Best University

7/28/2019 Best University

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What makes a top university top?

Ratings of universities, colleges, graduate schools, and professional schools have be-

come very common, for both good and bad. If we accept the characterizations of magazines

like Business Week and U.S. News and World Report as having value, is it possible to char-

acterize the properties of a school that determine its rating? School administrators would

be interested in this, of course, so that they could target efforts at improvement efficiently.

Prospective students would be interested also, as they could perhaps identify schools that

are better or worse than the magazine has decided.

The following data come from 2002 U.S. News and World Report Americas Best

Colleges. The data refer to the 249 national universitiesdoctoral in the United States

(universities that offer a full range of undergraduate majors, plus masters and Ph.D. de-

grees, and emphasize faculty research). For each university the following variables are

recorded:

(1) the SAT score at the 25th percentile for the Freshman class

(2) the SAT score at the 75th percentile for the Freshman class

(3) the percentage of the freshman class in top 10% of their high school class

(4) the acceptance rate (the percentage of applicants who are admitted to the university)

(5) the percentage of alumni who gave to the school

(6) the percentage of classes with 20 or fewer students

(7) the percentage of classes with 50 or more students

(8) the freshman retention rate (percentage of freshman entering in 19911994 who begin

their sophomore years)

(9) the graduation rate (within 6 years) of students entering 1986-1989

(10) the score of the institutions academic programs based on a survey of college presidents,

provosts, and deans of admission (on a scale from 1 to 5)

(11) the percentage of faculty who are fulltime

(12) whether or not the university was ranked Top 50, Tier 2, Tier 3, or Tier 4 by U.S.

News and World Report

Note that variables 14 refer to the quality of the entering class, while variables 511

c 2008, Jeffrey S. Simonoff 1


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refer to the quality and attractiveness of student life at the university. We will analyze these

two groups of variables separately. Note that many schools use the ACT exam, rather than

the SAT exam, as an admissions requirement. This is mostly a geographic effect (schools

in western states being more likely to use the ACT), although it is interesting to note that

only 4 of the schools in the top 50 use the ACT. For these schools an equivalent SAT score

is given, based on equating the observed percentiles for the ACT to the corresponding SAT

score. Is it possible to build a model for the probability of being a top 50 school from these

variables?

First, here are the entering class quality data. It is worth knowing also that the first 50

observations are in the rank order given by U.S. News and World Report (so, for example,

NYU was ranked #32; note also that the eight Ivy League schools are all in the top 16).

It is not possible to build an ordinary (least squares) regression model with rank as the

target variable here, since specific ranks were not given for any schools not in the top 50.

Row SAT 1st quartile

1 Princeton University (NJ) 1350

2 Harvard University (MA) 1410

3 Yale University (CT) 1380

4 California Institute of Technology 1450

5 Massachusetts Inst. of Technology 14106 Stanford University (CA) 1360

7 University of Pennsylvania 1310

8 Duke University (NC) 1300

9 Columbia University (NY) 1310

10 Dartmouth College (NH) 1330

11 University of Chicago 1280

12 Northwestern University (IL) 1290

13 Rice University (TX) 1310

14 Cornell University (NY) 1270

15 Washington University in St. Louis 129016 Brown University (RI) 1290

17 Johns Hopkins University (MD) 1300

18 Emory University (GA) 1300

19 University of Notre Dame (IN) 1270

20 University of California - Berkeley * 1200

21 University of Virginia * 1200

22 Vanderbilt University (TN) 1220



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23 Carnegie Mellon University (PA) 1280

24 Georgetown University (DC) 1280

25 University of Michigan - Ann Arbor * 1210

26 Univ. of California - Los Angeles * 1170

27 Wake Forest University (NC) 1220

28 Tufts University (MA) 124029 U. of North Carolina - Chapel Hill * 1130

30 College of William and Mary (VA)* 1230

31 Univ. of California - San Diego * 1160

32 New York University 1250

33 Univ. of Wisconsin - Madison * 1210

34 Brandeis University (MA) 1220

35 Univ. of Southern California 1210

36 U. of Illinois - Urbana-Champaign * 1170

37 University of Rochester (NY) 1240

38 Boston College 122039 Case Western Reserve Univ. (OH) 1240

40 Lehigh University (PA) 1183

41 Georgia Institute of Technology * 1250

42 University of California - Davis * 1060

43 University of California - Irvine * 1070

44 Yeshiva University (NY) 1130

45 University of Washington * 1050

46 Pennsylvania State U. - University Park * 1080

47 Tulane University (LA) 1210

48 Pepperdine University (CA) 1150

49 Rensselaer Polytechnic Inst. (NY) 118050 Texas A&M Univ. - College Station * 1070

51 Univ. of California - Santa Barbara * 1090

52 University of Texas - Austin * 1090

53 American University (DC) 1100

54 Auburn University (AL)* 1050

55 Baylor University (TX) 1080

56 Boston University 1210

57 Brigham Young Univ. - Provo (UT) 1210

58 Catholic University of America (DC) 1060

59 Clarkson University (NY) 109060 Clark University (MA) 1070

61 Clemson University (SC)* 1080

62 Colorado State University * 1090

63 Duquesne University (PA) 1070

64 Florida State University * 1060

65 Fordham University (NY) 1050

66 George Washington University (DC) 1150



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67 Howard University (DC) 830

68 Illinois Institute of Technology 1190

69 Indiana University - Bloomington * 970

70 Iowa State University * 1050

71 Loyola University Chicago 1090

72 Marquette University (WI) 113073 MCP Hahnemann University (PA) *

74 Miami University - Oxford (OH)* 1170

75 Michigan State University * 1050

76 Michigan Technological University * 1130

77 North Carolina State U. - Raleigh * 1080

78 Ohio State University - Columbus * 1090

79 Ohio University * 1050

80 Purdue Univ. - West Lafayette (IN)* 1010

81 Rutgers - Newark (NJ)* 960

82 Rutgers - New Brunswick (NJ)* 107083 Seton Hall University (NJ) 950

84 Southern Methodist University (TX) 1050

85 Stevens Institute of Technology (NJ) 1200

86 St. Louis University 1130

87 SUNY - Albany * 1040

88 SUNY - Binghamton * 1110

89 SUNY College Environmental Science and Forestry * 1040

90 SUNY - Stony Brook * 1020

91 SUNY - University at Buffalo * 1010

92 Syracuse University (NY) 1110

93 Texas Christian University 105094 University of Alabama * 1020

95 University of Arizona * 970

96 Univ. of California - Riverside * 940

97 Univ. of California - Santa Cruz * 1030

98 University of Colorado - Boulder * 1050

99 University of Connecticut * 1040

100 University of Dayton (OH) 1090

101 University of Delaware * 1040

102 University of Denver 1010

103 University of Florida * 1100104 University of Georgia * 1110

105 University of Iowa * 1090

106 University of Kansas * 1090

107 University of Kentucky * 1090

108 Univ. of Maryland - College Park * 1150

109 Univ. of Massachusetts - Amherst * 1010

110 University of Miami (FL) 1070



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111 Univ. of Minnesota - Twin Cities * 1090

112 Univ. of Missouri - Columbia * 1170

113 University of Missouri - Rolla * 1210

114 Univ. of Nebraska - Lincoln * 1050

115 University of New Hampshire * 1010

116 University of Oklahoma * 1090117 University of Oregon * 1000

118 University of Pittsburgh * 1060

119 University of San Diego 1070

120 University of San Francisco 1020

121 Univ. of South Carolina - Columbia * 1000

122 University of St. Thomas (MN) 1090

123 Univ. of Tennessee - Knoxville * 1050

124 University of the Pacific (CA) 1010

125 University of Tulsa (OK) 1130

126 University of Utah * 1050127 University of Vermont * 1040

128 Virginia Tech * 1080

129 Washington State University * 940

130 Worcester Polytechnic Inst. (MA) 1180

131 Adelphi University (NY) 970

132 Alabama A&M University * 830

133 Andrews University (MI) 970

134 Arizona State University * 970

135 Ball State University (IN)* 920

136 Bowling Green State University (OH)* 970

137 Clark Atlanta University *138 DePaul University (IL) 1050

139 Drexel University (PA) 1030

140 Florida Institute of Technology 1050

141 George Mason University (VA)* 970

142 Hofstra University (NY) 1010

143 Indiana Univ. of Pennsylvania * 970

144 Kansas State University * 970

145 Louisiana State U. - Baton Rouge * 1050

146 Mississippi State University * 970

147 Montana State University - Bozeman * 1010148 New Jersey Inst. of Technology * 1030

149 N.M. Inst. of Mining and Tech. * 1130

150 New Mexico State University * 930

151 New School University (NY) 1090

152 Northeastern University (MA) 1020

153 Oklahoma State University * 1050

154 Oregon State University * 960



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155 Pace University (NY) 950

156 Polytechnic University (NY) 1150

157 St. Johns University (NY) 890

158 Temple University (PA)* 920

159 Texas Tech University * 970

160 University of Alabama - Birmingham * 970161 University of Alabama - Huntsville * 1090

162 University of Alaska - Fairbanks * 880

163 University of Arkansas * 1090

164 University of Cincinnati * 970

165 University of Hawaii - Manoa * 980

166 University of Idaho * 1010

167 University of Illinois - Chicago * 1050

168 University of La Verne (CA) 870

169 University of Maine - Orono * 970

170 U. of Maryland - Baltimore County * 1080171 University of Mississippi * 1010

172 Univ. of Missouri - Kansas City * 1050

173 University of Nevada - Reno * 1010

174 University of New Mexico * 970

175 U. of North Carolina - Greensboro * 930

176 University of North Dakota * 1010

177 University of Rhode Island * 980

178 Univ. of Southern Mississippi * 930

179 University of South Florida * 970

180 University of Texas - Dallas * 1060

181 University of Wyoming * 1010182 Utah State University * 970

183 Virginia Commonwealth University * 920

184 Wayne State University (MI)* 880

185 Western Michigan University * 1010

186 West Virginia University * 1010

187 Widener University (PA) 1040

188 Biola University (CA) 1000

189 Central Michigan University * 1010

190 Cleveland State University * 830

191 East Carolina University (NC)* 940192 East Tennessee State University * 970

193 Florida Atlantic University * 950

194 Florida International University * 1020

195 Georgia State University * 940

196 Idaho State University * 930

197 Illinois State University * 1010

198 Indiana State University * 840



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199 Indiana U.-Purdue U. - Indianapolis * 840

