Problem Set 5
2024-11-18
Problem 1
For this problem, you will use the College dataset from the ISLR2 library. This dataset contains 18 features describing 777 colleges based on the 1995 issue of the US News and World Report Rankings. The goal of this analysis will be to cluster the colleges using the available information, and to identify similarities between public and private schools, as well as schools in certain categories.
Step 0
Separate the variable “Private” from the rest of the data, leaving you with 17 features. Scale the remaining features and save this dataframe as “college.scaled”. Extract the names of the colleges by creating a new variable that is equal to the row names of the College dataframe. Use the rownames() function in R to do this.
priv = College$Private
college_features = College[,-which(names(College)=="Private")]
college.scaled = scale(college_features)
college_names = rownames(College)Step 1
Set the random seed to 1 for this analysis. Make sure you run this command immediately before you conduct your analysis. To ensure consistency, only run each step one time so that the random seed is not changed.
Conduct K-Means clustering on the scaled data using 1 to 20 clusters. Create a plot of the within sum of squares relative to the number of clusters. Include the plot here. Comment on what you see.
set.seed(1)
wss = numeric(20)
for (k in 1:20) {
kmeans_result <- kmeans(college.scaled, centers = k, nstart = 20)
wss[k] <- kmeans_result$tot.withinss
}## Warning: did not converge in 10 iterationsplot(1:20, wss, type= "b")
*Answer: There seems to be an elbow around 3 and 5, indicating an optimal value for K.
Step 2
Run K-means with 5 clusters. What are the sizes of the clusters? Report the centers of the clusters. Do you observe any trends?
set.seed(1)
k5 = kmeans(college.scaled, centers = 5, nstart = 20)
cluster_sizes = k5$size
print(cluster_sizes)## [1] 115 263 62 76 261print(k5$centers)## Apps Accept Enroll Top10perc Top25perc F.Undergrad
## 1 0.2721386 0.3137047 0.5386923124 -0.5315055 -0.3660141 0.6450210
## 2 -0.3033590 -0.2667640 -0.3481553781 0.1838807 0.2786669 -0.3805719
## 3 2.4003436 2.6005038 2.6592525186 0.4556717 0.6064091 2.6728336
## 4 0.5032428 0.1287592 -0.0007639187 1.9397421 1.5722245 -0.1330025
## 5 -0.5309589 -0.5246515 -0.5180090031 -0.6241745 -0.7213954 -0.4969138
## P.Undergrad Outstate Room.Board Books Personal PhD
## 1 0.7729491 -0.9925279 -0.672522003 -0.004466688 0.73071917 0.2519379
## 2 -0.2930735 0.5000230 0.432729121 -0.079594076 -0.32701168 0.3461233
## 3 1.4738038 -0.4152657 -0.008509953 0.348983507 0.70634330 0.7991991
## 4 -0.4158054 1.8070466 1.225978520 0.279490402 -0.47979093 1.1530152
## 5 -0.2742737 -0.4940782 -0.494691490 -0.082112401 -0.02052799 -0.9853747
## Terminal S.F.Ratio perc.alumni Expend Grad.Rate
## 1 0.2848015 0.95205267 -0.7982466 -0.55654090 -0.74731926
## 2 0.3755673 -0.25781012 0.4315247 0.06203482 0.50293905
## 3 0.7454456 0.49319336 -0.4713327 0.09393798 -0.09175983
## 4 1.1105509 -1.22531645 1.3447607 2.12257943 1.15417907
## 5 -1.0043903 0.07993895 -0.3627273 -0.45767489 -0.49179983*Answer: The cluster sizes give a representation of how many colleges fall into the clusters generated, indicating that certain features may have more commonalities amongst colleges than others. Within cluster centers, the higher value’s in particular categories such as Top10Perc or GradRate contain schools that are more prestigious or stronger academically. #### Step 3
Identify the proportion of schools in each cluster that are private. Report that here. Compare these results to the centers of the clusters. What might you infer?
Private = College$Private
kmeans_5 = kmeans(college.scaled, centers = 5, nstart = 10)
private_table = table(Private, kmeans_5$cluster)
prop.table(private_table, margin=2)##
## Private 1 2 3 4 5
## No 0.04562738 0.14942529 0.90322581 0.90434783 0.01315789
## Yes 0.95437262 0.85057471 0.09677419 0.09565217 0.98684211*Answer: Based on the results, it can be inferred that a majority of schools in the 1, 2, and 5 cluster are private schools, whereas, clusters 3 and 4 contain primarily public schools.
