Abstract
This paper suggests an alternative to the standard practice of measuring the graduation rate performance using regression analysis. The alternative is production frontier analysis. Production frontier analysis is appealing because it compares an institution’s graduation rate to the best performance instead of the average performance. The paper explains the differences between these two types of analysis and provides examples of their application using data for 187 national universities.
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Notes
The first chapter of Burke (2002) provides a good summary of this movement.
See Christal (1998).
See http://www.2.edtrust.org/edtrust/collegeresults (accessed January 19, 2005).
Data Envelopment Analysis is growing in popularity among those interested in efficiency in higher education; see, for example, Johnes (2006) for a recent application of DEA to data from English institutions of higher education.
The adjusted R2 for this regression is .8, and the coefficient for X is statistically significant at the 1% level.
A convex combination is a linear combination of data points where all coefficients or weights assigned to each data point are non-negative and add up to 1. The set of all convex combinations constitutes the convex hull. With only two data points the value of the new point (formed by the convex combination) will lie on a straight line between the two points.
When Thanassoulis (1993) compares regression analysis with production frontier estimation, his first advantage for the production frontier estimation is its freedom from the requirement of a functional form.
One could criticize this linear example as being two simple. A curve with a declining slope might provide a better fit to the data. Yet observations such as firm 1 would still be possible. As our results will demonstrate, there will be points on the extremes that will be technically efficient but below the regression line.
Although the measure of technical efficiency for firm seven will be quite high, there may be a substantial amount of input slack. This is possible at the edge of a frontier if one (or more) input can be reduced without decreasing output. We discuss input slack more thoroughly later in the paper.
Estimating a stochastic frontier requires the researcher to impose a functional form on the relationship between input and output. Stochastic frontiers also allow deviations from the frontier to reflect random shocks as well as inefficiency. The case for a stochastic frontier is stronger if there is a lot of potential for measurement error in the variables.
One of the most commonly used free DEA programs is DEAP 2.1. See Coelli (1992, 1996). Another is EMS 1.3 (Efficiency measurement system) by H. Scheel. This can be downloaded from the net at http://www.wiso.uni-dortmund.de/
See Färe et al. (1994) for a full discussion of the costs and benefits of non-parametric techniques. In brief, we have no theory that tells us that inputs produce graduates with any particular functional form; so non-parametric techniques free us from having to impose an arbitrary structure on the data.
Assuming constant returns to scale would impose a theoretical restriction that doubling the inputs of an efficient school would double the graduation rate. We do not know that this is the case, so we use the less restrictive variable returns to scale assumption that allows the production functions to display either increasing returns (doubling input more than doubles output) or decreasing returns (doubling input less than doubles output). The first code for solving the linear programming problem to identify an efficient unit isoquant in the CRS case dates to Boles (1966). The method became more widely known following the work by Charnes et al. (1978). Banker et al. (1984) extended the CRS model to account for the possibility of variable returns to scale (VRS).
In this paper we are concerned with technical efficiency only. Our inputs have no clear price measure so we cannot determine if there exists any allocative inefficiency. On the other hand, Ferrier and Lovell (1990) argue that input slacks are a measure of allocative inefficiency.
Our use of the 25th percentile for SAT scores follows Mangold et al. (2003).
The technique we use for computing cost per full-time undergraduate is available from the authors on request.
See Pindyck and Rubinfeld (1991, pp. 260–262) for a discussion of this type of estimation. In cases in which the dependent variable is far from the boundaries of 0 and 1, little is gained from the logistic transformation. In our case, since graduation rates are as high as .98, ignoring the boundaries could lead to errors.
Robust standard errors have been adjusted for correlations of error terms across observations. This is an application of White’s correction for heteroskedasticity.
How we measure two of them differs. US News and World Report uses the mean SAT of the entering class. As mentioned earlier, we use the 25th percentile. More importantly, our measure of spending more accurately represents the undergraduate program since we include capital costs and exclude expenditures on graduate programs.