200 Jackson State University (MS)* 780

201 Kent State University (OH)* 930

202 Louisiana Tech University * 970

203 Middle Tennessee State Univ. * 970

204 National-Louis University (IL) 720205 North Dakota State University * 1050

206 Northern Arizona University * 920

207 Northern Illinois University * 1010

208 Nova Southeastern University (FL) 930

209 Oakland University (MI)* 930

210 Old Dominion University (VA)* 910

211 Portland State University (OR)* 890

212 San Diego State University * 940

213 South Carolina State University * 760

214 South Dakota State University * 970215 Southern Illinois U. - Carbondale * 1010

216 Tennessee State University * 780

217 Texas A&M University - Commerce * 870

218 Texas A&M Univ. - Kingsville * 800

219 Texas Southern University * *

220 Texas Womans University * 820

221 Union Institute (OH) *

222 United States International U. (CA) 820

223 University of Akron (OH)* 880

224 Univ. of Arkansas - Little Rock * *

225 University of Bridgeport (CT) 810226 University of Central Florida * 1050

227 University of Colorado - Denver * 970

228 University of Hartford (CT) 950

229 University of Houston * 920

230 University of Louisiana - Lafayette * 930

231 University of Louisville (KY)* 970

232 Univ. of Massachusetts - Boston * 920

233 Univ. of Massachusetts - Lowell * 970

234 University of Memphis * 970

235 Univ. of Missouri - St. Louis * 1000236 University of Montana * 970

237 University of Nevada - Las Vegas * 890

238 University of New Orleans * 930

239 University of Northern Colorado * 970

240 University of North Texas * 960

241 University of South Alabama * 1010

242 University of South Dakota * 970



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243 University of Texas - Arlington * 930

244 University of Texas - El Paso * 790

245 University of Toledo (OH)* 970

246 Univ. of Wisconsin - Milwaukee * 970

247 Wichita State University (KS)* 970

248 Wilmington College (DE) *249 Wright State University (OH)* 930

Row SAT 3rd quartile Top 10% Acceptance rate Tier

1 1540 92 12 Top 50

2 1580 90 11 Top 50

3 1550 95 16 Top 50

4 1580 98 13 Top 50

5 1560 97 16 Top 506 1560 89 13 Top 50

7 1490 92 23 Top 50

8 1500 86 26 Top 50

9 1510 87 13 Top 50

10 1520 86 21 Top 50

11 1490 82 44 Top 50

12 1470 84 33 Top 50

13 1510 84 23 Top 50

14 1460 82 31 Top 50

15 1450 85 30 Top 50

16 1490 87 16 Top 5017 1480 70 32 Top 50

18 1460 90 45 Top 50

19 1430 84 34 Top 50

20 1450 99 26 Top 50

21 1410 83 39 Top 50

22 1400 71 55 Top 50

23 1460 72 36 Top 50

24 1450 78 22 Top 50

25 1420 67 55 Top 50

26 1400 97 29 Top 5027 1380 67 49 Top 50

28 1410 73 26 Top 50

29 1340 65 37 Top 50

30 1410 79 41 Top 50

31 1370 99 38 Top 50

32 1420 70 29 Top 50

33 1370 48 72 Top 50



9/43

34 1420 61 48 Top 50

35 1400 80 34 Top 50

36 1370 53 64 Top 50

37 1410 53 50 Top 50

38 1390 68 32 Top 50

39 1450 71 71 Top 5040 1365 52 46 Top 50

41 1420 60 57 Top 50

42 1290 95 63 Top 50

43 1275 95 57 Top 50

44 1330 * 79 Top 50

45 1270 47 78 Top 50

46 1290 44 48 Top 50

47 1390 55 73 Top 50

48 1350 69 36 Top 50

49 1370 59 73 Top 5050 1290 53 66 Top 50

51 1290 95 47 Top 50

52 1310 47 62 Top 50

53 1290 28 72 Tier 2

54 1250 24 85 Tier 2

55 1280 38 84 Tier 2

56 1370 57 49 Tier 2

57 1420 56 64 Tier 2

58 1270 28 84 Tier 2

59 1280 40 83 Tier 2

60 1290 29 70 Tier 261 1260 37 64 Tier 2

62 1250 24 78 Tier 2

63 1300 54 84 Tier 2

64 1260 57 54 Tier 2

65 1250 27 63 Tier 2

66 1320 45 49 Tier 2

67 1350 21 56 Tier 2

68 1410 53 65 Tier 2

69 1220 21 82 Tier 2

70 1300 25 91 Tier 271 1300 29 80 Tier 2

72 1330 34 84 Tier 2

73 * * 69 Tier 2

74 1330 38 69 Tier 2

75 1250 24 69 Tier 2

76 1330 31 94 Tier 2

77 1290 37 65 Tier 2



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78 1330 32 72 Tier 2

79 1250 17 77 Tier 2

80 1240 27 78 Tier 2

81 1160 29 52 Tier 2

82 1290 32 58 Tier 2

83 1160 15 88 Tier 284 1270 31 82 Tier 2

85 1370 53 54 Tier 2

86 1330 36 67 Tier 2

87 1230 15 58 Tier 2

88 1310 53 42 Tier 2

89 1210 22 57 Tier 2

90 1240 25 56 Tier 2

91 1230 22 68 Tier 2

92 1300 40 58 Tier 2

93 1250 34 76 Tier 294 1250 28 86 Tier 2

95 1220 33 84 Tier 2

96 1200 94 85 Tier 2

97 1260 96 80 Tier 2

98 1260 21 86 Tier 2

99 1240 23 67 Tier 2

100 1330 22 79 Tier 2

101 1260 30 49 Tier 2

102 1220 33 78 Tier 2

103 1320 66 62 Tier 2

104 1300 41 62 Tier 2105 1300 20 83 Tier 2

106 1330 28 69 Tier 2

107 1300 31 80 Tier 2

108 1350 52 51 Tier 2

109 1240 19 67 Tier 2

110 1280 45 53 Tier 2

111 1330 30 75 Tier 2

112 1370 31 90 Tier 2

113 1420 46 92 Tier 2

114 1300 25 92 Tier 2115 1230 19 76 Tier 2

116 1330 31 86 Tier 2

117 1230 21 90 Tier 2

118 1270 32 62 Tier 2

119 1270 60 50 Tier 2

120 1220 23 80 Tier 2

121 1220 30 69 Tier 2



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122 1300 32 82 Tier 2

123 1250 26 62 Tier 2

124 1230 42 76 Tier 2

125 1370 50 75 Tier 2

126 1330 26 94 Tier 2

127 1230 18 80 Tier 2128 1270 37 61 Tier 2

129 1180 * 84 Tier 2

130 1360 45 71 Tier 2

131 1180 21 70 Tier 3

132 880 * 58 Tier 3

133 1210 18 54 Tier 3

134 1210 26 77 Tier 3

135 1140 16 79 Tier 3

136 1210 12 88 Tier 3

137 * * 70 Tier 3138 1250 18 73 Tier 3

139 1250 22 78 Tier 3

140 1260 34 82 Tier 3

141 1180 * 61 Tier 3

142 1180 12 80 Tier 3

143 1130 20 58 Tier 3

144 1250 24 61 Tier 3

145 1250 26 78 Tier 3

146 1300 26 71 Tier 3

147 1250 17 84 Tier 3

148 1230 22 53 Tier 3149 1370 41 68 Tier 3

150 1170 21 61 Tier 3

151 1310 24 * Tier 3

152 1230 16 70 Tier 3

153 1300 30 92 Tier 3

154 1200 22 87 Tier 3

155 1150 18 77 Tier 3

156 1300 44 66 Tier 3

157 1120 16 80 Tier 3

158 1140 17 71 Tier 3159 1180 21 70 Tier 3

160 1210 15 90 Tier 3

161 1300 32 87 Tier 3

162 1340 13 86 Tier 3

163 1340 33 90 Tier 3

164 1210 13 82 Tier 3

165 1180 30 71 Tier 3



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166 1250 19 83 Tier 3

167 1210 25 60 Tier 3

168 1070 * 74 Tier 3

169 1190 21 81 Tier 3

170 1280 34 65 Tier 3

171 1300 37 82 Tier 3172 1340 38 71 Tier 3

173 1210 * 90 Tier 3

174 1210 20 69 Tier 3

175 1130 13 74 Tier 3

176 1210 17 59 Tier 3

177 1190 17 75 Tier 3

178 1130 35 52 Tier 3

179 1190 22 58 Tier 3

180 1280 32 49 Tier 3

181 1250 23 97 Tier 3182 1210 * 98 Tier 3

183 1140 15 76 Tier 3

184 1170 * 78 Tier 3

185 1210 16 85 Tier 3

186 1210 21 94 Tier 3

187 1200 26 75 Tier 3

188 1250 38 82 Tier 4

189 1210 18 82 Tier 4

190 1090 7 86 Tier 4

191 1120 13 74 Tier 4

192 1170 11 79 Tier 4193 1150 * 72 Tier 4

194 1180 43 67 Tier 4

195 1130 10 59 Tier 4

196 1170 12 69 Tier 4

197 1210 10 76 Tier 4

198 1045 9 83 Tier 4

199 1060 7 80 Tier 4

200 970 * 46 Tier 4

201 1130 10 92 Tier 4

202 1170 36 93 Tier 4203 1170 14 79 Tier 4

204 1010 * 98 Tier 4

205 1250 10 78 Tier 4

206 1160 25 81 Tier 4

207 1210 10 65 Tier 4

208 1150 27 74 Tier 4

209 1170 * 78 Tier 4



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210 1120 19 70 Tier 4

211 1150 * 86 Tier 4

212 1143 * 65 Tier 4

213 960 6 40 Tier 4

214 1210 13 94 Tier 4

215 1170 10 71 Tier 4

216 1010 * 51 Tier 4

217 1100 14 60 Tier 4

218 1030 15 100 Tier 4

219 * * 79 Tier 4

220 1040 * 69 Tier 4

221 * * * Tier 4

222 1240 * 62 Tier 4

223 1170 11 90 Tier 4

224 * * 62 Tier 4

225 1100 17 80 Tier 4

226 1230 33 63 Tier 4

227 1210 17 73 Tier 4

228 1150 9 71 Tier 4

229 1150 21 81 Tier 4

230 1130 * 83 Tier 4

231 1210 19 88 Tier 4

232 1150 12 59 Tier 4

233 1160 * 67 Tier 4

234 1170 * 71 Tier 4

235 1240 20 57 Tier 4

236 1250 13 86 Tier 4

237 1140 22 79 Tier 4

238 1130 * 83 Tier 4

239 1170 12 79 Tier 4

240 1210 20 75 Tier 4

241 1170 * 93 Tier 4

242 1210 15 97 Tier 4

243 1160 20 86 Tier 4

244 1020 15 93 Tier 4

245 1210 14 99 Tier 4

246 1170 9 80 Tier 4

247 1210 10 75 Tier 4

248 * * * Tier 4

249 1170 14 89 Tier 4



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Just as was true in the twogroup situation, good way to get a feeling for the predictive

power of individual variables is to construct sidebyside boxplots. Here are the plots:



15/43

In order to make the pictures and output be in the correct order, the Tier number

variable has the coding of 4 for Top 50 schools, 3 for Tier 2, 2 for Tier 3, and 1 for Tier



16/43

4. Thus, we see the expected patterns of increasing SAT percentiles with increasing rating

quality, increasing percentages of students in the top 10% of their high school classes (with

a noticeable jump for Top 50 schools), and decreasing acceptance rate.