Step 4
Look at the school names in each cluster. Come up with a classification scheme for the groups of schools. Again, does this track with the centers from step 2? Identify three features that seem to be most associated with “elite” schools.
collage_names = rownames(College)
split(collage_names, kmeans_5$cluster)## $`1`
## [1] "Albertson College"
## [2] "Albertus Magnus College"
## [3] "Albion College"
## [4] "Albright College"
## [5] "Alfred University"
## [6] "Allegheny College"
## [7] "Allentown Coll. of St. Francis de Sales"
## [8] "Alma College"
## [9] "Arkansas College (Lyon College)"
## [10] "Assumption College"
## [11] "Augustana College IL"
## [12] "Austin College"
## [13] "Baldwin-Wallace College"
## [14] "Barry University"
## [15] "Beaver College"
## [16] "Bellarmine College"
## [17] "Beloit College"
## [18] "Bentley College"
## [19] "Berry College"
## [20] "Birmingham-Southern College"
## [21] "Bradford College"
## [22] "Bradley University"
## [23] "Bridgewater College"
## [24] "Buena Vista College"
## [25] "Butler University"
## [26] "Cabrini College"
## [27] "Caldwell College"
## [28] "California Lutheran University"
## [29] "Calvin College"
## [30] "Canisius College"
## [31] "Capital University"
## [32] "Carroll College"
## [33] "Carthage College"
## [34] "Castleton State College"
## [35] "Catholic University of America"
## [36] "Cedar Crest College"
## [37] "Centenary College of Louisiana"
## [38] "Central College"
## [39] "Chapman University"
## [40] "Chatham College"
## [41] "Chestnut Hill College"
## [42] "Christendom College"
## [43] "Clark University"
## [44] "Clarkson University"
## [45] "Coe College"
## [46] "Coker College"
## [47] "College Misericordia"
## [48] "College of Mount St. Vincent"
## [49] "College of Notre Dame of Maryland"
## [50] "College of Saint Benedict"
## [51] "College of Saint Catherine"
## [52] "College of Saint Elizabeth"
## [53] "College of Saint Rose"
## [54] "College of St. Scholastica"
## [55] "College of Wooster"
## [56] "Columbia College"
## [57] "Converse College"
## [58] "Cornell College"
## [59] "Creighton University"
## [60] "Denison University"
## [61] "DePauw University"
## [62] "Dickinson College"
## [63] "Drake University"
## [64] "Drury College"
## [65] "Earlham College"
## [66] "Eckerd College"
## [67] "Elizabethtown College"
## [68] "Elmira College"
## [69] "Elms College"
## [70] "Emory & Henry College"
## [71] "Erskine College"
## [72] "Eureka College"
## [73] "Fairfield University"
## [74] "Florida Institute of Technology"
## [75] "Fordham University"
## [76] "Franklin College"
## [77] "Furman University"
## [78] "Gonzaga University"
## [79] "Gordon College"
## [80] "Goucher College"
## [81] "Grove City College"
## [82] "Guilford College"
## [83] "Gustavus Adolphus College"
## [84] "Hamline University"
## [85] "Hampden - Sydney College"
## [86] "Hanover College"
## [87] "Harding University"
## [88] "Hartwick College"
## [89] "Hendrix College"
## [90] "Hillsdale College"
## [91] "Hiram College"
## [92] "Hobart and William Smith Colleges"
## [93] "Hofstra University"
## [94] "Hollins College"
## [95] "Hood College"
## [96] "Hope College"
## [97] "Houghton College"
## [98] "Illinois Benedictine College"
## [99] "Illinois Institute of Technology"
## [100] "Illinois Wesleyan University"
## [101] "Iona College"
## [102] "Ithaca College"
## [103] "John Carroll University"
## [104] "Juniata College"
## [105] "Keuka College"
## [106] "King's College"
## [107] "Knox College"
## [108] "La Salle University"
## [109] "Lake Forest College"
## [110] "Le Moyne College"
## [111] "Lebanon Valley College"
## [112] "Lenoir-Rhyne College"
## [113] "Lewis and Clark College"
## [114] "Linfield College"
## [115] "Loyola College"
## [116] "Loyola Marymount University"
## [117] "Loyola University"
## [118] "Loyola University Chicago"
## [119] "Luther College"
## [120] "Lycoming College"
## [121] "Manhattan College"
## [122] "Manhattanville College"
## [123] "Marietta College"
## [124] "Marist College"
## [125] "Marquette University"
## [126] "Mary Baldwin College"
## [127] "Mary Washington College"
## [128] "Marymount College Tarrytown"
## [129] "Marymount Manhattan College"
## [130] "Marymount University"
## [131] "Maryville College"
## [132] "Marywood College"
## [133] "Mercer University"
## [134] "Meredith College"
## [135] "Merrimack College"
## [136] "Messiah College"
## [137] "Michigan Technological University"
## [138] "Millikin University"
## [139] "Millsaps College"
## [140] "Monmouth College IL"
## [141] "Montana College of Mineral Sci. & Tech."
