Data information A study looks at factors that influence the decision of whether to apply to graduate school. College juniors are asked if they are unlikely, somewhat likely, or very likely to apply to graduate school. Hence, our outcome variable has three categories. Data on parental educational status, whether the undergraduate institution is public or private, and current GPA is also collected. The data set (ologit.sav) contains variables on 400 students. The outcome variable is apply (0=unlikely, 1=somewhat likely, 2 = very likely). There are three variables that we will use as predictors: pared, which is a 0=No,/1=Yes variable indicating whether at least one parent has a graduate degree; public, which is a 0/1 variable where 1 indicates that the undergraduate institution is public and 0 = private, and gpa, which is the student’s grade point average.
library(haven)
ologit <- read_sav("~/Box Sync/Courses/2019 Fall/BER642/10 Ordinal Logistic Model/SPSS/ologit.sav")
View(ologit)## Warning in system2("/usr/bin/otool", c("-L", shQuote(DSO)), stdout = TRUE):
## running command ''/usr/bin/otool' -L '/Library/Frameworks/R.framework/
## Resources/modules/R_de.so'' had status 1head(ologit)## # A tibble: 6 x 4
## apply pared public gpa
## <dbl> <dbl> <dbl> <dbl>
## 1 2 0 0 3.26
## 2 1 1 0 3.21
## 3 0 1 1 3.94
## 4 1 0 0 2.81
## 5 1 0 0 2.53
## 6 0 0 1 2.59OLR_data <- ologit
OLR_data$apply <- factor(OLR_data$apply,levels=c(0,1,2),labels=c("unlikely","somewhat likely","very likely"))
OLR_data$pared <- factor(OLR_data$pared,levels=c(0,1),labels=c("no degree","degree"))
OLR_data$public <- factor(OLR_data$public,levels=c(0,1),labels=c("private","public"))summary(OLR_data)## apply pared public gpa
## unlikely :220 no degree:337 private:343 Min. :1.900
## somewhat likely:140 degree : 63 public : 57 1st Qu.:2.720
## very likely : 40 Median :2.990
## Mean :2.999
## 3rd Qu.:3.270
## Max. :4.000table(OLR_data$apply, OLR_data$pared)##
## no degree degree
## unlikely 200 20
## somewhat likely 110 30
## very likely 27 13table(OLR_data$apply, OLR_data$public)##
## private public
## unlikely 189 31
## somewhat likely 124 16
## very likely 30 10table(OLR_data$gpa, OLR_data$apply)##
## unlikely somewhat likely very likely
## 1.89999997615814 1 0 0
## 1.98000001907349 1 0 0
## 2.02999997138977 1 0 0
## 2.08999991416931 1 0 0
## 2.13000011444092 1 0 0
## 2.15000009536743 1 0 0
## 2.17000007629395 1 0 0
## 2.22000002861023 1 1 0
## 2.23000001907349 1 0 0
## 2.25 2 1 0
## 2.25999999046326 2 0 0
## 2.27999997138977 1 0 0
## 2.28999996185303 1 0 0
## 2.30999994277954 1 0 0
## 2.3199999332428 1 0 0
## 2.33999991416931 1 0 0
## 2.34999990463257 0 1 0
## 2.38000011444092 1 1 0
## 2.40000009536743 3 0 0
## 2.41000008583069 1 0 1
## 2.42000007629395 1 0 0
## 2.4300000667572 0 1 0
## 2.44000005722046 4 1 0
## 2.45000004768372 0 1 0
## 2.47000002861023 2 2 0
## 2.48000001907349 1 0 0
## 2.49000000953674 2 1 0
## 2.5 1 1 0
## 2.50999999046326 1 1 0
## 2.52999997138977 2 2 0
## 2.54999995231628 2 1 1
## 2.55999994277954 0 3 0
## 2.5699999332428 2 2 1
## 2.57999992370605 0 1 0
## 2.58999991416931 1 0 0
## 2.59999990463257 2 1 0
## 2.60999989509583 1 1 0
## 2.61999988555908 2 0 1
## 2.63000011444092 4 0 0
## 2.64000010490417 1 2 0
## 2.65000009536743 0 4 0
## 2.67000007629395 3 0 0
## 2.6800000667572 1 2 0
## 2.70000004768372 2 1 1
## 2.71000003814697 0 1 1
## 2.72000002861023 3 2 0
## 2.73000001907349 1 2 0
## 2.74000000953674 4 1 0
## 2.75 1 1 1
## 2.75999999046326 3 0 0
## 2.76999998092651 2 1 0
## 2.77999997138977 0 2 0
## 2.78999996185303 3 1 1
## 2.79999995231628 1 2 0
## 2.80999994277954 1 1 0
## 2.82999992370605 