Suppose on average that public universities grade more leniently or offer a less rigorous curriculum than private universities. In this case public institutions could systematically achieve higher than predicted graduation rates, but the difference would not reflect a positive quality difference in favor of public institutions.
In this study we use the multi-stage DEAP Version 2.1 developed by Coelli (1992) to create our approximation of the production frontier for college graduation rates.
With four inputs and one output, each institution not on the efficient frontier can have a maximum of five peers that determine for it the local efficient surface.
The full output of the DEA analysis is available from the authors on request. The full output includes a listing of each school’s peers, the peer weights, a measure of scale efficiency, and all input slacks.
This possibility can be avoided by using truncated regression or tobit analysis instead of ordinary least squares.
Faculty in science departments, for instance, tend to earn more than the university average and the capital needs of most science departments exceeds that of humanities and social science departments.
Many studies of university performance use inputs (like research spending, or classroom hours) as proxies for output. We have focused instead on graduation rates as a clear output, recognizing that this single measure does not capture the full scope of a university’s function.
References
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An erratum to this article can be found at http://dx.doi.org/10.1007/s11162-007-9065-4
Appendices
Appendix 1—The Mathematics of Data Envelopment Analysis
The linear program underlying variable returns to scale output-oriented data envelopment analysis is:
The vector yi represents the outputs for the ith firm, xi is the vector of inputs, λ is an N × 1 vector of constants, N is the number of firms in the sample, and 1 ≤ ϕ < ∞. The proportional increase in output that the ith firm could have obtained if it were on the efficient frontier is ϕ − 1. The technical efficiency score is defined by 1/ϕ, which varies between zero and one.
To understand the results of this linear program note that a solution in which the vector λ has a one for a particular firm and zero’s otherwise and ϕ = 1 will always satisfy all of the constraints. Call this the default solution. If no better solution can be found, the firm is said to be technically efficient. Such an outcome can be the result of two possibilities. First, there may be no other λ vector that satisfies constraints (1), (3), and (4). In this case there is no convex combination of other firms’ outputs that is at least as large as this