Week6
2023-02-27
library(ggfortify)## Loading required package: ggplot2 library(ggplot2)
library(readr)
library(dplyr)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, union library(tidyr)
library(viridis)## Loading required package: viridisLite library(ggthemes)
library(ggalt)## Registered S3 methods overwritten by 'ggalt':
## method from
## fortify.table ggfortify
## grid.draw.absoluteGrob ggplot2
## grobHeight.absoluteGrob ggplot2
## grobWidth.absoluteGrob ggplot2
## grobX.absoluteGrob ggplot2
## grobY.absoluteGrob ggplot2Week6 <- read.table(file="C:/R/classwork/Assignment/wek6/data6.csv", header=TRUE, sep=",", stringsAsFactors = TRUE)
Week6## Expenditures Enrolled RVUs FTEs Quality.Score
## 1 114948144 25294 402703.73 954.91 0.67
## 2 116423140 42186 638251.99 949.25 0.58
## 3 119977702 23772 447029.54 952.51 0.52
## 4 19056531 2085 43337.26 199.98 0.93
## 5 246166031 67258 1579789.36 2162.15 0.96
## 6 152125186 23752 673036.55 1359.07 0.56
## 7 737556867 53781 2397334.12 5798.04 0.84
## 8 431714041 98763 2256730.04 3623.94 0.82
## 9 378230874 47986 1558835.73 3305.32 0.76
## 10 289246447 64634 1713157.92 2404.80 0.81
## 11 292001890 76714 1156318.42 1690.42 0.83
## 12 33290437 11163 161196.70 214.72 0.70
## 13 89918238 33266 550546.15 786.15 0.60
## 14 200289585 35008 1050922.34 1784.82 0.75
## 15 154264931 42984 1010128.12 1325.65 0.64
## 16 111397716 15821 372044.89 957.25 0.73
## 17 63691831 21657 485389.31 483.36 0.68
## 18 177511820 57006 838595.08 1323.09 0.83
## 19 155047333 18580 1030776.45 1326.46 0.83
## 20 421917863 40639 1357463.45 3495.57 0.82
## 21 49422291 17343 248057.53 337.61 0.76
## 22 704000406 108960 3017131.83 5345.62 0.85
## 23 231372781 48338 1665055.71 2032.83 0.75
## 24 92965171 27454 560475.02 762.67 0.89
## 25 148661984 20476 736191.93 1166.26 0.84
## 26 52500954 17125 254384.25 428.16 0.84
## 27 48990120 13781 240808.17 434.20 0.59
## 28 175601378 30571 994000.52 1474.55 0.93
## 29 627739846 65154 2707224.50 4784.79 0.88
## 30 56291987 12107 184173.46 500.96 0.74
## 31 384804579 65885 1648839.75 3444.47 0.70
## 32 211543524 57098 1102229.23 1744.00 0.90
## 33 481316889 115521 2552169.39 4003.16 0.73
## 34 123018246 30540 488432.15 753.12 0.84
## 35 17193213 4327 49293.75 177.65 0.56
## 36 30664899 13527 128738.76 283.17 0.75
## 37 19768937 7209 89148.62 206.42 0.69
## 38 55784977 17266 174289.61 586.06 0.71
## 39 37992105 14842 146174.89 348.40 0.71
## 40 39448397 11953 138345.36 373.62 0.70
## 41 49637051 25420 241419.44 497.49 0.74
## 42 28530205 12110 124162.92 260.72 0.55
## 43 30673348 11466 113732.83 298.50 0.76
## 44 25234710 11454 113222.94 265.81 0.53
## 45 49671099 16845 145258.22 413.01 0.40
## 46 22557139 8131 91710.45 217.08 0.65
## 47 15516577 4521 54406.39 204.56 0.71
## 48 47429842 8002 120720.78 426.67 0.76
## 49 43366439 13364 146786.49 406.80 0.74
## 50 20223890 10009 83916.19 202.41 0.62
## 51 18363510 6103 60862.37 188.40 0.75
## 52 47778415 22259 218827.23 410.02 0.63
## 53 45322490 22959 232476.04 366.63 0.80
## 54 41314090 7604 111419.60 464.88 0.73
## 55 41831991 10417 142963.42 378.73 0.81
## 56 20109657 6149 72473.83 229.39 0.65
## 57 50755308 7042 127018.26 517.54 0.91
## 58 69272642 24565 323225.69 703.60 0.67
## 59 40575888 14849 129274.86 343.20 0.41
## 60 20958223 2578 49777.27 207.06 0.68
## 61 25628795 7691 93237.75 277.07 0.53
## 62 44432989 16932 185326.68 329.76 0.59
## 63 33521880 12213 132282.28 352.28 0.69
## 64 46306286 14439 143826.20 307.96 0.76
## 65 15736197 3887 45836.14 169.74 0.72
## 66 91103765 22214 356917.60 947.74 0.65
## 67 29637497 11648 133160.97 288.03 0.64
## 68 32821893 11552 131678.51 332.10 0.56
## 69 28426761 11298 110997.40 288.20 0.76
## 70 37534503 7319 109734.18 457.09 0.62
## 71 35432093 11743 120563.35 352.10 0.65
## 72 82601055 28151 320705.78 746.06 0.85
## 73 64545467 27250 313899.05 546.20 0.66
## 74 28901543 6758 71855.26 268.14 0.67
## 75 671959058 54771 1865586.33 5697.77 0.81
## 76 29361928 12577 118062.13 300.97 0.89
## 77 320604335 40225 808455.56 2415.15 0.59
## 78 19790938 8265 75995.61 161.15 0.68
## 79 28382056 12188 125125.92 268.02 0.50
## 80 133789826 35570 550683.00 1239.03 0.77
## 81 11906768 1424 23581.51 137.55 0.72
## 82 19448328 6766 48223.16 170.92 0.61
## 83 150947798 37795 608182.20 1260.65 0.63
## 84 113604093 33319 406929.00 640.55 0.79
## 85 15487046 4227 44759.38 168.01 0.67
## 86 55312465 22109 256084.38 502.99 