## [142] "Moravian College"
## [143] "Morehouse College"
## [144] "Mount Saint Mary's College"
## [145] "Mount Saint Mary College"
## [146] "Mount St. Mary's College"
## [147] "Mount Union College"
## [148] "Muhlenberg College"
## [149] "Muskingum College"
## [150] "Nazareth College of Rochester"
## [151] "New Jersey Institute of Technology"
## [152] "Niagara University"
## [153] "North Central College"
## [154] "North Park College"
## [155] "Norwich University"
## [156] "Oglethorpe University"
## [157] "Ohio Northern University"
## [158] "Ohio Wesleyan University"
## [159] "Otterbein College"
## [160] "Pacific Lutheran University"
## [161] "Pacific University"
## [162] "Pitzer College"
## [163] "Polytechnic University"
## [164] "Presbyterian College"
## [165] "Providence College"
## [166] "Queens College"
## [167] "Quincy University"
## [168] "Quinnipiac College"
## [169] "Randolph-Macon College"
## [170] "Randolph-Macon Woman's College"
## [171] "Regis College"
## [172] "Rider University"
## [173] "Ripon College"
## [174] "Roanoke College"
## [175] "Rollins College"
## [176] "Rosary College"
## [177] "Sacred Heart University"
## [178] "Saint Anselm College"
## [179] "Saint John's University"
## [180] "Saint Joseph's University"
## [181] "Saint Joseph College"
## [182] "Saint Mary's College"
## [183] "Saint Michael's College"
## [184] "Saint Olaf College"
## [185] "Saint Peter's College"
## [186] "Saint Xavier University"
## [187] "Salem College"
## [188] "Samford University"
## [189] "Santa Clara University"
## [190] "Seattle Pacific University"
## [191] "Seattle University"
## [192] "Seton Hall University"
## [193] "Seton Hill College"
## [194] "Siena College"
## [195] "Simmons College"
## [196] "Simpson College"
## [197] "Skidmore College"
## [198] "Southern Methodist University"
## [199] "Southwestern University"
## [200] "Spelman College"
## [201] "St. Bonaventure University"
## [202] "St. John's College"
## [203] "St. John Fisher College"
## [204] "St. Lawrence University"
## [205] "St. Mary's College of California"
## [206] "St. Mary's College of Maryland"
## [207] "St. Mary's University of San Antonio"
## [208] "St. Norbert College"
## [209] "Stetson University"
## [210] "Stevens Institute of Technology"
## [211] "Stonehill College"
## [212] "SUNY College at Geneseo"
## [213] "Susquehanna University"
## [214] "Taylor University"
## [215] "Transylvania University"
## [216] "Trenton State College"
## [217] "Trinity College DC"
## [218] "Trinity College VT"
## [219] "University of Dallas"
## [220] "University of Dayton"
## [221] "University of Denver"
## [222] "University of Evansville"
## [223] "University of Hartford"
## [224] "University of Minnesota at Morris"
## [225] "University of Missouri at Rolla"
## [226] "University of New England"
## [227] "University of Portland"
## [228] "University of Puget Sound"
## [229] "University of Richmond"
## [230] "University of San Diego"
## [231] "University of San Francisco"
## [232] "University of Scranton"
## [233] "University of St. Thomas MN"
## [234] "University of St. Thomas TX"
## [235] "University of the Pacific"
## [236] "University of Tulsa"
## [237] "University of Vermont"
## [238] "Ursinus College"
## [239] "Valparaiso University"
## [240] "Villanova University"
## [241] "Virginia Wesleyan College"
## [242] "Wabash College"
## [243] "Wagner College"
## [244] "Wartburg College"
## [245] "Washington and Jefferson College"
## [246] "Washington College"
## [247] "West Virginia Wesleyan College"
## [248] "Western Maryland College"
## [249] "Westminster College"
## [250] "Westmont College"
## [251] "Wheaton College IL"
## [252] "Whitman College"
## [253] "Whittier College"
## [254] "Whitworth College"
## [255] "Widener University"
## [256] "Wilkes University"
## [257] "Willamette University"
## [258] "Wilson