Here is a nominal logistic regression fit on all four predicting variables:

Nominal Logistic Regression: Tier number versus SAT 1st quar,

SAT 3rd quar, ...

Response Information

Variable Value Count

Tier num 4 51 (Reference Event)

3 76

2 49

1 44Total 220

220 cases were used

29 cases contained missing values

Logistic Regression Table

Odds 95% CI

Predictor Coef SE Coef Z P Ratio Lower Upper

Logit 1: (3/4)

Constant 22.42 10.37 2.16 0.031

SAT 1st -0.01031 0.01638 -0.63 0.529 0.99 0.96 1.02

SAT 3rd -0.00646 0.01933 -0.33 0.738 0.99 0.96 1.03

Top 10% -0.08199 0.02303 -3.56 0.000 0.92 0.88 0.96

Acceptan 0.04780 0.02877 1.66 0.097 1.05 0.99 1.11

Logit 2: (2/4)

Constant 43.70 11.80 3.70 0.000

SAT 1st -0.02054 0.01708 -1.20 0.229 0.98 0.95 1.01

SAT 3rd -0.01491 0.02002 -0.74 0.456 0.99 0.95 1.02

Top 10% -0.12208 0.03110 -3.93 0.000 0.89 0.83 0.94

Acceptan 0.05507 0.03366 1.64 0.102 1.06 0.99 1.13

Logit 3: (1/4)

Constant 63.46 12.91 4.92 0.000

SAT 1st -0.02446 0.01809 -1.35 0.176 0.98 0.94 1.01

SAT 3rd -0.02957 0.02097 -1.41 0.159 0.97 0.93 1.01

Top 10% -0.18080 0.04493 -4.02 0.000 0.83 0.76 0.91

Acceptan 0.08871 0.03821 2.32 0.020 1.09 1.01 1.18



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Log-likelihood = -145.229

Test that all slopes are zero: G = 309.013, DF = 12, P-Value = 0.000

Goodness-of-Fit Tests

Method Chi-Square DF P

Pearson 388.204 642 1.000

Deviance 290.459 642 1.000

The output gives separate logistic model estimates for each group relative to the group

with largest code number (I wanted these to be relative to the Top 50 group, which is why

I coded the groups 4321). We can note that each of the coefficients gets farther from

zero as the quality levels get farther apart. The two SAT variables both have insignificant

Wald statistics for all three regressions, but this isnt such a surprise, given how strongly

correlated they are:

Here is the model using only the SAT third quartile variable:



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Nominal Logistic Regression: Tier number versus SAT 3rd quar, Top 10%, ...



Tier num 4 51 (Reference Event)3 76

2 49

1 44

Total 220

220 cases were used



Odds 95% CIPredictor Coef SE Coef Z P Ratio Lower Upper

Logit 1: (3/4)

Constant 25.831 9.292 2.78 0.005

SAT 3rd -0.018220 0.006483 -2.81 0.005 0.98 0.97 0.99

Top 10% -0.07908 0.02166 -3.65 0.000 0.92 0.89 0.96

Acceptan 0.05507 0.02676 2.06 0.040 1.06 1.00 1.11

Logit 2: (2/4)

Constant 44.16 10.70 4.13 0.000

SAT 3rd -0.032497 0.007914 -4.11 0.000 0.97 0.95 0.98

Top 10% -0.13314 0.03252 -4.09 0.000 0.88 0.82 0.93Acceptan 0.06365 0.03196 1.99 0.046 1.07 1.00 1.13

Logit 3: (1/4)

Constant 64.15 11.92 5.38 0.000

SAT 3rd -0.050803 0.009187 -5.53 0.000 0.95 0.93 0.97

Top 10% -0.19109 0.04502 -4.24 0.000 0.83 0.76 0.90

Acceptan 0.10108 0.03654 2.77 0.006 1.11 1.03 1.19





Pearson 381.486 645 1.000

Deviance 296.227 645 1.000



19/43

All coefficients are significantly different from zero (the test that any are nonzero is of

course highly significant). The results are as expected, with the coefficients for SAT and

percentage in the top 10% being negative (indicating greater probability of being a top 50

school with increasing values, given the other variables), and positive for acceptance rate

(indicating a smaller probability of being a top 50 school with increasing value, given the

other variables). The estimated odds ratios have the usual interpretation, relative to being

a top 50 school. Here are estimated probabilities of group membership (note that EPROB1

refers to the highest numbered group, or group 4):

Row Tier EPROB1 EPROB2 EPROB3 EPROB4

1 Top 50 0.999856 0.000144 0.000000 0.000000

2 Top 50 0.999923 0.000077 0.000000 0.000000

3 Top 50 0.999882 0.000118 0.000000 0.000000

4 Top 50 0.999954 0.000046 0.000000 0.000000

5 Top 50 0.999916 0.000084 0.000000 0.000000

6 Top 50 0.999866 0.000134 0.000000 0.000000

7 Top 50 0.999342 0.000658 0.000000 0.000000

8 Top 50 0.998961 0.001039 0.000001 0.000000

9 Top 50 0.999609 0.000391 0.000000 0.000000

10 Top 50 0.999452 0.000548 0.000000 0.000000

11 Top 50 0.995404 0.004592 0.000004 0.00000012 Top 50 0.996913 0.003084 0.000003 0.000000

13 Top 50 0.999140 0.000860 0.000000 0.000000

14 Top 50 0.996116 0.003880 0.000005 0.000000

15 Top 50 0.996519 0.003477 0.000004 0.000000

16 Top 50 0.999335 0.000664 0.000000 0.000000

17 Top 50 0.992657 0.007329 0.000013 0.000000

18 Top 50 0.995543 0.004453 0.000004 0.000000

19 Top 50 0.993258 0.006730 0.000012 0.000000

20 Top 50 0.999075 0.000924 0.000001 0.000000

21 Top 50 0.986252 0.013712 0.000036 0.00000022 Top 50 0.905238 0.094142 0.000620 0.000000

23 Top 50 0.988791 0.011184 0.000025 0.000000

24 Top 50 0.996103 0.003891 0.000006 0.000000

25 Top 50 0.909319 0.090127 0.000554 0.000000

26 Top 50 0.996827 0.003169 0.000004 0.000000

27 Top 50 0.870207 0.128465 0.001327 0.000001

28 Top 50 0.985178 0.014763 0.000059 0.000000



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29 Top 50 0.841402 0.155733 0.002862 0.000003

