Taller 2 Electiva en Estaadistica
David
20/2/2022
Estamos interesados en la relación entre ingresos y gastos en alimentos para una muestra de hogares belgas de clase trabajadora en 1857 (los datos de Engel).
library(quantreg)## Warning: package 'SparseM' was built under R version 4.1.1library(ggplot2)Datos
Leemos ahora el conjunto de datos engel:
datos=data(engel)
datos=engel
head(datos)## income foodexp
## 1 420.1577 255.8394
## 2 541.4117 310.9587
## 3 901.1575 485.6800
## 4 639.0802 402.9974
## 5 750.8756 495.5608
## 6 945.7989 633.7978#Diagrama me dispercion
Ejercicio
Use el conjunto de datos de engel para ajustar un modelo usando regresión lineal simple mediante el enfoque de mÃmimos cuadrados Ordinarios (MCO). Realice lo siguiente: a) Realice el ajuste a este conjunto de datos, explique y concluya b) Realice la prueba de hipótesis para la pendiente y concluya. c) Verifique los supuestos y concluya en cada caso.
##Diagrama de dispersion
En r base
plot(datos)
#Solucion:a)
AJUSTE DEL MODELO
ajuste de la linea de regresion
regresion <- lm (foodexp ~ income, data = datos)
summary(regresion)##
## Call:
## lm(formula = foodexp ~ income, data = datos)
##
## Residuals:
## Min 1Q Median 3Q Max
## -725.70 -60.24 -4.32 53.41 515.77
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 147.47539 15.95708 9.242 <2e-16 ***
## income 0.48518 0.01437 33.772 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 114.1 on 233 degrees of freedom
## Multiple R-squared: 0.8304, Adjusted R-squared: 0.8296
## F-statistic: 1141 on 1 and 233 DF, p-value: < 2.2e-16intervalos de confianza
confint(regresion,level=0.95)## 2.5 % 97.5 %
## (Intercept) 116.0367916 178.913985
## income 0.4568738 0.513483confint(regresion,level=0.99)## 0.5 % 99.5 %
## (Intercept) 106.0333557 188.9174213
## income 0.4478676 0.5224893muestra un anolisis de varianza
anova(regresion)## Analysis of Variance Table
##
## Response: foodexp
## Df Sum Sq Mean Sq F value Pr(>F)
## income 1 14850458 14850458 1140.5 < 2.2e-16 ***
## Residuals 233 3033805 13021
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Predicciones
datos$predicciones <- predict(regresion)
ggplot(datos, aes(x=income, y=foodexp)) +
geom_point(aes(colour = income)) + theme_light()+
geom_smooth(method='lm', formula=y~x, se=FALSE, col='red') +
geom_segment(aes(xend=income, yend=predicciones), col='red',
lty='dashed') +
geom_point(aes(y=predicciones), col='red') +
labs(x = "income",
y = "foodexp",
title = "diagrama de dispersion income vs foodexp")
intervalos de confianza y prediccion para todas las predicciones
futuro_y <- predict(object=regresion,
interval="prediction", level=0.95)## Warning in predict.lm(object = regresion, interval = "prediction", level = 0.95): predictions on current data refer to _future_ responsesnuevos_datos <- cbind(datos, futuro_y)
library(ggplot2)
ggplot(nuevos_datos, aes(x=income, y=foodexp)) +
geom_point() +
geom_smooth(method="lm", formula=y~x, se=TRUE, level=0.95,