College"
## [259] "Wittenberg University"
## [260] "Wofford College"
## [261] "Worcester Polytechnic Institute"
## [262] "Xavier University"
## [263] "York College of Pennsylvania"
##
## $`2`
## [1] "Adelphi University"
## [2] "Adrian College"
## [3] "Alaska Pacific University"
## [4] "Alderson-Broaddus College"
## [5] "Alverno College"
## [6] "American International College"
## [7] "Anderson University"
## [8] "Andrews University"
## [9] "Aquinas College"
## [10] "Arkansas Tech University"
## [11] "Augsburg College"
## [12] "Augustana College"
## [13] "Averett College"
## [14] "Baker University"
## [15] "Barat College"
## [16] "Belmont Abbey College"
## [17] "Belmont University"
## [18] "Bemidji State University"
## [19] "Benedictine College"
## [20] "Bennington College"
## [21] "Bethany College"
## [22] "Bethel College KS"
## [23] "Bethel College"
## [24] "Bethune Cookman College"
## [25] "Blackburn College"
## [26] "Bluefield College"
## [27] "Bluffton College"
## [28] "Brenau University"
## [29] "Brewton-Parker College"
## [30] "Briar Cliff College"
## [31] "Campbellsville College"
## [32] "Capitol College"
## [33] "Carson-Newman College"
## [34] "Catawba College"
## [35] "Cazenovia College"
## [36] "Cedarville College"
## [37] "Centenary College"
## [38] "Center for Creative Studies"
## [39] "Central Wesleyan College"
## [40] "Christian Brothers University"
## [41] "Claflin College"
## [42] "Clarke College"
## [43] "Clinch Valley Coll. of the Univ. of Virginia"
## [44] "College of Mount St. Joseph"
## [45] "College of Notre Dame"
## [46] "College of Santa Fe"
## [47] "College of St. Joseph"
## [48] "Concordia College at St. Paul"
## [49] "Concordia Lutheran College"
## [50] "Concordia University CA"
## [51] "Concordia University"
## [52] "Culver-Stockton College"
## [53] "Cumberland College"
## [54] "D'Youville College"
## [55] "Dana College"
## [56] "Daniel Webster College"
## [57] "Defiance College"
## [58] "Delta State University"
## [59] "Dickinson State University"
## [60] "Dillard University"
## [61] "Doane College"
## [62] "Dominican College of Blauvelt"
## [63] "Dordt College"
## [64] "Dowling College"
## [65] "East Texas Baptist University"
## [66] "Eastern College"
## [67] "Eastern Connecticut State University"
## [68] "Eastern Mennonite College"
## [69] "Eastern Nazarene College"
## [70] "Elon College"
## [71] "Embry Riddle Aeronautical University"
## [72] "Evergreen State College"
## [73] "Fayetteville State University"
## [74] "Ferrum College"
## [75] "Flagler College"
## [76] "Florida Southern College"
## [77] "Fontbonne College"
## [78] "Franciscan University of Steubenville"
## [79] "Franklin Pierce College"
## [80] "Freed-Hardeman University"
## [81] "Fresno Pacific College"
## [82] "Gannon University"
## [83] "Gardner Webb University"
## [84] "Geneva College"
## [85] "George Fox College"
## [86] "Georgetown College"
## [87] "Georgian Court College"
## [88] "Goldey Beacom College"
## [89] "Goshen College"
## [90] "Grace College and Seminary"
## [91] "Graceland College"
## [92] "Green Mountain College"
## [93] "Greensboro College"
## [94] "Greenville College"
## [95] "Gwynedd Mercy College"
## [96] "Hardin-Simmons University"
## [97] "Hastings College"
## [98] "Huntingdon College"
## [99] "Huntington College"
## [100] "Huron University"
## [101] "Husson College"
## [102] "Illinois College"
## [103] "Immaculata College"
## [104] "Incarnate Word College"
## [105] "Indiana Wesleyan University"
## [106] "Jamestown College"
## [107] "Jersey City State College"
## [108] "John Brown University"
## [109] "Johnson State College"
## [110] "Judson College"
## [111] "Kansas Wesleyan University"
## [112] "Keene State College"
## [113] "Kentucky Wesleyan College"
## [114] "King College"
## [115] "La Roche College"
## [116] "LaGrange College"
## [117] "Lakeland College"
## [118] "Lambuth University"
## [119] "Lander University"
## [120] "Lesley College"
## [121] "LeTourneau University"
## [122] "Lewis University"
## [123] "Lincoln Memorial University"
## [124] "Lincoln University"
## [125] "Livingstone College"
## [126] "Lock Haven University of Pennsylvania"
## [127] "Loras College"
## [128] "Louisiana College"