30 Top 50 0.979080 0.020851 0.000068 0.000000

31 Top 50 0.992349 0.007636 0.000015 0.000000

32 Top 50 0.981591 0.018333 0.000077 0.000000

33 Top 50 0.253852 0.716928 0.029056 0.000164

34 Top 50 0.901544 0.097675 0.000782 0.00000035 Top 50 0.984143 0.015803 0.000053 0.000000

36 Top 50 0.442547 0.541759 0.015645 0.000049

37 Top 50 0.782978 0.213924 0.003095 0.000003

38 Top 50 0.957025 0.042662 0.000313 0.000000

39 Top 50 0.908014 0.091647 0.000339 0.000000

40 Top 50 0.643605 0.346651 0.009725 0.000018

41 Top 50 0.837356 0.161173 0.001470 0.000001

42 Top 50 0.846490 0.152098 0.001409 0.000002

43 Top 50 0.853562 0.144856 0.001579 0.000002

44 Top 50 * * * *45 Top 50 0.030794 0.810000 0.152093 0.007114

46 Top 50 0.166216 0.737908 0.094687 0.001189

47 Top 50 0.451106 0.537551 0.011313 0.000031

48 Top 50 0.902724 0.096053 0.001222 0.000001

49 Top 50 0.438863 0.548721 0.012377 0.000038

50 Top 50 0.134782 0.791307 0.072846 0.001065

51 Top 50 0.930190 0.069251 0.000559 0.000000

52 Top 50 0.148133 0.778947 0.072030 0.000890

53 Tier 2 0.011933 0.703982 0.263549 0.020537

54 Tier 2 0.001307 0.448768 0.412729 0.137196

55 Tier 2 0.012209 0.758864 0.211551 0.01737756 Tier 2 0.714999 0.279280 0.005713 0.000008

57 Tier 2 0.717888 0.278756 0.003352 0.000004

58 Tier 2 0.003673 0.604088 0.333510 0.058729

59 Tier 2 0.015490 0.777931 0.192981 0.013598

60 Tier 2 0.014704 0.717923 0.250295 0.017078

61 Tier 2 0.022501 0.724414 0.238900 0.014185

62 Tier 2 0.002048 0.478030 0.414020 0.105903

63 Tier 2 0.070022 0.853017 0.075265 0.001696

64 Tier 2 0.191790 0.732081 0.075167 0.000963

65 Tier 2 0.007132 0.574933 0.372288 0.04564766 Tier 2 0.269472 0.676124 0.054020 0.000384

67 Tier 2 0.054963 0.783200 0.158444 0.003393

68 Tier 2 0.611855 0.381852 0.006283 0.000010

69 Tier 2 0.000435 0.277029 0.448158 0.274378

70 Tier 2 0.003694 0.655507 0.294495 0.046304

71 Tier 2 0.010337 0.729555 0.240259 0.019849

72 Tier 2 0.024185 0.829412 0.140574 0.005830



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73 Tier 2 * * * *

74 Tier 2 0.075298 0.823935 0.098912 0.001855

75 Tier 2 0.003596 0.511454 0.410059 0.074891

76 Tier 2 0.010643 0.802556 0.174296 0.012505

77 Tier 2 0.039820 0.784168 0.169962 0.006049

78 Tier 2 0.039394 0.817239 0.139230 0.00413779 Tier 2 0.000957 0.367693 0.460952 0.170399

80 Tier 2 0.002183 0.482327 0.409719 0.105771

81 Tier 2 0.001421 0.275115 0.525872 0.197592

82 Tier 2 0.037874 0.753286 0.201467 0.007372

83 Tier 2 0.000010 0.040645 0.225494 0.733852

84 Tier 2 0.005612 0.652133 0.300932 0.041323

85 Tier 2 0.579857 0.409271 0.010848 0.000023

86 Tier 2 0.071329 0.818899 0.107668 0.002104

87 Tier 2 0.001442 0.328149 0.518108 0.152302

88 Tier 2 0.465306 0.506075 0.028501 0.00011889 Tier 2 0.001988 0.354548 0.505799 0.137665

90 Tier 2 0.006843 0.527324 0.413255 0.052578

91 Tier 2 0.001749 0.397071 0.467887 0.133292

92 Tier 2 0.089643 0.789316 0.118765 0.002276

93 Tier 2 0.006707 0.635998 0.315368 0.041928

94 Tier 2 0.001935 0.511365 0.382118 0.104582

95 Tier 2 0.001691 0.465744 0.400679 0.131887

96 Tier 2 0.221626 0.746064 0.031853 0.000458

97 Tier 2 0.573799 0.419636 0.006541 0.000023

98 Tier 2 0.001149 0.440182 0.416317 0.142352

99 Tier 2 0.002782 0.460295 0.441677 0.095245100 Tier 2 0.011038 0.742451 0.230612 0.015898

101 Tier 2 0.027456 0.673116 0.284950 0.014478

102 Tier 2 0.002480 0.490903 0.401138 0.105478

103 Tier 2 0.499126 0.486851 0.013975 0.000048

104 Tier 2 0.078941 0.800487 0.118091 0.002481

105 Tier 2 0.003592 0.609340 0.334916 0.052152

106 Tier 2 0.033036 0.797156 0.164306 0.005502

107 Tier 2 0.012442 0.749670 0.221585 0.016302

108 Tier 2 0.508596 0.474153 0.017201 0.000051

109 Tier 2 0.001749 0.396913 0.472787 0.128552110 Tier 2 0.118389 0.767367 0.112317 0.001927

111 Tier 2 0.028203 0.808459 0.157459 0.005878

112 Tier 2 0.029627 0.864806 0.102522 0.003045

113 Tier 2 0.195795 0.783524 0.020571 0.000111

114 Tier 2 0.003480 0.652558 0.295696 0.048266

115 Tier 2 0.000707 0.315933 0.468916 0.214444

116 Tier 2 0.016693 0.810268 0.164301 0.008737



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117 Tier 2 0.000343 0.282766 0.424747 0.292144

118 Tier 2 0.019748 0.704796 0.259547 0.015909

119 Tier 2 0.316287 0.636781 0.046572 0.000360

120 Tier 2 0.000651 0.317322 0.452902 0.229125

121 Tier 2 0.003123 0.477354 0.424657 0.094865

122 Tier 2 0.012168 0.756273 0.215439 0.016121123 Tier 2 0.006824 0.563512 0.381870 0.047793

124 Tier 2 0.009233 0.669551 0.286648 0.034569

125 Tier 2 0.253106 0.719874 0.026868 0.000151

126 Tier 2 0.006761 0.757128 0.215457 0.020654

127 Tier 2 0.000469 0.282835 0.458575 0.258121

128 Tier 2 0.032934 0.749108 0.208734 0.009224

129 Tier 2 * * * *

130 Tier 2 0.189068 0.768705 0.041901 0.000326

131 Tier 3 0.000258 0.175924 0.454509 0.369310

132 Tier 3 * * * *133 Tier 3 0.001479 0.306732 0.529405 0.162384

134 Tier 3 0.000856 0.334512 0.456370 0.208263

135 Tier 3 0.000010 0.035785 0.231653 0.732552

136 Tier 3 0.000050 0.109282 0.349211 0.541457

137 Tier 3 * * * *

138 Tier 3 0.001406 0.400674 0.459817 0.138103

139 Tier 3 0.001628 0.445275 0.429681 0.123415

140 Tier 3 0.005965 0.655991 0.296898 0.041146

141 Tier 3 * * * *

142 Tier 3 0.000028 0.068062 0.311639 0.620271

143 Tier 3 0.000082 0.077409 0.388868 0.533641144 Tier 3 0.005870 0.537391 0.402281 0.054458

145 Tier 3 0.002561 0.510349 0.396719 0.090371

146 Tier 3 0.013085 0.713248 0.255718 0.017948

147 Tier 3 0.000595 0.336418 0.447843 0.215144

148 Tier 3 0.004544 0.451570 0.467864 0.076022

149 Tier 3 0.193487 0.762595 0.043600 0.000318

150 Tier 3 0.000353 0.175977 0.485455 0.338215

151 Tier 3 * * * *

152 Tier 3 0.000726 0.295758 0.490358 0.213158

153 Tier 3 0.005673 0.716340 0.247724 0.030264154 Tier 3 0.000170 0.189834 0.404051 0.405946

155 Tier 3 0.000027 0.059787 0.296032 0.644153

156 Tier 3 0.081364 0.811172 0.105305 0.002160

157 Tier 3 0.000004 0.019680 0.170956 0.809361

158 Tier 3 0.000025 0.051566 0.295271 0.653138

159 Tier 3 * * * *

160 Tier 3 0.000069 0.130777 0.361489 0.507665



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161 Tier 3 0.009142 0.748280 0.222503 0.020076

162 Tier 3 0.003893 0.653794 0.304093 0.038220

163 Tier 3 0.019457 0.837521 0.136758 0.006264

164 Tier 3 0.000096 0.138173 0.397316 0.464415

165 Tier 3 0.000830 0.293517 0.470214 0.235440

166 Tier 3 0.000819 0.373836 0.442845 0.182500167 Tier 3 0.002346 0.389222 0.484450 0.123982

168 Tier 3 * * * *

169 Tier 3 0.000161 0.167326 0.411864 0.420649

170 Tier 3 0.024869 0.744960 0.219031 0.011140

171 Tier 3 0.019061 0.797784 0.173442 0.009713

172 Tier 3 0.081449 0.829272 0.087801 0.001478

173 Tier 3 * * * *

174 Tier 3 0.000685 0.277160 0.488300 0.233855

175 Tier 3 0.000006 0.024427 0.205496 0.770071

176 Tier 3 0.000927 0.273947 0.520952 0.204175177 Tier 3 0.000141 0.145160 0.421296 0.433403

178 Tier 3 0.001172 0.243873 0.517235 0.237720

179 Tier 3 0.000997 0.270403 0.517663 0.210937

180 Tier 3 0.050454 0.733503 0.209471 0.006572

181 Tier 3 0.000520 0.374342 0.402801 0.222337

182 Tier 3 * * * *

183 Tier 3 0.000011 0.036225 0.241240 0.722523

184 Tier 3 * * * *

185 Tier 3 0.000119 0.159569 0.400322 0.439990

186 Tier 3 0.000124 0.183424 0.379373 0.437079

187 Tier 3 0.000713 0.299774 0.463717 0.235796188 Tier 4 0.006995 0.672744 0.282918 0.037343

189 Tier 4 0.000202 0.195490 0.429002 0.375306

190 Tier 4 0.000000 0.002389 0.054541 0.943070

191 Tier 4 0.000004 0.018392 0.178468 0.803137

192 Tier 4 0.000017 0.049969 0.276188 0.673826

193 Tier 4 * * * *

194 Tier 4 0.005099 0.517585 0.396791 0.080524

195 Tier 4 0.000014 0.031422 0.273353 0.695211

196 Tier 4 0.000047 0.074535 0.358206 0.567212

197 Tier 4 0.000098 0.128507 0.412775 0.458621198 Tier 4 0.000000 0.000813 0.030871 0.968316

199 Tier 4 0.000000 0.001204 0.040078 0.958718

200 Tier 4 * * * *

201 Tier 4 0.000001 0.008808 0.101695 0.889496

202 Tier 4 0.000270 0.239470 0.386384 0.373875

203 Tier 4 0.000028 0.065204 0.306442 0.628326

204 Tier 4 * * * *



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205 Tier 4 0.000362 0.255739 0.472086 0.271813

206 Tier 4 0.000092 0.120302 0.366077 0.513529

207 Tier 4 0.000230 0.164589 0.481074 0.354107

208 Tier 4 0.000148 0.135270 0.401307 0.463275

209 Tier 4 * * * *

210 Tier 4 0.000016 0.037978 0.257464 0.704543211 Tier 4 * * * *

212 Tier 4 * * * *

213 Tier 4 0.000000 0.000264 0.027483 0.972253

214 Tier 4 0.000036 0.099745 0.317912 0.582307

215 Tier 4 0.000029 0.059220 0.322584 0.618167

216 Tier 4 * * * *

217 Tier 4 0.000007 0.019537 0.211926 0.768530

218 Tier 4 0.000000 0.000453 0.017821 0.981726

219 Tier 4 * * * *

220 Tier 4 * * * *221 Tier 4 * * * *

222 Tier 4 * * * *

223 Tier 4 0.000006 0.033964 0.206300 0.759730

224 Tier 4 * * * *

225 Tier 4 0.000002 0.012121 0.132725 0.855152

226 Tier 4 0.008077 0.583247 0.363303 0.045373

227 Tier 4 0.000343 0.219051 0.469708 0.310898

228 Tier 4 0.000010 0.032203 0.246355 0.721431

229 Tier 4 0.000032 0.068434 0.298180 0.633354

230 Tier 4 * * * *

231 Tier 4 0.000147 0.183429 0.401494 0.414929232 Tier 4 0.000047 0.062165 0.364819 0.572968

233 Tier 4 * * * *

234 Tier 4 * * * *

235 Tier 4 0.003714 0.449081 0.465123 0.082082

236 Tier 4 0.000306 0.264568 0.444771 0.290355

237 Tier 4 0.000029 0.061979 0.290085 0.647908

238 Tier 4 * * * *

239 Tier 4 0.000020 0.054667 0.286253 0.659060

240 Tier 4 0.000448 0.251977 0.467378 0.280197

241 Tier 4 * * * *242 Tier 4 0.000038 0.107614 0.315870 0.576478

243 Tier 4 0.000027 0.068640 0.285681 0.645652

244 Tier 4 0.000000 0.000451 0.019257 0.980292

245 Tier 4 0.000028 0.093271 0.293975 0.612727

246 Tier 4 0.000011 0.040198 0.249682 0.710108

247 Tier 4 0.000106 0.131686 0.419374 0.448834

248 Tier 4 * * * *



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249 Tier 4 0.000012 0.046757 0.239430 0.713801

The top 32 schools are correctly classified as Top 50 schools. Here is a classification

table (see the end of this handout to see how to get Minitab to create a table like this).

