Stata: Linear Regression Stata 3, linear regression Hein Stigum Presentation, data and programs at: http://folk.uio.no/heins/ courses Apr-15 H.S. 1 Birth weight by gestational age SYNTHETIC Apr-15 DATA EXAMPLE H.S. 2 Linear regression Birth weight by gestational age Apr-15 H.S. 3 Regression idea 2500 3000 3500 4000 4500 5000 model: y b0 b1 x e y = outcome x = covariate b1 coefficient , effectof x e error,residual 250 260 270 280 290 gestational age (days) 300 310 model with manycofactors: y b0 b1 x1 b2 x2 e x1 , x 2 = covariate Apr-15 H.S. 4 Model, measure and assumptions • Model y 0 1x1 2 x2 , N (0, 2 ) • Association measure 1 = change in y for one unit increase in x1 • Assumptions – Independent errors – Linear effects – Constant error variance • Robustness – influence Apr-15 H.S. 5 Association measure Model: Start with: y β0 β1 x1 β2 x2 y1 y x1 2 y x1 1 β0 2 β1 β2 x2 β0 1 β1 β2 x2 Hence: Apr-15 β1 y1 β1 H.S. 6 Purpose of regression • Estimation – Estimate association between outcome and exposure adjusted for other covariates • Prediction – Use an estimated model to predict the outcome given covariates in a new dataset Apr-15 H.S. 7 Outcome distributions by exposure Exposed -3 0 1 Outcome 4 Exposed -3 0 Linear regression Unexposed 3 Quantile regression or cutoff, logistic regression 8 Linear regression or transform, linear regression Unexposed -2 -1 2 Apr-15 0 Outcome 4 Outcome 1 2 6 H.S. 8 Workflow C2 • DAG education C1 sex • Scatter- and densityplots • Bivariate analysis E D gest age birth weight • Regression – Model estimation – Test of assumptions • Independent errors • Linear effects • Constant error variance – Robustness • Influence Apr-15 H.S. 9 Scatter and density plots Distribution of birth weight for low/high gestational age 4000 6000 Scatter of birth weight by gestational age gest>=280 days 2000 gest<280 days 4962 0 370 240 260 280 300 Gestational age 320 340 0 Look for deviations from linearity and outliers Apr-15 6000 4000 2000 Birth weight (gr) Look for shift in shape H.S. 10 Syntax • Estimation – regress y x1 x2 – regress y c.age i.sex – regress y c.age##i.sex linear regression continuous age, categorical sex main+interaction • Compare models – estimates store m1 – estimates table m1 m2 – estimates stats m1 m2 save model compare coefficients compare model fit • Post estimation – predict res, residuals Apr-15 predict residuals H.S. 11 Model 1: outcome+exposure regress bw gest crude model estimates store m1 store model results Apr-15 H.S. 12 Model 2 and 3: Add covariates regress bw gest i.educ sex estimates table m1 m2 m3 add covariates compare coefs Estimate association: m1 is biased, m2=m3 m3 more precise? estimates stats m1 m2 m3 compare fit Prediction: m3 is best Apr-15 H.S. 13 Factor (categorical) variables • Variable – educ = 1, 2, 3 for low, medium and high education • Built in – i.educ – ib3.educ use educ=1 as base (reference) use educ=3 as base (reference) • Manual “dummies” – educ=1 as base, make dummies for 2 and 3 – generate Medium =(educ==2) if educ<. – generate High =(educ==3) if educ<. Apr-15 H.S. 14 Create meaningful constant Expected birth weight: E(bw) 0 1 gest 2 educ2 3 educ3 4 sex Expected birth weight at: gest= 