col='blue', fill='green') +
labs(x = "income",
y = "foodexp",
title = "diagrama de dispersion income vs foodexp") +
theme_light()
VALIDACIoN DEL MODELO
residuos = rstandard(regresion)
valores.ajustados = fitted(regresion)#prueba de normalidad
library(car)## Loading required package: carDataqqPlot(regresion)
## [1] 105 138shapiro.test(residuos) ##
## Shapiro-Wilk normality test
##
## data: residuos
## W = 0.85318, p-value = 3.54e-14homogeneidad
datos2=cbind(datos, residuos,valores.ajustados)
datos2## income foodexp predicciones residuos valores.ajustados
## 1 420.1577 255.8394 351.3268 -0.840721146 351.3268
## 2 541.4117 310.9587 410.1567 -0.872542485 410.1567
## 3 901.1575 485.6800 584.6975 -0.869651204 584.6975
## 4 639.0802 402.9974 457.5433 -0.479491418 457.5433
## 5 750.8756 495.5608 511.7840 -0.142538992 511.7840
## 6 945.7989 633.7978 606.3566 0.241000412 606.3566
## 7 829.3979 630.7566 549.8813 0.710406048 549.8813
## 8 979.1648 700.4409 622.5450 0.684107890 622.5450
## 9 1309.8789 830.9586 783.0004 0.421545252 783.0004
## 10 1492.3987 815.3602 871.5551 -0.494546713 871.5551
## 11 502.8390 338.0014 391.4420 -0.470195398 391.4420
## 12 616.7168 412.3613 446.6931 -0.301834677 446.6931
## 13 790.9225 520.0006 531.2139 -0.098507836 531.2139
## 14 555.8786 452.4015 417.1757 0.309813639 417.1757
## 15 713.4412 512.7201 493.6217 0.167825324 493.6217
## 16 838.7561 658.8395 554.4218 0.917182919 554.4218
## 17 535.0766 392.5995 407.0830 -0.127402447 407.0830
## 18 596.4408 443.5586 436.8556 0.058938275 436.8556
## 19 924.5619 640.1164 596.0529 0.386990943 596.0529
## 20 487.7583 333.8394 384.1252 -0.442489779 384.1252
## 21 692.6397 466.9583 483.5292 -0.145628868 483.5292
## 22 997.8770 543.3969 631.6238 -0.774839624 631.6238
## 23 506.9995 317.7198 393.4606 -0.666381160 393.4606
## 24 654.1587 424.3209 464.8591 -0.356325894 464.8591
## 25 933.9193 518.9617 600.5929 -0.716926370 600.5929
## 26 433.6813 338.0014 357.8882 -0.175073009 357.8882
## 27 587.5962 419.6412 432.5644 -0.113637199 432.5644
## 28 896.4746 476.3200 582.4255 -0.931909169 582.4255
## 29 454.4782 386.3602 367.9784 0.161794127 367.9784
## 30 584.9989 423.2783 431.3042 -0.070574904 431.3042
## 31 800.7990 503.3572 536.0058 -0.286806598 536.0058
## 32 502.4369 354.6389 391.2469 -0.322095578 391.2469
## 33 713.5197 497.3182 493.6597 0.032148113 493.6597
## 34 906.0006 588.5195 587.0473 0.012929245 587.0473
## 35 880.5969 654.5971 574.7220 0.701548358 574.7220
## 36 796.8289 550.7274 534.0796 0.146247086 534.0796
## 37 854.8791 528.3770 562.2443 -0.297472486 562.2443
## 38 1167.3716 640.4813 713.8589 -0.644601900 713.8589
## 39 523.8000 401.3204 401.6119 -0.002564425 401.6119
## 40 670.7792 435.9990 472.9230 -0.324529711 472.9230
## 41 377.0584 276.5606 330.4160 -0.474361937 330.4160
## 42 851.5430 588.3488 560.6257 0.243507207 560.6257
## 43 1121.0937 664.1978 691.4059 -0.238986919 691.4059
## 44 625.5179 444.8602 450.9632 -0.053653289 450.9632
## 