3 1 0
## 2.83999991416931 2 2 0
## 2.84999990463257 3 2 0
## 2.85999989509583 1 3 0
## 2.86999988555908 2 2 0
## 2.88000011444092 3 4 0
## 2.89000010490417 2 2 0
## 2.90000009536743 3 0 0
## 2.91000008583069 3 0 0
## 2.92000007629395 3 0 0
## 2.9300000667572 2 0 0
## 2.94000005722046 2 1 1
## 2.95000004768372 2 1 1
## 2.96000003814697 0 1 2
## 2.97000002861023 1 2 1
## 2.98000001907349 5 1 1
## 2.99000000953674 3 1 1
## 3 2 1 1
## 3.01999998092651 3 0 0
## 3.02999997138977 1 0 0
## 3.03999996185303 3 1 1
## 3.04999995231628 7 1 0
## 3.05999994277954 2 0 0
## 3.0699999332428 0 1 1
## 3.07999992370605 2 3 1
## 3.08999991416931 2 2 0
## 3.09999990463257 4 0 0
## 3.10999989509583 2 3 1
## 3.11999988555908 3 2 0
## 3.13000011444092 1 2 0
## 3.14000010490417 3 1 0
## 3.15000009536743 1 2 1
## 3.16000008583069 3 2 0
## 3.17000007629395 0 1 0
## 3.1800000667572 0 1 0
## 3.19000005722046 1 0 0
## 3.20000004768372 1 3 0
## 3.21000003814697 1 2 2
## 3.22000002861023 4 2 0
## 3.23000001907349 1 1 0
## 3.24000000953674 1 1 1
## 3.25 2 0 1
## 3.25999999046326 3 1 1
## 3.26999998092651 2 1 0
## 3.27999997138977 2 1 1
## 3.28999996185303 2 1 0
## 3.29999995231628 3 0 0
## 3.30999994277954 1 0 0
## 3.3199999332428 1 1 0
## 3.32999992370605 0 1 0
## 3.33999991416931 2 1 0
## 3.35999989509583 2 1 1
## 3.36999988555908 2 1 0
## 3.38000011444092 1 2 1
## 3.39000010490417 4 1 0
## 3.40000009536743 0 1 1
## 3.41000008583069 1 0 0
## 3.42000007629395 0 0 1
## 3.4300000667572 0 1 0
## 3.44000005722046 0 1 0
## 3.45000004768372 2 0 1
## 3.46000003814697 0 1 0
## 3.48000001907349 0 1 0
## 3.49000000953674 1 1 0
## 3.5 2 1 0
## 3.50999999046326 3 2 0
## 3.51999998092651 1 0 1
## 3.52999997138977 1 0 1
## 3.53999996185303 0 3 0
## 3.54999995231628 1 0 0
## 3.55999994277954 1 2 1
## 3.5699999332428 2 2 1
## 3.59999990463257 1 0 0
## 3.60999989509583 1 0 1
## 3.61999988555908 1 1 0
## 3.63000011444092 0 0 1
## 3.65000009536743 3 1 0
## 3.67000007629395 1 0 0
## 3.6800000667572 0 1 0
## 3.69000005722046 0 1 0
## 3.70000004768372 1 1 0
## 3.71000003814697 1 0 0
## 3.72000002861023 1 1 0
## 3.73000001907349 0 1 0
## 3.75 0 1 0
## 3.76999998092651 1 0 0
## 3.80999994277954 1 0 1
## 3.90000009536743 0 1 1
## 3.91000008583069 0 1 0
## 3.94000005722046 1 0 0
## 4 0 1 0library(MASS)
OLRmodel <- polr(apply ~ pared + public + gpa , data = OLR_data, Hess = TRUE)
summary(OLRmodel)## Call:
## polr(formula = apply ~ pared + public + gpa, data = OLR_data,
## Hess = TRUE)
##
## Coefficients:
## Value Std. Error t value
## pareddegree 1.04769 0.2658 3.9418
## publicpublic -0.05879 0.2979 -0.1974
## gpa 0.61594 0.2606 2.3632
##
## Intercepts:
## Value Std. Error t value
## unlikely|somewhat likely 2.2039 0.7795 2.8272
## somewhat likely|very likely 4.2994 0.8043 5.3453
##
## Residual Deviance: 717.0249
## AIC: 727.0249OIM <- polr(apply ~ 1, data = OLR_data)
summary(OIM)##
## Re-fitting to get Hessian## Call:
## polr(formula = apply ~ 1, data = OLR_data)
##
## No coefficients
##
## Intercepts:
## Value Std. Error t value
## unlikely|somewhat likely 0.2007 0.1005 1.9967
## somewhat likely|very likely 2.1972 0.1667 13.1833
##
## Residual Deviance: 741.2053
## AIC: 745.2053anova(OIM,OLRmodel)## Likelihood ratio tests of ordinal regression models
##
## Response: apply
## Model Resid. df Resid. Dev Test Df LR stat.
## 1 1 398 741.2053
## 2 pared + public + gpa 395 717.0249 1 vs 2 3 24.18041
## Pr(Chi)
## 1
## 2 2.290456e-05From the table, we can see that the chi-square is 24.18 and p < .001. This means that we can reject the null hypothesis that the model without predictors is as good as the model with the predictors.