Note that the top 50 schools are best separated from the others (84% correctly classified),

while the Tier 3 schools (coded 2 here) are basically indistinguishable from the Tier 2 and

Tier 4 schools.

Rows: Tier num Columns: Predict

1 2 3 4 All

1 33 8 3 0 4415.07 3.65 1.37 -- 20.09

2 16 17 15 0 48

7.31 7.76 6.85 -- 21.92

3 1 11 57 7 76

0.46 5.02 26.03 3.20 34.70

4 0 0 8 43 51

-- -- 3.65 19.63 23.29

All 50 36 83 50 219

22.83 16.44 37.90 22.83 100.00

Cell Contents --

Count

% of Tbl

About 68% (150/219) of the schools are correctly classified, far higher than Cmax = 34.7%and Cpro = 33.3%.

These logistic regression results seem pretty good, especially when you consider that

weve ignored a key aspect of these data the ordering of the groups. Here are the results

of a logit proportional odds fit:



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Ordinal Logistic Regression: Tier number versus SAT 1st quar,

SAT 3rd quar, ...

Link Function: Logit



Tier num 1 44

2 49

3 76

4 51

Total 220

220 cases were used



Odds 95% CI


Const(1) 21.674 3.617 5.99 0.000

Const(2) 23.946 3.707 6.46 0.000

Const(3) 29.103 3.966 7.34 0.000

SAT 1st -0.008999 0.003907 -2.30 0.021 0.99 0.98 1.00

SAT 3rd -0.012005 0.004292 -2.80 0.005 0.99 0.98 1.00

Top 10% -0.07493 0.01475 -5.08 0.000 0.93 0.90 0.96

Acceptan 0.03005 0.01316 2.28 0.022 1.03 1.00 1.06





Pearson 416.820 650 1.000

Deviance 295.877 650 1.000

Measures of Association:

(Between the Response Variable and Predicted Probabilities)

Pairs Number Percent Summary Measures

Concordant 16522 92.6% Somers D 0.85

Discordant 1300 7.3% Goodman-Kruskal Gamma 0.85

Ties 21 0.1% Kendalls Tau-a 0.63



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Total 17843 100.0%

The variables have a highly significant relationship with probability of group mem-

bership. The negative signs are consistent with decreasing odds of being a lowerrated

school with increasing SAT quartiles and percentage in the top 10% of the high school

class, given the other variables; the positive sign is consistent with increasing odds of a

lowerrated school with increasing acceptance rate. The measures of association are high,

reinforcing the worth of the model. The pattern in the constants suggests that the highest

(Top 50) group is slightly more removed from the Tier 3 schools than the Tier 2 schools

are from the Tier 4 schools. Somers D is reasonably high; in this context, a set of objects

(one from each group) is concordant if the probability of being in group 1 is highest for the

actual member of group 1, the probability of being in group 1 or 2 is highest for the actual

member of group 2, the probability of being in groups 1, 2, or 3 is highest for the actual

member of group 3, and so on, where the groups are ordered from lowest to highest.

Here are the estimated group probabilities from this model, along with a classification

matrix:


1 Top 50 0.000000 0.000002 0.000312 0.999686

2 Top 50 0.000000 0.000001 0.000127 0.999872

3 Top 50 0.000000 0.000001 0.000190 0.999809

4 Top 50 0.000000 0.000000 0.000052 0.999948

5 Top 50 0.000000 0.000001 0.000111 0.999888

6 Top 50 0.000000 0.000002 0.000289 0.999709

7 Top 50 0.000001 0.000006 0.001133 0.998860

8 Top 50 0.000001 0.000010 0.001885 0.998104

9 Top 50 0.000001 0.000005 0.000960 0.999034

10 Top 50 0.000001 0.000005 0.000975 0.99901911 Top 50 0.000004 0.000031 0.005875 0.994091

12 Top 50 0.000003 0.000022 0.004229 0.995746

13 Top 50 0.000001 0.000008 0.001622 0.998368

14 Top 50 0.000004 0.000033 0.006232 0.993731

15 Top 50 0.000003 0.000024 0.004557 0.995417

16 Top 50 0.000001 0.000008 0.001598 0.998393

17 Top 50 0.000006 0.000050 0.009447 0.990498



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18 Top 50 0.000002 0.000021 0.003988 0.995989

19 Top 50 0.000005 0.000044 0.008398 0.991553

20 Top 50 0.000002 0.000017 0.003186 0.996796

21 Top 50 0.000015 0.000132 0.024691 0.975163

22 Top 50 0.000057 0.000492 0.086505 0.912946

23 Top 50 0.000008 0.000073 0.013895 0.98602324 Top 50 0.000004 0.000035 0.006611 0.993351

25 Top 50 0.000066 0.000572 0.099067 0.900296

26 Top 50 0.000006 0.000050 0.009606 0.990338

27 Top 50 0.000081 0.000705 0.119405 0.879809

28 Top 50 0.000015 0.000131 0.024660 0.975194

29 Top 50 0.000239 0.002071 0.284432 0.713259

30 Top 50 0.000017 0.000144 0.026949 0.972890

31 Top 50 0.000010 0.000089 0.016872 0.983028

32 Top 50 0.000017 0.000146 0.027312 0.972525

33 Top 50 0.000829 0.007151 0.574838 0.41718334 Top 50 0.000076 0.000663 0.113185 0.886076

35 Top 50 0.000017 0.000146 0.027338 0.972499

36 Top 50 0.000642 0.005552 0.513603 0.480203

37 Top 50 0.000139 0.001208 0.188405 0.810248

38 Top 50 0.000040 0.000348 0.062793 0.936819

39 Top 50 0.000042 0.000365 0.065610 0.933983

40 Top 50 0.000381 0.003300 0.387172 0.609147

41 Top 50 0.000082 0.000715 0.120965 0.878237

42 Top 50 0.000188 0.001636 0.239073 0.759104

43 Top 50 0.000172 0.001495 0.223118 0.775216

44 Top 50 * * * *45 Top 50 0.014783 0.112269 0.834893 0.038055

46 Top 50 0.004559 0.037971 0.842721 0.114749

47 Top 50 0.000398 0.003446 0.397412 0.598744

48 Top 50 0.000127 0.001104 0.175146 0.823622

49 Top 50 0.000491 0.004249 0.447927 0.547334

50 Top 50 0.004365 0.036428 0.839965 0.119242

51 Top 50 0.000089 0.000773 0.129409 0.869730

52 Top 50 0.003988 0.033401 0.833512 0.129099

53 Tier 2 0.025432 0.176560 0.775767 0.022241

54 Tier 2 0.116559 0.444797 0.434166 0.00447955 Tier 2 0.023324 0.164749 0.787674 0.024254

56 Tier 2 0.000212 0.001837 0.260794 0.737157

57 Tier 2 0.000196 0.001705 0.246757 0.751341

58 Tier 2 0.063846 0.334296 0.593229 0.008629

59 Tier 2 0.017905 0.132363 0.818200 0.031532

60 Tier 2 0.028999 0.195618 0.755894 0.019488

61 Tier 2 0.017625 0.130601 0.819747 0.032027



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62 Tier 2 0.069412 0.350372 0.572321 0.007895

63 Tier 2 0.006159 0.050540 0.855884 0.087418

64 Tier 2 0.003541 0.029776 0.823538 0.143145

65 Tier 2 0.051594 0.293817 0.643795 0.010794

66 Tier 2 0.001624 0.013910 0.717125 0.267341

67 Tier 2 0.130926 0.462778 0.402372 0.00392568 Tier 2 0.000342 0.002966 0.362349 0.634343

69 Tier 2 0.307740 0.504005 0.186922 0.001334

70 Tier 2 0.074447 0.363819 0.554408 0.007326

71 Tier 2 0.029013 0.195689 0.755819 0.019479

72 Tier 2 0.011150 0.087436 0.851402 0.050013

73 Tier 2 * * * *

74 Tier 2 0.003700 0.031073 0.827428 0.137799

75 Tier 2 0.075421 0.366305 0.551050 0.007224

76 Tier 2 0.018710 0.137365 0.813732 0.030193

77 Tier 2 0.012733 0.098454 0.844812 0.04400178 Tier 2 0.012921 0.099742 0.843957 0.043381

79 Tier 2 0.149147 0.480520 0.366957 0.003375

80 Tier 2 0.121257 0.451110 0.423350 0.004283

81 Tier 2 0.182210 0.501451 0.313682 0.002657

82 Tier 2 0.016359 0.122550 0.826629 0.034462

83 Tier 2 0.672492 0.279700 0.047519 0.000289

84 Tier 2 0.053140 0.299347 0.637047 0.010466

85 Tier 2 0.000363 0.003147 0.376086 0.620403

86 Tier 2 0.005790 0.047675 0.854030 0.092505

87 Tier 2 0.137961 0.470243 0.388101 0.003695

88 Tier 2 0.001168 0.010050 0.652148 0.33663389 Tier 2 0.104629 0.426649 0.463668 0.005054

90 Tier 2 0.070316 0.352861 0.569035 0.007787

91 Tier 2 0.143517 0.475580 0.377373 0.003530

92 Tier 2 0.005619 0.046344 0.852976 0.095061

93 Tier 2 0.045424 0.270376 0.671879 0.012321

94 Tier 2 0.116588 0.444838 0.434096 0.004478

95 Tier 2 0.161134 0.489603 0.346182 0.003081

96 Tier 2 0.003400 0.028628 0.819751 0.148221

97 Tier 2 0.000547 0.004731 0.474306 0.520416

98 Tier 2 0.131173 0.463055 0.401856 0.00391699 Tier 2 0.092676 0.405001 0.496544 0.005778

100 Tier 2 0.033047 0.215921 0.733960 0.017072

101 Tier 2 0.026937 0.184739 0.767331 0.020993

102 Tier 2 0.100644 0.419846 0.474234 0.005276

103 Tier 2 0.000781 0.006743 0.560835 0.431640

104 Tier 2 0.005878 0.048361 0.854524 0.091237

105 Tier 2 0.060313 0.323379 0.607145 0.009164



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106 Tier 2 0.015885 0.119490 0.829156 0.035470

