MGMT.6600 MKT.6800 15 0 ACTG.6920 MKT.6800 80 0 ACTG.6920 MGMT.6600 MGMT.6650 14 0 ACTG.6920 MGMT.6600 72 0 ACTG.6920 ECON.6000 MGMT.6650 13 0 FIN.6710 MKT.6800 70 0 ACTG.6920 ECON.6000 MKT.6800 13 1 INFS.6610 MKT.6800 70 1 ECON.6000 FIN.6710 MGMT.6650 11 1 BUAD.6980 FIN.6710 64 0 ECON.6000 FIN.6710 MGMT.6650 11 1 ACTG.6920 ECON.6000 60 0 ECON.6000 INFS.6610 MGMT.6600 11 0 BUAD.6980 MGMT.6650 60 0 ECON.6000 INFS.6610 MGMT.6650 11 0 ACTG.6920 INFS.6610 50 1 ECON.6000 MGMT.6600 MGMT.6650 11 0 ACTG.6910 ECON.6000 45 1 ACTG.6920 INFS.6610 MKT.6800 10 1 ACTG.6910 MGMT.6600 39 0 FIN.6710 MGMT.6650 MKT.6800 10 1 ACTG.6920 FIN.6710 39 0 FIN.6710 MGMT.6650 MKT.6800 10 1 BUAD.6980 MKT.6800 38 0 INFS.6610 MGMT.6650 MKT.6800 10 0 BUAD.6980 MGMT.6600 37 0 BUAD.6980 FIN.6710 MGMT.6650 9 1 ACTG.6910 MKT.6800 33 0 BUAD.6980 FIN.6710 MGMT.6650 9 1 BUAD.6980 ECON.6000 31 0 ACTG.6910 ECON.6000 MGMT.6600 8 1 BUAD.6980 INFS.6610 31 0 ACTG.6910 MGMT.6600 MKT.6800 8 0 MGMT.6650 QM.6770 26 1 ACTG.6920 ECON.6000 INFS.6610 8 0 BUAD.6980 QM.6770 23 0 ACTG.6920 ECON.6000 MGMT.6600 8 0 ACTG.6910 MGMT.6650 22 0 BUAD.6980 MGMT.6650 MKT.6800 8 0 ACTG.6910 INFS.6610 21 1 ACTG.6910 MGMT.6650 MKT.6800 7 0 INFS.6610 QM.6770 19 0 BUAD.6980 FIN.6710 INFS.6610 7 0 ACTG.6910 FIN.6710 18 0 BUAD.6980 INFS.6610 MGMT.6650 7 0 ACTG.6920 BUAD.6980 18 0 BUAD.6980 MGMT.6600 MGMT.6650 7 0 FIN.6710 QM.6770 17 0 FIN.6710 INFS.6610 MKT.6800 7 1 ECON.6000 QM.6770 16 0 ACTG.6910 INFS.6610 MGMT.6600 6 1 MGMT.6600 QM.6770 12 0 ACTG.6920 ECON.6000 FIN.6710 6 0 ACTG.6910 BUAD.6980 10 0 ACTG.6920 FIN.6710 MGMT.6600 6 1 ACTG.6910 QM.6770 9 0 ACTG.6920 FIN.6710 INFS.6610 6 0 MKT.6800 QM.6770 9 0 ACTG.6920 FIN.6710 MGMT.6650 6 1 ACTG.6920 QM.6770 6 0 ACTG.6920 FIN.6710 MGMT.6650 6 1 ACTG.6910 ACTG.6920 2 0 ACTG.6920 FIN.6710 MKT.6800 6 0 Notes: Frequency of course association from all students taking two or more courses in a semester. Where the value in the column titled “opt” equals 1, two of the courses are directly linked in Figure 3. 41 | J o u r n a l f o r E c o n o m i c E d u c a t o r s , 1 7 ( 2 ) , 2 0 1 7 41 Table 8: Actual and optimal course sequences. course description t-test regression composite actual ACTG.6910 Accounting and Business Decisions 8 9 8.5 1 MGMT.6600 Study of Organizations 7 8 10 2 MKT.6800 Marketing Management 9.5 4 8.5 3 ECON.6000 Managerial Economics 5 4 3 4 ACTG.6920 Financial Statement Analysis 6 7 7 5 INFS.6610 Information Systems Management and Applications 2 1 1.5 6 MGMT.6650 Operations Management 3.5 4 5 7 FIN.6710 Financial Analysis 3.5 4 5 8 QM.6770 Computer-Based Decision Modeling 1 4 1.5 9 BUAD.6980 Strategic Management 9.5 10 5 10 Notes: Ranks calculated from Equation 1. Actual based on the top matrix in Table 2. Summary and Conclusion The point of this exercise was to seek insight into the effect of course sequencing on student performance. We used two different methods—t-tests and fixed-effects OLS models—to identify course-pairs for which taking a prior course led to a significant improvement in student performance in a subsequent course. We constructed graphical representations of the results and produced a proposed course sequence based on those graphical representations. We also tried