## [129] "Lynchburg College"
## [130] "Lyndon State College"
## [131] "MacMurray College"
## [132] "Malone College"
## [133] "Manchester College"
## [134] "Marian College of Fond du Lac"
## [135] "Maryville University"
## [136] "Mayville State University"
## [137] "McKendree College"
## [138] "McMurry University"
## [139] "McPherson College"
## [140] "Mercyhurst College"
## [141] "Mesa State College"
## [142] "MidAmerica Nazarene College"
## [143] "Milligan College"
## [144] "Milwaukee School of Engineering"
## [145] "Mississippi College"
## [146] "Mississippi University for Women"
## [147] "Missouri Southern State College"
## [148] "Missouri Valley College"
## [149] "Monmouth College"
## [150] "Montreat-Anderson College"
## [151] "Morningside College"
## [152] "Morris College"
## [153] "Mount Marty College"
## [154] "Mount Mary College"
## [155] "Mount Mercy College"
## [156] "Mount Saint Clare College"
## [157] "Mount Vernon Nazarene College"
## [158] "National-Louis University"
## [159] "Newberry College"
## [160] "North Adams State College"
## [161] "North Carolina Wesleyan College"
## [162] "Northwest Nazarene College"
## [163] "Northwestern College"
## [164] "Notre Dame College"
## [165] "Oklahoma Baptist University"
## [166] "Oklahoma Christian University"
## [167] "Ouachita Baptist University"
## [168] "Our Lady of the Lake University"
## [169] "Pacific Union College"
## [170] "Pembroke State University"
## [171] "Peru State College"
## [172] "Pfeiffer College"
## [173] "Philadelphia Coll. of Textiles and Sci."
## [174] "Phillips University"
## [175] "Piedmont College"
## [176] "Pikeville College"
## [177] "Point Loma Nazarene College"
## [178] "Point Park College"
## [179] "Rivier College"
## [180] "Rockhurst College"
## [181] "Rocky Mountain College"
## [182] "Roger Williams University"
## [183] "Saint Ambrose University"
## [184] "Saint Francis College IN"
## [185] "Saint Francis College"
## [186] "Saint Joseph's College IN"
## [187] "Saint Joseph's College"
## [188] "Saint Mary's College of Minnesota"
## [189] "Saint Mary-of-the-Woods College"
## [190] "Saint Vincent College"
## [191] "Salem-Teikyo University"
## [192] "Savannah Coll. of Art and Design"
## [193] "Schreiner College"
## [194] "Shorter College"
## [195] "Siena Heights College"
## [196] "Sioux Falls College"
## [197] "Southeastern Oklahoma State Univ."
## [198] "Southern California College"
## [199] "Southwest Baptist University"
## [200] "Southwest State University"
## [201] "Southwestern Adventist College"
## [202] "Southwestern College"
## [203] "Spalding University"
## [204] "Spring Arbor College"
## [205] "St. Martin's College"
## [206] "St. Paul's College"
## [207] "St. Thomas Aquinas College"
## [208] "Stephens College"
## [209] "SUNY College at Potsdam"
## [210] "Tabor College"
## [211] "Talladega College"
## [212] "Tennessee Wesleyan College"
## [213] "Texas Lutheran College"
## [214] "Texas Wesleyan University"
## [215] "Thiel College"
## [216] "Tiffin University"
## [217] "Tri-State University"
## [218] "Tusculum College"
## [219] "Tuskegee University"
## [220] "Union College KY"
## [221] "University of Charleston"
## [222] "University of Detroit Mercy"
## [223] "University of Dubuque"
## [224] "University of Indianapolis"
## [225] "University of La Verne"
## [226] "University of Maine at Farmington"
## [227] "University of Maine at Machias"
## [228] "University of Maine at Presque Isle"
## [229] "University of Mobile"
## [230] "University of Montevallo"
## [231] "University of North Carolina at Asheville"
## [232] "University of South Carolina at Aiken"
## [233] "University of the Arts"
## [234] "University of Wisconsin-Superior"
## [235] "Upper Iowa University"
## [236] "Ursuline College"
## [237] "Valley City State University"
## [238] "Virginia State University"
## [239] "Virginia Union University"
## [240] "Viterbo College"
## [241] "Voorhees College"
## [242] "Walsh University"
## [243] "Warren Wilson College"
## [244] "Wayne State College"
## [245] "Waynesburg College"
## [246] "Webber College"
## [247] "Webster University"
## [248] "Wentworth Institute of Technology"
## [249] "Wesley College"
## [250] "West Liberty State College"
## [251] "Westfield