firm’s output. Second, there may be a different λ vector that satisfies constraints (1), (3), and (4), but, given that λ vector, the 1/ϕ required to satisfy constraint (2) is greater than one. Such a solution is dominated by the default solution. Clearly, these two outcomes will not always occur. There will be firms that have feasible solutions for values of 1/ϕ less than one. These firms are below the production frontier; they are technically inefficient.
Appendix 2—Table of Results
Rank | School | Resid. | TE | Input slacks | ||||
|---|---|---|---|---|---|---|---|---|
DEA | RES | SAT | Top10 | Full | Cost | |||
1 | 8 | Auburn Univ. | 0.555 | 1 | 0 | 0 | 0 | 0 |
1 | 13 | Bowling Green State Univ. | 0.485 | 1 | 0 | 0 | 0 | 0 |
1 | 5 | Brown Univ. | 0.830 | 1 | 0 | 0 | 0 | 0 |
1 | 51 | Florida State Univ. | 0.195 | 1 | 0 | 0 | 0 | 0 |
1 | 4 | Fordham Univ. | 0.840 | 1 | 0 | 0 | 0 | 0 |
1 | 1 | Harvard Univ. | 1.374 | 1 | 0 | 0 | 0 | 0 |
1 | 10 | Howard Univ. | 0.500 | 1 | 0 | 0 | 0 | 0 |
1 | 34 | Illinois State Univ. | 0.268 | 1 | 0 | 0 | 0 | 0 |
1 | 84 | Indiana State Univ. | 0.049 | 1 | 0 | 0 | 0 | 0 |
1 | 20 | Indiana Univ.—Bloomington | 0.390 | 1 | 0 | 0 | 0 | 0 |
1 | 68 | Johns Hopkins Univ. | 0.109 | 1 | 0 | 0 | 0 | 0 |
1 | 107 | Louisiana State Univ.—Baton Rouge | −0.053 | 1 | 0 | 0 | 0 | 0 |
1 | 56 | Louisiana Tech Univ. | 0.170 | 1 | 0 | 0 | 0 | 0 |
1 | 72 | Mississippi State Univ. | 0.090 | 1 | 0 | 0 | 0 | 0 |
1 | 73 | New School Univ. | 0.086 | 1 | 0 | 0 | 0 | 0 |
1 | 121 | Oklahoma State Univ. | −0.133 | 1 | 0 | 0 | 0 | 0 |
1 | 15 | Pace Univ. | 0.433 | 1 | 0 | 0 | 0 | 0 |
1 | 19 | Pennsylvania State Univ. | 0.396 | 1 | 0 | 0 | 0 | 0 |
1 | 2 | St. John’s Univ. | 0.879 | 1 | 0 | 0 | 0 | 0 |
1 | 33 | SUNY—Albany | 0.274 | 1 | 0 | 0 | 0 | 0 |
1 | 16 | Syracuse Univ. | 0.431 | 1 | 0 | 0 | 0 | 0 |
1 | 118 | Texas A&M Univ.—Commerce | −0.112 | 1 | 0 | 0 | 0 | 0 |
1 | 76 | Univ. of Akron | 0.074 | 1 | 0 | 0 | 0 | 0 |
1 | 35 | Univ. of California—Davis | 0.266 | 1 | 0 | 0 | 0 | 0 |
1 | 54 | Univ. of California—Irvine | 0.181 | 1 | 0 | 0 | 0 | 0 |
1 | 57 | Univ. of Colorado—Denver | 0.167 | 1 | 0 | 0 | 0 | 0 |
1 | 113 | Univ. of Georgia | −0.083 | 1 | 0 | 0 | 0 | 0 |
1 | 40 | Univ. of Illinois—Urbana-Champagne | 0.243 | 1 | 0 | 0 | 0 | 0 |
1 | 131 | Univ. of Louisville | −0.165 | 1 | 0 | 0 | 0 | 0 |
1 | 95 | Univ. of Missouri—St. Louis | −0.013 | 1 | 0 | 0 | 0 | 0 |
1 | 14 | Univ. of New Hampshire | 0.433 | 1 | 0 | 0 | 0 | 0 |
1 | 6 | Univ. of Notre Dame | 0.825 | 1 | 0 | 0 | 0 | 0 |
1 | 55 | Univ. of Vermont | 0.172 | 1 | 0 | 0 | 0 | 0 |
1 | 3 | Univ. of Virginia | 0.843 | 1 | 0 | 0 | 0 | 0 |
1 | 101 | Univ. of Wisconsin—Milwaukee | −0.032 | 1 | 0 | 0 | 0 | 0 |