0.60
## 87 49645517 17442 242765.52 433.78 0.78
## 88 151611402 27667 556371.89 1373.80 0.63
## 89 47833729 15449 181309.06 429.29 0.76
## 90 31467898 10324 112718.98 311.40 0.68
## 91 221899340 27207 710440.86 1824.36 0.68
## 92 51656073 10054 199461.51 492.75 0.72
## 93 93746534 20601 231366.13 919.47 0.78
## 94 45093715 16176 185456.99 421.68 0.70
## 95 290852934 36549 855724.09 2271.85 0.83
## 96 16275452 2062 32058.03 185.23 0.49
## 97 23880752 8922 101135.34 228.50 0.72
## 98 30422168 10949 108488.17 300.02 0.41
## 99 179510966 33125 684679.01 1587.88 0.81
## 100 15868393 4943 55961.81 151.10 0.71
## 101 191489002 45613 738618.70 1649.62 0.55
## 102 33946305 9001 90746.40 389.07 0.82
## 103 112886022 26776 503561.98 966.04 0.81
## 104 118474189 10184 584607.15 1129.51 0.78
## 105 170396460 34609 714332.05 1315.37 0.83
## 106 242465456 31301 1212558.88 2432.32 0.79
## 107 279086160 48049 1468797.93 2317.23 0.83
## 108 90695660 10976 280721.41 811.31 0.73
## 109 33692909 1218 52387.28 290.49 0.73
## 110 281085750 56812 1387650.41 2353.20 0.81
## 111 70232710 14086 297280.10 659.61 0.72
## 112 67058602 5628 154626.03 556.60 0.78
## 113 57406032 14897 327289.34 473.52 0.69
## 114 133421055 15611 734445.95 1318.31 0.78
## 115 239806816 46403 1030474.35 2031.98 0.78
## 116 44222933 2642 90139.68 398.66 0.82
## 117 55797686 7852 167676.80 503.27 0.64
## 118 74134323 14783 337025.73 636.82 0.64
## 119 129861050 21585 480628.61 1306.36 0.58
## 120 38680530 13153 243534.20 332.89 0.76
## 121 49129643 16577 231843.71 419.97 0.91
## 122 49575746 14431 279027.99 401.68 0.79
## 123 66972157 12728 169068.75 471.70 0.73
## 124 75793328 29339 525417.05 664.01 0.86
## 125 38437460 15428 177338.98 321.46 0.86
## 126 59443879 21742 347643.62 551.92 0.71
## 127 806367625 101454 3434703.16 6658.61 0.92
## 128 785079539 40402 1844285.41 4655.55 0.92
## 129 129946379 25540 460637.39 910.46 0.72
## 130 117827082 42206 763481.71 974.07 0.53
## 131 124633496 24435 471401.75 941.42 0.60
## 132 15335594 1906 45977.13 186.63 0.70
## 133 280918530 71993 1810996.19 2094.17 0.93
## 134 142671518 29449 680826.00 1227.45 0.67
## 135 1052311021 57397 2892975.46 7518.63 0.79
## 136 452667260 100569 2390290.31 3587.36 0.76
## 137 370586067 45085 1624727.69 3107.84 0.76
## 138 294389493 68214 1836855.21 2311.60 0.85
## 139 478981007 81497 1736067.57 3542.01 0.69
## 140 34827530 10631 150177.82 208.10 0.64
## 141 94258566 34222 568984.13 788.41 0.49
## 142 204873437 32113 945113.43 1598.94 0.73
## 143 164690465 45838 1157782.51 1363.49 0.65
## 144 105635716 13241 338120.18 811.58 0.63
## 145 73110154 21077 512124.61 501.03 0.73
## 146 178424383 59407 989358.63 1164.10 0.84
## 147 168267964 20236 1071997.55 1297.82 0.85
## 148 453595967 41256 1453868.97 3361.97 0.73
## 149 55868770 17757 264847.07 338.83 0.83
## 150 672113787 116389 3093666.05 5004.26 0.79
## 151 263617995 54261 1692788.53 2050.02 0.74
## 152 88213119 25920 587833.28 725.47 0.91
## 153 154460753 21819 723014.94 1105.41 0.79
## 154 54996100 16242 261201.56 395.37 0.86
## 155 48828341 12078 252941.14 394.23 0.94
## 156 187119684 30279 1015301.82 1422.28 0.82
## 157 674563921 66212 2673082.99 4685.47 0.74
## 158 59942400 11141 213605.00 483.70 0.83
## 159 417756027 72259 1852559.13 3478.57 0.81
## 160 214429435 58625 1290205.03 1671.86 0.85
## 161 522108466 116298 2616782.68 3879.58 0.79
## 162 122543052 29016 479062.73 720.95 0.77
## 163 18386529 4199 56064.49 156.28 0.59
## 164 33333082 13781 133997.55 270.14 0.83
## 165 20523439 7050 91612.91 176.79 0.76
## 166 59426076 17134 172765.34 532.25 0.57
## 167 40355791 13909 153028.08 351.94 0.78
## 168 41684323 12776 147494.34 373.30 0.67
## 169 53590324 23901 247909.39 463.56 0.78
## 170 29439049 11834 134265.66 249.42 0.71
## 171 30939512 10869 121982.43 297.32 0.82
## 172 26985594 11279 122499.26 256.12 0.64
## 173 51174054 16994 185507.03 394.24 0.39
## 174 23753605 7853 94789.14 210.53 0.59
## 175 15659058 4749 63977.31 187.68 0.72
## 176 50888100 7850 136429.27 412.04 0.85
## 177 43768308 12351 154733.89 350.68 0.81
## 178 23082323 9982 90140.88 241.51 0.66
## 179 18261275 5937 67479.03 186.47 0.66
## 180 49348318 19519 227044.07 394.49 0.68
## 181 43254472 24654 258675.34 330.13 0.85
## 182 43335191 7259 128745.31 398.97 0.70
## 183 42773074 10141 165737.75 374.74 0.67
## 184 21323162 6142 62422.39 193.86 0.49
## 185 56032849 6812 143776.97 459.01 0.63
## 186 