REGRESSION MODELS
DEVINE FATHY MAE S. GRINO
2023-10-03
PAGE 16
- Take the Galton dataset and find the mean, standard deviation and correlation between the parental and child heights.
library(UsingR)## Warning: package 'UsingR' was built under R version 4.2.3## Loading required package: MASS## Loading required package: HistData## Warning: package 'HistData' was built under R version 4.2.3## Loading required package: Hmisc## Warning: package 'Hmisc' was built under R version 4.2.3##
## Attaching package: 'Hmisc'## The following objects are masked from 'package:base':
##
## format.pval, unitsdata("galton")head(galton)## child parent
## 1 61.7 70.5
## 2 61.7 68.5
## 3 61.7 65.5
## 4 61.7 64.5
## 5 61.7 64.0
## 6 62.2 67.5summary(galton)## child parent
## Min. :61.70 Min. :64.00
## 1st Qu.:66.20 1st Qu.:67.50
## Median :68.20 Median :68.50
## Mean :68.09 Mean :68.31
## 3rd Qu.:70.20 3rd Qu.:69.50
## Max. :73.70 Max. :73.00cor(galton )## child parent
## child 1.0000000 0.4587624
## parent 0.4587624 1.0000000- Center the parent and child variables and verify that the centered variable means are 0.
galton## child parent
## 1 61.7 70.5
## 2 61.7 68.5
## 3 61.7 65.5
## 4 61.7 64.5
## 5 61.7 64.0
## 6 62.2 67.5
## 7 62.2 67.5
## 8 62.2 67.5
## 9 62.2 66.5
## 10 62.2 66.5
## 11 62.2 66.5
## 12 62.2 64.5
## 13 63.2 70.5
## 14 63.2 69.5
## 15 63.2 68.5
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## 43 63.2 64.0
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## 765 70.2 67.5
## 766 70.2 67.5
## 767 70.2 67.5
## 768 70.2 67.5
## 769 70.2 67.5
## 770 70.2 67.5
## 771 70.2 67.5
## 772 70.2 67.5
## 773 70.2 67.5
## 774 70.2 67.5
## 775 70.2 67.5
## 776 70.2 67.5
## 777 70.2 67.5
## 778 70.2 67.5
## 779 70.2 67.5
## 780 70.2 67.5
## 781 70.2 67.5
## 782 70.2 67.5
## 783 70.2 67.5
## 784 70.2 66.5
## 785 70.2 66.5
## 786 70.2 66.5
## 787 70.2 66.5
## 788 70.2 65.5
## 789 70.2 65.5
## 790 70.2 65.5
## 791 70.2 65.5
## 792 70.2 65.5
## 793 71.2 72.5
## 794 71.2 72.5
## 795 71.2 71.5
## 796 71.2 71.5
## 797 71.2 71.5
## 798 71.2 71.5
## 799 71.2 70.5
## 800 71.2 70.5
## 801 71.2 70.5
## 802 71.2 70.5
## 803 71.2 70.5
## 804 71.2 70.5
## 805 71.2 70.5
## 806 71.2 69.5
## 807 71.2 69.5
## 808 71.2 69.5
## 809 71.2 69.5
## 810 71.2 69.5
## 811 71.2 69.5
## 812 71.2 69.5
## 813 