107 Tier 2 0.025076 0.174597 0.777770 0.022557

108 Tier 2 0.000713 0.006158 0.538894 0.454235

109 Tier 2 0.152947 0.483603 0.360173 0.003277

110 Tier 2 0.006055 0.049736 0.855418 0.088791

111 Tier 2 0.016368 0.122609 0.826580 0.034443112 Tier 2 0.007245 0.058862 0.858672 0.075221

113 Tier 2 0.000963 0.008302 0.609665 0.381070

114 Tier 2 0.076545 0.369133 0.547212 0.007110

115 Tier 2 0.210625 0.510678 0.276477 0.002220

116 Tier 2 0.021036 0.151441 0.800641 0.026882

117 Tier 2 0.276815 0.510996 0.210641 0.001548

118 Tier 2 0.025427 0.176534 0.775794 0.022245

119 Tier 2 0.002036 0.017365 0.755181 0.225419

120 Tier 2 0.186846 0.503441 0.307137 0.002577

121 Tier 2 0.104728 0.426814 0.463409 0.005049122 Tier 2 0.024716 0.172597 0.779800 0.022887

123 Tier 2 0.053835 0.301793 0.634047 0.010325

124 Tier 2 0.045449 0.270478 0.671760 0.012314

125 Tier 2 0.001602 0.013725 0.714661 0.270011

126 Tier 2 0.053900 0.302018 0.633770 0.010312

127 Tier 2 0.198452 0.507561 0.291595 0.002392

128 Tier 2 0.014331 0.109267 0.837177 0.039225

129 Tier 2 * * * *

130 Tier 2 0.001486 0.012744 0.700634 0.285137

131 Tier 3 0.333781 0.495560 0.169476 0.001183

132 Tier 3 * * * *133 Tier 3 0.212939 0.511142 0.273730 0.002189

134 Tier 3 0.228714 0.513309 0.255980 0.001998

135 Tier 3 0.707682 0.251473 0.040601 0.000245

136 Tier 3 0.540923 0.378620 0.079953 0.000504

137 Tier 3 * * * *

138 Tier 3 0.126039 0.457095 0.412767 0.004099

139 Tier 3 0.129438 0.461092 0.405494 0.003976

140 Tier 3 0.048109 0.280853 0.659429 0.011609

141 Tier 3 * * * *

142 Tier 3 0.480966 0.418917 0.099477 0.000640143 Tier 3 0.406953 0.462431 0.129752 0.000864

144 Tier 3 0.116424 0.444609 0.434482 0.004485

145 Tier 3 0.084273 0.387367 0.521952 0.006409

146 Tier 3 0.077534 0.371583 0.543870 0.007013

147 Tier 3 0.236678 0.513791 0.247620 0.001911

148 Tier 3 0.081876 0.381929 0.529583 0.006612

149 Tier 3 0.002546 0.021614 0.787160 0.188680



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150 Tier 3 0.381878 0.475112 0.142050 0.000960

151 Tier 3 * * * *

152 Tier 3 0.203160 0.508905 0.285612 0.002323

153 Tier 3 0.053915 0.302072 0.633703 0.010309

154 Tier 3 0.400040 0.466047 0.133023 0.000889

155 Tier 3 0.570565 0.357427 0.071561 0.000447156 Tier 3 0.003702 0.031084 0.827460 0.137755

157 Tier 3 0.806007 0.169780 0.024070 0.000143

158 Tier 3 0.638487 0.306359 0.054818 0.000336

159 Tier 3 * * * *

160 Tier 3 0.499840 0.406646 0.092921 0.000594

161 Tier 3 0.028609 0.193587 0.758047 0.019757

162 Tier 3 0.327022 0.497951 0.173807 0.001220

163 Tier 3 0.018164 0.133980 0.816768 0.031089

164 Tier 3 0.477219 0.421303 0.100828 0.000650

165 Tier 3 0.193804 0.506053 0.297680 0.002463166 Tier 3 0.205724 0.509563 0.282427 0.002287

167 Tier 3 0.085375 0.389804 0.518502 0.006319

168 Tier 3 * * * *

169 Tier 3 0.382107 0.475002 0.141932 0.000959

170 Tier 3 0.017882 0.132217 0.818329 0.031572

171 Tier 3 0.034556 0.223153 0.725978 0.016314

172 Tier 3 0.010194 0.080633 0.854679 0.054493

173 Tier 3 * * * *

174 Tier 3 0.267686 0.512320 0.218373 0.001621

175 Tier 3 0.728842 0.234219 0.036718 0.000221

176 Tier 3 0.191224 0.505135 0.301136 0.002504177 Tier 3 0.389081 0.471596 0.138393 0.000931

178 Tier 3 0.210663 0.510686 0.276432 0.002219

179 Tier 3 0.223266 0.512750 0.261924 0.002061

180 Tier 3 0.015415 0.116424 0.831632 0.036530

181 Tier 3 0.226197 0.513074 0.258703 0.002027

182 Tier 3 * * * *

183 Tier 3 0.704523 0.254031 0.041197 0.000249

184 Tier 3 * * * *

185 Tier 3 0.357599 0.486137 0.155198 0.001065

186 Tier 3 0.334032 0.495469 0.169318 0.001182187 Tier 3 0.143615 0.475670 0.377188 0.003527

188 Tier 4 0.062111 0.329007 0.599998 0.008884

189 Tier 4 0.304529 0.504895 0.189222 0.001354

190 Tier 4 0.960039 0.035688 0.004248 0.000025

191 Tier 4 0.734741 0.229375 0.035670 0.000214

192 Tier 4 0.610335 0.327909 0.061377 0.000379

193 Tier 4 * * * *



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194 Tier 4 0.053161 0.299422 0.636955 0.010462

195 Tier 4 0.662128 0.287893 0.049676 0.000303

196 Tier 4 0.606646 0.330694 0.062275 0.000385

197 Tier 4 0.399714 0.466216 0.133180 0.000891

198 Tier 4 0.967368 0.029166 0.003446 0.000020

199 Tier 4 0.963347 0.032746 0.003885 0.000023200 Tier 4 * * * *

201 Tier 4 0.852517 0.129960 0.017420 0.000103

202 Tier 4 0.268173 0.512259 0.217951 0.001617

203 Tier 4 0.555747 0.368115 0.075664 0.000474

204 Tier 4 * * * *

205 Tier 4 0.233848 0.513664 0.250548 0.001941

206 Tier 4 0.507437 0.401593 0.090394 0.000576

207 Tier 4 0.323611 0.499107 0.176043 0.001239

208 Tier 4 0.425454 0.452337 0.121408 0.000801

209 Tier 4 * * * *210 Tier 4 0.672381 0.279788 0.047542 0.000289

211 Tier 4 * * * *

212 Tier 4 * * * *

213 Tier 4 0.983080 0.015148 0.001761 0.000010

214 Tier 4 0.566954 0.360049 0.072544 0.000453

215 Tier 4 0.480829 0.419004 0.099526 0.000641

216 Tier 4 * * * *

217 Tier 4 0.801101 0.173941 0.024811 0.000147

218 Tier 4 0.981847 0.016251 0.001892 0.000011

219 Tier 4 * * * *

220 Tier 4 * * * *221 Tier 4 * * * *

222 Tier 4 * * * *

223 Tier 4 0.830505 0.148888 0.020486 0.000121

224 Tier 4 * * * *

225 Tier 4 0.909650 0.080214 0.010077 0.000059

226 Tier 4 0.042254 0.257434 0.687035 0.013276

227 Tier 4 0.340428 0.493082 0.165341 0.001149

228 Tier 4 0.685304 0.269494 0.044930 0.000273

229 Tier 4 0.610529 0.327762 0.061330 0.000379

230 Tier 4 * * * *231 Tier 4 0.410843 0.460357 0.127949 0.000851

232 Tier 4 0.613680 0.325375 0.060571 0.000374

233 Tier 4 * * * *

234 Tier 4 * * * *

235 Tier 4 0.119504 0.448807 0.427335 0.004355

236 Tier 4 0.389082 0.471595 0.138392 0.000931

237 Tier 4 0.669189 0.282318 0.048200 0.000293



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238 Tier 4 * * * *

239 Tier 4 0.592374 0.341383 0.065835 0.000408

240 Tier 4 0.323859 0.499024 0.175880 0.001238

241 Tier 4 * * * *

242 Tier 4 0.552238 0.370619 0.076662 0.000481

243 Tier 4 0.614125 0.325037 0.060465 0.000373244 Tier 4 0.981841 0.016256 0.001892 0.000011

245 Tier 4 0.585345 0.346593 0.067641 0.000420

246 Tier 4 0.652181 0.295702 0.051801 0.000316

247 Tier 4 0.480823 0.419008 0.099528 0.000641

248 Tier 4 * * * *

249 Tier 4 0.707733 0.251431 0.040591 0.000245

Rows: Tier num Columns: Predict

1 2 3 4 All

1 30 11 3 0 44

13.70 5.02 1.37 -- 20.09

2 10 24 14 0 48

4.57 10.96 6.39 -- 21.92

3 1 16 54 5 76

0.46 7.31 24.66 2.28 34.70

4 0 0 6 45 51

-- -- 2.74 20.55 23.29

All 41 51 77 50 219

18.72 23.29 35.16 22.83 100.00

Cell Contents --

Count

% of Tbl

The model correctly classifies 69.9% of the schools, slightly better than the nominal

logistic regression model, and does so using fewer parameters (the nominal model based

on three predictors estimates three logistic regressions, each with four parameters, for 12

total parameters, while the proportional odds model uses only 7 parameters). This is the



34/43

benefit that comes from taking advantage of the natural ordering in the categories. The

ordinal model does a noticeably better job of differentiating the Tier 3 schools from the

others, and correctly classifies the top 43 schools.

Here is a probit proportional odds fit. Nothing much changes, although note that the

probit regression output doesnt include any estimated odds ratio information.

Ordinal Logistic Regression: Tier number versus SAT 1st quar,

SAT 3rd quar, ...