State College"
## [252] "Westminster College MO"
## [253] "Westminster College of Salt Lake City"
## [254] "Wheeling Jesuit College"
## [255] "William Jewell College"
## [256] "William Woods University"
## [257] "Wingate College"
## [258] "Winona State University"
## [259] "Wisconsin Lutheran College"
## [260] "Worcester State College"
## [261] "Xavier University of Louisiana"
##
## $`3`
## [1] "Arizona State University Main campus"
## [2] "Auburn University-Main Campus"
## [3] "Boston University"
## [4] "Bowling Green State University"
## [5] "Brigham Young University at Provo"
## [6] "Clemson University"
## [7] "Colorado State University"
## [8] "Florida State University"
## [9] "Georgia Institute of Technology"
## [10] "Indiana University at Bloomington"
## [11] "Iowa State University"
## [12] "James Madison University"
## [13] "Louisiana State University at Baton Rouge"
## [14] "Miami University at Oxford"
## [15] "Michigan State University"
## [16] "North Carolina State University at Raleigh"
## [17] "Northeastern University"
## [18] "Northern Illinois University"
## [19] "Ohio University"
## [20] "Pennsylvania State Univ. Main Campus"
## [21] "Purdue University at West Lafayette"
## [22] "Rutgers at New Brunswick"
## [23] "San Diego State University"
## [24] "SUNY at Albany"
## [25] "SUNY at Binghamton"
## [26] "SUNY at Buffalo"
## [27] "SUNY at Stony Brook"
## [28] "Syracuse University"
## [29] "Texas A&M Univ. at College Station"
## [30] "University of California at Berkeley"
## [31] "University of California at Irvine"
## [32] "University of Cincinnati"
## [33] "University of Connecticut at Storrs"
## [34] "University of Delaware"
## [35] "University of Florida"
## [36] "University of Georgia"
## [37] "University of Illinois - Urbana"
## [38] "University of Illinois at Chicago"
## [39] "University of Kansas"
## [40] "University of Maryland at College Park"
## [41] "University of Massachusetts at Amherst"
## [42] "University of Michigan at Ann Arbor"
## [43] "University of Minnesota Twin Cities"
## [44] "University of Missouri at Columbia"
## [45] "University of Nebraska at Lincoln"
## [46] "University of New Hampshire"
## [47] "University of North Carolina at Chapel Hill"
## [48] "University of Oregon"
## [49] "University of Pittsburgh-Main Campus"
## [50] "University of Rhode Island"
## [51] "University of South Carolina at Columbia"
## [52] "University of South Florida"
## [53] "University of Southern California"
## [54] "University of Tennessee at Knoxville"
## [55] "University of Texas at Austin"
## [56] "University of Utah"
## [57] "University of Virginia"
## [58] "University of Washington"
## [59] "University of Wisconsin at Madison"
## [60] "Virginia Tech"
## [61] "Washington State University"
## [62] "Western Michigan University"
##
## $`4`
## [1] "Abilene Christian University"
## [2] "Angelo State University"
## [3] "Appalachian State University"
## [4] "Baylor University"
## [5] "Bloomsburg Univ. of Pennsylvania"
## [6] "California Polytechnic-San Luis"
## [7] "California State University at Fresno"
## [8] "Campbell University"
## [9] "Central Connecticut State University"
## [10] "Central Missouri State University"
## [11] "Central Washington University"
## [12] "Christopher Newport University"
## [13] "College of Charleston"
## [14] "Columbia College MO"
## [15] "East Carolina University"
## [16] "East Tennessee State University"
## [17] "Eastern Illinois University"
## [18] "Emporia State University"
## [19] "Florida International University"
## [20] "Fort Lewis College"
## [21] "Francis Marion University"
## [22] "George Mason University"
## [23] "Georgia State University"
## [24] "Grand Valley State University"
## [25] "Hampton University"
## [26] "Illinois State University"
## [27] "Indiana State University"
## [28] "Kansas State University"
## [29] "Lamar University"
## [30] "Lindenwood College"
## [31] "Longwood College"
## [32] "Louisiana Tech University"
## [33] "Mankato State University"
## [34] "Marshall University"
## [35] "Millersville University of Penn."
## [36] "Mississippi State University"
## [37] "Montana State University"
## [38] "Montclair State University"
## [39] "Moorhead State University"
## [40] "Murray State University"
## [41] "New Mexico Institute of Mining and Tech."