71.2 69.5
## 814 71.2 69.5
## 815 71.2 69.5
## 816 71.2 69.5
## 817 71.2 69.5
## 818 71.2 69.5
## 819 71.2 69.5
## 820 71.2 69.5
## 821 71.2 69.5
## 822 71.2 69.5
## 823 71.2 69.5
## 824 71.2 69.5
## 825 71.2 69.5
## 826 71.2 68.5
## 827 71.2 68.5
## 828 71.2 68.5
## 829 71.2 68.5
## 830 71.2 68.5
## 831 71.2 68.5
## 832 71.2 68.5
## 833 71.2 68.5
## 834 71.2 68.5
## 835 71.2 68.5
## 836 71.2 68.5
## 837 71.2 68.5
## 838 71.2 68.5
## 839 71.2 68.5
## 840 71.2 68.5
## 841 71.2 68.5
## 842 71.2 68.5
## 843 71.2 68.5
## 844 71.2 67.5
## 845 71.2 67.5
## 846 71.2 67.5
## 847 71.2 67.5
## 848 71.2 67.5
## 849 71.2 67.5
## 850 71.2 67.5
## 851 71.2 67.5
## 852 71.2 67.5
## 853 71.2 67.5
## 854 71.2 67.5
## 855 71.2 65.5
## 856 71.2 65.5
## 857 72.2 73.0
## 858 72.2 72.5
## 859 72.2 72.5
## 860 72.2 72.5
## 861 72.2 72.5
## 862 72.2 72.5
## 863 72.2 72.5
## 864 72.2 72.5
## 865 72.2 71.5
## 866 72.2 71.5
## 867 72.2 71.5
## 868 72.2 71.5
## 869 72.2 71.5
## 870 72.2 71.5
## 871 72.2 71.5
## 872 72.2 71.5
## 873 72.2 71.5
## 874 72.2 70.5
## 875 72.2 70.5
## 876 72.2 70.5
## 877 72.2 70.5
## 878 72.2 69.5
## 879 72.2 69.5
## 880 72.2 69.5
## 881 72.2 69.5
## 882 72.2 69.5
## 883 72.2 69.5
## 884 72.2 69.5
## 885 72.2 69.5
## 886 72.2 69.5
## 887 72.2 69.5
## 888 72.2 69.5
## 889 72.2 68.5
## 890 72.2 68.5
## 891 72.2 68.5
## 892 72.2 68.5
## 893 72.2 67.5
## 894 72.2 67.5
## 895 72.2 67.5
## 896 72.2 67.5
## 897 72.2 65.5
## 898 73.2 73.0
## 899 73.2 73.0
## 900 73.2 73.0
## 901 73.2 72.5
## 902 73.2 72.5
## 903 73.2 71.5
## 904 73.2 71.5
## 905 73.2 70.5
## 906 73.2 70.5
## 907 73.2 70.5
## 908 73.2 69.5
## 909 73.2 69.5
## 910 73.2 69.5
## 911 73.2 69.5
## 912 73.2 68.5
## 913 73.2 68.5
## 914 73.2 68.5
## 915 73.7 72.5
## 916 73.7 72.5
## 917 73.7 72.5
## 918 73.7 72.5
## 919 73.7 71.5
## 920 73.7 71.5
## 921 73.7 70.5
## 922 73.7 70.5
## 923 73.7 70.5
## 924 73.7 69.5
## 925 73.7 69.5
## 926 73.7 69.5
## 927 73.7 69.5
## 928 73.7 69.5x = galton$parenty = galton$childx[1 : 10]## [1] 70.5 68.5 65.5 64.5 64.0 67.5 67.5 67.5 66.5 66.5y[1 : 10]## [1] 61.7 61.7 61.7 61.7 61.7 62.2 62.2 62.2 62.2 62.2xc = x - mean(x)xc = x - mean(x, na.rm = TRUE)mean(xc)## [1] 9.775954e-16- Rescale the parent and child variables and verify that the scaled variable standard deviations are 1.