Link Function: Normit



Tier num 1 44

2 49

3 76

4 51

Total 220

220 cases were used



Predictor Coef SE Coef Z P

Const(1) 12.322 1.988 6.20 0.000

Const(2) 13.629 2.026 6.73 0.000

Const(3) 16.415 2.139 7.67 0.000

SAT 1st -0.004881 0.002251 -2.17 0.030

SAT 3rd -0.006962 0.002476 -2.81 0.005

Top 10% -0.041432 0.008005 -5.18 0.000

Acceptan 0.016093 0.007422 2.17 0.030







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Pearson 400.465 650 1.000

Deviance 295.539 650 1.000







Total 17843 100.0%

Just for fun, I also took a look at a proportional odds fit using the quality/reputation

variables. Probably not surprisingly, it does even better than the model based on incoming

student quality, correctly classifying 86% of the universities (including the top 46); after all,

these variables more directly represent the actual quality of the experience at the school,

rather than the raw material of incoming students. Three Tier 2 schools are misclassified

as Top 50 George Washington University, Indiana University, and Syracuse University

these might be up and comers for the future.

Ordinal Logistic Regression: Tier number versus Alumni givin,

Pct under 20, ...

Link Function: Logit



Tier num 1 53

2 53

3 77

4 52

Total 235

235 cases were used



Odds 95% CI



36/43


Const(1) 52.608 7.899 6.66 0.000

Const(2) 59.891 8.798 6.81 0.000

Const(3) 71.74 10.20 7.04 0.000

Alumni g -0.10911 0.03973 -2.75 0.006 0.90 0.83 0.97

Pct unde -0.12117 0.02624 -4.62 0.000 0.89 0.84 0.93Freshman -0.18775 0.06565 -2.86 0.004 0.83 0.73 0.94

Graduati -0.22979 0.04341 -5.29 0.000 0.79 0.73 0.87

Reputati -8.809 1.293 -6.81 0.000 0.00 0.00 0.00




Method Chi-Square DF PPearson 137.218 697 1.000

Deviance 128.965 697 1.000







Total 20487 100.0%


1 Top 50 0.00000 0.000000 0.000000 1.00000

2 Top 50 0.00000 0.000000 0.000000 1.00000

3 Top 50 0.00000 0.000000 0.000000 1.00000

4 Top 50 0.00000 0.000000 0.000000 1.00000

5 Top 50 0.00000 0.000000 0.000000 1.000006 Top 50 0.00000 0.000000 0.000000 1.00000

7 Top 50 0.00000 0.000000 0.000000 1.00000

8 Top 50 0.00000 0.000000 0.000000 1.00000

9 Top 50 0.00000 0.000000 0.000000 1.00000

10 Top 50 0.00000 0.000000 0.000000 1.00000

11 Top 50 0.00000 0.000000 0.000000 1.00000

12 Top 50 0.00000 0.000000 0.000000 1.00000



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13 Top 50 0.00000 0.000000 0.000000 1.00000

14 Top 50 0.00000 0.000000 0.000000 1.00000

15 Top 50 0.00000 0.000000 0.000000 1.00000

16 Top 50 0.00000 0.000000 0.000000 1.00000

17 Top 50 0.00000 0.000000 0.000000 1.00000

18 Top 50 0.00000 0.000000 0.000003 1.0000019 Top 50 0.00000 0.000000 0.000001 1.00000

20 Top 50 0.00000 0.000000 0.000000 1.00000

21 Top 50 0.00000 0.000000 0.000000 1.00000

22 Top 50 0.00000 0.000000 0.000003 1.00000

23 Top 50 0.00000 0.000000 0.000013 0.99999

24 Top 50 0.00000 0.000000 0.000002 1.00000

25 Top 50 0.00000 0.000000 0.000002 1.00000

26 Top 50 0.00000 0.000000 0.000072 0.99993

27 Top 50 0.00000 0.000000 0.000932 0.99907

28 Top 50 0.00000 0.000000 0.000033 0.9999729 Top 50 0.00000 0.000000 0.000158 0.99984

30 Top 50 0.00000 0.000000 0.000131 0.99987

31 Top 50 0.00000 0.000000 0.005367 0.99463

32 Top 50 0.00000 0.000000 0.012775 0.98723

33 Top 50 0.00000 0.000000 0.000705 0.99930

34 Top 50 0.00000 0.000000 0.000226 0.99977

35 Top 50 0.00000 0.000000 0.002432 0.99757

36 Top 50 0.00000 0.000000 0.004522 0.99548

37 Top 50 0.00000 0.000001 0.122112 0.87789

38 Top 50 0.00000 0.000000 0.016972 0.98303

39 Top 50 0.00000 0.000000 0.054862 0.9451440 Top 50 0.00000 0.000001 0.073599 0.92640

41 Top 50 0.00000 0.000001 0.141683 0.85832

42 Top 50 0.00000 0.000001 0.146120 0.85388

43 Top 50 0.00000 0.000002 0.179780 0.82022

44 Top 50 0.00000 0.000002 0.252763 0.74723

45 Top 50 0.00000 0.000000 0.062867 0.93713

46 Top 50 0.00000 0.000000 0.029337 0.97066

47 Top 50 0.00000 0.000012 0.627161 0.37283

48 Top 50 0.00000 0.000044 0.859737 0.14022

49 Top 50 0.00000 0.000002 0.241206 0.7587950 Top 50 0.00000 0.000011 0.600679 0.39931

51 Top 50 0.00000 0.000050 0.876629 0.12332

52 Top 50 0.00000 0.000000 0.027835 0.97216

53 Tier 2 0.00001 0.016829 0.982745 0.00041

54 Tier 2 0.00001 0.017989 0.981611 0.00039

55 Tier 2 0.00000 0.003715 0.994381 0.00190

56 Tier 2 0.00000 0.000021 0.751128 0.24885



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57 Tier 2 0.00000 0.000775 0.990145 0.00908

58 Tier 2 0.00001 0.009788 0.989487 0.00072

59 Tier 2 0.00009 0.119619 0.880235 0.00005

60 Tier 2 0.00000 0.001775 0.994244 0.00398

61 Tier 2 0.00001 0.019471 0.980158 0.00036

62 Tier 2 0.00010 0.131609 0.868240 0.0000563 Tier 2 0.00002 0.034896 0.964883 0.00020

64 Tier 2 0.00002 0.030800 0.968954 0.00022

65 Tier 2 0.00000 0.000294 0.976152 0.02355

66 Tier 2 0.00000 0.000001 0.135109 0.86489

67 Tier 2 0.00007 0.095668 0.904192 0.00007

68 Tier 2 0.00002 0.027481 0.972248 0.00025

69 Tier 2 0.00000 0.000004 0.365555 0.63444

70 Tier 2 0.00000 0.001842 0.994322 0.00383

71 Tier 2 0.00003 0.046812 0.953009 0.00014

72 Tier 2 0.00000 0.001681 0.994116 0.0042073 Tier 2 * * * *

74 Tier 2 0.00000 0.000044 0.860519 0.13944

75 Tier 2 0.00000 0.000561 0.986944 0.01249

76 Tier 2 0.00006 0.084820 0.915040 0.00008

77 Tier 2 0.00001 0.010249 0.989058 0.00069

78 Tier 2 0.00000 0.000133 0.949174 0.05069

79 Tier 2 0.00000 0.003169 0.994599 0.00223

80 Tier 2 0.00000 0.000013 0.649601 0.35039

81 Tier 2 0.00027 0.279139 0.720576 0.00002

82 Tier 2 0.00000 0.000197 0.964923 0.03488

83 Tier 2 0.00009 0.117156 0.882699 0.0000584 Tier 2 0.00000 0.000314 0.977593 0.02209

85 Tier 2 0.00073 0.516101 0.483158 0.00001

86 Tier 2 0.00000 0.003905 0.994283 0.00181

87 Tier 2 0.00023 0.246499 0.753254 0.00002

88 Tier 2 0.00000 0.000055 0.885548 0.11440

89 Tier 2 0.00000 0.001931 0.994409 0.00366

90 Tier 2 0.00012 0.147666 0.852174 0.00004

91 Tier 2 0.00006 0.081305 0.918554 0.00008

92 Tier 2 0.00000 0.000001 0.152573 0.84743

93 Tier 2 0.00002 0.025593 0.974118 0.0002794 Tier 2 0.00001 0.007223 0.991796 0.00098

95 Tier 2 0.00000 0.005877 0.992919 0.00120

96 Tier 2 0.00000 0.003873 0.994300 0.00182

97 Tier 2 0.00000 0.003466 0.994493 0.00204

98 Tier 2 0.00000 0.000046 0.866436 0.13352

99 Tier 2 0.00000 0.000382 0.981380 0.01824

100 Tier 2 0.00032 0.315915 0.683752 0.00002



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101 Tier 2 0.00000 0.000222 0.968763 0.03101

102 Tier 2 0.00001 0.011614 0.987774 0.00060

103 Tier 2 0.00000 0.000040 0.850082 0.14988

104 Tier 2 0.00000 0.000047 0.868526 0.13143

105 Tier 2 0.00000 0.000033 0.822234 0.17773

106 Tier 2 0.00003 0.046344 0.953476 0.00015107 Tier 2 0.00042 0.379489 0.620079 0.00001

108 Tier 2 0.00000 0.000040 0.848745 0.15122

109 Tier 2 0.00001 0.009512 0.989743 0.00074

110 Tier 2 0.00000 0.006616 0.992314 0.00107

111 Tier 2 0.00000 0.000316 0.977690 0.02199

112 Tier 2 0.00000 0.000725 0.989579 0.00970

113 Tier 2 0.00197 0.739664 0.258364 0.00000

114 Tier 2 0.00013 0.160416 0.839416 0.00004

115 Tier 2 0.00000 0.004521 0.993914 0.00156

116 Tier 2 0.00073 0.515468 0.483792 0.00001117 Tier 2 0.00000 0.005358 0.993321 0.00132

118 Tier 2 0.00000 0.006269 0.992602 0.00112

119 Tier 2 0.00001 0.011158 0.988206 0.00063

120 Tier 2 0.00000 0.005116 0.993501 0.00138

121 Tier 2 0.00011 0.136064 0.863783 0.00005

122 Tier 2 0.00008 0.107246 0.892612 0.00006

123 Tier 2 0.00007 0.093060 0.906800 0.00007

124 Tier 2 0.00084 0.549711 0.449442 0.00001

125 Tier 2 0.00533 0.880965 0.113705 0.00000

126 Tier 2 0.00022 0.238119 0.761643 0.00002

127 Tier 2 0.00000 0.002915 0.994659 0.00242128 Tier 2 0.00000 0.000379 0.981224 0.01840

129 Tier 2 0.00001 0.013917 0.985571 0.00050

130 Tier 2 0.00000 0.000063 0.898900 0.10104

131 Tier 3 0.76257 0.237217 0.000214 0.00000

132 Tier 3 * * * *

133 Tier 3 0.50034 0.498979 0.000686 0.00000

134 Tier 3 0.00112 0.619623 0.379249 0.00000

135 Tier 3 0.42271 0.576355 0.000938 0.00000

136 Tier 3 0.01228 0.935338 0.052379 0.00000

137 Tier 3 * * * *138 Tier 3 0.00018 0.208827 0.790964 0.00003

139 Tier 3 0.00009 0.115169 0.884687 0.00005

140 Tier 3 0.04902 0.937820 0.013159 0.00000

141 Tier 3 0.01634 0.943919 0.039746 0.00000

142 Tier 3 0.00503 0.875274 0.119695 0.00000

143 Tier 3 0.92291 0.077029 0.000057 0.00000

144 Tier 3 0.00033 0.323247 0.676409 0.00001



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145 Tier 3 0.00139 0.668447 0.330157 0.00000