70268334 24571 344199.65 657.36 0.69
## 187 42586104 12020 139945.55 320.77 0.31
## 188 21902259 2662 55472.38 190.79 0.74
## 189 27534164 7644 99528.47 271.41 0.68
## 190 47356861 16513 192851.20 307.29 0.60
## 191 33675604 11456 146346.45 316.67 0.64
## 192 49966638 14648 168293.14 277.44 0.78
## 193 15484896 3810 47317.01 150.47 0.70
## 194 95034213 17811 359659.92 811.18 0.59
## 195 30276033 11075 129988.91 281.42 0.76
## 196 33343818 11080 136237.85 324.82 0.48
## 197 28896477 11666 112823.36 265.84 0.79
## 198 40787554 7357 115452.21 427.65 0.85
## 199 38388629 8548 109251.92 330.95 0.61
## 200 80916337 27326 361224.99 685.37 0.74
## 201 68113499 26144 330031.76 518.75 0.66
## 202 29372038 7087 64318.29 267.16 0.73
## 203 651743770 50687 1601793.92 5048.47 0.88
## 204 29696092 12393 127030.34 276.22 0.80
## 205 348797885 39078 863863.99 2340.06 0.58
## 206 20663099 8438 82060.19 164.62 0.58
## 207 30201767 11945 144498.15 272.21 0.54
## 208 150320044 37600 620445.23 1284.54 0.76
## 209 12085971 1445 23218.01 120.30 0.79
## 210 20375822 6716 58518.34 165.32 0.68
## 211 160608494 37162 641388.93 1228.19 0.58
## 212 96365310 33829 415452.70 581.08 0.68
## 213 14780257 4145 53073.19 145.87 0.67
## 214 57948488 20244 262939.04 454.48 0.54
## 215 50683600 17047 256896.61 396.00 0.72
## 216 148969823 28338 558736.60 1130.04 0.63
## 217 49950740 13792 189081.27 420.35 0.80
## 218 32146051 10634 99266.92 288.41 0.83
## 219 235165404 26731 740702.89 1684.47 0.76
## 220 49943851 9517 195731.34 444.26 0.70
## 221 93550306 20120 235349.05 839.14 0.65
## 222 46805669 16892 213023.68 408.52 0.61
## 223 322738262 37171 903695.93 2031.06 0.87
## 224 17519828 2423 34588.14 184.65 0.60
## 225 25110314 8647 92861.20 213.82 0.57
## 226 32623104 10319 118254.98 293.35 0.48
## 227 191736477 34861 721944.14 1549.64 0.84
## 228 16646964 5166 60638.99 146.90 0.66
## 229 223513993 47105 869628.09 1659.27 0.58
## 230 35964730 8748 97346.03 361.02 0.83
## 231 121800528 26994 519196.24 965.89 0.70
## 232 121524734 10564 564985.07 1078.21 0.84
## 233 182373869 34255 747807.29 1319.79 0.79
## 234 256575567 31548 1331724.66 2282.88 0.76
## 235 288406519 48251 1483892.38 2229.13 0.76
## 236 95104995 10983 301374.21 806.66 0.67
## 237 36287531 1602 49394.58 280.38 0.61
## 238 287374111 56569 1358605.37 2252.70 0.71
## 239 73206957 13163 303746.49 615.57 0.77
## 240 63328607 5319 153975.67 531.35 0.81
## 241 59419192 14653 332710.38 427.31 0.77
## 242 144359005 16182 783710.24 1265.67 0.67
## 243 246761080 45429 1075503.19 1973.93 0.87
## 244 49838003 2700 99966.78 404.84 0.87
## 245 59641487 8344 186116.14 494.66 0.66
## 246 78802547 14439 353629.69 643.62 0.66
## 247 141710583 20243 525514.30 1281.47 0.80
## 248 44037424 13693 244664.82 316.48 0.67
## 249 50511046 16514 213393.96 349.72 0.87
## 250 49336614 14441 294497.96 401.89 0.79
## 251 61325036 12573 191410.10 462.72 0.69
## 252 84252197 29155 520728.07 642.60 0.83
## 253 43864242 15582 178586.90 312.64 0.74
## 254 61412502 21966 349811.35 548.66 0.68
## 255 818656771 100584 3574006.15 6613.28 0.89
## 256 1301993928 50927 2822798.99 7264.44 0.80
## 257 94047315 25065 490004.41 877.76 0.68
## 258 78707019 37484 685670.00 882.74 0.61
## 259 99843160 23433 464193.99 930.94 0.57
## 260 10411614 2146 52575.69 168.81 0.55
## 261 210852507 69862 1497851.52 2074.33 0.93
## 262 107067124 31989 670977.65 1365.40 0.87
## 263 782762374 57259 2694416.86 7410.52 0.82
## 264 337731158 101429 2209837.83 3334.43 0.78
## 265 282567119 44940 1330621.99 3016.47 0.71
## 266 229521482 70900 1782559.31 2330.10 0.83
## 267 380317663 82717 1649419.95 3447.24 0.64
## 268 26083915 11517 146801.47 191.09 0.76
## 269 69635722 32476 531543.72 779.58 0.42
## 270 150516975 32330 838022.94 1538.72 0.66
## 271 123626592 42568 955567.97 1324.11 0.67
## 272 76813305 13182 245494.03 714.26 0.60
## 273 46084228 21118 432508.24 482.78 0.71
## 274 140235269 61512 840078.05 1169.56 0.77
## 275 122357396 22131 992598.16 1285.99 0.78
## 276 360490554 52361 1519461.14 3440.47 0.67
## 277 37884197 17669 254492.22 318.90 0.78
## 278 524079263 119076 2860646.31 4868.98 0.74
## 279 186727705 55480 1469704.84 1985.09 0.57
## 280 63568011 25112 494032.79 670.45 0.90
## 281 110928482 23827 699243.04 1034.51 0.77
## 282 39233373 16640 260642.92 397.81 0.87
## 283 35323954 12072 228236.18 345.64 0.91
## 284 133895187 30483 931018.64 1341.27 0.78
## 285 529484041 67509 2382358.81 