2260:
1387:
2255:{\displaystyle {\begin{aligned}&{\widehat {\operatorname {var} }}({\hat {\beta }}_{j})=s^{2}_{jj}=s^{2}r_{1,1}^{-1}\\={}&s^{2}^{-1}\\={}&s^{2}^{-1}\\={}&s^{2}^{-1}\\={}&s^{2}{\frac {1}{\mathrm {RSS} _{j}}}\\={}&{\frac {s^{2}}{(n-1){\widehat {\operatorname {var} }}(X_{j})}}\cdot {\frac {1}{1-R_{j}^{2}}}\end{aligned}}}
401:
2894:
then multicollinearity is high (a cutoff of 5 is also commonly used). However, there is no value of VIF greater than 1 in which the variance of the slopes of predictors isn't inflated. As a result, including two or more variables in a multiple regression that are not orthogonal (i.e. have correlation
2913:
If the variance inflation factor of a predictor variable were 5.27 (√5.27 = 2.3), this means that the standard error for the coefficient of that predictor variable is 2.3 times larger than if that predictor variable had 0 correlation with the other predictor variables.
1188:
36:) of the variance of a parameter estimate when fitting a full model that includes other parameters to the variance of the parameter estimate if the model is fit with only the parameter on its own. The VIF provides an index that measures how much the
2566:
959:
254:
744:
2892:
2735:
2838:
558:
2906:
The square root of the variance inflation factor indicates how much larger the standard error increases compared to if that variable had 0 correlation to other predictor variables in the model.
1392:
967:
2664:
2302:
469:
2895:= 0), will alter each other's slope, SE of the slope, and P-value, because there is shared variance between the predictors that can't be uniquely attributed to any one of them.
2394:
1376:
2450:
821:
2616:
498:
2596:
193:
783:
2359:
1226:
2778:
2329:
814:
396:{\displaystyle {\widehat {\operatorname {var} }}({\hat {\beta }}_{j})={\frac {s^{2}}{(n-1){\widehat {\operatorname {var} }}(X_{j})}}\cdot {\frac {1}{1-R_{j}^{2}}},}
3300:
2898:
Some software instead calculates the tolerance which is just the reciprocal of the VIF. The choice of which to use is a matter of personal preference.
637:
3148:
3030:
573:) is the VIF. It reflects all other factors that influence the uncertainty in the coefficient estimates. The VIF equals 1 when the vector
2953:
613:
on the other covariates. Finally, note that the VIF is invariant to the scaling of the variables (that is, we could scale each variable
2843:
2619:
2672:
3047:
2795:
560:: greater variability in a particular covariate leads to proportionately less variance in the corresponding coefficient estimate
3290:
507:: greater scatter in the data around the regression surface leads to proportionately more variance in the coefficient estimates
518:
3080:
2965:
3000:
2750:
416:
3204:
Marquardt, D. W. (1970). "Generalized
Inverses, Ridge Regression, Biased Linear Estimation, and Nonlinear Estimation".
3295:
132:
1183:{\displaystyle r_{j,j}=X_{j}^{T}X_{j},r_{j,-j}=X_{j}^{T}X_{-j},r_{-j,j}=X_{-j}^{T}X_{j},r_{-j,-j}=X_{-j}^{T}X_{-j}}
500:. This identity separates the influences of several distinct factors on the variance of the coefficient estimate:
165:
2943:
2930:
2780:
on the left hand side, and all other predictor variables (all the other X variables) on the right hand side.
50:
claims to have invented the concept behind the variance inflation factor, but did not come up with the name.
2397:
2633:
2268:
2561:{\displaystyle X_{1}=\alpha _{0}+\alpha _{2}X_{2}+\alpha _{3}X_{3}+\cdots +\alpha _{k}X_{k}+\varepsilon }
438:
2364:
1234:
954:{\displaystyle r^{-1}={\begin{bmatrix}r_{j,j}&r_{j,-j}\\r_{-j,j}&r_{-j,-j}\end{bmatrix}}^{-1}}
3072:
2601:
474:
239:. It turns out that the square of this standard error, the estimated variance of the estimate of
3257:
2574:
171:
41:
3144:
3086:
3076:
3026:
2789:
752:
3247:
3213:
3064:
2980:
2334:
1201:
1195:
2756:
2307:
792:
513:: greater sample size results in proportionately less variance in the coefficient estimates
47:
3105:
3065:
2975:
3284:
3275:
3252:
3235:
200:
3261:
3217:
59:
431:
on the other covariates (a regression that does not involve the response variable
168:(note that RMSE is a consistent estimator of the true variance of the error term,
739:{\displaystyle {\widehat {\operatorname {var} }}({\hat {\beta }}_{j})=s^{2}_{jj}}
2997:
595:
on the other covariates. By contrast, the VIF is greater than 1 when the vector
44:) of an estimated regression coefficient is increased because of collinearity.
583:
21:
3090:
2984:
604:
is not orthogonal to all columns of the design matrix for the regression of
235:, the predictor vector associated with the intercept term, equals 1 for all
3021:
James, Gareth; Witten, Daniela; Hastie, Trevor; Tibshirani, Robert (2017).