## [42] "North Carolina A. & T. State University"
## [43] "North Dakota State University"
## [44] "Northeast Missouri State University"
## [45] "Northern Arizona University"
## [46] "Northwest Missouri State University"
## [47] "Oakland University"
## [48] "Oklahoma State University"
## [49] "Pace University"
## [50] "Prairie View A. and M. University"
## [51] "Radford University"
## [52] "Ramapo College of New Jersey"
## [53] "Rowan College of New Jersey"
## [54] "Rutgers State University at Camden"
## [55] "Rutgers State University at Newark"
## [56] "Saint Cloud State University"
## [57] "Saint Louis University"
## [58] "Salisbury State University"
## [59] "Shippensburg University of Penn."
## [60] "South Dakota State University"
## [61] "Southeast Missouri State University"
## [62] "Southern Illinois University at Edwardsville"
## [63] "Southwest Missouri State University"
## [64] "Stockton College of New Jersey"
## [65] "SUNY College at Brockport"
## [66] "SUNY College at Oswego"
## [67] "SUNY College at Buffalo"
## [68] "SUNY College at Cortland"
## [69] "SUNY College at Fredonia"
## [70] "SUNY College at New Paltz"
## [71] "SUNY College at Plattsburgh"
## [72] "SUNY College at Purchase"
## [73] "Texas A&M University at Galveston"
## [74] "Texas Christian University"
## [75] "Texas Southern University"
## [76] "The Citadel"
## [77] "Univ. of Wisconsin at OshKosh"
## [78] "University of Alabama at Birmingham"
## [79] "University of Arkansas at Fayetteville"
## [80] "University of Central Florida"
## [81] "University of Hawaii at Manoa"
## [82] "University of Louisville"
## [83] "University of Maryland at Baltimore County"
## [84] "University of Massachusetts at Dartmouth"
## [85] "University of Minnesota at Duluth"
## [86] "University of Mississippi"
## [87] "University of Missouri at Saint Louis"
## [88] "University of North Carolina at Charlotte"
## [89] "University of North Carolina at Greensboro"
## [90] "University of North Carolina at Wilmington"
## [91] "University of North Dakota"
## [92] "University of North Florida"
## [93] "University of North Texas"
## [94] "University of Northern Colorado"
## [95] "University of Northern Iowa"
## [96] "University of Oklahoma"
## [97] "University of Sci. and Arts of Oklahoma"
## [98] "University of Southern Colorado"
## [99] "University of Southern Indiana"
## [100] "University of Southern Mississippi"
## [101] "University of Texas at Arlington"
## [102] "University of Texas at San Antonio"
## [103] "University of West Florida"
## [104] "University of Wisconsin-Stout"
## [105] "University of Wisconsin-Whitewater"
## [106] "University of Wisconsin at Green Bay"
## [107] "University of Wisconsin at Milwaukee"
## [108] "University of Wyoming"
## [109] "Virginia Commonwealth University"
## [110] "West Chester University of Penn."
## [111] "Western Carolina University"
## [112] "Western New England College"
## [113] "Western State College of Colorado"
## [114] "Western Washington University"
## [115] "Winthrop University"
##
## $`5`
## [1] "Agnes Scott College"
## [2] "Amherst College"
## [3] "Antioch University"
## [4] "Bard College"
## [5] "Barnard College"
## [6] "Bowdoin College"
## [7] "Brandeis University"
## [8] "Brown University"
## [9] "Bryn Mawr College"
## [10] "Bucknell University"
## [11] "Carleton College"
## [12] "Carnegie Mellon University"
## [13] "Case Western Reserve University"
## [14] "Centre College"
## [15] "Claremont McKenna College"
## [16] "Colby College"
## [17] "Colgate University"
## [18] "College of the Holy Cross"
## [19] "College of William and Mary"
## [20] "Colorado College"
## [21] "Columbia University"
## [22] "Connecticut College"
## [23] "Dartmouth College"
## [24] "Davidson College"
## [25] "Drew University"
## [26] "Duke University"
## [27] "Emory University"
## [28] "George Washington University"
## [29] "Georgetown University"
## [30] "Gettysburg College"
## [31] "Grinnell College"
## [32] "Hamilton College"
## [33] "Harvard University"
## [34] "Harvey Mudd College"
## [35] "Johns Hopkins University"
## [36] "Kenyon College"
## [37] "Lafayette College"