4505.88 0.72
## 286 43736668 10878 180151.59 499.05 0.65
## 287 328821735 70252 1773661.16 3362.13 0.78
## 288 160042102 59168 1061118.57 1671.68 0.78
## 289 389431356 119850 2296382.01 3842.06 0.78
## 290 88037818 28051 448635.12 683.56 0.68
## 291 12643654 4176 62945.84 138.28 0.64
## 292 24252878 13885 128449.03 247.60 0.89
## 293 13805046 7456 88515.40 161.19 0.81
## 294 42703747 16813 157896.70 484.02 0.63
## 295 29590277 14476 138303.63 322.78 0.75
## 296 29827614 12618 163418.40 359.91 0.78
## 297 41077626 24956 250186.86 480.69 0.77
## 298 20069865 11401 122452.70 254.62 0.73
## 299 21702658 10566 104959.54 271.87 0.77
## 300 19829246 11045 110138.68 246.52 0.79
## 301 35631699 17212 174017.19 383.20 0.45
## 302 17309855 7802 88910.56 193.31 0.62
## 303 12232590 5041 58002.30 171.03 0.75
## 304 38724750 7633 126546.43 444.72 0.78
## 305 28129131 12915 134342.38 296.44 0.89
## 306 16982095 9906 96588.69 211.67 0.61
## 307 13481673 5633 55652.31 163.42 0.55
## 308 40115495 20690 219136.01 369.63 0.69
## 309 28374756 24649 230505.80 328.96 0.89
## 310 35739680 7267 114423.35 406.87 0.59
## 311 30151930 10063 145334.13 385.38 0.68
## 312 16189811 5999 67858.67 202.65 0.44
## 313 43005541 7012 129690.83 430.12 0.59
## 314 54299717 24207 315507.42 630.25 0.70
## 315 29156276 11793 133477.57 305.89 0.31
## 316 18592043 2500 44197.16 185.50 0.82
## 317 20081127 7446 88526.54 248.49 0.69
## 318 32138620 16419 170237.89 306.84 0.58
## 319 22877488 11449 127905.57 301.17 0.68
## 320 33859690 14823 144092.99 265.25 0.91
## 321 10180284 3685 52157.86 125.00 0.75
## 322 70781594 17746 320445.13 814.42 0.66
## 323 20969209 10753 109821.43 272.34 0.70
## 324 23225910 10673 117906.07 288.43 0.50
## 325 19876545 11915 106202.23 246.63 0.77
## 326 29423436 7619 107960.68 398.87 0.75
## 327 27852806 7943 87815.68 315.10 0.66
## 328 58803092 27118 331616.59 648.77 0.69
## 329 48609297 25552 310370.02 497.67 0.58
## 330 21187801 7384 74864.60 242.31 0.74
## 331 463611180 52490 1426871.64 4665.43 0.88
## 332 20926931 12575 128328.62 261.39 0.75
## 333 248413335 39215 810171.13 2261.38 0.48
## 334 16339169 8284 65142.62 170.92 0.60
## 335 22953113 12253 134131.67 269.36 0.64
## 336 111205662 38545 598097.42 1297.28 0.65
## 337 7839563 1390 23391.25 116.29 0.86
## 338 16286538 6901 47320.66 135.83 0.58
## 339 119293449 37124 623674.43 1186.71 0.63
## 340 68360435 36696 385697.33 578.16 0.59
## 341 10836186 4120 47847.89 135.89 0.71
## 342 40189939 20393 243788.74 441.88 0.55
## 343 35401236 17116 223210.61 343.91 0.72
## 344 107506235 28789 479689.96 1146.34 0.70
## 345 34917728 15227 159420.23 406.61 0.75
## 346 22436001 9892 98620.22 264.85 0.75
## 347 164036058 26372 666386.35 1580.19 0.71
## 348 33399114 9693 176535.27 382.26 0.71
## 349 70201713 19743 197356.55 799.58 0.65
## 350 32754540 17299 207190.31 384.97 0.50
## 351 206886756 37266 783489.55 1870.42 0.81
## 352 13779195 2232 28972.31 162.39 0.55
## 353 17940731 8509 86070.80 205.99 0.44
## 354 21996524 10731 111598.06 279.85 0.40
## 355 145052995 36714 651147.15 1554.25 0.75
## 356 11599674 5117 53306.14 139.95 0.70
## 357 157480675 48077 774026.56 1467.97 0.57
## 358 25699503 9269 94087.39 338.05 0.78
## 359 87606611 26811 460717.48 931.23 0.77
## 360 88268292 10638 584343.14 1032.31 0.85
## 361 137687522 34157 659398.59 1331.56 0.77
## 362 209778567 28452 1207717.46 2291.80 0.77
## 363 217234362 54088 1468947.94 2256.08 0.73
## 364 72748524 12578 272805.82 807.40 0.67
## 365 27017111 2495 45743.93 273.63 0.65
## 366 211706640 55780 1186891.23 2197.67 0.65
## 367 53091253 12886 283437.31 602.52 0.73
## 368 50101230 5085 137424.07 548.62 0.72
## 369 44076052 14108 294641.54 457.95 0.75
## 370 114704162 17808 758965.24 1361.03 0.64
## 371 180743712 47608 997600.45 1919.63 0.89
## 372 41248792 2627 86193.90 436.02 0.91
## 373 43070966 7366 171945.87 492.47 0.74
## 374 57893841 14046 333834.43 632.80 0.68
## 375 115901469 20703 482809.71 1259.61 0.74
## 376 30655008 12915 225910.92 301.84 0.79
## 377 33484177 16302 181430.97 372.68 0.88
## 378 36840773 14143 311830.18 414.35 0.73
## 379 44320061 12909 165571.09 448.58 0.73
## 380 62746031 29229 468078.61 652.88 0.75
## 381 27700261 15128 159782.94 295.72 0.68
## 382 42906055 20627 286767.83 523.77 0.53
## 383 613026612 102325 3313522.61 6392.68 0.82
## 384 973911640 50913 2384670.02 6666.90 0.71str(Week6)## 'data.frame': 384 obs. of 5 variables:
## $ Expenditures : num 1.15e+08 1.16e+08 1.20e+08 1.91e+07 2.46e+08 ...
## $ Enrolled : int 25294 42186 23772 2085 67258 23752 53781 98763 47986 64634 ...
## $ RVUs : num 402704 638252 447030 43337 1579789 ...
## $ FTEs : num 955 949 953 200 2162 ...
## $ Quality.Score: num 0.67 0.58 0.52 0.93 0.96 0.56 0.84 0.82 0.76 0.81 ...## x<-model.matrix(~ Expenditures + Enrolled + RVUs + FTEs + Quality.Score,Week6)
## x
## y<-Week6$Enrolled
## y
##xtxi<-solve(t(x) %*% x)