2940:
2438:
as a function of all the other explanatory variables in the first equation.
2927:
37:
33:
3049:
Origins of the
Variance Inflation Factor as Recalled by Cuthbert Daniel
3236:"A protocol for data exploration to avoid common statistical problems"
3063:
Rawlings, John O.; Pantula, Sastry G.; Dickey, David A. (1998).
2887:{\displaystyle \operatorname {VIF} ({\hat {\alpha }}_{i})>10}
3177:
Hair, J. F.; Anderson, R.; Tatham, R. L.; Black, W. C. (2006).
2962:
2730:{\displaystyle \mathrm {VIF} _{i}={\frac {1}{1-R_{i}^{2}}}}
785:, and without losing generality, we reorder the columns of
3025:(8th ed.). Springer Science+Business Media New York.
2833:{\displaystyle \operatorname {VIF} ({\hat {\alpha }}_{i})}
2429:
First we run an ordinary least square regression that has
586:
to each column of the design matrix for the regression of
471:
are the coefficient estimates, id est, the estimates of
3229:(5th ed.). Pearson International. pp. 258–259.
553:{\displaystyle {\widehat {\operatorname {var} }}(X_{j})}
2304:
is the coefficient of regression of dependent variable
564:
The remaining term, 1 / (1 −
847:
2846:
2798:
2759:
2675:
2636:
2604:
2577:
2453:
2367:
2337:
2310:
2271:
1390:
1237:
1204:
970:
824:
795:
755:
640:
521:
477:
441:
257:
174:
3186:
Kutner, M. H.; Nachtsheim, C. J.; Neter, J. (2004).
3124:
Kutner, M. H.; Nachtsheim, C. J.; Neter, J. (2004).
3172:. Thousand Oaks, CA: Pine Forge Press. p. 142.
1198:, the element in the first row and first column in
3067:Applied regression analysis : a research tool
2886:
2832:
2772:
2729:
2658:
2610:
2590:
2560:
2388:
2353:
2323:
2296:
2254:
1370:
1220:
1182:
953:
808:
777:
738:
552:
492:
463:
395:
187:
3071:(Second ed.). New York: Springer. pp.
3234:Zuur, A.F.; Ieno, E.N.; Elphick, C.S (2010).
2753:of the regression equation in step one, with
8:
3251:
2869:
2858:
2857:
2845:
2821:
2810:
2809:
2797:
2764:
2758:
2718:
2713:
2697:
2688:
2677:
2674:
2650:
2639:
2638:
2635:
2603:
2582:
2576:
2546:
2536:
2517:
2507:
2494:
2484:
2471:
2458:
2452:
2380:
2369:
2366:
2342:
2336:
2315:
2309:
2285:
2274:
2273:
2270:
2239:
2234:
2218:
2203:
2185:
2184:
2162:
2156:
2152:
2137:
2126:
2120:
2114:
2106:
2090:
2077:
2066:
2065:
2052:
2042:
2034:
2021:
2013:
2002:
2001:
1991:
1981:
1976:
1963:
1955:
1939:
1929:
1919:
1911:
1898:
1885:
1875:
1867:
1848:
1838:
1830:
1814:
1801:
1791:
1783:
1767:
1757:
1752:
1739:
1729:
1724:
1711:
1703:
1687:
1677:
1667:
1659:
1646:
1633:
1623:
1615:
1599:
1589:
1584:
1571:
1561:
1556:
1543:
1535:
1519:
1508:
1498:
1482:
1469:
1456:
1440:
1424:
1413:
1412:
1397:
1396:
1391:
1389:
1359:
1340:
1327:
1310:
1291:
1272:
1253:
1242:
1236:
1209:
1203:
1171:
1161:
1153:
1128:
1115:
1105:
1097:
1075:
1059:
1049:
1044:
1022:
1009:
999:
994:
975:
969:
942:
916:
895:
872:
854:
842:
829:
823:
800:
794:
766:
754:
727:
714:
701:
685:
669:
658:
657:
642:
641:
639:
541:
523:
522:
520:
484:
479:
476:
455:
444:
443:
440:
381:
376:
360:
345:
327:
326:
304:
298:
286:
275:
274:
259:
258:
256:
179:
173:
16:Statistical measure in mathematical model
3181:. Upper Saddle River, NJ: Prentice Hall.
3023:An Introduction to Statistical Learning
3013:
3195:Longnecker, M. T.; Ott, R. L. (2004).
3141:A modern approach to regression with R
3107:Practical Regression and Anova using R
3227:Using Econometrics: A Practical Guide
3197:A First Course in Statistical Methods
2444:= 1, for example, equation would be
7:
3301:Statistical deviation and dispersion
3052:(Technical report). Snee Associates.
2659:{\displaystyle {\hat {\alpha }}_{i}}
2297:{\displaystyle {\hat {\beta }}_{*j}}
248:, can be equivalently expressed as:
3199:. Thomson Brooks/Cole. p. 615.