36 | 27 | Clemson Univ. | 0.330 | 0.999 | 10 | 0 | 0 | 0 |
36 | 58 | Univ. of California—Riverside | 0.148 | 0.999 | 0 | 0.463 | 0.065 | 0 |
36 | 87 | Univ. of Northern Colorado | 0.029 | 0.999 | 0 | 0 | 0.104 | 0 |
39 | 24 | Univ. of Wisconsin—Madison | 0.366 | 0.995 | 0 | 0 | 0.041 | 0 |
39 | 53 | Miami Univ.—Oxford | 0.182 | 0.995 | 17 | 0 | 0.129 | 0 |
41 | 39 | Tufts Univ. | 0.243 | 0.994 | 3 | 0 | 0.035 | 0 |
42 | 9 | Georgetown Univ. | 0.520 | 0.992 | 0 | 0 | 0 | 705 |
42 | 52 | Ohio Univ. | 0.186 | 0.992 | 0 | 0 | 0.018 | 0 |
44 | 32 | Lehigh Univ. | 0.282 | 0.991 | 0 | 0 | 0.04 | 2411 |
45 | 11 | College of William and Mary | 0.494 | 0.989 | 0 | 0 | 0.036 | 0 |
46 | 21 | Yale Univ. | 0.387 | 0.987 | 0 | 0.047 | 0 | 19196 |
47 | 12 | Northwestern Univ. | 0.489 | 0.985 | 0 | 0.052 | 0 | 4553 |
48 | 17 | Pepperdine Univ. | 0.412 | 0.984 | 0 | 0.218 | 0 | 4664 |
49 | 31 | Univ. of Michigan—Ann Arbor | 0.291 | 0.981 | 0 | 0 | 0.073 | 0 |
50 | 25 | Seton Hall Univ. | 0.356 | 0.980 | 0 | 0 | 0 | 2823 |
50 | 46 | Virginia Tech | 0.222 | 0.980 | 0 | 0 | 0.006 | 0 |
52 | 41 | Dartmouth Univ. | 0.239 | 0.979 | 0 | 0 | 0 | 16089 |
53 | 7 | Univ. of Missouri—Kansas City | 0.578 | 0.976 | 0 | 0.089 | 0 | 1192 |
53 | 30 | Univ. of Delaware | 0.313 | 0.976 | 0 | 0 | 0.088 | 492 |
55 | 23 | Stanford Univ. | 0.371 | 0.975 | 0 | 0.069 | 0 | 2133 |
56 | 22 | Michigan State Univ. | 0.376 | 0.973 | 0 | 0 | 0 | 0 |
56 | 43 | DePaul Univ. | 0.227 | 0.973 | 4 | 0 | 0 | 0 |
58 | 29 | Duquesne Univ. | 0.319 | 0.972 | 0 | 0.224 | 0 | 0 |
59 | 82 | Colorado State Univ. | 0.057 | 0.970 | 51 | 0 | 0 | 0 |
60 | 28 | Columbia Univ. | 0.320 | 0.969 | 0 | 0.081 | 0 | 3159 |
61 | 143 | Wake Forest Univ. | −0.237 | 0.968 | 0 | 0 | 0.031 | 16484 |
62 | 70 | Univ. of California—Los Angeles | 0.098 | 0.967 | 0 | 0.142 | 0 | 0 |
63 | 59 | Univ. of Pennsylvania | 0.134 | 0.964 | 0 | 0.109 | 0 | 11255 |
64 | 26 | Univ. of Connecticut | 0.342 | 0.963 | 0 | 0 | 0 | 0 |
65 | 36 | Marquette Univ. | 0.257 | 0.961 | 0 | 0 | 0.05 | 0 |
65 | 65 | Duke Univ. | 0.121 | 0.961 | 0 | 0.048 | 0 | 9686 |
67 | 127 | Univ. of Houston | −0.152 | 0.960 | 0 | 0.049 | 0.012 | 0 |
68 | 71 | Univ. of North Carolina—Chapel Hill | 0.097 | 0.958 | 0 | 0.022 | 0 | 25 |
68 | 96 | Univ. of California—Santa Barbara | −0.016 | 0.958 | 0 | 0.283 | 0 | 0 |
70 | 120 | Washington Univ. in St. Louis | −0.130 | 0.954 | 0 | 0.06 | 0 | 24288 |
71 | 18 | Univ. of Denver | 0.410 | 0.952 | 0 | 0.019 | 0 | 0 |
71 | 135 | Univ. of Chicago | −0.170 | 0.952 | 0 | 0.035 | 0 | 12593 |
73 | 145 | Massachusetts Inst. of Technology | −0.245 | 0.948 | 0 | 0.101 | 0 | 5192 |
74 | 60 | SUNY—Binghamton | 0.134 | 0.945 | 0 | 0 | 0.007 | 0 |