OLRmodel$fitted.values## unlikely somewhat likely very likely
## 1 0.5488310 0.3593310 0.09183798
## 2 0.3055632 0.4759496 0.21848725
## 3 0.2293835 0.4781951 0.29242138
## 4 0.6161224 0.3126888 0.07118879
## 5 0.6560149 0.2833901 0.06059505
## 6 0.6609240 0.2797117 0.05936430
## 7 0.6518332 0.2865114 0.06165546
## 8 0.6277081 0.3042932 0.06799872
## 9 0.5880943 0.3325798 0.07932590
## 10 0.2690265 0.4804611 0.25051246
## 11 0.2624695 0.4806557 0.25687482
## 12 0.6117427 0.3158370 0.07242031
## 13 0.4651895 0.4109079 0.12390270
## 14 0.6348763 0.2990510 0.06607274
## 15 0.2606326 0.4806729 0.25869445
## 16 0.5746035 0.3419265 0.08347005
## 17 0.5267017 0.3737593 0.09953904
## 18 0.7041885 0.2466752 0.04913633
## 19 0.5481355 0.3597920 0.09207247
## 20 0.5274017 0.3733107 0.09928762
## 21 0.6248248 0.3063916 0.06878359
## 22 0.3068717 0.4756909 0.21743736
## 23 0.6684138 0.2740709 0.05751533
## 24 0.6132046 0.3147877 0.07200763
## 25 0.6022571 0.3226055 0.07513737
## 26 0.5618257 0.3506323 0.08754201
## 27 0.3147880 0.4739974 0.21121460
## 28 0.5320050 0.3703478 0.09764730
## 29 0.6327950 0.3005767 0.06662826
## 30 0.6029296 0.3221279 0.07494245
## 31 0.5304711 0.3713375 0.09819138
## 32 0.6096106 0.3173643 0.07302504
## 33 0.5910752 0.3304940 0.07843087
## 34 0.2438432 0.4800238 0.27613296
## 35 0.6262676 0.3053423 0.06839011
## 36 0.6433997 0.2927718 0.06382850
## 37 0.6102788 0.3168861 0.07283516
## 38 0.6601726 0.2802757 0.05955171
## 39 0.6248248 0.3063916 0.06878359
## 40 0.6925119 0.2556961 0.05179204
## 41 0.7218296 0.2329099 0.04526050
## 42 0.3545254 0.4624880 0.18298661
## 43 0.6291463 0.3032443 0.06760940
## 44 0.6044033 0.3210801 0.07451656
## 45 0.4097138 0.4397370 0.15054923
## 46 0.5821135 0.3367425 0.08114402
## 47 0.5730972 0.3429603 0.08394247
## 48 0.6504340 0.2875532 0.06201277
## 49 0.7143486 0.2387668 0.04688457
## 50 0.5970166 0.3263150 0.07666847
## 51 0.5197197 0.3782043 0.10207597
## 52 0.6233798 0.3074410 0.06917916
## 53 0.6081438 0.3184131 0.07344308
## 54 0.6044033 0.3210801 0.07451656
## 55 0.5925631 0.3294501 0.07798682
## 56 0.4636574 0.4117697 0.12457286
## 57 0.6219326 0.3084905 0.06957684
## 58 0.6587894 0.2813130 0.05989760
## 59 0.5518794 0.3573049 0.09081570
## 60 0.6684138 0.2740709 0.05751533
## 61 0.6204833 0.3095401 0.06997663
## 62 0.5970166 0.3263150 0.07666847
## 63 0.3042578 0.4762014 0.21954080
## 64 0.5120284 0.3830381 0.10493350
## 65 0.2606326 0.4806729 0.25869445
## 66 0.5625169 0.3501652 0.08731795
## 67 0.5678814 0.3465247 0.08559390
## 68 0.6014541 0.3231753 0.07537058
## 69 0.5457789 0.3613506 0.09287059
## 70 0.5542303 0.3557362 0.09003348
## 71 0.4935528 0.3943661 0.11208103
## 72 0.6532297 0.2854702 0.06130008
## 73 0.5925631 0.3294501 0.07798682
## 74 0.2535742 0.4805812 0.26584462
## 75 0.5993026 0.3246995 0.07599792
## 76 0.5113267 0.3834758 0.10519756
## 77 0.5955338 0.3273606 0.07710563
## 78 0.2438432 0.4800238 0.27613296
## 79 0.3788350 0.4533165 0.16784856
## 80 0.2432259 0.4799708 0.27680330
## 81 0.6615530 0.2792393 0.05920768
## 82 0.6291463 0.3032443 0.06760940
## 83 0.4428406 0.4231318 0.13402766
## 84 0.6560149 0.2833901 0.06059505
## 85 0.5488310 0.3593310 0.09183798
## 86 0.6504340 0.2875532 0.06201277
## 87 0.6029296 0.3221279 0.07494245
## 88 0.5806144 0.3377812 0.08160445
## 89 0.6102788 0.3168861 0.07283516
## 90 0.4354012 0.4270264 0.13757240
## 91 0.2813099 0.4795637 0.23912638
## 92 0.5663693 0.3475535 0.08607721
## 93 0.5251660 0.3747415 0.10009248
## 94 0.6532297 0.2854702 0.06130008
## 95 0.5791138 0.3388189 0.08206725
## 96 0.5880943 0.3325798 0.07932590
## 97 0.5685704 0.3460553 0.08537435
## 98 0.5640321 0.3491396 0.08682833
## 99 0.6643057 0.2771691 0.05852521
## 100 0.6044033 0.3210801 0.07451656
## 101 0.6262676 0.3053423 0.06839011
## 102 0.5640321 0.3491396 0.08682833
## 103 0.5999767 0.3242223 0.07580095
## 104 0.5012517 0.3896957 0.10905257
## 105 0.2702395 0.4804028 0.24935777
## 106 0.6650526 0.2766066 0.05834084
## 107 0.3174512 0.4733793 0.20916953
## 108 0.3831929 0.4515237 0.16528343
## 109 0.6587894 0.2813130 0.05989760
## 110 0.5761083 0.3408916 0.08300005
## 111 0.6391489 0.2959095 0.06494160
## 112 0.5895856 0.3315372 0.07887722
## 113 0.6419852 0.2938172 0.06419755
## 114 0.6175781 0.3116393 0.07078260
## 115 0.5328392 0.3698084 0.09735243
## 116 0.4950925 0.3934380 0.11146952
## 117 0.5761083 0.3408916 0.08300005
## 118 0.2964917 0.4775697 0.22593859
## 119 0.5557515 0.3547184 0.08953013
## 