45 805.5377 462.8995 538.3049 -0.662400663 538.3049
## 46 558.5812 377.7792 418.4869 -0.358021171 418.4869
## 47 884.4005 553.1504 576.5674 -0.205671650 576.5674
## 48 1257.4989 810.8962 757.5867 0.468463736 757.5867
## 49 2051.1789 1067.9541 1142.6632 -0.662167809 1142.6632
## 50 1466.3330 1049.8788 858.9085 1.680299671 858.9085
## 51 730.0989 522.7012 501.7036 0.184501491 501.7036
## 52 800.7990 572.0807 536.0058 0.316904972 536.0058
## 53 1245.6964 907.3969 751.8604 1.366728505 751.8604
## 54 1201.0002 811.5776 730.1748 0.715178946 730.1748
## 55 634.4002 427.7975 455.2727 -0.241529191 455.2727
## 56 956.2315 649.9985 611.4183 0.338826239 611.4183
## 57 1148.6010 860.6002 704.7518 1.369013438 704.7518
## 58 1768.8236 1143.4211 1005.6705 1.215771099 1005.6705
## 59 2822.5330 2032.6792 1516.9075 4.656920794 1516.9075
## 60 922.3548 590.6183 594.9821 -0.038324812 594.9821
## 61 2293.1920 1570.3911 1260.0827 2.763281894 1260.0827
## 62 627.4726 483.4800 451.9116 0.277523511 451.9116
## 63 889.9809 600.4804 579.2749 0.186246392 579.2749
## 64 1162.2000 696.2021 711.3497 -0.133065778 711.3497
## 65 1197.0794 774.7962 728.2725 0.408736867 728.2725
## 66 530.7972 390.5984 405.0067 -0.126744542 405.0067
## 67 1142.1526 612.5619 701.6232 -0.782325095 701.6232
## 68 1088.0039 708.7622 675.3514 0.293450542 675.3514
## 69 484.6612 296.9192 382.6225 -0.754164767 382.6225
## 70 1536.0201 1071.4627 892.7192 1.573628472 892.7192
## 71 678.8974 496.5976 476.8618 0.173453972 476.8618
## 72 671.8802 503.3974 473.4571 0.263147914 473.4571
## 73 690.4683 357.6411 482.4757 -1.097084316 482.4757
## 74 860.6948 430.3376 565.0659 -1.183369679 565.0659
## 75 873.3095 624.6990 571.1863 0.470011514 571.1863
## 76 894.4598 582.5413 581.4480 0.009601879 581.4480
## 77 1148.6470 580.2215 704.7741 -1.094103422 704.7741
## 78 926.8762 543.8807 597.1757 -0.468066024 597.1757
## 79 839.0414 588.6372 554.5602 0.299325001 554.5602
## 80 829.4974 627.9999 549.9296 0.685767144 549.9296
## 81 1264.0043 712.1012 760.7430 -0.427458870 760.7430
## 82 1937.9771 968.3949 1087.7401 -1.055830154 1087.7401
## 83 698.8317 482.5816 486.5335 -0.034729025 486.5335
## 84 920.4199 593.1694 594.0433 -0.007675020 594.0433
## 85 1897.5711 1033.5658 1068.1360 -0.305651365 1068.1360
## 86 891.6824 693.6795 580.1004 0.997554918 580.1004
## 87 889.6784 693.6795 579.1281 1.006097599 579.1281
## 88 1221.4818 761.2791 740.1120 0.185980843 740.1120
## 89 544.5991 361.3981 411.7031 -0.442472040 411.7031
## 90 1031.4491 628.4522 647.9122 -0.170907794 647.9122
## 91 1462.9497 771.4486 857.2670 -0.755075307 857.2670
## 92 830.4353 757.1187 550.3847 1.815942036 550.3847
## 93 975.0415 821.5970 620.5445 1.765712054 620.5445
## 94 1337.9983 1022.3202 796.6433 1.983967503 796.6433
## 95 867.6427 679.4407 568.4369 0.974975421 568.4369
## 96 725.7459 538.7491 499.5917 0.344074601 499.5917
## 97 989.0056 679.9981 627.3196 0.462640809 627.3196
## 98 1525.0005 977.0033 887.3727 0.789016516 