## xtxi %*% t(x) %*% y#loading psych package
require(psych)## Loading required package: psych##
## Attaching package: 'psych'## The following objects are masked from 'package:ggplot2':
##
## %+%, alphaWeek6Desc <- describe(Week6) # Summary Statistics
Week6Desc## vars n mean sd median trimmed
## Expenditures 1 384 124765816.51 173435868.06 52796103.84 84600936.46
## Enrolled 2 384 24713.82 22756.42 16466.00 20647.21
## RVUs 3 384 546857.77 680047.21 246701.71 398680.18
## FTEs 4 384 1060.86 1336.29 483.07 750.11
## Quality.Score 5 384 0.71 0.11 0.73 0.72
## mad min max range skew kurtosis
## Expenditures 47444814.57 7839562.52 1.301994e+09 1.294154e+09 3.04 11.27
## Enrolled 13092.10 1218.00 1.198500e+05 1.186320e+05 1.94 4.14
## RVUs 237760.93 23218.01 3.574006e+06 3.550788e+06 2.10 4.27
## FTEs 401.38 116.29 7.518630e+03 7.402340e+03 2.55 6.94
## Quality.Score 0.10 0.31 9.600000e-01 6.500000e-01 -0.55 0.40
## se
## Expenditures 8850612.08
## Enrolled 1161.28
## RVUs 34703.51
## FTEs 68.19
## Quality.Score 0.01The RVUs values, are greater than 3 standard deviations from the mean. This could influence any interpretations made from the outcome of the correlation analysis.
Week6$outlier_Enrolled <- ( Week6$Enrolled - Week6Desc["Enrolled","mean"] ) / Week6Desc["Enrolled","sd"]
sum(abs((Week6$outlier_Enrolled)>3))## [1] 12Week6$outlier_RVUs <- ( Week6$RVUs - Week6Desc["RVUs","mean"]) / (Week6Desc["RVUs","sd"] )
sum(abs((Week6$outlier_RVUs)>3)) ## [1] 12- for a bivariate regression takes the following form: y = mx + c ** y is the response (dependent) variable, m is the gradient (slope), x is the predictor (independent) variable, and c is the intercept.
slope <-cor(Week6$RVUs,Week6$Enrolled) * (sd(Week6$Enrolled)/sd(Week6$RVUs))
slope## [1] 0.03062542intercept <- mean(Week6$Enrolled) - (slope * mean(Week6$RVUs))
intercept## [1] 7966.066library('tidyverse')## ── Attaching packages ─────────────────────────────────────── tidyverse 1.3.2 ──
## ✔ tibble 3.1.8 ✔ stringr 1.5.0
## ✔ purrr 1.0.1 ✔ forcats 1.0.0
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ psych::%+%() masks ggplot2::%+%()
## ✖ psych::alpha() masks ggplot2::alpha()
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag() masks stats::lag()library(ggplot2)
library(dplyr)
library(ggplot2)
Week6 %>%
ggplot(aes(x = Week6$RVUs, y = Week6$Enrolled)) +
geom_point(colour = "red")## Warning: Use of `Week6$RVUs` is discouraged.
## ℹ Use `RVUs` instead.## Warning: Use of `Week6$Enrolled` is discouraged.
## ℹ Use `Enrolled` instead.
cor(Week6)## Expenditures Enrolled RVUs FTEs Quality.Score
## Expenditures 1.0000000 0.7707756 0.9217239 0.9796506 0.2749501
## Enrolled 0.7707756 1.0000000 0.9152024 0.8148491 0.2526991
## RVUs 0.9217239 0.9152024 1.0000000 0.9504093 0.3075742
## FTEs 0.9796506 0.8148491 0.9504093 1.0000000 0.2769058
## Quality.Score 0.2749501 0.2526991 0.3075742 0.2769058 1.0000000
## outlier_Enrolled 0.7707756 1.0000000 0.9152024 0.8148491 0.2526991
## outlier_RVUs 0.9217239 0.9152024 1.0000000 0.9504093 0.3075742
## outlier_Enrolled outlier_RVUs
## Expenditures 0.7707756 0.9217239
## Enrolled 1.0000000 0.9152024
## RVUs 0.9152024 1.0000000
## FTEs 0.8148491 0.9504093
## Quality.Score 0.2526991 0.3075742
## outlier_Enrolled 1.0000000 0.9152024
## outlier_RVUs 0.9152024 1.0000000plot(Week6)
** The strength of the relationship can be quantified using the Pearson correlation coefficient.
cor(Week6$RVUs,Week6$Enrolled)## [1] 0.9152024Week6 %>%
ggplot(aes(x = sqrt(RVUs), y = sqrt(Enrolled))) +
geom_point(colour = "orangered") 
cor(sqrt(Week6$RVUs), sqrt(Week6$Enrolled))## [1] 0.9317273*determine whether the relationship is statistically significant . data fit well to the line, then the relationship is likely to be a real effect. The goodness of fit can be quantified using the root mean squared error (RMSE) and R-squared metrics.. A rule of thumb for OLS linear regression is that at least 20 data points are required for a valid model.