120 0.6504340 0.2875532 0.06201277
## 121 0.5289367 0.3723252 0.09873814
## 122 0.6925119 0.2556961 0.05179204
## 123 0.6711384 0.2720104 0.05685119
## 124 0.6462210 0.2906826 0.06309634
## 125 0.2882650 0.4787662 0.23296876
## 126 0.5910752 0.3304940 0.07843087
## 127 0.3214684 0.4724022 0.20612934
## 128 0.5511848 0.3577673 0.09104784
## 129 0.6117427 0.3158370 0.07242031
## 130 0.6161224 0.3126888 0.07118879
## 131 0.6190318 0.3105897 0.07037855
## 132 0.6738516 0.2699542 0.05619426
## 133 0.5851070 0.3346627 0.08023026
## 134 0.6391489 0.2959095 0.06494160
## 135 0.6532297 0.2854702 0.06130008
## 136 0.3174512 0.4733793 0.20916953
## 137 0.5910752 0.3304940 0.07843087
## 138 0.2559129 0.4806398 0.26344726
## 139 0.5910752 0.3304940 0.07843087
## 140 0.5594830 0.3522123 0.08830468
## 141 0.5104893 0.3839973 0.10551342
## 142 0.5730972 0.3429603 0.08394247
## 143 0.5043313 0.3878072 0.10786143
## 144 0.5910752 0.3304940 0.07843087
## 145 0.5895856 0.3315372 0.07887722
## 146 0.5806144 0.3377812 0.08160445
## 147 0.5895856 0.3315372 0.07887722
## 148 0.6845870 0.2617757 0.05363731
## 149 0.6601726 0.2802757 0.05955171
## 150 0.6169146 0.3121178 0.07096754
## 151 0.3088983 0.4752782 0.21582341
## 152 0.5955338 0.3273606 0.07710563
## 153 0.5058710 0.3868588 0.10727015
## 154 0.5715896 0.3439931 0.08441732
## 155 0.3003604 0.4769163 0.22272330
## 156 0.6684138 0.2740709 0.05751533
## 157 0.7092947 0.2427073 0.04799791
## 158 0.5549240 0.3552723 0.08980367
## 159 0.5640321 0.3491396 0.08682833
## 160 0.5289367 0.3723252 0.09873814
## 161 0.6320160 0.3011471 0.06683697
## 162 0.5746035 0.3419265 0.08347005
## 163 0.5640321 0.3491396 0.08682833
## 164 0.6898825 0.2577171 0.05240036
## 165 0.6014541 0.3231753 0.07537058
## 166 0.4889347 0.3971321 0.11393320
## 167 0.6088128 0.3179350 0.07325220
## 168 0.5435548 0.3628164 0.09362885
## 169 0.4797050 0.4025774 0.11771760
## 170 0.5503557 0.3583188 0.09132555
## 171 0.5670588 0.3470846 0.08585654
## 172 0.6724964 0.2709817 0.05652183
## 173 0.5761083 0.3408916 0.08300005
## 174 0.5993026 0.3246995 0.07599792
## 175 0.5655461 0.3481127 0.08634120
## 176 0.5955338 0.3273606 0.07710563
## 177 0.6977332 0.2516714 0.05059537
## 178 0.6088128 0.3179350 0.07325220
## 179 0.3503083 0.4639263 0.18576537
## 180 0.6433997 0.2927718 0.06382850
## 181 0.5320050 0.3703478 0.09764730
## 182 0.5104893 0.3839973 0.10551342
## 183 0.4889347 0.3971321 0.11393320
## 184 0.2788261 0.4797990 0.24137493
## 185 0.6724964 0.2709817 0.05652183
## 186 0.5709017 0.3444636 0.08463464
## 187 0.5549240 0.3552723 0.08980367
## 188 0.6560149 0.2833901 0.06059505
## 189 0.6405683 0.2948631 0.06456858
## 190 0.5970166 0.3263150 0.07666847
## 191 0.6518332 0.2865114 0.06165546
## 192 0.5844251 0.3351371 0.08043773
## 193 0.5366027 0.3673661 0.09603120
## 194 0.5549240 0.3552723 0.08980367
## 195 0.5806144 0.3377812 0.08160445
## 196 0.5442515 0.3623577 0.09339080
## 197 0.6305823 0.3021955 0.06722216
## 198 0.6073449 0.3189836 0.07367144
## 199 0.5442515 0.3623577 0.09339080
## 200 0.5335382 0.3693559 0.09710592
## 201 0.3680898 0.4575489 0.17436123
## 202 0.6190318 0.3105897 0.07037855
## 203 0.4827803 0.4007755 0.11644418
## 204 0.5027916 0.3887529 0.10845556
## 205 0.5089500 0.3849538 0.10609616
## 206 0.2959062 0.4776636 0.22643013
## 207 0.5859203 0.3340964 0.07998331
## 208 0.3068717 0.4756909 0.21743736
## 209 0.6132046 0.3147877 0.07200763
## 210 0.5761083 0.3408916 0.08300005
## 211 0.6066750 0.3194617 0.07386333
## 212 0.6405683 0.2948631 0.06456858
## 213 0.6643057 0.2771691 0.05852521
## 214 0.5776118 0.3398557 0.08253245
## 215 0.5579644 0.3532337 0.08880182
## 216 0.6532297 0.2854702 0.06130008
## 217 0.6629307 0.2782037 0.05886552
## 218 0.5442515 0.3623577 0.09339080
## 219 0.6405683 0.2948631 0.06456858
## 220 0.5549240 0.3552723 0.08980367
## 221 0.3616040 0.4599687 0.17842726
## 222 0.4935528 0.3943661 0.11208103
## 223 0.6433997 0.2927718 0.06382850
## 224 0.5963407 0.3267917 0.07686750
## 225 0.6745876 0.2693956 0.05601679
## 226 0.5874139 0.3330549 0.07953124
## 227 0.3134610 0.4742964 0.21224260
## 228 0.6391489 0.2959095 0.06494160
## 229 0.5821135 0.3367425 0.08114402
## 230 0.5948572 0.3278371 0.07730571
## 231 0.6058751 0.3200320 0.07409290
## 232 0.6088128 0.3179350 0.07325220
## 233 0.5880943 0.3325798 0.07932590
## 234 0.6964326 0.2526754 0.05089206
## 235 0.5814301 0.3372162 0.08135366
## 236 0.6363029 0.2980034 0.06569367
## 237 0.5715896 0.3439931 0.08441732