146 Tier 3 0.33113 0.667479 0.001386 0.00000

147 Tier 3 0.18946 0.807607 0.002932 0.00000

148 Tier 3 0.00398 0.849340 0.146676 0.00000

149 Tier 3 0.34739 0.651317 0.001290 0.00000

150 Tier 3 0.27012 0.728025 0.001854 0.00000151 Tier 3 * * * *

152 Tier 3 0.00727 0.906946 0.085781 0.00000

153 Tier 3 0.05028 0.936903 0.012817 0.00000

154 Tier 3 0.00023 0.252663 0.747084 0.00002

155 Tier 3 0.46786 0.531355 0.000781 0.00000

156 Tier 3 0.00324 0.822085 0.174677 0.00000

157 Tier 3 0.00012 0.153229 0.846607 0.00004

158 Tier 3 0.02174 0.948260 0.029995 0.00000

159 Tier 3 0.03461 0.946580 0.018814 0.00000

160 Tier 3 0.54071 0.458708 0.000584 0.00000161 Tier 3 0.62305 0.376531 0.000416 0.00000

162 Tier 3 * * * *

163 Tier 3 0.03039 0.948149 0.021457 0.00000

164 Tier 3 0.11835 0.876557 0.005094 0.00000

165 Tier 3 0.00463 0.866699 0.128667 0.00000

166 Tier 3 0.00156 0.692339 0.306102 0.00000

167 Tier 3 0.15216 0.844020 0.003815 0.00000

168 Tier 3 0.58284 0.416667 0.000492 0.00000

169 Tier 3 0.00694 0.903544 0.089512 0.00000

170 Tier 3 0.00619 0.894402 0.099409 0.00000

171 Tier 3 0.06355 0.926421 0.010027 0.00000172 Tier 3 0.13807 0.857661 0.004273 0.00000

173 Tier 3 0.36633 0.632486 0.001188 0.00000

174 Tier 3 0.09295 0.900383 0.006663 0.00000

175 Tier 3 0.30891 0.689556 0.001535 0.00000

176 Tier 3 0.28669 0.711602 0.001707 0.00000

177 Tier 3 0.00484 0.871394 0.123762 0.00000

178 Tier 3 0.31304 0.685456 0.001506 0.00000

179 Tier 3 0.68596 0.313727 0.000315 0.00000

180 Tier 3 0.16877 0.827858 0.003374 0.00000

181 Tier 3 0.06482 0.925359 0.009819 0.00000182 Tier 3 0.52755 0.471831 0.000615 0.00000

183 Tier 3 0.02069 0.947802 0.031505 0.00000

184 Tier 3 0.11295 0.881683 0.005369 0.00000

185 Tier 3 0.02136 0.948109 0.030532 0.00000

186 Tier 3 0.01761 0.945460 0.036930 0.00000

187 Tier 3 0.01879 0.946556 0.034659 0.00000

188 Tier 4 0.61904 0.380538 0.000423 0.00000



41/43

189 Tier 4 0.83065 0.169213 0.000140 0.00000

190 Tier 4 0.99998 0.000017 0.000000 0.00000

191 Tier 4 0.37105 0.627784 0.001164 0.00000

192 Tier 4 0.99960 0.000404 0.000000 0.00000

193 Tier 4 0.99422 0.005779 0.000004 0.00000

194 Tier 4 0.75086 0.248914 0.000228 0.00000195 Tier 4 0.84109 0.158777 0.000130 0.00000

196 Tier 4 0.99799 0.002013 0.000001 0.00000

197 Tier 4 0.38041 0.618468 0.001118 0.00000

198 Tier 4 0.56272 0.436751 0.000534 0.00000

199 Tier 4 0.99485 0.005142 0.000004 0.00000

200 Tier 4 * * * *

201 Tier 4 0.31630 0.682213 0.001484 0.00000

202 Tier 4 0.43901 0.560112 0.000878 0.00000

203 Tier 4 0.99790 0.002098 0.000001 0.00000

204 Tier 4 * * * *205 Tier 4 0.29319 0.705152 0.001654 0.00000

206 Tier 4 0.74971 0.250056 0.000229 0.00000

207 Tier 4 0.63347 0.366129 0.000398 0.00000

208 Tier 4 0.99798 0.002019 0.000001 0.00000

209 Tier 4 0.96338 0.036596 0.000026 0.00000

210 Tier 4 0.80464 0.195195 0.000167 0.00000

211 Tier 4 0.98057 0.019414 0.000014 0.00000

212 Tier 4 0.83908 0.160784 0.000132 0.00000

213 Tier 4 0.65765 0.341988 0.000358 0.00000

214 Tier 4 0.19043 0.806657 0.002914 0.00000

215 Tier 4 0.58495 0.414558 0.000487 0.00000216 Tier 4 0.99491 0.005085 0.000004 0.00000

217 Tier 4 0.99919 0.000805 0.000001 0.00000

218 Tier 4 0.99995 0.000046 0.000000 0.00000

219 Tier 4 1.00000 0.000000 0.000000 0.00000

220 Tier 4 0.92896 0.070985 0.000053 0.00000

221 Tier 4 0.76114 0.238649 0.000216 0.00000

222 Tier 4 0.93258 0.067370 0.000050 0.00000

223 Tier 4 0.99558 0.004421 0.000003 0.00000

224 Tier 4 * * * *

225 Tier 4 * * * *226 Tier 4 0.79828 0.201544 0.000174 0.00000

227 Tier 4 * * * *

228 Tier 4 0.12587 0.869380 0.004751 0.00000

229 Tier 4 0.85457 0.145310 0.000117 0.00000

230 Tier 4 0.99999 0.000009 0.000000 0.00000

231 Tier 4 0.98875 0.011238 0.000008 0.00000

232 Tier 4 0.97217 0.027806 0.000020 0.00000



42/43

233 Tier 4 * * * *

234 Tier 4 0.98806 0.011931 0.000008 0.00000

235 Tier 4 0.99291 0.007086 0.000005 0.00000

236 Tier 4 0.22876 0.768933 0.002312 0.00000

237 Tier 4 0.98664 0.013352 0.000009 0.00000

238 Tier 4 0.99976 0.000242 0.000000 0.00000239 Tier 4 0.99050 0.009495 0.000007 0.00000

240 Tier 4 0.99876 0.001242 0.000001 0.00000

241 Tier 4 0.99765 0.002351 0.000002 0.00000

242 Tier 4 * * * *

243 Tier 4 0.98935 0.010642 0.000007 0.00000

244 Tier 4 0.99998 0.000021 0.000000 0.00000

245 Tier 4 0.96171 0.038260 0.000027 0.00000

246 Tier 4 * * * *

247 Tier 4 0.99538 0.004614 0.000003 0.00000

248 Tier 4 * * * *

249 Tier 4 0.98993 0.010067 0.000007 0.00000

Minitab commands

Nominal (multigroup, or polytomous) logistic regression is obtained by clicking on

Stat Regression Nominal Logistic Regression. Enter the variable that defines

the groups under Response:, and enter the model in the same was as is done for binary

logistic regression. Estimated probabilities of membership for each group are obtained by

clicking on Storage, entering the number of groups in the box next to Enter the number

of events:, and clicking the box next to Event probabilities.

A proportional odds model is fit by clicking on Stat Regression Ordinal

Logistic Regression. All commands are entered in the same way as is done for nominal

logistic regression.

Creating a classification matrix when there are more than two groups

Creating a classification matrix from a logistic regression when there are more than

two groups is not easy in Minitab, but it can be done, as follows:

(1) Say the grouping (response) variable takes on the values 1, 2, 3, 4, and 5, and is

called Group (change these instructions in the obvious way for a different number



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of groups). When fitting a nominal or ordinal logistic regression, click on Storage,

enter the number of groups under Enter the number of events: and click on Event

probabilities. This will create variables EPROB1, EPROB2, etc., up to EPROB5. If you

have fit a nominal logistic regression the variable EPROB5 is the estimated probability

that the observation belongs to group 1, EPROB4 is the probability of belonging to

group 2, etc., while if you have fit an ordinal logistic regression the variable EPROB1

is the estimated probability that the observation belongs to group 1, EPROB2 is the

probability of belonging to group 2, etc.

(2) Click on Calc Row Statistics. Click the radio button for Maximum, enter the

EPROB variables under Input variables:, and enter Maxvar under Store result

in:.

(3) Go to the calculator, and create a variable Predict as follows. If you are running a

nominal logistic regression, enter

5 (EPROB1 = Maxvar)+4 (EPROB2 = Maxvar)

+3 (EPROB3 = Maxvar)+2 (EPROB4 = Maxvar) + (EPROB5 = Maxvar)

If you are running an ordinal logistic regression, enter

5 (EPROB5 = Maxvar)+4 (EPROB4 = Maxvar)

+3 (EPROB3 = Maxvar)+2 (EPROB2 = Maxvar) + (EPROB1 = Maxvar)

(4) Go to Stat Tables Cross Tabulation and Chi-Square, enter Group and Pre-

dict under Categorical variables:, and click on Counts and Total percents.

One point about the table: any observations that have missing values in any of the

predictors will have as the entry under Predict the sum of the numbers of the groups

(in this case 1 + 2 + 3 + 4 + 5 = 15). This column in the classification table should be

ignored and/or removed. Note also that once again Minitab will combine all observations

with the same covariate pattern, so to get a classification matrix from a nominal or ordinal

logistic regression with replications at different covariate patterns youll want to jitter the

predictor variables before getting the estimated event probabilities.

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