Week6 %>%
ggplot(aes(x = sqrt(RVUs), y = sqrt(Enrolled))) +
geom_point(colour = "maroon") +
geom_smooth(method = "lm", fill = NA)## `geom_smooth()` using formula = 'y ~ x'
hist(x = Week6$RVUs, xlab = "", main = "Outpatients (RVU)")
hist(x=Week6$Enrolled,xlab = "",main = "Week6 Enrolled")
plot(x = Week6$RVUs, y = Week6$Enrolled, xlab = "RVUs", ylab = "Enrolled")
univariate_reg = lm(Week6$Enrolled ~ Week6$RVUs)
summary(univariate_reg)##
## Call:
## lm(formula = Week6$Enrolled ~ Week6$RVUs)
##
## Residuals:
## Min 1Q Median 3Q Max
## -43488 -4115 -436 3998 41556
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 7.966e+03 6.016e+02 13.24 <2e-16 ***
## Week6$RVUs 3.063e-02 6.900e-04 44.39 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 9183 on 382 degrees of freedom
## Multiple R-squared: 0.8376, Adjusted R-squared: 0.8372
## F-statistic: 1970 on 1 and 382 DF, p-value: < 2.2e-16options(scipen = 999)
summary(univariate_reg)##
## Call:
## lm(formula = Week6$Enrolled ~ Week6$RVUs)
##
## Residuals:
## Min 1Q Median 3Q Max
## -43488 -4115 -436 3998 41556
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 7966.06606 601.62884 13.24 <0.0000000000000002 ***
## Week6$RVUs 0.03063 0.00069 44.39 <0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 9183 on 382 degrees of freedom
## Multiple R-squared: 0.8376, Adjusted R-squared: 0.8372
## F-statistic: 1970 on 1 and 382 DF, p-value: < 0.00000000000000022?abline## starting httpd help server ... doneabline(reg = univariate_reg, col="blue")
model<-lm(Enrolled ~ RVUs,data=Week6)
summary(model)##
## Call:
## lm(formula = Enrolled ~ RVUs, data = Week6)
##
## Residuals:
## Min 1Q Median 3Q Max
## -43488 -4115 -436 3998 41556
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 7966.06606 601.62884 13.24 <0.0000000000000002 ***
## RVUs 0.03063 0.00069 44.39 <0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 9183 on 382 degrees of freedom
## Multiple R-squared: 0.8376, Adjusted R-squared: 0.8372
## F-statistic: 1970 on 1 and 382 DF, p-value: < 0.00000000000000022lmodel <- lm(sqrt(RVUs) ~ sqrt(Enrolled), data = Week6)
lmodel##
## Call:
## lm(formula = sqrt(RVUs) ~ sqrt(Enrolled), data = Week6)
##
## Coefficients:
## (Intercept) sqrt(Enrolled)
## -173.623 5.609lmodel$coefficients## (Intercept) sqrt(Enrolled)
## -173.62320 5.60907summary(lmodel)##
## Call:
## lm(formula = sqrt(RVUs) ~ sqrt(Enrolled), data = Week6)
##
## Residuals:
## Min 1Q Median 3Q Max
## -304.61 -90.46 -21.67 61.43 587.94
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -173.6232 17.5848 -9.873 <0.0000000000000002 ***
## sqrt(Enrolled) 5.6091 0.1119 50.145 <0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 139.2 on 382 degrees of freedom
## Multiple R-squared: 0.8681, Adjusted R-squared: 0.8678
## F-statistic: 2514 on 1 and 382 DF, p-value: < 0.00000000000000022Fvalue<-fitted.values(model)
Fvalue## 1 2 3 4 5 6 7
## 20299.037 27512.801 21656.534 9293.288 56347.779 28578.093 81385.430
## 8 9 10 11 12 13 14
## 77079.371 55706.065 60432.247 43378.803 12902.783 24826.773 40151.004
## 15 16 17 18 19 20 21
## 38901.664 19360.097 22831.318 33648.393 39534.028 49538.954 15562.932
## 22 23 24 25 26 27 28
## 100366.996 58959.097 25130.849 30512.253 15756.691 15340.917 38407.750
## 29 30 31 32 33 34 35
## 90875.954 13606.456 58462.476 41722.299 86127.326 22924.506 9475.708
## 36 37 38 39 40 41 42
## 11908.745 10696.280 13303.759 12442.733 12202.951 15359.638 11768.608
## 43 44 45 46 47 48 49
## 11449.182 11433.566 12414.660 10774.737 9632.285 11663.191 12461.464
## 50 51 52 53 54 55 56
## 10536.035 9830.002 14667.742 15085.742 11378.338 12344.381 10185.608
## 57 58 59 60 61 62 63
## 11856.054 17864.989 11925.163 9490.516 10821.511 13641.773 12017.266
## 64 65 66 67 68 69 70
## 12370.804 9369.817 18896.817 12044.177 11998.776 11365.408 11326.721
## 71 72 73 74 75 76 77
## 11658.369 17787.815 17579.356 10166.664 65100.431 11581.768 32725.357
## 78 79 80 81 82 83 84
## 10293.464 11798.100 24830.964 8688.260 9442.921 26591.901 20428.438
## 85 86 87 88 89 90 91
## 9336.841 15808.758 15400.862 25005.189 13518.732 11418.132 29723.616
## 92 93 94 95 96 97 98
## 14074.659 15051.751 13645.764 34172.976 8947.857 11063.378 11288.562
## 99 100 101 102 103 104 105
## 28934.648 9679.920 30586.574 10745.213 23387.863 25869.906 29842.785
## 106 107 108 109 110 111 112
## 45101.191 52948.620 16563.277 9570.449 50463.443 17070.394 12701.553
## 113 114 115 116 117 118 119
## 17989.440 