## 238 0.4766311 0.4043658 0.11900307
## 239 0.6670473 0.2751026 0.05785013
## 240 0.5821135 0.3367425 0.08114402
## 241 0.6738516 0.2699542 0.05619426
## 242 0.5227938 0.3762538 0.10095239
## 243 0.6504340 0.2875532 0.06201277
## 244 0.6117427 0.3158370 0.07242031
## 245 0.6363029 0.2980034 0.06569367
## 246 0.2870030 0.4789258 0.23407122
## 247 0.3503083 0.4639263 0.18576537
## 248 0.6204833 0.3095401 0.06997663
## 249 0.5549240 0.3552723 0.08980367
## 250 0.2939287 0.4779710 0.22810030
## 251 0.3531172 0.4629736 0.18390926
## 252 0.6132046 0.3147877 0.07200763
## 253 0.4812425 0.4016782 0.11707939
## 254 0.5655461 0.3481127 0.08634120
## 255 0.3357958 0.4684999 0.19570434
## 256 0.5999767 0.3242223 0.07580095
## 257 0.6044033 0.3210801 0.07451656
## 258 0.2654083 0.4805939 0.25399782
## 259 0.5730972 0.3429603 0.08394247
## 260 0.2863178 0.4790097 0.23467252
## 261 0.6262676 0.3053423 0.06839011
## 262 0.6277081 0.3042932 0.06799872
## 263 0.3730548 0.4556271 0.17131805
## 264 0.5074106 0.3859077 0.10668174
## 265 0.2751257 0.4800997 0.24477468
## 266 0.5910752 0.3304940 0.07843087
## 267 0.6058751 0.3200320 0.07409290
## 268 0.3055632 0.4759496 0.21848725
## 269 0.5791138 0.3388189 0.08206725
## 270 0.6738516 0.2699542 0.05619426
## 271 0.5012517 0.3896957 0.10905257
## 272 0.3503083 0.4639263 0.18576537
## 273 0.5473054 0.3603417 0.09235299
## 274 0.5999767 0.3242223 0.07580095
## 275 0.6320160 0.3011471 0.06683697
## 276 0.4008067 0.4438566 0.15533665
## 277 0.5304711 0.3713375 0.09819138
## 278 0.5640321 0.3491396 0.08682833
## 279 0.6462210 0.2906826 0.06309634
## 280 0.6233798 0.3074410 0.06917916
## 281 0.5027916 0.3887529 0.10845556
## 282 0.6073449 0.3189836 0.07367144
## 283 0.3350488 0.4687189 0.19623236
## 284 0.6348763 0.2990510 0.06607274
## 285 0.5895856 0.3315372 0.07887722
## 286 0.5940493 0.3284056 0.07754508
## 287 0.3796239 0.4529951 0.16738101
## 288 0.5457789 0.3613506 0.09287059
## 289 0.6399215 0.2953401 0.06473840
## 290 0.7376186 0.2204577 0.04192370
## 291 0.5806144 0.3377812 0.08160445
## 292 0.6813115 0.2642781 0.05441037
## 293 0.5381340 0.3663682 0.09549784
## 294 0.5174809 0.3796183 0.10290085
## 295 0.7067481 0.2446879 0.04856396
## 296 0.6132046 0.3147877 0.07200763
## 297 0.6190318 0.3105897 0.07037855
## 298 0.6291463 0.3032443 0.06760940
## 299 0.3282227 0.4706420 0.20113531
## 300 0.5978225 0.3257459 0.07643158
## 301 0.6058751 0.3200320 0.07409290
## 302 0.5564447 0.3542538 0.08930148
## 303 0.6262676 0.3053423 0.06839011
## 304 0.5012517 0.3896957 0.10905257
## 305 0.6697775 0.2730401 0.05718235
## 306 0.5715896 0.3439931 0.08441732
## 307 0.6233798 0.3074410 0.06917916
## 308 0.5685704 0.3460553 0.08537435
## 309 0.6248248 0.3063916 0.06878359
## 310 0.6462210 0.2906826 0.06309634
## 311 0.4858570 0.3989603 0.11518274
## 312 0.5488310 0.3593310 0.09183798
## 313 0.5258662 0.3742940 0.09983981
## 314 0.5518794 0.3573049 0.09081570
## 315 0.5806144 0.3377812 0.08160445
## 316 0.4231846 0.4332190 0.14359649
## 317 0.6448116 0.2917269 0.06346143
## 318 0.3221998 0.4722187 0.20558152
## 319 0.5366027 0.3673661 0.09603120
## 320 0.6117427 0.3158370 0.07242031
## 321 0.6175781 0.3116393 0.07078260
## 322 0.5564447 0.3542538 0.08930148
## 323 0.6419852 0.2938172 0.06419755
## 324 0.6859154 0.2607591 0.05332551
## 325 0.6885632 0.2587297 0.05270705
## 326 0.5104893 0.3839973 0.10551342
## 327 0.6073449 0.3189836 0.07367144
## 328 0.5746035 0.3419265 0.08347005
## 329 0.3503083 0.4639263 0.18576537
## 330 0.4097138 0.4397370 0.15054923
## 331 0.4928509 0.3947883 0.11236083
## 332 0.5579644 0.3532337 0.08880182
## 333 0.6504340 0.2875532 0.06201277
## 334 0.5806144 0.3377812 0.08160445
## 335 0.5473054 0.3603417 0.09235299
## 336 0.5427233 0.3633631 0.09391361
## 337 0.6277081 0.3042932 0.06799872
## 338 0.3861087 0.4503005 0.16359087
## 339 0.6219326 0.3084905 0.06957684
## 340 0.5655461 0.3481127 0.08634120
## 341 0.6102788 0.3168861 0.07283516
## 342 0.6938219 0.2546877 0.05149038
## 343 0.5027916 0.3887529 0.10845556
## 344 0.6320160 0.3011471 0.06683697
## 345 0.5970166 0.3263150 0.07666847
## 346 0.3934336 0.4471460 0.15942047
## 347 0.5925631 0.3294501 0.07798682
## 348 0.5304711 0.3713375 0.09819138
## 349 0.5940493 0.3284056 0.07754508
## 350 0.6684138 0.2740709 0.05751533
## 351 0.4882330 0.3975500 0.11421701
## 352 0.5776118 0.3398557 0.08253245
## 353 0.6291463 0.3032443 0.06760940
## 354 0.5685704 0.3460553 0.08537435
## 355 0.5289367 0.3723252 