x / sd(x)## [1] 39.44424 38.32525 36.64677 36.08728 35.80753 37.76576 37.76576 37.76576
## [9] 37.20626 37.20626 37.20626 36.08728 39.44424 38.88474 38.32525 38.32525
## [17] 38.32525 38.32525 38.32525 38.32525 38.32525 37.76576 37.76576 37.76576
## [25] 37.76576 37.76576 37.20626 37.20626 37.20626 36.64677 36.64677 36.64677
## [33] 36.64677 36.64677 36.64677 36.64677 36.64677 36.64677 36.08728 36.08728
## [41] 36.08728 36.08728 35.80753 35.80753 38.88474 38.88474 38.88474 38.88474
## [49] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474
## [57] 38.88474 38.88474 38.88474 38.88474 38.32525 38.32525 38.32525 38.32525
## [65] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 37.76576
## [73] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [81] 37.76576 37.76576 37.76576 37.76576 37.76576 37.20626 37.20626 37.20626
## [89] 37.20626 37.20626 36.64677 36.64677 36.64677 36.64677 36.64677 36.08728
## [97] 36.08728 36.08728 36.08728 35.80753 35.80753 35.80753 35.80753 40.00373
## [105] 39.44424 38.88474 38.88474 38.88474 38.88474 38.32525 38.32525 38.32525
## [113] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [121] 38.32525 38.32525 38.32525 38.32525 38.32525 37.76576 37.76576 37.76576
## [129] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [137] 37.76576 37.76576 37.76576 37.76576 37.20626 37.20626 36.64677 36.64677
## [145] 36.64677 36.64677 36.64677 36.64677 36.64677 36.08728 35.80753 40.00373
## [153] 40.00373 40.00373 39.44424 38.88474 38.88474 38.88474 38.88474 38.88474
## [161] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474
## [169] 38.88474 38.88474 38.88474 38.88474 38.32525 38.32525 38.32525 38.32525
## [177] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [185] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [193] 38.32525 38.32525 38.32525 38.32525 38.32525 37.76576 37.76576 37.76576
## [201] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [209] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [217] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [225] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [233] 37.76576 37.20626 37.20626 37.20626 37.20626 37.20626 37.20626 37.20626
## [241] 37.20626 37.20626 37.20626 37.20626 37.20626 37.20626 37.20626 37.20626
## [249] 37.20626 37.20626 36.64677 36.64677 36.64677 36.64677 36.64677 36.64677
## [257] 36.64677 36.64677 36.64677 36.64677 36.64677 36.08728 36.08728 36.08728
## [265] 36.08728 36.08728 35.80753 35.80753 40.00373 40.00373 40.00373 40.00373
## [273] 39.44424 39.44424 39.44424 38.88474 38.88474 38.88474 38.88474 38.88474
## [281] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474
## [289] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474
## [297] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.32525 38.32525
## [305] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [313] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [321] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [329] 38.32525 38.32525 38.32525 38.32525 38.32525 37.76576 37.76576 37.76576
## [337] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [345] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [353] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [361] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [369] 37.76576 37.76576 37.76576 37.20626 37.20626 37.20626 37.20626 37.20626
## [377] 37.20626 37.20626 37.20626 37.20626 37.20626 37.20626 37.20626 37.20626
## [385] 37.20626 37.20626 37.20626 37.20626 36.64677 36.64677 36.64677 36.64677
## [393] 36.64677 36.64677 36.64677 36.64677 36.64677 36.64677 36.64677 36.08728
## [401] 36.08728 36.08728 36.08728 36.08728 35.80753 35.80753 40.56322 40.00373
## [409] 40.00373 40.00373 39.44424 39.44424 39.44424 39.44424 39.44424 39.44424
## [417] 39.44424 39.44424 39.44424 39.44424 39.44424 39.44424 38.88474 38.88474
## [425] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474
## [433] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474
## [441] 38.88474 38.88474 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [449] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [457] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [465] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [473] 38.32525 38.32525 38.32525 38.32525 37.76576 37.76576 37.76576 37.76576
## [481] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [489] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [497] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [505] 37.20626 37.20626 37.20626 37.20626 37.20626 37.20626 37.20626 37.20626
## [513] 