30458.782 39524.776 10726.632 13101.238 18287.621 22685.519
## 120 121 122 123 124 125 126
## 15424.403 15066.377 16511.415 13143.868 24057.184 13397.147 18612.798
## 127 128 129 130 131 132 133
## 113155.293 64448.081 22073.280 31348.014 22402.943 9374.135 63428.585
## 134 135 136 137 138 139 140
## 28816.648 96564.655 81169.711 57724.034 64220.528 61133.865 12565.325
## 141 142 143 144 145 146 147
## 25391.444 36910.562 43423.642 18321.139 23650.097 38265.590 40796.441
## 148 149 150 151 152 153 154
## 52491.414 16077.119 102710.888 59808.426 25968.707 30108.702 15965.474
## 155 156 157 158 159 160 161
## 15712.495 39060.111 89830.355 14507.809 64701.468 47479.137 88106.135
## 162 163 164 165 166 167 168
## 22637.563 9683.065 12069.797 10771.750 13257.077 12652.615 12483.142
## 169 170 171 172 173 174 175
## 15558.395 12078.008 11701.829 11717.657 13647.297 10869.023 9925.398
## 176 177 178 179 180 181 182
## 12144.270 12704.856 10726.668 10032.640 14919.386 15888.107 11908.945
## 183 184 185 186 187 188 189
## 13041.854 9877.778 12369.296 18507.325 12251.957 9664.931 11014.167
## 190 191 192 193 194 195 196
## 13872.215 12447.988 13120.114 9415.169 18980.802 11947.031 12138.407
## 197 198 199 200 201 202 203
## 11421.329 11501.838 11311.952 19028.733 18073.427 9935.841 57021.678
## 204 205 206 207 208 209 210
## 11856.424 34422.264 10479.194 12391.383 26967.462 8677.127 9758.215
## 211 212 213 214 215 216 217
## 27608.871 20689.480 9591.455 16018.685 15833.633 25077.609 13756.759
## 218 219 220 221 222 223 224
## 11006.157 30650.403 13960.421 15173.730 14490.006 35642.134 9025.342
## 225 226 227 228 229 230 231
## 10809.979 11587.674 30075.909 9823.161 34598.792 10947.329 23866.669
## 232 233 234 235 236 237 238
## 25268.971 30867.978 48750.693 53410.894 17195.778 9478.796 49573.926
## 239 240 241 242 243 244 245
## 17268.430 12681.636 18155.461 31967.521 40903.803 11027.591 13665.951
## 246 247 248 249 250 251 252
## 18796.124 24060.162 15459.029 14501.346 16985.190 13828.081 23913.582
## 253 254 255 256 257 258 259
## 13435.365 18679.186 117421.506 94415.471 22972.657 28964.998 22182.202
## 260 261 262 263 264 265 266
## 9576.219 53838.398 28515.038 90483.714 75643.278 48716.923 62557.694
## 267 268 269 270 271 272 273
## 58480.245 12461.923 24244.816 33630.871 37230.737 15484.424 21211.813
## 274 275 276 277 278 279 280
## 33693.809 38364.802 54500.202 15759.997 95574.561 52976.394 23096.028
## 281 282 283 284 285 286 287
## 29380.678 15948.365 14955.895 36478.903 80926.805 13483.284 62285.184
## 288 289 290 291 292 293 294
## 40463.268 78293.730 21705.705 9893.809 11899.872 10676.887 12801.719
## 295 296 297 298 299 300 301
## 12201.673 12970.823 15628.144 11716.231 11180.496 11339.109 13295.416
## 302 303 304 305 306 307 308
## 10688.989 9742.411 11841.604 12080.358 10924.135 9670.441 14677.198
## 309 310 311 312 313 314 315
## 15025.403 11470.329 12416.985 10044.266 11937.902 17628.613 12053.873
## 316 317 318 319 320 321 322
## 9319.623 10677.229 13179.673 11883.228 12378.974 9563.422 17779.833
## 323 324 325 326 327 328 329
## 11329.393 11576.989 11218.554 11272.407 10655.458 18121.963 17471.278
## 330 331 332 333 334 335 336
## 10258.826 51664.609 11896.184 32777.897 9961.086 12073.905 26283.051
## 337 338 339 340 341 342 343
## 8682.433 9415.281 27066.357 19778.209 9431.428 15432.199 14801.985
## 344 345 346 347 348 349 350
## 22656.773 12848.378 10986.352 28374.428 13372.533 14010.193 14311.356
## 351 352 353 354 355 356 357
## 31960.763 8853.355 10602.020 11383.804 27907.721 9598.589 31670.955
## 358 359 360 361 362 363 364
## 10847.532 22075.732 25861.820 28160.425 44952.921 52953.214 16320.859
## 365 366 367 368 369 370 371
## 9366.993 44315.109 16646.453 12174.736 16989.587 31209.695 38517.999
## 372 373 374 375 376 377 378
## 10605.790 13231.981 18189.886 22752.316 14884.683 13522.466 17515.996
## 379 380 381 382 383 384
## 13036.750 22301.170 12859.486 16748.451 109444.088 80997.587Res<-lm(formula = model, data = Week6)
Res##
## Call:
## lm(formula = model, data = Week6)
##
## Coefficients:
## (Intercept) RVUs
## 7966.06606 0.03063# Residual Analysis
# plot(fitted(model),resid(model))
## plot(fitted(model),Res)
# abline(0,0)
qqnorm(resid(model))
qqline(resid(model))
par(mfrow=c(2,2))
plot(model)
Interpret the linear model - Expenditures~RVUs. Enrolled=7966.06606 +
0.03063.1∗RVUs
1 unit increase in RVUs is associated with 235.1 units increase in Expenditures.