to provide some insight into the actual sequences in which courses have been taken. Figure 3 and Table 8 summarize our findings. These show that some courses, both directly and indirectly, draw on student knowledge obtained in earlier courses, whereas other courses function primarily to provide that knowledge. Hence, they provide a basis for developing a course sequence. Courses such as INFS 6610 and QM 6770 clearly belong at the beginning, even though students have tended to take them in the latter half of the sequence. Courses that students often take first (and benefit from Economics and Finance), such as ACTG 6910, MGMT 6600, and MKT 6800, actually belong near the end. Surprisingly, the capstone course BUAD 6980 appears not to draw heavily on knowledge obtained in earlier courses. This may indicate insufficient integration of courses into a cumulative learning process. References Chevan, A., Sutherland, J. (1991). Hierarchical partitioning. The American Statistician, 45, 90- 96. Christensen, Gene Donald, William R. Nance and Darin W. White (2012) “Academic Performance in MBA Programs: Do Prerequisites Really Matter?” Journal of Education for Business, 87: 42-47. Dailey, Lynn C. (2011) “Marketing Part-Time MBA Programs: Understanding the Need for and Dimensions of Flexibility,” Journal of Marketing Development and Competitiveness, 5(2):122-129. Grömping, Ulrike (2006). Relative Importance for Linear Regression in R: The Package relaimpo. Journal of Statistical Software, 17(1), 1--27. Kolluri, Bharat and Rao Singamsett (2007), “Teaching Managerial Economics in MBA Programs: A Survey of AACSB Colleges,” Journal of College Teaching and Learning, 4(9): 47-54. 42 | J o u r n a l f o r E c o n o m i c E d u c a t o r s , 1 7 ( 2 ) , 2 0 1 7 42 Krausz, Joshua, Allen Schiff, Jonathan Schiff, and Joan Van Hise (2002) “Predicting Success in Graduate Financial Statement Analysis Courses – Do Traditional Predictors of Accounting Success Apply?” Accounting Educators’ Journal, 14: 1-8. Malik, Ali Asghar (2011) “Students’ Prior Degree Performance as Predictor of Their Performance at MBA Level,” Pakistan Business Review, October, 459-487. MacMillan-Capehart, Amy and Tope Adeyemi-Bello (2008) “Prerequisite Coursework as a Predictor of Performance in a Graduate Management Course,” Journal of College Teaching and Learning, 5(7): 11-16. Satterthwaite, F. E. (1946), "An Approximate Distribution of Estimates of Variance Components.", Biometrics Bulletin 2: 110–114, doi:10.2307/3002019 Sulaiman, Ainin and Suhana Muhezar (2006) “Student Success Factors: Identifying Key Predictors,” Journal of Education for Business, July/August, 328-333. Truell, Allen D., Jensen J. Zhao, Melody W. Alexander, and Inga B. Hill. (2006) “Predicting Final Student Performance in a Graduate Business Program: The MBA,” The Delta Pi Epsilon Journal, 48(3):144-152. Welch, B. L. (1947), "The generalization of "student's" problem when several different population variances are involved." Biometrika 34: 28–35, doi:10.1093/biomet/34.1-2.28 http://en.wikipedia.org/wiki/Digital_object_identifier http://dx.doi.org/10.2307%2F3002019 http://en.wikipedia.org/wiki/Digital_object_identifier http://dx.doi.org/10.1093%2Fbiomet%2F34.1-2.28