37.20626 37.20626 37.20626 37.20626 37.20626 37.20626 36.64677 36.64677
## [521] 36.64677 36.64677 36.64677 36.64677 36.64677 35.80753 40.56322 40.56322
## [529] 40.00373 40.00373 40.00373 40.00373 40.00373 39.44424 39.44424 39.44424
## [537] 39.44424 39.44424 39.44424 39.44424 39.44424 39.44424 39.44424 39.44424
## [545] 39.44424 39.44424 39.44424 39.44424 39.44424 39.44424 39.44424 38.88474
## [553] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474
## [561] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474
## [569] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474
## [577] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474
## [585] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [593] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [601] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [609] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [617] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [625] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [633] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [641] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [649] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [657] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [665] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.20626 37.20626
## [673] 37.20626 37.20626 37.20626 37.20626 37.20626 37.20626 37.20626 37.20626
## [681] 37.20626 37.20626 37.20626 36.64677 36.64677 36.64677 36.64677 36.64677
## [689] 36.64677 36.64677 36.08728 36.08728 35.80753 40.56322 40.00373 40.00373
## [697] 40.00373 40.00373 40.00373 40.00373 40.00373 40.00373 40.00373 40.00373
## [705] 39.44424 39.44424 39.44424 39.44424 39.44424 39.44424 39.44424 39.44424
## [713] 39.44424 39.44424 39.44424 39.44424 39.44424 39.44424 38.88474 38.88474
## [721] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474
## [729] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474
## [737] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.32525
## [745] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [753] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [761] 38.32525 38.32525 38.32525 38.32525 37.76576 37.76576 37.76576 37.76576
## [769] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576
## [777] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 37.20626
## [785] 37.20626 37.20626 37.20626 36.64677 36.64677 36.64677 36.64677 36.64677
## [793] 40.56322 40.56322 40.00373 40.00373 40.00373 40.00373 39.44424 39.44424
## [801] 39.44424 39.44424 39.44424 39.44424 39.44424 38.88474 38.88474 38.88474
## [809] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474
## [817] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474
## [825] 38.88474 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [833] 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525 38.32525
## [841] 38.32525 38.32525 38.32525 37.76576 37.76576 37.76576 37.76576 37.76576
## [849] 37.76576 37.76576 37.76576 37.76576 37.76576 37.76576 36.64677 36.64677
## [857] 40.84297 40.56322 40.56322 40.56322 40.56322 40.56322 40.56322 40.56322
## [865] 40.00373 40.00373 40.00373 40.00373 40.00373 40.00373 40.00373 40.00373
## [873] 40.00373 39.44424 39.44424 39.44424 39.44424 38.88474 38.88474 38.88474
## [881] 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474 38.88474
## [889] 38.32525 38.32525 38.32525 38.32525 37.76576 37.76576 37.76576 37.76576
## [897] 36.64677 40.84297 40.84297 40.84297 40.56322 40.56322 40.00373 40.00373
## [905] 39.44424 39.44424 39.44424 38.88474 38.88474 38.88474 38.88474 38.32525
## [913] 38.32525 38.32525 40.56322 40.56322 40.56322 40.56322 40.00373 40.00373
## [921] 39.44424 39.44424 39.44424 38.88474 38.88474 38.88474 38.88474 38.88474xs = x / sd(x)sd(xs)## [1] 1- Normalize the parental and child heights. Verify that the normalized variables have mean 0 and standard deviation 1 and take the correlation between them.
x[1 : 10]## [1] 70.5 68.5 65.5 64.5 64.0 67.5 67.5 67.5 66.5 66.5y[1 : 10]## [1] 61.7 61.7 61.7 61.7 61.7 62.2 62.2 62.2 62.2 62.2xn = (x - mean(x)) / sd(x)yn = (y - mean(y)) / sd(y)mean(xn)## [1] 5.501733e-16mean(yn)## [1] 2.183943e-16sd(xn)## [1] 1sd(yn)## [1] 1cor(xn, yn)## [1] 0.4587624cor(x, y)## [1] 0.4587624xn[1]## [1] 1.226302yn[1]## [1] -2.53718PAGE 21
- Install and load the package UsingR and load the father.son data with data(father.son). Get the linear regression fit where the son’s height is the outcome and the father’s height is the predictor. Give the intercept and the slope, plot the data and overlay the fitted regression line.