75 | 112 | Univ. of California—San Diego | −0.081 | 0.944 | 0 | 0.149 | 0 | 0 |
76 | 42 | West Virginia Univ. | 0.232 | 0.942 | 0 | 0.034 | 0.011 | 0 |
76 | 115 | Rensselaer Polytechnic Institute | −0.098 | 0.942 | 0 | 0 | 0.061 | 0 |
78 | 45 | Univ. of South Carolina—Columbia | 0.223 | 0.940 | 0 | 0.113 | 0 | 0 |
78 | 119 | Univ. of Central Florida | −0.113 | 0.940 | 28 | 0 | 0 | 0 |
80 | 63 | Univ. of the Pacific | 0.127 | 0.935 | 0 | 0.035 | 0.008 | 0 |
80 | 77 | Texas A&M Univ.—College Station | 0.069 | 0.935 | 0 | 0 | 0 | 0 |
82 | 141 | Rice Univ. | −0.232 | 0.934 | 0 | 0.101 | 0 | 13462 |
82 | 147 | Brandeis Univ. | −0.265 | 0.934 | 0 | 0.01 | 0 | 414 |
82 | 153 | Vanderbilt Univ. | −0.284 | 0.934 | 0 | 0 | 0.013 | 0 |
85 | 90 | Univ. of Southern California | 0.018 | 0.933 | 0 | 0.017 | 0 | 0 |
86 | 105 | North Dakota State Univ. | −0.047 | 0.931 | 0 | 0 | 0.102 | 0 |
87 | 74 | Univ. of Massachusetts—Amherst | 0.083 | 0.929 | 0 | 0 | 0 | 0 |
88 | 75 | Univ. of San Francisco | 0.083 | 0.925 | 0 | 0 | 0 | 0 |
89 | 104 | Univ. of New Mexico | −0.041 | 0.924 | 0 | 0.008 | 0 | 0 |
90 | 149 | Emory Univ. | −0.271 | 0.923 | 0 | 0.171 | 0 | 12594 |
91 | 50 | Univ. of South Dakota | 0.198 | 0.917 | 0 | 0 | 0 | 0 |
92 | 47 | Purdue Univ.—West Lafayette | 0.205 | 0.916 | 0 | 0.027 | 0 | 0 |
93 | 126 | Univ. of California—Berkeley | −0.148 | 0.915 | 0 | 0.205 | 0 | 0 |
94 | 81 | Univ. of Iowa | 0.059 | 0.914 | 0 | 0 | 0 | 0 |
94 | 99 | Univ. of Florida | −0.030 | 0.914 | 0 | 0.129 | 0 | 0 |
96 | 78 | Univ. of Washington | 0.062 | 0.913 | 0 | 0.001 | 0 | 0 |
97 | 80 | George Washington Univ. | 0.059 | 0.909 | 0 | 0.097 | 0 | 0 |
98 | 151 | Worcester Polytechnic Institute | −0.282 | 0.908 | 0 | 0.017 | 0 | 2462 |
98 | 186 | California Institute of Technology | −0.905 | 0.908 | 0 | 0.211 | 0 | 21722 |
100 | 83 | Univ. of Oregon | 0.054 | 0.905 | 0 | 0 | 0 | 0 |
101 | 44 | Western Michigan Univ. | 0.223 | 0.902 | 0 | 0.005 | 0 | 0 |
101 | 86 | Iowa State Univ. | 0.039 | 0.902 | 0 | 0 | 0 | 0 |
103 | 69 | Northern Illinois Univ. | 0.105 | 0.899 | 0 | 0 | 0 | 0 |
103 | 88 | American Univ. | 0.025 | 0.899 | 0 | 0.037 | 0 | 0 |
105 | 116 | Univ. of Texas—Austin | −0.108 | 0.898 | 0 | 0.039 | 0 | 0 |
106 | 48 | Univ. of Alabama | 0.203 | 0.896 | 0 | 0.077 | 0 | 0 |
107 | 166 | New York Univ. | −0.388 | 0.893 | 0 | 0.187 | 0 | 2550 |
108 | 64 | Loyola Univ. Chicago | 0.121 | 0.889 | 0 | 0.051 | 0 | 1430 |
108 | 109 | Kent State Univ. | −0.062 | 0.889 | 0 | 0 | 0.032 | 0 |
110 | 181 | Univ. of Rochester | −0.568 | 0.888 | 0 | 0.088 | 0 | 6140 |
111 | 38 | St. Louis Univ. | 0.245 | 0.887 | 0 | 0.044 | 0 | 0 |
112 | 117 | Univ. of San Diego | −0.109 | 0.886 | 0 | 0.166 | 0 | 0 |
113 | 183 | Carnegie Mellon Univ. | −0.654 | 0.884 | 0 | 0.165 | 0 | 4520 |