887.3727
## 99 672.1960 561.2015 473.6104 0.769844744 473.6104
## 100 923.3977 728.3997 595.4880 1.167308168 595.4880
## 101 472.3215 372.3186 376.6356 -0.037992050 376.6356
## 102 590.7601 361.5210 434.0994 -0.638188714 434.0994
## 103 940.9218 517.9196 603.9903 -0.755912730 603.9903
## 104 643.3571 459.8177 459.6184 0.001751561 459.6184
## 105 2551.6615 863.9199 1385.4865 -4.673078260 1385.4865
## 106 1795.3226 831.4407 1018.5272 -1.651766183 1018.5272
## 107 1165.7734 534.7610 713.0835 -1.566506986 713.0835
## 108 815.6212 392.0502 543.1972 -1.327717817 543.1972
## 109 1264.2066 934.9752 760.8411 1.530271211 760.8411
## 110 1095.4056 813.3081 678.9426 1.180163412 678.9426
## 111 447.4479 263.7100 364.5674 -0.887789188 364.5674
## 112 1178.9742 769.0838 719.4882 0.435699497 719.4882
## 113 975.8023 630.5863 620.9136 0.084948763 620.9136
## 114 1017.8522 645.9874 641.3153 0.041032250 641.3153
## 115 423.8798 319.5584 353.1327 -0.295596356 353.1327
## 116 558.7767 348.4518 418.5818 -0.616787115 418.5818
## 117 943.2487 614.5068 605.1193 0.082445108 605.1193
## 118 1348.3002 662.0096 801.6416 -1.227603940 801.6416
## 119 2340.6174 1504.3708 1283.0924 1.972517025 1283.0924
## 120 587.1792 406.2180 432.3621 -0.229891869 432.3621
## 121 1540.9741 692.1689 895.1228 -1.786851222 895.1228
## 122 1115.8481 588.1371 688.8608 -0.884715304 688.8608
## 123 1044.6843 511.2609 654.3337 -1.256552242 654.3337
## 124 1389.7929 700.5600 821.7729 -1.065941368 821.7729
## 125 2497.7860 1301.1451 1359.3472 -0.520757052 1359.3472
## 126 1585.3809 879.0660 916.6680 -0.331193083 916.6680
## 127 1862.0438 912.8851 1050.8989 -1.219617357 1050.8989
## 128 2008.8546 1509.7812 1122.1283 3.433409428 1122.1283
## 129 697.3099 484.0605 485.7951 -0.015243545 485.7951
## 130 571.2517 399.6703 424.6344 -0.219538806 424.6344
## 131 598.3465 444.1001 437.7802 0.055568893 437.7802
## 132 461.0977 248.8101 371.1901 -1.077115085 371.1901
## 133 977.1107 527.8014 621.5484 -0.823318768 621.5484
## 134 883.9849 500.6313 576.3658 -0.665176900 576.3658
## 135 718.3594 436.8107 496.0079 -0.520178158 496.0079
## 136 543.8971 374.7990 411.3625 -0.321605450 411.3625
## 137 1587.3480 726.3921 917.6224 -1.684361234 917.6224
## 138 4957.8130 1827.2000 2552.8993 -7.367024307 2552.8993
## 139 969.6838 523.4911 617.9450 -0.829527560 617.9450
## 140 419.9980 334.9998 351.2494 -0.143069732 351.2494
## 141 561.9990 473.2009 420.1452 0.466610420 420.1452
## 142 689.5988 581.2029 482.0539 0.871355964 482.0539
## 143 1398.5203 929.7540 826.0072 0.912396742 826.0072
## 144 820.8168 591.1974 545.7180 0.399498616 545.7180
## 145 875.1716 637.5483 572.0898 0.574931394 572.0898
## 146 1392.4499 674.9509 823.0620 -1.302505543 823.0620
## 147 1256.3174 776.7589 757.0135 0.173514596 757.0135
## 148 1362.8590 959.5170 808.7052 1.326008132 808.7052
## 149 1999.2552 1250.9643 1117.4709 1.182152788 1117.4709
## 150 1209.4730 737.8201 734.2856 0.031053835 734.2856
## 151 1125.0356 