library(UsingR)data(father.son)fit = lm(sheight ~ fheight, data = father.son)y = father.son$sheightx = father.son$fheightb1 = cor(y, x) * sd(y) / sd(x)b0 = mean(y) - b1 * mean(x)rbind(coef(fit), c(b0, b1))## (Intercept) fheight
## [1,] 33.8866 0.514093
## [2,] 33.8866 0.514093library(ggplot2)
g = ggplot(father.son, aes(x = fheight, y = sheight))
g = g + geom_point()
g = g + geom_smooth(method = lm, se = FALSE, lwd = 2)
g## `geom_smooth()` using formula = 'y ~ x'
PAGE 32
- Fit a linear regression model to the father.son dataset with the father as the predictor and the son as the outcome. Give a p-value for the slope coefficient and perform the relevant hypothesis test.
fit = lm(sheight ~ fheight, data = father.son)
summary(fit)$coef## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 33.886604 1.83235382 18.49348 1.604044e-66
## fheight 0.514093 0.02704874 19.00618 1.121268e-69Hypothesis test SH = B0 + B1 FH + ε H0 : β1 = 0;HA : β1 ≠0
The p-value of the null hypothesis is 1.1212675^{-69} which is very small. So we reject the null hypothesis, which demonstrates that does exist a linear relation between fheight and sheight.
- Refer to question 1. Interpret both parameters. Recenter for the intercept if necessary.
fit <- lm(sheight ~ I(fheight - mean(fheight)), father.son)
summary(fit)$coef## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 68.684070 0.07421078 925.52689 0.000000e+00
## I(fheight - mean(fheight)) 0.514093 0.02704874 19.00618 1.121268e-69Now, the slope did no change. Centering or shifting the regressor will not have impact on the slope. Intercept is now 68.68, the sheight at the mean of fheight (67.69).
- Refer to question 1. Predict the son’s height if the father’s height is 80 inches. Would you recommend this prediction? Why or why not?
fit = lm(sheight ~ fheight, data = father.son)
predict(fit, newdata = data.frame(fheight = 80))## 1
## 75.01405summary(father.son)## fheight sheight
## Min. :59.01 Min. :58.51
## 1st Qu.:65.79 1st Qu.:66.93
## Median :67.77 Median :68.62
## Mean :67.69 Mean :68.68
## 3rd Qu.:69.60 3rd Qu.:70.47
## Max. :75.43 Max. :78.36- Load the mtcars dataset. Fit a linear regression with miles per gallon as the outcome and horsepower as the predictor. Interpret your coefficients, recenter for the intercept if necessary.
data(mtcars)
head(mtcars)## mpg cyl disp hp drat wt qsec vs am gear carb
## Mazda RX4 21.0 6 160 110 3.90 2.620 16.46 0 1 4 4
## Mazda RX4 Wag 21.0 6 160 110 3.90 2.875 17.02 0 1 4 4
## Datsun 710 22.8 4 108 93 3.85 2.320 18.61 1 1 4 1
## Hornet 4 Drive 21.4 6 258 110 3.08 3.215 19.44 1 0 3 1
## Hornet Sportabout 18.7 8 360 175 3.15 3.440 17.02 0 0 3 2
## Valiant 18.1 6 225 105 2.76 3.460 20.22 1 0 3 1?mtcars## starting httpd help server ... donesummary(lm(mpg ~ hp, data = mtcars))$coef## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 30.09886054 1.6339210 18.421246 6.642736e-18
## hp -0.06822828 0.0101193 -6.742389 1.787835e-07summary(lm(mpg ~ I(hp - mean(hp)), data = mtcars))$coef## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 20.09062500 0.6828817 29.420360 1.101810e-23
## I(hp - mean(hp)) -0.06822828 0.0101193 -6.742389 1.787835e-07- Refer to question 4. Overlay the fit onto a scatterplot.