114 | 124 | Baylor Univ. | −0.144 | 0.882 | 0 | 0.017 | 0 | 0 |
115 | 100 | Clark Univ. | −0.031 | 0.881 | 0 | 0.079 | 0 | 410 |
115 | 144 | Univ. of Montana | −0.238 | 0.881 | 0 | 0 | 0.124 | 0 |
117 | 138 | Univ. of Texas—Dallas | −0.203 | 0.880 | 17 | 0 | 0 | 0 |
118 | 123 | Univ. of Maryland—College Park | −0.144 | 0.878 | 0 | 0.053 | 0 | 0 |
119 | 62 | Univ. of Maine—Orono | 0.127 | 0.876 | 0 | 0.018 | 0 | 0 |
119 | 171 | Tulane Univ. | −0.428 | 0.876 | 0 | 0.15 | 0 | 26 |
121 | 125 | Hofstra Univ. | −0.144 | 0.873 | 0 | 0 | 0 | 4265 |
122 | 66 | Univ. of Toledo | 0.112 | 0.872 | 0 | 0 | 0 | 0 |
123 | 91 | Univ. of Nebraska—Lincoln | 0.016 | 0.871 | 0 | 0.038 | 0 | 0 |
124 | 140 | Univ. of Southern Mississippi | −0.227 | 0.870 | 0 | 0.186 | 0.041 | 0 |
124 | 173 | Case Western Reserve Univ. | −0.454 | 0.870 | 33 | 0.292 | 0 | 0 |
126 | 98 | Univ. of Pittsburgh | −0.024 | 0.865 | 0 | 0 | 0 | 0 |
127 | 178 | Boston Univ. | −0.493 | 0.865 | 0 | 0.177 | 0 | 4242 |
128 | 177 | Clarkson Univ. | −0.490 | 0.864 | 0 | 0.006 | 0 | 4291 |
129 | 103 | Catholic Univ. of America | −0.037 | 0.861 | 0 | 0.117 | 0 | 0 |
130 | 165 | Brigham-Young Univ.—Provo | −0.371 | 0.859 | 0 | 0.155 | 0 | 0 |
131 | 61 | Univ. of North Carolina—Greensboro | 0.131 | 0.858 | 0 | 0 | 0 | 0 |
132 | 97 | Washington State Univ. | −0.019 | 0.856 | 0 | 0.216 | 0.004 | 0 |
133 | 146 | Univ. of Maryland—Baltimore County | −0.257 | 0.855 | 0 | 0 | 0 | 0 |
134 | 85 | Oregon State Univ. | 0.043 | 0.854 | 0 | 0.054 | 0 | 0 |
135 | 94 | Northeastern Univ. | −0.013 | 0.848 | 0 | 0 | 0 | 1318 |
136 | 160 | Univ. of Miami | −0.322 | 0.846 | 0 | 0.219 | 0 | 1146 |
137 | 155 | Stevens Institute of Technology | −0.296 | 0.844 | 15 | 0.238 | 0 | 0 |
138 | 67 | Univ. of Rhode Island | 0.112 | 0.842 | 0 | 0.031 | 0 | 0 |
139 | 169 | Univ. of Missouri—Columbia | −0.401 | 0.841 | 0 | 0.021 | 0 | 0 |
140 | 110 | Ohio State Univ.—Columbus | −0.067 | 0.837 | 0 | 0.079 | 0 | 0 |
140 | 176 | Illinois Institute of Technology | −0.487 | 0.837 | 76 | 0.281 | 0 | 0 |
142 | 158 | Texas Christian Univ. | −0.305 | 0.836 | 0 | 0.125 | 0 | 0 |
143 | 37 | Andrews Univ. | 0.253 | 0.828 | 0 | 0 | 0 | 2370 |
144 | 106 | Univ. of Tennessee—Knoxville | −0.047 | 0.825 | 0 | 0.054 | 0 | 0 |
145 | 79 | Univ. of Idaho | 0.062 | 0.819 | 0 | 0.035 | 0 | 0 |
145 | 179 | Univ. of California—Santa Cruz | −0.535 | 0.819 | 0 | 0.679 | 0 | 110 |
147 | 130 | Wichita State Univ. | −0.162 | 0.818 | 0 | 0 | 0 | 256 |
147 | 133 | North Carolina State Univ.—Raleigh | −0.167 | 0.818 | 0 | 0.018 | 0 | 0 |
149 | 92 | Univ. of Wyoming | −0.006 | 0.816 | 0 | 0.095 | 0 | 0 |
150 | 49 | Temple Univ. | 0.200 | 0.809 | 0 | 0.031 | 0 | 2687 |