0, educ=1, sex=0, not meaningful 0 1572gr 0 1 280 3426gr gest=280, educ=1, sex=0 Margins: margins, at(gest= 0 educ=1 sex=0) = -1572 not meaningful margins, at(gest= 280 educ=1 sex=0) = 3426 Apr-15 H.S. 15 Results so far coeff 95% conf. Int. 3426 (3385 , 3467) Birth weight at ref Gestational age per day 17.9 Education Low 0 Medium 71.5 High 99.1 Sex Boy 0 Girl -154.3 (16 , 20) (25 , 118) (51 , 148) (-187 , -121) Would normally check for interaction now! Apr-15 H.S. 16 ASSUMPTIONS Apr-15 H.S. 17 Test of assumptions • Assumptions estat hettest p=0.9 no heteroskedasticity Apr-15 plot residuals versus predicted y 0 -3000 -2000 -1000 predict res, residuals predict pred, xb scatter res pred discuss 1000 2000 – Independent residuals: – Linear effects: – Constant variance: 2500 H.S. 3000 3500 Linear prediction 4000 4500 18 Violations of assumptions • Dependent residuals .5 1 Use mixed models or GEE -.5 0 • Non linear effects -1 Add square term or spline 220 240 260 gest 280 300 2 200 0 -1 -2 Use robust variance estimation res 1 • Non-constant variance 3400 Apr-15 H.S. 3500 3600 p 3700 19 3800 Measures of influence ROBUSTNESS Apr-15 H.S. 20 6000 Influence idea 4000 sion s e r reg ut o t u o with lier 2000 outlier h t i w ion regress 0 outlier 250 Apr-15 300 350 Gestational age (days) H.S. 400 21 .2 Measures of influence -.6 -.4 -.2 0 Remove obs 1, see change remove obs 2, see change 1 2 10 Id • Measure change in: – Predicted outcome – Deviance – Coefficients (beta) • Delta beta Apr-15 H.S. 22 Leverage versus residuals2 .015 lvr2plot, mlabel(id) .005 .01 370 529 2725 3273 2630 1723 1851 31684738 4178 64 2442 1768 2494 1164 818 4221 393 2472 3947 163 3552 567 4732 3164 4659 7 2110 2157 820 1375 1021 3037 4763 3285 843 2746 1172 4346 3715 4252 4353 490 4026 2878 342 4614 3774 2981 183 2739 1248 435 636 1269 1674 2790 2358 2634 3136 939 3414 4770 3875 2452 1516 3626 3878 888 4883 2424 1424 1286 2108 2672 3359 3677 3440 4449 2546 4310 1469 4047 2024 2295 2064 1872 1365 3137 3703 4618 4742 1308 4304 3710 4023 2738 822 4020 4788 1065 85 4683 2621 4680 3 278 212 68 2404 1704 4022 3437 3589 3664 4479 1804 4532 4254 3807 809 4341 2900 568 915 1113 610 639 658 570 3277 1729 623 177 2293 3574 391 1028 1459 573 1098 361 3970 2232 349 1669 4546 3933 771 2285 618 33 1161 4087 3689 4815 2573 97 159 1211 602 4749 4784 3422 711 2450 2928 988 3352 1313 4662 4199 3675 3275 205 3680 1058 1831 3526 2119 3852 1106 847 2881 2048 1457 4948 3336 4913 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3364 3481 4091 1049 1708 1997 4862 1073 2845 3428 4118 4996 1852 2511 4251 4253 2334 3139 240 767 849 2406 1057 2041 148 1382 1791 2777 3054 3091 3224 3573 193 1025 3342 955 1793 3351 41 1228 1364 1059 2797 4302 3547 4083 750 624 2481 2936 4836 3924 4051 4791 294 2361 4164 4986 4337 587 4169 251 2832 3461 4528 2047 2 4297 2831 3505 1940 1088 3693 758 3784 2345 1983 1 2159 3407 03 3529 786 2612 580 1538 4067 20 382 258 3187 195 1035 2240 4311 1578 250 3078 2027 3870 4616 989 3283 1751 533 1623 774 996 1789 3857 63 690 3095 341 503 582 676 744 855 998 1282 1479 1552 1625 1643 1754 1779 1792 1854 1897 1907 1912 1934 2087 2097 2162 2186 2432 2454 2644 2670 2821 2963 3208 3276 3392 3465 3511 3525 