0.09873814
## 356 0.6892814 0.2581786 0.05253998
## 357 0.6073449 0.3189836 0.07367144
## 358 0.6154579 0.3131674 0.07137471
## 359 0.3003604 0.4769163 0.22272330
## 360 0.2795004 0.4797378 0.24076182
## 361 0.2512497 0.4804945 0.26825582
## 362 0.5715896 0.3439931 0.08441732
## 363 0.6305823 0.3021955 0.06722216
## 364 0.5806144 0.3377812 0.08160445
## 365 0.5984976 0.3252689 0.07623358
## 366 0.6805835 0.2648335 0.05458304
## 367 0.6615530 0.2792393 0.05920768
## 368 0.5488310 0.3593310 0.09183798
## 369 0.4704890 0.4079005 0.12161046
## 370 0.5282369 0.3727748 0.09898833
## 371 0.5289367 0.3723252 0.09873814
## 372 0.5320050 0.3703478 0.09764730
## 373 0.7279710 0.2280808 0.04394818
## 374 0.3433319 0.4661994 0.19046877
## 375 0.5335382 0.3693559 0.09710592
## 376 0.5700807 0.3450247 0.08489461
## 377 0.5534022 0.3562894 0.09030841
## 378 0.4651895 0.4109079 0.12390270
## 379 0.6320160 0.3011471 0.06683697
## 380 0.6819210 0.2638130 0.05426606
## 381 0.5427233 0.3633631 0.09391361
## 382 0.5715896 0.3439931 0.08441732
## 383 0.5670588 0.3470846 0.08585654
## 384 0.2477895 0.4803120 0.27189848
## 385 0.6977332 0.2516714 0.05059537
## 386 0.6448116 0.2917269 0.06346143
## 387 0.5297716 0.3717881 0.09844033
## 388 0.5761083 0.3408916 0.08300005
## 389 0.6190318 0.3105897 0.07037855
## 390 0.5700807 0.3450247 0.08489461
## 391 0.6497952 0.2880285 0.06217633
## 392 0.5058710 0.3868588 0.10727015
## 393 0.5910752 0.3304940 0.07843087
## 394 0.2913786 0.4783449 0.23027654
## 395 0.2642091 0.4806242 0.25516669
## 396 0.4812425 0.4016782 0.11707939
## 397 0.6419852 0.2938172 0.06419755
## 398 0.6938219 0.2546877 0.05149038
## 399 0.5488310 0.3593310 0.09183798
## 400 0.5089500 0.3849538 0.10609616predict(OLRmodel)## [1] unlikely somewhat likely somewhat likely unlikely
## [5] unlikely unlikely unlikely unlikely
## [9] unlikely somewhat likely somewhat likely unlikely
## [13] unlikely unlikely somewhat likely unlikely
## [17] unlikely unlikely unlikely unlikely
## [21] unlikely somewhat likely unlikely unlikely
## [25] unlikely unlikely somewhat likely unlikely
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## [33] unlikely somewhat likely unlikely unlikely
## [37] unlikely unlikely unlikely unlikely
## [41] unlikely somewhat likely unlikely unlikely
## [45] somewhat likely unlikely unlikely unlikely
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## [53] unlikely unlikely unlikely unlikely
## [57] unlikely unlikely unlikely unlikely
## [61] unlikely unlikely somewhat likely unlikely
## [65] somewhat likely unlikely unlikely unlikely
## [69] unlikely unlikely unlikely unlikely
## [73] unlikely somewhat likely unlikely unlikely
## [77] unlikely somewhat likely somewhat likely somewhat likely
## [81] unlikely unlikely unlikely unlikely
## [85] unlikely unlikely unlikely unlikely
## [89] unlikely unlikely somewhat likely unlikely
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## [97] unlikely unlikely unlikely unlikely
## [101] unlikely unlikely unlikely unlikely
## [105] somewhat likely unlikely somewhat likely somewhat likely
## [109] unlikely unlikely unlikely unlikely
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## [117] unlikely somewhat likely unlikely unlikely
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## [125] somewhat likely unlikely somewhat likely unlikely
## [129] unlikely unlikely unlikely unlikely
## [133] unlikely unlikely unlikely somewhat likely
## [137] unlikely somewhat likely unlikely unlikely
## [141] unlikely unlikely unlikely unlikely
## [145] unlikely unlikely unlikely unlikely
## [149] unlikely unlikely somewhat likely unlikely
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## [161] unlikely unlikely unlikely unlikely
## [165] unlikely unlikely unlikely unlikely
## [169] unlikely unlikely unlikely unlikely
## [173] unlikely unlikely unlikely unlikely
## [177] unlikely unlikely somewhat likely unlikely
## [181] unlikely unlikely unlikely somewhat likely
## [185] unlikely unlikely unlikely unlikely
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## [193] unlikely unlikely unlikely unlikely
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## [201] somewhat likely unlikely unlikely unlikely
## [205] unlikely somewhat likely unlikely somewhat likely
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## [213] unlikely unlikely unlikely unlikely