library(ggplot2)
g = ggplot(mtcars, aes(x=hp, y=mpg))
g = g + geom_point(cex=5, alpha= 0.5)
g = g + geom_smooth(method = 'lm', se=FALSE, lwd = 2)
g## `geom_smooth()` using formula = 'y ~ x'
- Refer to question 4. Test the hypothesis of no linear relationship between horsepower and miles per gallon.
summary(lm(mpg ~ hp, data = mtcars))$coef## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 30.09886054 1.6339210 18.421246 6.642736e-18
## hp -0.06822828 0.0101193 -6.742389 1.787835e-07- Refer to question 4. Predict the miles per gallon for a horsepower of 111.
fit = lm(mpg ~ hp, data = mtcars)
summary(fit)##
## Call:
## lm(formula = mpg ~ hp, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.7121 -2.1122 -0.8854 1.5819 8.2360
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 30.09886 1.63392 18.421 < 2e-16 ***
## hp -0.06823 0.01012 -6.742 1.79e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.863 on 30 degrees of freedom
## Multiple R-squared: 0.6024, Adjusted R-squared: 0.5892
## F-statistic: 45.46 on 1 and 30 DF, p-value: 1.788e-07predict(fit, newdata = data.frame(hp = 111))## 1
## 22.5255230.09886 + -0.06823 * 111## [1] 22.52533PAGE 45
- Fit a linear regression model to the father.son dataset with the father as the predictor and the son as the outcome. Plot the son’s height (horizontal axis) versus the residuals (vertical axis).
fit = lm(sheight ~ fheight, data = father.son)
plot(fit)
plot(father.son$fheight, resid(fit))
abline(v = 0, col = "red")
- Refer to question 1. Directly estimate the residual variance and compare this estimate to the output of lm.
Residual Variance
fit = lm(sheight ~ fheight, data = father.son)
sum(resid(fit))## [1] 1.206535e-13sum(resid(fit)^2) / (nrow(father.son) - 2)## [1] 5.936804summary(fit)$sigma^2## [1] 5.936804- Refer to question 1. Give the R squared for this model.
fit = lm(sheight ~ fheight, data = father.son)
summary(fit)##
## Call:
## lm(formula = sheight ~ fheight, data = father.son)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.8772 -1.5144 -0.0079 1.6285 8.9685
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 33.88660 1.83235 18.49 <2e-16 ***
## fheight 0.51409 0.02705 19.01 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.437 on 1076 degrees of freedom
## Multiple R-squared: 0.2513, Adjusted R-squared: 0.2506
## F-statistic: 361.2 on 1 and 1076 DF, p-value: < 2.2e-16summary(fit)$r.squared## [1] 0.2513401- Load the mtcars dataset. Fit a linear regression with miles per gallon as the outcome and horsepower as the predictor. Plot horsepower versus the residuals.
fit = lm(mpg ~ hp, data = mtcars)
temp = mtcars; temp$resid = resid(fit)
g = ggplot(temp, aes (x = hp, y = resid))
g = g + geom_hline(yintercept = 0, col = "red")
g = g + geom_point(alpha = .5, cex = 5)
g
qplot(hp, mpg, data = mtcars)## Warning: `qplot()` was deprecated in ggplot2 3.4.0.
- Refer to question 4. Directly estimate the residual variance and compare this estimate to the output of lm.
fit = lm(mpg ~ hp, data = mtcars)
sum(resid(fit)^2) / (nrow(mtcars) - 2)## [1] 14.92248summary(fit)$sigma^2## [1] 14.92248- Refer to question 4. Give the R squared for this model.
summary(fit)##
## Call:
## lm(formula = mpg ~ hp, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.7121 -2.1122 -0.8854 1.5819 8.2360
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 30.09886 1.63392 18.421 < 2e-16 ***
## hp -0.06823 0.01012 -6.742 1.79e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.863 on 30 degrees of freedom
## Multiple R-squared: 0.6024, Adjusted R-squared: 0.5892
## F-statistic: 45.46 on 1 and 30 DF, p-value: 1.788e-07