810.6772 693.3184 1.030851740 693.3184
## 152 1827.4010 983.0009 1034.0909 -0.451261586 1034.0909
## 153 1014.1540 708.8968 639.5210 0.609286827 639.5210
## 154 880.3944 633.1200 574.6238 0.513776502 574.6238
## 155 2432.3910 1424.8047 1327.6190 0.868167778 1327.6190
## 156 1177.8547 830.9586 718.9451 0.984039958 718.9451
## 157 1222.5939 925.5795 740.6516 1.624845422 740.6516
## 158 1519.5811 1162.0024 884.7434 2.440593149 884.7434
## 159 687.6638 383.4580 481.1150 -0.858251138 481.1150
## 160 953.1192 621.1173 609.9083 0.098442413 609.9083
## 161 953.1192 621.1173 609.9083 0.098442413 609.9083
## 162 953.1192 621.1173 609.9083 0.098442413 609.9083
## 163 939.0418 548.6002 603.0782 -0.478451237 603.0782
## 164 1283.4025 745.2353 770.1546 -0.219007423 770.1546
## 165 1511.5789 837.8005 880.8608 -0.379016073 880.8608
## 166 1342.5821 795.3402 798.8673 -0.031007570 798.8673
## 167 511.7980 418.5976 395.7887 0.200669071 395.7887
## 168 689.7988 508.7974 482.1509 0.234178766 482.1509
## 169 1532.3074 883.2780 890.9179 -0.067258036 890.9179
## 170 1056.0808 742.5276 659.8630 0.726019359 659.8630
## 171 387.3195 242.3202 335.3945 -0.819723993 335.3945
## 172 387.3195 242.3202 335.3945 -0.819723993 335.3945
## 173 410.9987 266.0010 346.8831 -0.712187641 346.8831
## 174 832.7554 614.7588 551.5104 0.555568382 551.5104
## 175 614.9986 385.3184 445.8594 -0.532264334 445.8594
## 176 887.4658 515.6200 578.0547 -0.548361912 578.0547
## 177 1024.8177 708.4787 644.6948 0.560179691 644.6948
## 178 1006.4353 734.2356 635.7761 0.864708859 635.7761
## 179 726.0000 433.0010 499.7149 -0.586211346 499.7149
## 180 494.4174 327.4188 387.3560 -0.527390030 387.3560
## 181 748.6413 429.0399 510.7000 -0.717478711 510.7000
## 182 987.6417 619.6408 626.6578 -0.061625800 626.6578
## 183 788.0961 400.7990 529.8426 -1.133645695 529.8426
## 184 831.7983 620.8006 551.0460 0.612719725 551.0460
## 185 1139.4945 819.9964 700.3335 1.051126539 700.3335
## 186 507.5169 360.8780 393.7116 -0.288875088 393.7116
## 187 576.1972 395.7608 427.0338 -0.275011993 427.0338
## 188 696.5991 442.0001 485.4502 -0.381842531 485.4502
## 189 650.8180 404.0384 463.2382 -0.520368439 463.2382
## 190 949.5802 670.7993 608.1912 0.549850182 608.1912
## 191 497.1193 297.5702 388.6669 -0.801547111 388.6669
## 192 570.1674 353.4882 424.1083 -0.621050907 424.1083
## 193 724.7306 383.9376 499.0990 -1.011922049 499.0990
## 194 408.3399 284.8008 345.5931 -0.535304955 345.5931
## 195 638.6713 431.1000 457.3449 -0.230709464 457.3449
## 196 1225.7890 801.3518 742.2018 0.519720216 742.2018
## 197 715.3701 448.4513 494.5575 -0.405150874 494.5575
## 198 800.4708 577.9111 535.8465 0.369522482 535.8465
## 199 975.5974 570.5210 620.8142 -0.441691924 620.8142
## 200 1613.7565 865.3205 930.4352 -0.573682038 930.4352
## 201 608.5019 444.5578 442.7074 0.016268901 442.7074
## 202 958.6634 680.4198 612.5982 0.595635375 612.5982
## 203 835.9426 576.2779 553.0567 0.203971057 553.0567
## 204 