2630:Then, calculate the VIF factor for
464:{\displaystyle {\hat {\beta }}_{j}}
225:case or observation, and such that
148: + 1 diagonal element of
3190:(4th ed.). McGraw-Hill Irwin.
3128:(4th ed.). McGraw-Hill Irwin.
2684:
2681:
2678:
2666:with the following formula :
2389:{\displaystyle \mathrm {RSS} _{j}}
2376:
2373:
2370:
2133:
2130:
2127:
1371:{\displaystyle r_{1,1}^{-1}=^{-1}}
14:
3253:10.1111/j.2041-210X.2009.00001.x
3240:Methods in Ecology and Evolution
3188:Applied Linear Regression Models
3126:Applied Linear Regression Models
2996:(categorical data) functions in
2792:by considering the size of the
3218:10.1080/00401706.1970.10488699
2875:
2863:
2853:
2827:
2815:
2805:
2644:
2279:
2209:
2196:
2181:
2169:
2087:
2071:
2061:
2027:
2007:
1969:
1936:
1895:
1860:
1857:
1823:
1811:
1776:
1717:
1684:
1643:
1608:
1549:
1479:
1466:
1449:
1446:
1430:
1418:
1408:
1356:
1265:
789:to set the first column to be
724:
711:
694:
691:
675:
663:
653:
547:
534:
449:
351:
338:
323:
311:
292:
280:
270:
166:root mean squared error (RMSE)
40:(the square of the estimate's
1:
3170:Multiple Regression: A Primer
2840:. A rule of thumb is that if
2412:different VIFs (one for each
221:independent variable for the
2751:coefficient of determination
2611:{\displaystyle \varepsilon }
493:{\displaystyle {\beta }_{j}}
3104:Faraway, Julian J. (2002).
2591:{\displaystyle \alpha _{0}}
631:without changing the VIF).
203:— a matrix such that
188:{\displaystyle \sigma ^{2}}
3317:
3225:Studenmund, A. H. (2006).
3179:Multivariate Data Analysis
3143:. New York, NY: Springer.
144:is the square root of the
2959:variance_inflation_factor
2788:Analyze the magnitude of
26:variance inflation factor
3139:Sheather, Simon (2009).
2404:Calculation and analysis
778:{\displaystyle r=X^{T}X}
3168:Allison, P. D. (1999).
2398:residual sum of squares
66:independent variables:
58:Consider the following
3291:Regression diagnostics
2992:(non categorical) and
2888:
2834:
2774:
2731:
2660:
2612:
2592:
2562:
2390:
2355:
2354:{\displaystyle X_{-j}}
2325:
2298:
2256:
1372:
1222:
1221:{\displaystyle r^{-1}}
1184:
955:
810:
779:
740:
554:
494:
465:
422:for the regression of
397:
189:
2889:
2835:
2775:
2773:{\displaystyle X_{i}}
2732:
2661:
2613:
2593:
2563:
2396:is the corresponding
2391:
2356:
2326:
2324:{\displaystyle X_{j}}
2299:
2257:
1373:
1223:
1185:
956:
811:
809:{\displaystyle X_{j}}
780:
741:
555:
495:
466:
398:
190:
3113:. pp. 117, 118.
2844:
2796:
2757:
2673:
2634:
2602:
2575:
2451:
2365:
2335:
2308:
2269:
1388:
1235:
1202:
968:
822:
793:
753:
638:
519:
475:
439:
255:
217:is the value of the
172:
3003:programing language
2723:
2244:
2047:
2026:
1986:
1924:
1880:
1843:
1796:
1762:
1734:
1672:
1628:
1594:
1566:
1527:
1335:
1261:
1166:
1110:
1054:
1004:
386:
135:of the estimate of
3296:Statistical ratios
3046:Snee, Ron (1981).