150 | 93 | Northern Arizona Univ. | −0.012 | 0.809 | 0 | 0.077 | 0 | 0 |
152 | 111 | Univ. of Kansas | −0.079 | 0.807 | 0 | 0.071 | 0 | 0 |
153 | 134 | Univ. of Mississippi | −0.169 | 0.804 | 0 | 0.165 | 0.003 | 0 |
154 | 142 | Univ. of Kentucky | −0.235 | 0.801 | 0 | 0.055 | 0 | 0 |
155 | 89 | Univ. of Utah | 0.025 | 0.799 | 0 | 0.044 | 0 | 0 |
156 | 136 | Univ. of Oklahoma | −0.181 | 0.794 | 0 | 0.119 | 0 | 0 |
157 | 154 | SUNY—Buffalo | −0.291 | 0.792 | 0 | 0.035 | 0 | 700 |
158 | 162 | Univ. of Alabama—Huntsville | −0.326 | 0.791 | 0 | 0.228 | 0 | 0 |
159 | 108 | Old Dominion Univ. | −0.059 | 0.790 | 0 | 0.052 | 0 | 0 |
160 | 150 | Drexel Univ. | −0.277 | 0.789 | 0 | 0.019 | 0 | 0 |
161 | 161 | Montana State Univ. | −0.325 | 0.788 | 0 | 0.018 | 0.107 | 0 |
162 | 122 | Nova Southeastern Univ. | −0.133 | 0.779 | 0 | 0.118 | 0 | 0 |
163 | 129 | Texas Tech Univ. | −0.161 | 0.774 | 0 | 0.048 | 0 | 0 |
164 | 132 | Florida Institute of Technology | −0.167 | 0.765 | 0 | 0.09 | 0 | 0 |
165 | 182 | Michigan Technological Univ. | −0.577 | 0.762 | 0 | 0.126 | 0 | 0 |
166 | 114 | Ball State Univ. | −0.085 | 0.759 | 0 | 0 | 0 | 144 |
166 | 159 | SUNY—Stony Brook | −0.307 | 0.759 | 0 | 0.064 | 0 | 369 |
168 | 102 | Virginia Commonwealth Univ. | −0.037 | 0.758 | 0 | 0 | 0 | 356 |
169 | 152 | Univ. of Arizona | −0.284 | 0.755 | 0 | 0.139 | 0 | 0 |
170 | 139 | Middle Tennessee State Univ. | −0.217 | 0.744 | 0 | 0 | 0.039 | 0 |
171 | 137 | Univ. of South Florida | −0.184 | 0.736 | 0 | 0.102 | 0 | 100 |
171 | 148 | Univ. of North Texas | −0.269 | 0.736 | 0 | 0.044 | 0 | 0 |
173 | 163 | Univ. of Hawaii—Manoa | −0.366 | 0.726 | 0 | 0.137 | 0 | 74 |
174 | 157 | Arizona State Univ. | −0.303 | 0.719 | 0 | 0.117 | 0 | 0 |
175 | 156 | Southern Illinois Univ.—Carbondale | −0.302 | 0.712 | 0 | 0 | 0 | 663 |
176 | 164 | Univ. of North Dakota | −0.371 | 0.706 | 0 | 0.061 | 0 | 1175 |
177 | 187 | Univ. of Missouri—Rolla | −1.126 | 0.702 | 0 | 0.209 | 0 | 0 |
178 | 175 | Univ. of Arkansas—Fayetteville | −0.475 | 0.701 | 0 | 0.122 | 0.011 | 0 |
179 | 172 | Univ. of Minnesota—Twin Cities | −0.441 | 0.700 | 0 | 0.077 | 0 | 0 |
180 | 170 | Indiana Univ. of Pennsylvania | −0.412 | 0.693 | 0 | 0.062 | 0 | 633 |
181 | 128 | Wright State Univ. | −0.157 | 0.681 | 0 | 0 | 0 | 844 |
182 | 185 | Univ. of Tulsa | −0.860 | 0.679 | 0 | 0.199 | 0 | 1094 |
183 | 184 | Polytechnic Univ. | −0.675 | 0.676 | 0 | 0.152 | 0 | 1952 |
184 | 168 | New Jersey Institute of Technology | −0.398 | 0.667 | 0 | 0.084 | 0 | 0 |
185 | 167 | Univ. of Texas—Arlington | −0.392 | 0.660 | 0 | 0.068 | 0 | 0 |
186 | 180 | Univ. of Illinois—Chicago | −0.545 | 0.647 | 0 | 0.096 | 0 | 1256 |
187 | 174 | Univ. of Alabama—Birmingham | −0.469 | 0.564 | 0 | 0.048 | 0 | 1015 |