3623 3853 3938 3963 3966 4099 4324 4350 4426 4498 4613 4696 374 914 1229 1290 1813 1886 2004 2045 2081 2540 2741 2871 3197 3681 42874288 4611 4673 4746 348 434 436 507 535 1668 1684 1811 2301 2536 2555 2996 3393 3395 3684 4172 656 870 1502 1986 2123 2335 2841 2870 3284 3329 4308 4552 238 1014 1356 2074 2922 3716 3729 3810 4077 4176 4351 718 1734 2122 2942 3035 3317 3439 3941 3955 4261 4693 4730 4905 79 358 452 668 1868 2252 2768 3309 4648 542 1030 1446 2104 4588 1488 1611 3238 4296 886 1553 1694 3000 3021 4403 453 1530 2527 3222 3898 4563 4786 874 1010 1107 1498 2744 454 1104 1133 1280 1981 2146 2647 2655 2709 2994 736 2440 3620 4574 566 1142 2042 2203 2350 2495 3014 3217 4359 2216 4450 116 299 918 2615 3989 2057 2864 1191 2322 71 353 499 1411 3270 3682 2416 4005 4443 289 2773 1225 4798 506 2726 4736 1644 8 904905 1435 1443 4028 4400 2993 3846 677 3234 4281 851 1524 3539 4008 431 603 1163 2933 3606 764 3538 935 2153 3408 3594 4841 378 3711 1167 2517 213 2952 3189 723 953 1062 1066 1832 3051 4889 326 1344 157 1585 4054 4550 3570 2972 654 0 2491 4182 172 3101 4558 1802 3366 3469 3565 4190 28 1603 4394 306 2510 956 2769 4 207 1764 2561 569 829 4934 541 2067 2467 4652 595 105 4147 1616 1460 2734 4695 2737 3952 4029 1746 2665 4063 1472 1222 1453 1574 2030 4766 1039 1226 4722 4755 3482 3603 2289 510 4076 1148 1548 712 4284 2205 3957 2968 265 209 4781 530 1614 30 59 155 180 248 312 318 355 404 424 450 457 486 596 598 621 620 784 907 1048 1070 1190 1240 1321 1363 1492 1520 1569 1806 1838 1861 1895 1914 1931 2020 2151 2161 2178 2180 2244 23082309 2338 2343 2413 2431 2508 2581 2593 2638 2723 2779 2890 2944 3337 3566 3641 3736 3752 3808 4025 4068 4217 4269 4358 4369 4634 4637 4650 4863 4894 4917 4949 4955 4958 4980 4988 4987 149 650 1036 1111 1181 1292 1397 1474 1473 1505 1545 1618 1904 2079 2116 2156 2369 2575 2608 3268 3741 3866 3973 4482 4936 4971 5000 128 792 982 1046 1243 1316 1519 1518 1960 1988 2396 2698 2798 2839 2885 2892 3069 3184 3206 3379 3483 3630 4197 4620 4754 4989 214 896 921 928 938 951 1291 1676 1945 2170 2305 2710 2713 3269 3389 3632 3835 4003 4441 4463 4512 4861 194 255 830 1350 1696 2325 2544 3041 3400 3470 4356 4373 4585 317 423 651652 977 1355 1562 2277 3089 3442 3571 3948 47 1491 1782 3663 3740 3826 4372 96 416 531 558 903 1031 1342 2135 2179 2693 3340 3508 3527 3833 3867 3915 4666 4702 339340 372 743 2833 4413 4674 4864 25 973 1042 2374 3515 4183 4560 4824 4840 4938 175 1138 1840 2265 2531 2700 3019 3356 3501 4138 4437 4542 4638 4725 4750 4916 489 1003 1539 1638 1979 3750 4021 4290 4653 1129 2766 3142 3990 4148 4397 4447 4990 284 414 1199 2913 3907 4831 34 327 485 627 1722 2107 3913 437 1011 1203 1347 3624 4776 114 382 2089 2127 2165 4031 4238 727 1196 1607 3227 3444 1650 3484 3676 3787 66 740 1089 1246 1396 1726 3150 4223 4765 538 974 3114 4058 44 536 4711 4950 480 2215 2307 432 2498 3146 3432 1333 2548 2962 4112 4157 13 451 733 1551 2288 2765 2976 916 2058 2923 3253 3335 3912 4342 1849 1946 1973 2279 3249 3369 4555 546 913 3105 1312 819 3338 3512 4709 70 1663 2249 2503 3133 3490 4524 2530 2605 3883 4211 1951 308 641 699 4235 748 1935 2093 3894 4594 4598 7 1675 4179 4780 4796 456 2697 1385 2652 3688 4125 2712 4376 1252 1270 4015 4670 4719 