2884:
2830:
2770:
2727:
2709:
2656:
2608:
2598:is a constant and
2588:
2558:
2421:) in three steps:
2386:
2351:
2321:
2294:
2252:
2250:
2230:
2030:
2000:
1972:
1907:
1863:
1826:
1779:
1748:
1720:
1655:
1611:
1580:
1552:
1504:
1368:
1306:
1238:
1218:
1180:
1149:
1093:
1040:
990:
951:
936:
806:
775:
736:
550:
490:
461:
393:
372:
199:is the regression
185:
42:standard deviation
3150:978-0-387-09607-0
3032:978-1-4614-7138-7
2866:
2818:
2790:multicollinearity
2725:
2647:
2408:We can calculate
2282:
2246:
2213:
2193:
2143:
2074:
2010:
1421:
1405:
666:
650:
531:
452:
388:
355:
335:
283:
267:
3308:
3265:
3255:
3230:
3221:
3200:
3191:
3182:
3173:
3155:
3154:
3136:
3130:
3129:
3121:
3115:
3114:
3112:
3101:
3095:
3094:
3070:
3060:
3054:
3053:
3043:
3037:
3036:
3018:
2995:
2991:
2973:
2960:
2951:
2939:function in the
2938:
2926:function in the
2925:
2893:
2891:
2890:
2885:
2874:
2873:
2868:
2867:
2859:
2839:
2837:
2836:
2831:
2826:
2825:
2820:
2819:
2811:
2779:
2777:
2776:
2771:
2769:
2768:
2736:
2734:
2733:
2728:
2726:
2724:
2722:
2717:
2698:
2693:
2692:
2687:
2665:
2663:
2662:
2657:
2655:
2654:
2649:
2648:
2640:
2617:
2615:
2614:
2609:
2597:
2595:
2594:
2589:
2587:
2586:
2567:
2565:
2564:
2559:
2551:
2550:
2541:
2540:
2522:
2521:
2512:
2511:
2499:
2498:
2489:
2488:
2476:
2475:
2463:
2462:
2395:
2393:
2392:
2387:
2385:
2384:
2379:
2360:
2358:
2357:
2352:
2350:
2349:
2330:
2328:
2327:
2322:
2320:
2319:
2303:
2301:
2300:
2295:
2293:
2292:
2284:
2283:
2275:
2261:
2259:
2258:
2253:
2251:
2247:
2245:
2243:
2238:
2219:
2214:
2212:
2208:
2207:
2195:
2194:
2186:
2167:
2166:
2157:
2153:
2144:
2142:
2141:
2136:
2121:
2119:
2118:
2107:
2098:
2097:
2085:
2084:
2076:
2075:
2067:
2060:
2059:
2046:
2041:
2025:
2020:
2012:
2011:
2003:
1996:
1995:
1985:
1980:
1968:
1967:
1956:
1947:
1946:
1934:
1933:
1923:
1918:
1906:
1905:
1893:
1892:
1879:
1874:
1856:
1855:
1842:
1837:
1822:
1821:
1809:
1808:
1795:
1790:
1775:
1774:
1761:
1756:
1744:
1743:
1733:
1728:
1716:
1715:
1704:
1695:
1694:
1682:
1681:
1671:
1666:
1654:
1653:
1641:
1640:
1627:
1622:
1607:
1606:
1593:
1588:
1576:
1575:
1565:
1560:
1548:
1547:
1536:
1526:
1518:
1503:
1502:
1490:
1489:
1477:
1476:
1461:
1460:
1445:
1444:
1429:
1428:
1423:
1422:
1414:
1407:
1406:
1398:
1394:
1377:
1375:
1374:
1369:
1367:
1366:
1354:
1353:
1334:
1326:
1305:
1304:
1283:
1282:
1260:
1252:
1227:
1225:
1224:
1219:
1217:
1216:
1196:Schur complement
1189:
1187:
1186:
1181:
1179:
1178:
1165:
1160:
1145:
1144:
1120:
1119:
1109:
1104:
1089:
1088:
1067:
1066:
1053:
1048:
1036:
1035:
1014:
1013:
1003:
998:
986:
985:
960:
958:
957:
952:
950:
949:
941:
940:
933:
932:
909:
908:
886:
885:
865:
864:
837:
836:
815:
813:
812:
807:
805:
804:
784:
782:
781:
776:
771:
770:
745:
743:
742:
737:
735:
734:
722:
721:
706:
705:
690:
689:
674:
673:
668:
667:
659:
652:
651:
643:
559:
557:
556:
551:
546:
545:
533:
532:
524:
499:
497:
496:
491:
489:
488:
483:
470:
468:
467:
462:
460:
459:
454:
453:
445:
402:
400:
399:
394:
389:
387:
385:
380:
361:
356:
354:
350:
349:
337:
336:
328:
309:
308:
299:
291:
290:
285:
284:
276:
269:
268:
260:
194:
192:
191:
186:
184:
183:
32:) is the ratio (
3316:
3315:
3311:
3310:
3309:
3307:
3306:
3305:
3281:
3280:
3272:
3233:
3224:
3212:(3): 591–612 .