## [217] unlikely unlikely unlikely unlikely
## [221] somewhat likely unlikely unlikely unlikely
## [225] unlikely unlikely somewhat likely unlikely
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## [233] unlikely unlikely unlikely unlikely
## [237] unlikely unlikely unlikely unlikely
## [241] unlikely unlikely unlikely unlikely
## [245] unlikely somewhat likely somewhat likely unlikely
## [249] unlikely somewhat likely somewhat likely unlikely
## [253] unlikely unlikely somewhat likely unlikely
## [257] unlikely somewhat likely unlikely somewhat likely
## [261] unlikely unlikely somewhat likely unlikely
## [265] somewhat likely unlikely unlikely somewhat likely
## [269] unlikely unlikely unlikely somewhat likely
## [273] unlikely unlikely unlikely somewhat likely
## [277] unlikely unlikely unlikely unlikely
## [281] unlikely unlikely somewhat likely unlikely
## [285] unlikely unlikely somewhat likely unlikely
## [289] unlikely unlikely unlikely unlikely
## [293] unlikely unlikely unlikely unlikely
## [297] unlikely unlikely somewhat likely unlikely
## [301] unlikely unlikely unlikely unlikely
## [305] unlikely unlikely unlikely unlikely
## [309] unlikely unlikely unlikely unlikely
## [313] unlikely unlikely unlikely somewhat likely
## [317] unlikely somewhat likely unlikely unlikely
## [321] unlikely unlikely unlikely unlikely
## [325] unlikely unlikely unlikely unlikely
## [329] somewhat likely somewhat likely unlikely unlikely
## [333] unlikely unlikely unlikely unlikely
## [337] unlikely somewhat likely unlikely unlikely
## [341] unlikely unlikely unlikely unlikely
## [345] unlikely somewhat likely unlikely unlikely
## [349] unlikely unlikely unlikely unlikely
## [353] unlikely unlikely unlikely unlikely
## [357] unlikely unlikely somewhat likely somewhat likely
## [361] somewhat likely unlikely unlikely unlikely
## [365] unlikely unlikely unlikely unlikely
## [369] unlikely unlikely unlikely unlikely
## [373] unlikely somewhat likely unlikely unlikely
## [377] unlikely unlikely unlikely unlikely
## [381] unlikely unlikely unlikely somewhat likely
## [385] unlikely unlikely unlikely unlikely
## [389] unlikely unlikely unlikely unlikely
## [393] unlikely somewhat likely somewhat likely unlikely
## [397] unlikely unlikely unlikely unlikely
## Levels: unlikely somewhat likely very likelychisq.test(OLR_data$apply,predict(OLRmodel))##
## Pearson's Chi-squared test
##
## data: OLR_data$apply and predict(OLRmodel)
## X-squared = 20.496, df = 2, p-value = 3.543e-05The chi-square test above yeilds a result of 20.496(df=2), p < .001. This means that the observed and predicted values are not similar, indicating a bad model fit of the data.
predictOLR <- predict(OLRmodel,OLR_data)
cTab <- table(OLR_data$apply, predictOLR)
mean(as.character(OLR_data$apply) != as.character(predictOLR))## [1] 0.4225(CCR <- sum(diag(cTab)) / sum(cTab))## [1] 0.5775We can find that the misclassification error for our model is 42.25%, and classification rate is 57.75%.
library("DescTools")
PseudoR2(OLRmodel, which = c("CoxSnell","Nagelkerke","McFadden"))## CoxSnell Nagelkerke McFadden
## 0.05866013 0.06956550 0.03262310There is the relationship of 5.90%, 7.00%, and 3.26% between independent variables and dependent variable based on CoxSnell’s, Nagelkerke’s, and McFadden’s R^2 respectively, indicating a very low power.
(OLRestimates <- coef(summary(OLRmodel)))## Value Std. Error t value
## pareddegree 1.04769010 0.2657894 3.9418050
## publicpublic -0.05878572 0.2978614 -0.1973593
## gpa 0.61594057 0.2606340 2.3632399
## unlikely|somewhat likely 2.20391473 0.7795455 2.8271792
## somewhat likely|very likely 4.29936315 0.8043267 5.3452947p <- pnorm(abs(OLRestimates[, "t value"]), lower.tail = FALSE) * 2
(OLRestimates_P <- cbind(OLRestimates, "p value" = p))## Value Std. Error t value p value
## pareddegree 1.04769010 0.2657894 3.9418050 8.087072e-05
## publicpublic -0.05878572 0.2978614 -0.1973593 8.435464e-01
## gpa 0.61594057 0.2606340 2.3632399 1.811594e-02
## unlikely|somewhat likely 2.20391473 0.7795455 2.8271792 4.696004e-03
## somewhat likely|very likely 4.29936315 0.8043267 5.3452947 9.027008e-08\(logit[P(Y≤j)]=α_j-β_1Pared-β_2Public-β_3GPA,j=1, 2\)
\(logit[P(Y≤1)]=logit{P(unlikely)/{P(somewhat likely)+P(very likely)}}=2.20-1.05Pared+0.06Public-0.62GPA\)
\(logit[P(Y≤2)]=logit{{P(unlikely)+P(somewhat likely)}/P(very likely)}=4.30-1.05Pared+0.06Public-0.62GPA\)
The estimates \(-β_1=-1.05\) suggest that the cumulative probability starting at the ‘unlikely’ end of the scale increases as parent degree decreases.