## [38] "Lawrence University"
## [39] "Lehigh University"
## [40] "Macalester College"
## [41] "Massachusetts Institute of Technology"
## [42] "Mount Holyoke College"
## [43] "New York University"
## [44] "Northwestern University"
## [45] "Oberlin College"
## [46] "Occidental College"
## [47] "Pepperdine University"
## [48] "Princeton University"
## [49] "Reed College"
## [50] "Rensselaer Polytechnic Institute"
## [51] "Rhodes College"
## [52] "Sarah Lawrence College"
## [53] "Scripps College"
## [54] "Smith College"
## [55] "Sweet Briar College"
## [56] "Trinity College CT"
## [57] "Trinity University"
## [58] "Tulane University"
## [59] "Union College NY"
## [60] "University of Chicago"
## [61] "University of Miami"
## [62] "University of Notre Dame"
## [63] "University of Pennsylvania"
## [64] "University of Rochester"
## [65] "University of the South"
## [66] "Vanderbilt University"
## [67] "Vassar College"
## [68] "Wake Forest University"
## [69] "Washington and Lee University"
## [70] "Washington University"
## [71] "Wellesley College"
## [72] "Wells College"
## [73] "Wesleyan University"
## [74] "Westminster College PA"
## [75] "Williams College"
## [76] "Yale University"*Answer: The categories seem to be classified based on the prestige of each school, with elite schools having higher Top10Perc, Outstate, and PHD values. Cluster 3 has many Universities, while Cluster 5 has many notable Ivy Leagues such as Yale and Harvard.
Problem 2
For this problem we will use the Caravan dataset from PS 4. This dataset contains 86 features describing 5822 customer records as it relates to them purchasing insurance; DV is “Purchase”.
Step 0:
Separate the variable “Purchase” from the rest of the data, leaving you with only features; call this dataframe X.caravan.
data(Caravan)
Purchase = Caravan$Purchase
x.caravan = Caravan[,-which(names(Caravan) == "Purchase")]Step 1:
Conduct PCA on the dataframe X.caravan. Make sure the input features are scaled. Using a skree plot, identify the optimal number of principal components. Explain why this is the best value.
x.caravan.scaled = scale(x.caravan)
pca_result = prcomp(x.caravan.scaled, center = TRUE, scale.= TRUE)
skree_values = pca_result$sdev^2 / sum(pca_result$sdev^2)
plot(skree_values, type = "b")
*Answer: Based on the plot, the elbow is likely to be somewhere between 3 and 6, indicating the optimal k value.
Step 2:
Create a new dataframe caravan.new that combines the transformed data with the dependent variable Purchase. Only include the number of principal components you identified as optimal in step 1 (i.e., if you determine that three is the best number, make a new data frame that includes Purchase and the first three principal components).
optimal_k = 3
caravan.new = data.frame(pca_result$x[, 1:optimal_k], Purchase)Step 3:
Conduct logistic regression predicting Purchase with the principal components. Calculate the MCR and confusion matrix.
logistic_model_pc = glm(Purchase ~ ., data = caravan.new, family=binomial)
predictions_pc = predict(logistic_model_pc, type = "response")
predicted_classes_pc = ifelse(predictions_pc > 0.5, "Yes", "No")
mcr_pc = mean(predicted_classes_pc != Purchase)
table_pc = table(Predicted = predicted_classes_pc, Actual = Purchase)
print(table_pc)## Actual
## Predicted No Yes
## No 5474 348print(mcr_pc)## [1] 0.05977327Step 4:
Conduct logistic regression predicting Purchase with the original 86 features. Calculate the MCR and confusion matrix. Is this better or worse than Step 3?
logistic_model_orig = glm(Purchase ~ ., data = Caravan, family = binomial)## Warning: glm.fit: fitted probabilities numerically 0 or 1 occurredpredictions_orig = predict(logistic_model_orig, type = "response")
predicted_classes_orig = ifelse(predictions_orig > 0.5, "Yes", "No")
mcr_orig = mean(predicted_classes_orig != Purchase)
table_orig = table(Predicted = predicted_classes_orig, Actual = Purchase)
print(table_orig)## Actual
## Predicted No Yes
## No 5466 341
## Yes 8 7print(mcr_orig)## [1] 0.05994504*Answer:PCA model has a lower MCr which may indicate that it is a better model. However, the confusion matrix for the original logistic model is more accurate for predicint Yes and No.