Appendix 3—Ranking Efficient Schools Using Super Efficiency Analysis
One of the difficulties of using production frontier analysis is that it generates a large group of efficient institutions with identical technical efficiency scores of 1.00. This leaves us unable to produce a ranking among efficient schools or make any efficiency comparisons among them. In our case 35 schools would be ranked as number one. The notion of “super efficiency scores” has been designed as a partial solution to this problem.
Andersen and Petersen (1993) developed super efficiency scores as a measure of how much the efficient boundary is moved because a particular firm is present in the data. It is easy to illustrate the calculation of super efficiency scores by using the example in Fig. 2. If we eliminated firm 3, the production frontier would contain a segment between firm 2 and firm 7. The measure of super efficiency for firm 3 would be firm 3’s output over the output for firm 3’s inputs on the altered production frontier. If we eliminated firm 2, firm 5 would become efficient, and the super efficiency score for firm 2 would be its output over the output for its inputs on the new segment of the altered production frontier between the points for firm 1 and firm 5. Point 1 presents a problem. If we eliminate firm 1, the new production frontier will be vertical at the input of firm 2. There is no way to project firm 1’s output on to this altered production frontier, and as a result it is impossible to define a super efficiency score in this case.
Table A1 gives the super efficiency scores for the efficient institutions with the institutions with undefined super efficiency scores in alphabetical order followed by the other in order of their super efficiency scores. The institutions with undefined super efficiency scores have an average graduation rate of 41.7%, well below the average. Those for which we could calculate super efficiency scores have much more varied graduation rates, which are generally higher. It is not surprising that St. Johns University and Fordham University, two institutions that are frequent peers of institutions in the lower right quadrant of Fig. 3 also have very high super efficiency scores. These two institutions push out the production frontier more than do most of the other efficient institutions.
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Archibald, R.B., Feldman, D.H. Graduation Rates and Accountability: Regressions Versus Production Frontiers. Res High Educ 49, 80–100 (2008). https://doi.org/10.1007/s11162-007-9063-6
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DOI: https://doi.org/10.1007/s11162-007-9063-6