When public = 0 (private), gpa = 0, and pared = 0 (no graduate degree), \(logit[P(Y≤1)]=2.20, Odds=exp(2.20)=e^{2.20}=9.03\)
When public = 0 (private), gpa = 0, and pared = 1 (with graduate degree), \(logit[P(Y≤1)]=2.20-1.05=1.15, Odds=exp(1.15)=e^{1.15}=3.16\)
\(Odds Ratio=9.03/3.16=2.86\)
\(Exp(-1.05) =2.86\) means that when the number of pared is raised by one unit the odds ratio is 2.86 times as high and therefore students who have more parents with a graduate degree are less likely to belong to ‘unlikely (Y=1)’ category’. In other words, students who who have parents without a graduate degree tend to belong to ‘unlikely to apply’ group.
The estimates \(-β_2=0.06\) suggest that the cumulative probability starting at the ‘unlikely’ end of the scale increases as public increases (1=public).
When public = 0 (private), gpa = 0, and pared = 0 (no graduate degree), \(logit[P(Y≤1)]=2.20, Odds=exp(2.20)=e^{2.20}=9.03\)
When public = 1 (private), gpa = 0, and pared = 0 (with graduate degree), \(logit[P(Y≤1)]=2.20+0.06=2.26, Odds=exp(2.26)=e^{2.26}=9.58\)
\(Odds Ratio=9.03/9.58=.94\)
\(Exp(0.06) =.94\) means that when the number of public is raised by one unit the odds ratio is .94 times as high and therefore students who go to private schools are less likely to belong to ‘unlikely (Y=1)’ category’. In other words, students who go to public schools tend to belong to ‘unlikely to apply’ group.
The estimates \(-β_3=-.62\) suggest that the cumulative probability starting at the ‘unlikely’ end of the scale increases as gpa decreases.
When public = 0 (private), gpa = 0, and pared = 0 (no graduate degree), \(logit[P(Y≤1)]=2.20, Odds=exp(2.20)=e^{2.20}=9.03\)
When public = 0 (private), gpa = 1, and pared = 0 (with graduate degree), \(logit[P(Y≤1)]=2.20-0.62=1.58, Odds=exp(1.58)=e^{1.58}=4.85\)
\(Odds Ratio=9.03/4.85=1.86\)
\(Exp(-.62) =1.86\) means that when the number of gpa is raised by one unit the odds ratio is 1.86 times as high and therefore students with higher gpa are less likely to belong to ‘unlikely (Y=1)’ category’. In other words, students with lower gpa tend to belong to ‘unlikely to apply’ group.
We calclulate the predicted probabilities when pared=0, public=0, and gpa=3
When \(α_1=2.20, prob[apply0(unlikely)]=1/{1+e^{-2.20+0.62*3}}=0.5842\)
When \(α_2=4.30\), prob[apply0(unlikely) or 1(somewhat likely)]\(=1/{1+e^{-4.30+0.62*3}}=0.9198\)
Prob (apply0(unlikely) or 1(somewhat likely) or 2(very likely)) = 1
From above, we have
prob(apply = 0) = 0.5842
prob(apply = 1) = prob(apply = 0 or 1) - prob(apply = 0) = 0.9198 - 0.5842 = 0.3356
prob(apply = 2) = prob(apply = 0 or 1 or 2) - prob(apply = 0 or 1) = 1 - 0.9198 = 0.0802
This means that when pared = 0, public= 0 and GPA =3.0, the probability of unlikely to apply is 0.5842 (58.42%), the probability of somewhat likely to apply is 0.3356 (33.56%), and the probability of verly likely to apply is 0.0802 (8.02%).
We calclulate the predicted probabilities when pared=1, public=0, and gpa=2.25
When \(α_1=2.20, prob[apply0(unlikely)]=1/{1+e^{-2.20+1.05*1+0.62*2.25}}=0.4391\)
When \(α_2=4.30\), prob[apply0(unlikely) or 1(somewhat likely)]\(=1/{1+e^{-4.30+1.05*1+0.62*3}}=0.8647\)
Prob (apply0(unlikely) or 1(somewhat likely) or 2(very likely)) = 1
From above, we have
prob(apply = 0) = 0.4391
prob(apply = 1) = prob(apply = 0 or 1) - prob(apply = 0) = 0.8647 - 0.4391 = 0.4256
prob(apply = 2) = prob(apply = 0 or 1 or 2) - prob(apply = 0 or 1) = 1 - 0.8647 = 0.1353
This means that for Jane, (when pared = 1, public= 0 and GPA =2.25) the probability for apply = 0 (unlikely) is 0.4391 (43.91%), the probability of apply = 1 (somewhat likely) is 0.4256 (42.56%), and the probability of apply = 2 (verly likely) is 0.1353 (13.53%).