plot(x = univariate_reg)
qqnorm(y=Week6$Enrolled)
qqnorm( y = log(Week6$Enrolled))
Transformed regression - ln(Enrolled)~RVU
hist(x = log(Week6$Enrolled),xlab = "", main = "Log Enrolled Details" )
hist(x=log(Week6$RVUs), xlab = "", main = "Log Outpatient Details (RVU)")
plot(x = Week6$RVUs, y = log (Week6$Enrolled) , xlab = "RVUs", ylab = "Log of Enrolled")
univariate_reg_transformedY = lm( formula = log(Week6$Enrolled) ~ Week6$RVUs)
?abline
abline(reg = univariate_reg_transformedY, col="blue")
summary(univariate_reg)##
## Call:
## lm(formula = Week6$Enrolled ~ Week6$RVUs)
##
## Residuals:
## Min 1Q Median 3Q Max
## -43488 -4115 -436 3998 41556
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 7966.06606 601.62884 13.24 <0.0000000000000002 ***
## Week6$RVUs 0.03063 0.00069 44.39 <0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 9183 on 382 degrees of freedom
## Multiple R-squared: 0.8376, Adjusted R-squared: 0.8372
## F-statistic: 1970 on 1 and 382 DF, p-value: < 0.00000000000000022summary(univariate_reg_transformedY)##
## Call:
## lm(formula = log(Week6$Enrolled) ~ Week6$RVUs)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.13708 -0.24629 0.08705 0.39173 0.98413
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 9.18878091966 0.03716184238 247.26 <0.0000000000000002 ***
## Week6$RVUs 0.00000101667 0.00000004262 23.86 <0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5672 on 382 degrees of freedom
## Multiple R-squared: 0.5983, Adjusted R-squared: 0.5973
## F-statistic: 569.1 on 1 and 382 DF, p-value: < 0.00000000000000022plot(univariate_reg_transformedY)
Constant variability The plot of residuals versus fitted observations
shows that the variability of errors around the predicted values is
slightly better.
Scale-Location plot - For OLS, the trend line is even and the residuals are uniformly scattered.
plot( univariate_reg , which = 3)
plot( univariate_reg_transformedY , which = 3)
hist(x = log(Week6$Enrolled), xlab = "", main = "Log Enrolled Details" )
hist(x = log(Week6$RVUs) , xlab = "", main = "Log Outpatient Details (RVU)")
log_log_reg = lm( formula = log(Week6$Enrolled) ~ log(Week6$RVUs))
summary(log_log_reg)##
## Call:
## lm(formula = log(Week6$Enrolled) ~ log(Week6$RVUs))
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.3934 -0.1784 0.0763 0.2389 0.5527
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.5550 0.1856 2.99 0.00297 **
## log(Week6$RVUs) 0.7310 0.0147 49.71 < 0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3275 on 382 degrees of freedom
## Multiple R-squared: 0.8661, Adjusted R-squared: 0.8658
## F-statistic: 2472 on 1 and 382 DF, p-value: < 0.00000000000000022plot(log_log_reg)
plot(x = Week6$RVUs, y = log (Week6$Enrolled) , xlab = "RVUs", ylab = "Log of Enrolled") 
Week6$RVUs2 <- Week6$RVUs^2
univariate_reg_transformedY2 = lm( formula = log(Week6$Enrolled) ~ Week6$RVUs+ Week6$RVUs2)
# abline(reg = univariate_reg_transformedY2, col="blue")
# summary(univariate_reg)
# summary(univariate_reg_transformedY)
summary(univariate_reg_transformedY2)##
## Call:
## lm(formula = log(Week6$Enrolled) ~ Week6$RVUs + Week6$RVUs2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.88905 -0.22091 0.09647 0.33631 0.91031
##
## Coefficients:
## Estimate Std. Error t value
## (Intercept) 8.87524472896051719 0.03858990397976500 229.99
## Week6$RVUs 0.00000229359534853 0.00000010161348123 22.57
## Week6$RVUs2 -0.00000000000050606 0.00000000000003778 -13.39
## Pr(>|t|)
## (Intercept) <0.0000000000000002 ***
## Week6$RVUs <0.0000000000000002 ***
## Week6$RVUs2 <0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4683 on 381 degrees of freedom
## Multiple R-squared: 0.7269, Adjusted R-squared: 0.7255
## F-statistic: 507.1 on 2 and 381 DF, p-value: < 0.00000000000000022plot(univariate_reg_transformedY2)
Some ways to fix heteroskedasticity - Transform the dependent variable. sqrt() will have larger penalty, but interpretation is not as easy/standard as when taking log.
Alternative models - sqrt(Enrolled)~RVU
plot(x = Week6$RVUs, y = sqrt(Week6$Enrolled) , xlab = "RVUs", ylab = "Square Root of Enrolled")
univariate_reg_transformedY_sqrt = lm( formula = sqrt(Week6$Enrolled) ~ Week6$RVUs)
?abline
abline(reg = univariate_reg_transformedY_sqrt, col="blue")
summary(univariate_reg)##
## Call:
## lm(formula = Week6$Enrolled ~ Week6$RVUs)
##
## Residuals:
## Min 1Q Median 3Q Max
## -43488 -4115 -436 3998 41556
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 7966.06606 601.62884 13.24 <0.0000000000000002 ***
## Week6$RVUs 0.03063 0.00069 44.39 <0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 9183 on 382 degrees of freedom
## Multiple R-squared: 0.8376, Adjusted R-squared: 0.8372
## F-statistic: 1970 on 1 and 382 DF, p-value: < 0.00000000000000022summary(univariate_reg_transformedY)##
## Call:
## lm(formula = log(Week6$Enrolled) ~ Week6$RVUs)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.13708 -0.24629 0.08705 0.39173 0.98413
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 9.18878091966 0.03716184238 247.26 <0.0000000000000002 ***
## Week6$RVUs 0.00000101667 0.00000004262 23.86 <0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5672 on 382 degrees of freedom
## Multiple R-squared: 0.5983, Adjusted R-squared: 0.5973
## F-statistic: 569.1 on 1 and 382 DF, p-value: < 0.00000000000000022summary(univariate_reg_transformedY_sqrt)##
## Call:
## lm(formula = sqrt(Week6$Enrolled) ~ Week6$RVUs)
##
## Residuals:
## Min 1Q Median 3Q Max
## -105.941 -18.065 -0.457 20.797 82.874
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 98.68414087 1.96206332 50.30 <0.0000000000000002 ***
## Week6$RVUs 0.00008252 0.00000225 36.67 <0.0000000000000002 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 29.95 on 382 degrees of freedom
## Multiple R-squared: 0.7788, Adjusted R-squared: 0.7782
## F-statistic: 1345 on 1 and 382 DF, p-value: < 0.00000000000000022plot( univariate_reg_transformedY_sqrt) # much better - 