873.7375 631.7982 571.3940 0.530540034 571.3940
## 205 951.4432 608.6419 609.0951 -0.003980669 609.0951
## 206 473.0022 300.9999 376.9658 -0.668541031 376.9658
## 207 601.0030 377.9984 439.0691 -0.536965714 439.0691
## 208 713.9979 397.0015 493.8918 -0.851411823 493.8918
## 209 829.2984 588.5195 549.8331 0.339820536 549.8331
## 210 959.7953 681.7616 613.1473 0.602595855 613.1473
## 211 1212.9613 807.3603 735.9780 0.627168165 735.9780
## 212 958.8743 696.8011 612.7005 0.738603049 612.7005
## 213 1129.4431 811.1962 695.4568 1.016637433 695.4568
## 214 1943.0419 1305.7201 1090.1974 1.906849799 1090.1974
## 215 539.6388 442.0001 409.2965 0.287663157 409.2965
## 216 463.5990 353.6013 372.4036 -0.165483453 372.4036
## 217 562.6400 468.0008 420.4562 0.418139960 420.4562
## 218 736.7584 526.7573 504.9347 0.191746390 504.9347
## 219 1415.4461 890.2390 834.2193 0.492720162 834.2193
## 220 2208.7897 1318.8033 1219.1325 0.886012281 1219.1325
## 221 636.0009 331.0005 456.0493 -1.099271449 456.0493
## 222 759.4010 416.4015 515.9204 -0.874354639 515.9204
## 223 1078.8382 596.8406 670.9044 -0.650501877 670.9044
## 224 499.7510 408.4992 389.9438 0.163263072 389.9438
## 225 1020.0225 775.0209 642.3683 1.165012977 642.3683
## 226 1595.1611 1138.1620 921.4131 1.909275582 921.4131
## 227 776.5958 485.5198 524.2629 -0.340370254 524.2629
## 228 1230.9235 772.7611 744.6929 0.246625790 744.6929
## 229 1807.9520 993.9630 1024.6547 -0.271018651 1024.6547
## 230 415.4407 305.4390 349.0383 -0.383887517 349.0383
## 231 440.5174 306.5191 361.2049 -0.481396458 361.2049
## 232 541.2006 299.1993 410.0542 -0.975077778 410.0542
## 233 581.3599 468.0008 429.5387 0.338220873 429.5387
## 234 743.0772 522.6019 508.0004 0.128293603 508.0004
## 235 1057.6767 750.3202 660.6373 0.787660632 660.6373library(ggplot2)
ggplot(datos2, aes(x=valores.ajustados, y=residuos)) +
geom_hline(yintercept = 0)+
geom_point(col='red') + theme_light()+
labs(x = "valores.ajustados",
y = "residuos",
title = "Residuos Vs Ajustados")
library(lmtest) ## Warning: package 'lmtest' was built under R version 4.1.1## Loading required package: zoo##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numericbptest(regresion) #prueba de homogeneidad de Breusch-Pagan##
## studentized Breusch-Pagan test
##
## data: regresion
## BP = 109.26, df = 1, p-value < 2.2e-16independencia - AUTOcorrelacion DURWIN WATSON
library(lmtest)
dwtest(regresion,alternative = "two.sided")##
## Durbin-Watson test
##
## data: regresion
## DW = 1.4108, p-value = 5.354e-06
## alternative hypothesis: true autocorrelation is not 0acf(residuos)
plot(as.ts(residuos),col="red")
abline(h=0,col="blue")
Residuals, Fitted, leverage…
par(mfrow = c(2,2))
plot(regresion,which=1:3)
library (car)
residualPlots(regresion, las=1)
## Test stat Pr(>|Test stat|)
## income -8.01 5.546e-14 ***
## Tukey test -8.01 1.147e-15 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1marginalModelPlots(regresion)
#Solucion:b)
#Solucion:c)