3203:
3194:
3185:
3176:
3167:
3164:
3162:Further reading
3159:
3158:
3151:
3138:
3137:
3133:
3123:
3122:
3118:
3110:
3103:
3102:
3098:
3083:
3062:
3061:
3057:
3045:
3044:
3040:
3033:
3020:
3019:
3015:
3010:
2993:
2989:
2971:
2958:
2949:
2936:
2923:
2920:
2912:
2904:
2856:
2842:
2841:
2808:
2794:
2793:
2786:
2760:
2755:
2754:
2748:
2702:
2676:
2671:
2670:
2637:
2632:
2631:
2628:
2600:
2599:
2578:
2573:
2572:
2542:
2532:
2513:
2503:
2490:
2480:
2467:
2454:
2449:
2448:
2439:
2437:
2427:
2420:
2406:
2368:
2363:
2362:
2338:
2333:
2332:
2331:over covariate
2311:
2306:
2305:
2272:
2267:
2266:
2249:
2248:
2223:
2199:
2168:
2158:
2154:
2146:
2145:
2125:
2110:
2108:
2100:
2099:
2086:
2064:
2048:
1987:
1959:
1957:
1949:
1948:
1935:
1925:
1894:
1881:
1844:
1810:
1797:
1763:
1735:
1707:
1705:
1697:
1696:
1683:
1673:
1642:
1629:
1595:
1567:
1539:
1537:
1529:
1528:
1494:
1478:
1465:
1452:
1436:
1411:
1386:
1385:
1381:Then we have,
1355:
1336:
1287:
1268:
1233:
1232:
1205:
1200:
1199:
1167:
1124:
1111:
1071:
1055:
1018:
1005:
971:
966:
965:
935:
934:
912:
910:
891:
888:
887:
868:
866:
850:
843:
841:
825:
820:
819:
796:
791:
790:
762:
751:
750:
723:
710:
697:
681:
656:
636:
635:
630:
621:
612:
603:
594:
581:
572:
537:
517:
516:
478:
473:
472:
442:
437:
436:
430:
414:
365:
341:
310:
300:
273:
253:
252:
247:
234:
216:
175:
170:
169:
143:
122:
114:
105:
99:
92:
86:
79:
56:
48:Cuthbert Daniel
19:
17:
12:
11:
5:
3314:
3312:
3304:
3303:
3298:
3293:
3283:
3282:
3279:
3278:
3271:
3268:
3267:
3266:
3231:
3222:
3201:
3192:
3183:
3174:
3163:
3160:
3157:
3156:
3149:
3131:
3116:
3096:
3081:
3055:
3038:
3031:
3012:
3011:
3009:
3006:
3005:
3004:
2987:
2978:
2969:
2956:
2947:
2934:
2919:
2918:Implementation
2916:
2903:
2902:Interpretation
2900:
2883:
2880:
2877:
2872:
2865:
2862:
2855:
2852:
2849:
2829:
2824:
2817:
2814:
2807:
2804:
2801:
2785:
2782:
2767:
2763:
2744:
2738:
2737:
2721:
2716:
2712:
2708:
2705:
2701:
2696:
2691:
2686:
2683:
2680:
2653:
2646:
2643:
2627:
2624:
2607:
2585:
2581:
2569:
2568:
2557:
2554:
2549:
2545:
2539:
2535:
2531:
2528:
2525:
2520:
2516:
2510:
2506:
2502:
2497:
2493:
2487:
2483:
2479:
2474:
2470:
2466:
2461:
2457:
2433:
2426:
2423:
2416:
2405:
2402:
2383:
2378:
2375:
2372:
2348:
2345:
2341:
2318:
2314:
2291:
2288:
2281:
2278:
2263:
2262:
2242:
2237:
2233:
2229:
2226:
2222:
2217:
2211:
2206:
2202:
2198:
2192:
2189:
2183:
2180:
2177:
2174:
2171:
2165:
2161:
2155:
2151:
2148:
2147:
2140:
2135:
2132:
2129:
2124:
2117:
2113:
2109:
2105:
2102:
2101:
2096:
2093:
2089:
2083:
2080:
2073:
2070:
2063:
2058:
2055:
2051:
2045:
2040:
2037:
2033:
2029:
2024:
2019:
2016:
2009:
2006:
1999:
1994:
1990:
1984:
1979:
1975:
1971:
1966:
1962:
1958:
1954:
1951:
1950:
1945:
1942:
1938:
1932:
1928:
1922:
1917:
1914:
1910:
1904:
1901:
1897:
1891:
1888:
1884:
1878:
1873:
1870:
1866:
1862:
1859:
1854:
1851:
1847:
1841:
1836:
1833:
1829:
1825:
1820:
1817:
1813:
1807:
1804:
1800:
1794:
1789:
1786:
1782:
1778:
1773:
1770:
1766:
1760:
1755:
1751:
1747:
1742:
1738:
1732:
1727:
1723:
1719:
1714:
1710:
1706:
1702:
1699:
1698:
1693:
1690:
1686:
1680:
1676:
1670:
1665:
1662:
1658:
1652:
1649:
1645:
1639:
1636:
1632:
1626:
1621:
1618:
1614:
1610:
1605:
1602:
1598:
1592:
1587:
1583:
1579:
1574:
1570:
1564:
1559:
1555:
1551:
1546:
1542:
1538:
1534:
1531:
1530:
1525:
1522:
1517:
1514:
1511:
1507:
1501:
1497:
1493:
1488:
1485:
1481:
1475:
1472:
1468:
1464:
1459:
1455:
1451:
1448:
1443:
1439:
1435:
1432:
1427:
1420:
1417:
1410:
1404:
1401:
1395:
1393:
1379:
1378:
1365:
1362:
1358:
1352:
1349:
1346:
1343:
1339:
1333:
1330:
1325:
1322:
1319:
1316:
1313:
1309:
1303:
1300:
1297:
1294:
1290:
1286:
1281:
1278:
1275:
1271:
1267:
1264:
1259:
1256:
1251:
1248:
1245:
1241:
1215:
1212:
1208:
1192:
1191:
1177:
1174:
1170:
1164:
1159:
1156:
1152:
1148:
1143:
1140:
1137:
1134:
1131:
1127:
1123:
1118:
1114:
1108:
1103:
1100:
1096:
1092:
1087:
1084:
1081:
1078:
1074:
1070:
1065:
1062:
1058:
1052:
1047:
1043:
1039:
1034:
1031:
1028:
1025:
1021:
1017:
1012:
1008:
1002:
997:
993:
989:
984:
981:
978:
974:
962:
961:
948:
945:
939:
931:
928:
925:
922:
919:
915:
911:
907:
904:
901:
898:
894:
890:
889:
884:
881:
878:
875:
871:
867:
863:
860:
857:
853:
849:
848:
846:
840:
835:
832:
828:
803:
799:
774:
769:
765:
761:
758:
747:
746:
733:
730:
726:
720:
717:
713:
709:
704:
700:
696:
693:
688:
684:
680:
677:
672:
665:
662:
655:
649:
646:
626:
622:by a constant
617:
608:
599:
590:
577:
568:
562:
561:
549:
544:
540:
536:
530:
527:
514:
508:
487:
482:
458:
451:
448:
426:
410:
404:
403:
392:
384:
379:
375:
371:
368:
364:
359:
353:
348:
344:
340:
334:
331:
325:
322:
319:
316:
313:
307:
303:
297:
294:
289:
282:
279:
272:
266:
263:
243:
229:
207:
182:
178:
139:
133:standard error
129:
128:
118:
110:
103:
97:
90:
84:
77:
55:
52:
15:
13:
10:
9:
6:
4:
3:
2:
3313:
3302:
3299:
3297:
3294:
3292:
3289:
3288:
3286:
3277:
3276:Design effect
3274:
3273:
3269:
3263:
3259:
3254:
3249:
3245:
3241:
3237:
3232:
3228:
3223:
3219:
3215:
3211:
3207:
3206:Technometrics
3202:
3198:
3193:
3189:
3184:
3180:
3175:
3171:
3166:
3165:
3161:
3152:
3146:
3142:
3135:
3132:
3127:
3120:
3117:
3109:
3108:
3100:
3097:
3092:
3088:
3084:
3078:
3074:
3069:
3068:
3059:
3056:
3051:
3050:
3042:
3039:
3034:
3028:
3024:
3017:
3014:
3007:
3002:
2999:
2988:
2986:
2982:
2979:
2977:
2970:
2967:
2964:
2957:
2955:
2948:
2945:
2942:
2935:
2932:
2929:
2922:
2921:
2917:
2915:
2911:
2907:
2901:
2899:
2896:
2881:
2878:
2870:
2860:
2850:
2847:
2822:
2812:
2802:
2799:
2791:
2783:
2781:
2765:
2761:
2752:
2747:
2743:
2719:
2714:
2710:
2706:
2703:
2699:
2694:
2689:
2669:
2668:
2667:
2651:
2641:
2625:
2623:
2621:
2605:
2583:
2579:
2555:
2552:
2547:
2543:
2537:
2533:
2529:
2526:
2523:
2518:
2514:
2508:
2504:
2500:
2495:
2491:
2485:
2481:
2477:
2472:
2468:
2464:
2459:
2455:
2447:
2446:
2445:
2443:
2436:
2432:
2424:
2422:
2419:
2415:
2411:
2403:
2401:
2399:
2381:
2346:
2343:
2339:
2316:
2312:
2289:
2286:
2276:
2240:
2235:
2231:
2227:
2224:
2220:
2215:
2204:
2200:
2190:
2187:
2178:
2175:
2172:
2163:
2159:
2149:
2138:
2122:
2115:
2111:
2103:
2094:
2091:
2081:
2078:
2068:
2056:
2053:
2049:
2043:
2038:
2035:
2031:
2022:
2017:
2014:
2004:
1997:
1992:
1988:
1982:
1977:
1973:
1964:
1960:
1952:
1943:
1940:
1930:
1926:
1920:
1915:
1912:
1908:
1902:
1899:
1889:
1886:
1882:
1876:
1871:
1868:
1864:
1852:
1849:
1845:
1839:
1834:
1831:
1827:
1818:
1815:
1805:
1802:
1798:
1792:
1787:
1784:
1780:
1771:
1768:
1764:
1758:
1753:
1749:
1745:
1740:
1736:
1730:
1725:
1721:
1712:
1708:
1700:
1691:
1688:
1678:
1674:
1668:
1663:
1660:
1656:
1650:
1647:
1637:
1634:
1630:
1624:
1619:
1616:
1612:
1603:
1600:
1596:
1590:
1585:
1581:
1577:
1572:
1568:
1562:
1557:
1553:
1544:
1540:
1532:
1523:
1520:
1515:
1512:
1509:
1505:
1499:
1495:
1491:
1486:
1483:
1473:
1470:
1462:
1457:
1453:
1441:
1437:
1433:
1425:
1415:
1402:
1399:
1384:
1383:
1382:
1363:
1360:
1350:
1347:
1344:
1341:
1337:
1331:
1328:
1323:
1320:
1317:
1314:
1311:
1307:
1301:
1298:
1295:
1292:
1288:
1284:
1279:
1276:
1273:
1269:
1262:
1257:
1254:
1249:
1246:
1243:
1239:
1231:
1230:
1229:
1213:
1210:
1206:
1197:
1175:
1172:
1168:
1162:
1157:
1154:
1150:
1146:
1141:
1138:
1135:
1132:
1129:
1125:
1121:
1116:
1112:
1106:
1101:
1098:
1094:
1090:
1085:
1082:
1079:
1076:
1072:
1068:
1063:
1060:
1056:
1050:
1045:
1041:
1037:
1032:
1029:
1026:
1023:
1019:
1015:
1010:
1006:
1000:
995:
991:
987:
982:
979:
976:
972:
964:
963:
946:
943:
937:
929:
926:
923:
920:
917:
913:
905:
902:
899:
896:
892:
882:
879:
876:
873:
869:
861:
858:
855:
851:
844:
838:
833:
830:
826:
818:
817:
816:
801:
797:
788:
772:
767:
763:
759:
756:
731:
728:
718:
715:
707:
702:
698:
686:
682:
678:
670:
660:
647:
644:
634:
633:
632:
629:
625:
620:
616:
611:
607:
602:
598:
593:
589:
585:
580:
576:
571:
567:
542:
538:
528:
525:
515:
512:
509:
506:
503:
502:
501:
485:
480:
456:
446:
434:
429:
425:
421:
420:
413:
409:
390:
382:
377:
373:
369:
366:
362:
357:
346:
342:
332:
329:
320:
317:
314:
305:
301:
295:
287:
277:
264:
261:
251:
250:
249:
246:
242:
238:
232:
228:
224:
220:
214:
210:
206:
202:
201:design matrix
198:
180:
176:
167:
163:
159:
155:
151:
147:
142:
138:
134:
126:
121:
117:
113:
109:
102:
96:
89:
83:
76:
72:
69:
68:
67:
65:
61:
53:
51:
49:
45:
43:
39:
35:
31:
27:
23:
3243:
3239:
3226:
3209:
3205:
3196:
3187:
3178:
3169:
3140:
3134:
3125:
3119:
3106:
3099:
3066:
3058:
3048:
3041:
3022:
3016:
2961:function in
2909:
2908:
2905:
2897:
2787:
2745:
2741:
2739:
2629:
2570:
2441:
2434:
2430:
2428:
2417:
2413:
2409:
2407:
2264:
1380:
1193:
786:
748:
627:
623:
618:
614:
609:
605:
600:
596:
591:
587:
578:
574:
569:
565:
563:
510:
504:
432:
427:
423:
418:
411:
407:
405:
244:
240:
236:
230:
226:
222:
218:
212:
208:
204:
196:
161:
157:
153:
149:
145:
140:
136:
130:
124:
119:
115:
111:
107:
100:
94:
87:
81:
74:
70:
63:
60:linear model
57:
46:
29:
25:
18:
2998:StatsModels
2963:statsmodels
2937:ols_vif_tol
3285:Categories
3082:0387227539
3008:References
2983:addon for
2784:Step three
2620:error term
584:orthogonal
54:Definition
22:statistics
2985:GRASS GIS
2972:estat vif
2864:^
2861:α
2851:
2816:^
2813:α
2803:
2707:−
2645:^
2642:α
2606:ε
2580:α
2556:ε
2534:α
2527:⋯
2505:α
2482:α
2469:α
2344:−
2287:∗
2280:^
2277:β
2228:−
2216:⋅
2191:^
2176:−
2092:−
2079:∗
2072:^
2069:β
2054:−
2036:−
2015:∗
2008:^
2005:β
1998:−
1941:−
1913:−
1900:−
1887:−
1869:−
1850:−
1832:−
1816:−
1803:−
1785:−
1769:−
1746:−
1689:−
1661:−
1648:−
1635:−
1617:−
1601:−
1578:−
1521:−
1471:−
1419:^
1416:β
1403:^
1361:−
1342:−
1329:−
1321:−
1312:−
1299:−
1285:−
1255:−
1211:−
1194:By using
1173:−
1155:−
1139:−
1130:−
1099:−
1077:−
1061:−
1030:−
944:−
927:−
918:−
897:−
880:−
831:−
716:−
664:^
661:β
648:^
529:^
481:β
450:^
447:β
417:multiple
370:−
358:⋅
333:^
318:−
281:^
278:β
265:^
177:σ
160:), where
3270:See also
3262:18814132
3246:: 3–14.
3091:54851769
2950:PROC REG
2626:Step two
2425:Step one
749:Now let
106:+ ... +
38:variance
34:quotient
3075:, 373.
2968:package
2952:in SAS
2946:package
2933:package
2910:Example
2749:is the
2618:is the
435:) and
415:is the
164:is the
156:′
3260:
3147:
3089:
3079:
3029:
2966:Python
2954:System
2740:where
2571:where
406:where
24:, the
3258:S2CID
3111:(PDF)
3001:Julia
2981:r.vif
2976:Stata
2941:olsrr
2265:Here
1228:is,
62:with
3145:ISBN
3087:OCLC
3077:ISBN
3027:ISBN
2994:gvif
2879:>
131:The
3248:doi
3214:doi
3073:372
2990:vif
2974:in
2928:car
2924:vif
2848:VIF
2800:VIF
2440:If
2188:var
1400:var
645:var
582:is
526:var
330:var
262:var
195:);
30:VIF
20:In
3287::
3256:.