115 1900 3441 4572 277 547 670 671 263 1370 1532 2895 2930 1407 4810 1067 3092 4687 362 2802 3691 8 2043 3160 3696 47 106 3 2 2973 4855 2342 992 182 230 1908 1664 2362 23 4876 526 1180 4720 2509 4033 2438 2241 995 222 1632 12 484 2327 4180 685 954 1665 4 1287 4225 895 1264 1478 993 3599 1795 4497 3104 3355 2419 1937 412 1449 1427 3862 997 1911 2400 2015 4969 1087 2 174 2493 4878 1193 778 168 1336 229 883 1205 564 2114 4381 2519 3016 4191 2437 3775 1368 4540 3665 165 3295 1267 1362 3267 1456 661 2059 4192 67 271 737 4514 3365 4332 2915 862 188 593 49 78 714 876 901 948 978 1112 1218 1550 1671 1859 1964 1978 2125 2499 2523 2757 3122 3263 3280 3363 3586 3608 3778 3877 3998 4111 4420 4429 4608 4672 4677 4866 4911 366 779 1085 1099 1249 1476 1507 2223 2792 3075 3534 3561 3660 3770 3865 4259 4427 4733 218 1744 1821 3754 3836 3855 3949 40064007 4106 4241 4691 4880 4926 2306 2989 3097 3179 4277 4349 257 2083 2227 2927 2983 3712 4037 2018 2055 2435 2730 1108 4283 189 2194 2304 3094 3579 4276 735 1244 2100 2829 3471 50 286 1093 1959 2148 2365 2462 2666 2872 2883 720 1678 3723 3743 3964 3981 4123 4809 1965 2091 2852 4438 4939 135 1582 2101 3010 3661 4877 1283 1994 2144 2492 2794 3899 4320 906 1288 2458 3658 4700 15 1737 3869 1194 1438 4049 4401 762 576 836 1032 2251 2559 751 3353 4380 4462 1858 1566 1750 2359 1513 4838 111 765 2049 4140 143 3291 3748 1915 4935 1827 2838 3827 494 1503 1631 1417 1707 2817 878 3950 4543 1748 4919 3785 4216 4601 4858 2735 2784 10901091 1406 5 2463 846 1628 2606 3993 233 150 227 3191 3228 398 941 880 3745 4468 4729 861 2788 2801 4923 309 662 97 1544 845 2785 591 2184 1728 3488 4407 267 4220 2377 3493 4541 4516 637 1272 4782 100 3063 3318 4052 2689 2237 3048 589 4912 61 1373 2436 2643 1262 3737 4386 2473 2796 2121 4599 884 2861 458 241 403 440 521 648 826 877 899 942 963 1126 1149 1331 1376 1466 1661 1995 2065 2183 2310 2461 2637 2728 3026 3155 3158 3240 3522 3839 3887 4081 4136 4198 4200 4465 4536 4692 4826 4867 4882 580 1866 2061 2112 2256 2388 3334 3387 4624 4860 825 1146 1736 1783 2502 2601 4282 4737 4895 734 746 1903 2541 2618 2623 2642 3196 3361 4323 4578 515 813 910 933 969 2588 2910 3922 4175 4526 4748 470 2128 3911 4072 4129 4326 517 958 2847 4556 1315 1847 3655 216 545 3070 3401 3777 4036 488 984 1081 2003 2040 2427 2949 3005 87 347 540 3426 3976 4034 1165 1977 2701 2939 3251 3435 3799 4886 1906 4436 2284 2551 2639 3052 3430 4812 108 2383 3038 3695 4458 201 791 871 1324 1742 202 3986 3818 295 2140 2337 2818 4999 1325 3549 12 1692 2964 3281 300 1204 1233 1819 2425 2803 307 1609 1982 2348 2149 2405 1687 1885 2022 4069 112 768 4354 4718 4071 4509 1383 2339 3600 242 467 2995 1305 3324 4908 2177 4571 4819 1786 2117 3901 279 3258 26902691 1924 1265 2552 3431 36 577 1227 1956 2909 3779 3255 4525 3604 478 2667 3380 4417 4173 563 1 2673 3670 4760 474 1763 1686 1156 4965 1778 2516 2675 4747 4227 2434 3254 4188 4379 4266 2075 780 3563 4529 862 2188 3633 4873 4567 3326 1379 1475 2210 1210 4383 2272 3153 1517 3419 4161 4250 1219 2222 3812 1621 427 1529 2291 4424 2830 335 3764 4410 1677 60 632 3902 4094 3022 3751 3274 4195 4365 1589 4993 2254 0 4962 0 Apr-15 .002 .004 .006 Normalized residual squared H.S. .008 23 Delta-beta for gestational age .2 dfbeta(gest) scatter _dfbeta_1 id -.4 -.2 0 4962 -.6 OBS, variable specific -.8 370 0 1000 2000 3000 id 4000 5000 If obs nr 370 is removed, beta will change from 17.9 to 18.6 beta(gest)= 17.9 Apr-15 H.S. 24 Removing outlier regress bw gest i.educ sex if id!