3242:.
3238:.
3210:12
3208:.
3085:.
2882:10
2622:.
2400:.
2361:.
233:,1
215:+1
211:,
123:+
93:+
80:+
73:=
3264:.
3250::
3244:1
3220:.
3216::
3153:.
3093:.
3035:.
2944:R
2931:R
2876:)
2871:i
2854:(
2828:)
2823:i
2806:(
2766:i
2762:X
2746:i
2742:R
2720:2
2715:i
2711:R
2704:1
2700:1
2695:=
2690:i
2685:F
2682:I
2679:V
2652:i
2584:0
2553:+
2548:k
2544:X
2538:k
2530:+
2524:+
2519:3
2515:X
2509:3
2501:+
2496:2
2492:X
2486:2
2478:+
2473:0
2465:=
2460:1
2456:X
2442:i
2435:i
2431:X
2418:i
2414:X
2410:k
2382:j
2377:S
2374:S
2371:R
2347:j
2340:X
2317:j
2313:X
2290:j
2241:2
2236:j
2232:R
2225:1
2221:1
2210:)
2205:j
2201:X
2197:(
2182:)
2179:1
2173:n
2170:(
2164:2
2160:s
2150:=
2139:j
2134:S
2131:S
2128:R
2123:1
2116:2
2112:s
2104:=
2095:1
2088:]
2082:j
2062:)
2057:j
2050:X
2044:T
2039:j
2032:X
2028:(
2023:T
2018:j
1993:j
1989:X
1983:T
1978:j
1974:X
1970:[
1965:2
1961:s
1953:=
1944:1
1937:]
1931:j
1927:X
1921:T
1916:j
1909:X
1903:1
1896:)
1890:j
1883:X
1877:T
1872:j
1865:X
1861:(
1858:)
1853:j
1846:X
1840:T
1835:j
1828:X
1824:(
1819:1
1812:)
1806:j
1799:X
1793:T
1788:j
1781:X
1777:(
1772:j
1765:X
1759:T
1754:j
1750:X
1741:j
1737:X
1731:T
1726:j
1722:X
1718:[
1713:2
1709:s
1701:=
1692:1
1685:]
1679:j
1675:X
1669:T
1664:j
1657:X
1651:1
1644:)
1638:j
1631:X
1625:T
1620:j
1613:X
1609:(
1604:j
1597:X
1591:T
1586:j
1582:X
1573:j
1569:X
1563:T
1558:j
1554:X
1550:[
1545:2
1541:s
1533:=
1524:1
1516:1
1513:,
1510:1
1506:r
1500:2
1496:s
1492:=
1487:j
1484:j
1480:]
1474:1
1467:)
1463:X
1458:T
1454:X
1450:(
1447:[
1442:2
1438:s
1434:=
1431:)
1426:j
1409:(
1364:1
1357:]
1351:j
1348:,
1345:j
1338:r
1332:1
1324:j
1318:,
1315:j
1308:r
1302:j
1296:,
1293:j
1289:r
1280:j
1277:,
1274:j
1270:r
1266:[
1263:=
1258:1
1250:1
1247:,
1244:1
1240:r
1214:1
1207:r
1190:.
1176:j
1169:X
1163:T
1158:j
1151:X
1147:=
1142:j
1136:,
1133:j
1126:r
1122:,
1117:j
1113:X
1107:T
1102:j
1095:X
1091:=
1086:j
1083:,
1080:j
1073:r
1069:,
1064:j
1057:X
1051:T
1046:j
1042:X
1038:=
1033:j
1027:,
1024:j
1020:r
1016:,
1011:j
1007:X
1001:T
996:j
992:X
988:=
983:j
980:,
977:j
973:r
947:1
938:]
930:j
924:,
921:j
914:r
906:j
903:,
900:j
893:r
883:j
877:,
874:j
870:r
862:j
859:,
856:j
852:r
845:[
839:=
834:1
827:r
802:j
798:X
787:X
773:X
768:T
764:X
760:=
757:r
732:j
729:j
725:]
719:1
712:)
708:X
703:T
699:X
695:(
692:[
687:2
683:s
679:=
676:)
671:j
654:(
628:j
624:c
619:j
615:X
610:j
606:X
601:j
597:X
592:j
588:X
579:j
575:X
570:j
566:R
548:)
543:j
539:X
535:(
511:n
505:s
486:j
457:j
433:Y
428:j
424:X
419:R
412:j
408:R
391:,
383:2
378:j
374:R
367:1
363:1
352:)
347:j
343:X
339:(
324:)
321:1
315:n
312:(
306:2
302:s
296:=
293:)
288:j
271:(
245:j
241:β
237:i
231:i
227:X
223:i
219:j
213:j
209:i
205:X
197:X
181:2
162:s
158:X
154:X
152:(
150:s
146:j
141:j
137:β
127:.
125:ε
120:k
116:X
112:k
108:β
104:2
101:X
98:2
95:β
91:1
88:X
85:1
82:β
78:0
75:β
71:Y
64:k
28:(
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