=370 est store m4 est table m3 m4, b(%8.1f) Apr-15 H.S. 25 Removing outlier Full model N=5000 Outlier removed N=4999 coeff 95% conf. Int. 3426 (3385 , 3467) Birth weight at ref Gestational age per day 17.9 Education Low 0 Medium 71.5 High 99.1 Sex Boy 0 Girl -154.3 coeff 95% conf. Int. 3433 (3391 , 3474) Birth weight at ref Gestational age 18.5 per day Education 0 Low 64.2 Medium 88.6 High Sex 0 Boy -152.7 Girl (16 , 20) (25 , 118) (51 , 148) (-187 , -121) One outlier affected several estimates Apr-15 (17 , 20) (18 , 110) (40 , 137) (-185 , -120) Final model H.S. 26 Help • Linear regression – help regress • syntax and options – help regress postestimation • • • • • Apr-15 dfbeta estat hettest lvr2plot predict margins H.S. 27 bw2 NON-LINEAR EFFECTS Apr-15 H.S. 28 5000 6000 bw2: Non-linear effects 1000 2000 3000 4000 Handle: add polynomial or spline 240 Apr-15 260 280 Gestational age 300 H.S. 320 29 Non-linear effects: polynomial regress bw2 c.gest##c.gest i.educ sex 2. order polynomial in gest 3500 3000 2500 Linear Prediction 4000 margins, at(gest=(250(10)310)) predicted bw2 by gest marginsplot plot Predictive Margins with 95% CIs 250 Apr-15 260 270 280 290 Gestational age H.S. 300 310 30 Non-linear effects: spline • Qubic spline mkspline g=gest, cubic nknots(4) regress bw2 g1 g2 g3 i.educ sex make spline with 4 knots regression with spline • Plot Apr-15 3000 2500 • Linear spline mkspline g1 280 g2=gest regress bw2 g1 g2 i.educ sex 3500 4000 5-year integer values of gest Predictive Margins with 95% CIs predicted bw by gest Linear Prediction gen igest=5*round(gest/5) margins, over(igest) marginsplot 250 260 270 280 igest 290 300 310 make linear spline with knot at 280 regression with spline H.S. 31 bw3 INTERACTION Apr-15 H.S. 32 Interaction definitions • Interaction: combined effect of two variables • Scale – Linear models additive • y=b0+b1x1+b2x2 both x1 and x2 = b1+b2 – Logistic, Poisson, Cox multiplicative • Interaction – deviation from additivity (multiplicativity) – effect of x1 depends on x2 Apr-15 H.S. 33 bw3: Interaction (only linear effects) • Add interaction terms regress bw3 c.gest##i.sex i.educ gest-sex interaction • Show results margins, dydx(gest) at(sex=0) margins, dydx(gest) at(sex=1) Apr-15 effect of gest for boys effect of gest for girls H.S. 34 Summing up 1 • Build model – – – – regress bw gest est store m1 regress bw gest i.educ sex est store m2 crude model store full model – est table m1 m2 compare coefficients • Interaction – regress bw3 c.gest##i.sex i.educ – margins, dydx(gest) at(sex=0) test interaction gest for boys • Assumptions – predict res, residuals – predict pred, xb – scatter res pred Apr-15 residuals predicted plot H.S. 35 Summing up 2 • Non-linearity (linear spline) – mkspline g1 280 g2=gest – regress bw2 g1 g2 i.educ sex spline with knot at 280 regression with spline • Robustness – dfbeta(gest) – scatter _dfbeta_1 id Apr-15 delta-beta plot versus id H.S. 36