1968:
804:
1963:{\displaystyle {\begin{aligned}&\mathbb {P} (Y_{i1}=1,Y_{i2}=0|X_{i1},X_{i2},Y_{i1}+Y_{i2}=1)\\&={\frac {\mathbb {P} (Y_{i1}=1|X_{i1})\mathbb {P} (Y_{i2}=0|X_{i2})}{\mathbb {P} (Y_{i1}=1|X_{i1})\mathbb {P} (Y_{i2}=0|X_{i2})+\mathbb {P} (Y_{i1}=0|X_{i1})\mathbb {P} (Y_{i2}=1|X_{i2})}}\\\ &={\frac {{\frac {\exp(\alpha _{i}+{\boldsymbol {\beta }}^{\top }X_{i1})}{1+\exp(\alpha _{i}+{\boldsymbol {\beta }}^{\top }X_{i1})}}\times {\frac {1}{1+\exp(\alpha _{i}+{\boldsymbol {\beta }}^{\top }X_{i2})}}}{{\frac {\exp(\alpha _{i}+{\boldsymbol {\beta }}^{\top }X_{i1})}{1+\exp(\alpha _{i}+{\boldsymbol {\beta }}^{\top }X_{i1})}}\times {\frac {1}{1+\exp(\alpha _{i}+{\boldsymbol {\beta }}^{\top }X_{i2})}}+{\frac {1}{1+\exp(\alpha _{i}+{\boldsymbol {\beta }}^{\top }X_{i1})}}\times {\frac {\exp(\alpha _{i}+{\boldsymbol {\beta }}^{\top }X_{i2})}{1+\exp(\alpha _{i}+{\boldsymbol {\beta }}^{\top }X_{i2})}}}}\\\ &={\frac {\exp({\boldsymbol {\beta }}^{\top }X_{i1})}{\exp({\boldsymbol {\beta }}^{\top }X_{i1})+\exp({\boldsymbol {\beta }}^{\top }X_{i2})}}.\\\end{aligned}}}
2363:
775:, so the number of parameters is of the same order as the number of datapoints. In these settings, as we increase the amount of data, the asymptotic results on which maximum likelihood estimation is based on are not valid and the resulting estimates are biased. Conditional logistic regression fixes this issue. In fact, it can be shown that the unconditional analysis of matched pair data results in an estimate of the
2019:
400:
2358:{\displaystyle \mathbb {P} (Y_{ij}=1{\text{ for }}j\leq k,Y_{ij}=0{\text{ for }}k<j\leq m|X_{i1},...,X_{im},\sum _{j=1}^{m}Y_{ij}=k)={\frac {\exp(\sum _{j=1}^{k}{\boldsymbol {\beta }}^{\top }X_{ij})}{\sum _{J\in {\mathcal {C}}_{k}^{m}}\exp(\sum _{j\in J}{\boldsymbol {\beta }}^{\top }X_{ij})}},}
459:
For example, consider estimating the impact of exercise on the risk of cardiovascular disease. If people who exercise more are younger, have better access to healthcare, or have other differences that improve their health, then a logistic regression of cardiovascular disease incidence on minutes
223:
2536:
allows to test the association between a binary outcome and a binary predictor while taking into account stratification with arbitrary strata size. When its conditions of application are verified, it is identical to the conditional logistic regression
786:. However, they did not allow for the analysis of continuous predictors with arbitrary stratum size. All of those procedures also lack the flexibility of conditional logistic regression and in particular the possibility to control for covariates.
794:
Conditional logistic regression uses a conditional likelihood approach that deals with the above pathological behavior by conditioning on the number of cases in each stratum. This eliminates the need to estimate the strata parameters.
720:
Pathological behavior, however, occurs when we have many small strata because the number of parameters grow with the amount of data. For example, if each stratum contains two datapoints, then the number of parameters in a model with
667:
Logistic regression as described above works satisfactorily when the number of strata is small relative to the amount of data. If we hold the number of strata fixed and increase the amount of data, estimates of the model parameters
809:
460:
spent exercising may overestimate the impact of exercise on health. To address this, we can group people based on demographic characteristics like age and zip code of their home residence. Each stratum
215:
395:{\displaystyle \mathbb {P} (Y_{i\ell }=1|X_{i\ell })={\frac {\exp(\alpha _{i}+{\boldsymbol {\beta }}^{\top }X_{i\ell })}{1+\exp(\alpha _{i}+{\boldsymbol {\beta }}^{\top }X_{i\ell })}}}
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629:(which, in this example, is just a scalar) is the quantity of interest --- the impact of exercise on cardiovascular disease. We can also include control variables within
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package because the log likelihood of a conditional logistic model is the same as the log likelihood of a Cox model with a particular data structure.
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In addition to tests based on logistic regression, several other tests existed before conditional logistic regression for matched data as shown in
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Day, N. E., Byar, D. P. (1979). "Testing hypotheses in case-control studies-equivalence of Mantel-Haenszel statistics and logit score tests".
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The full conditional log likelihood is then simply the sum of the log likelihoods for each stratum. The estimator is then defined as the
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67:
27:
798:
When the strata are pairs, where the first observation is a case and the second is a control, this can be seen as follows
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allows to test the association between a binary outcome and a continuous predictor while taking into account pairing.
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contains information about the variable of interest (in this case, minutes spent exercising) for individual
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can account for stratification by having a different constant term for each stratum. Let us denote
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the values of the corresponding predictors. We then take the likelihood of one observation to be
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Statistical
Methods in Cancer Research. Volume 1-The Analysis of Case-Control Studies
39:
2561:"Estimation of multiple relative risk functions in matched case-control studies"
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58:
and C. Sabai. It is the most flexible and general procedure for matched data.
607:, which is assumed to be the same for all people in the stratum. The vector
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With similar computations, the conditional likelihood of a stratum of size
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Conditional logistic regression is available in R as the function
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is the impact of demographics on cardiovascular disease incidence
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th stratum. The parameters in this model can be estimated using
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Breslow NE, Day NE, Halvorsen KT, Prentice RL, Sabai C (1978).
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is a group of people with similar demographics. The vector
2649:"statsmodels.discrete.conditional_models.ConditionalLogit"
2604:. Lyon, France: IARC. pp. 249–251. Archived from
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which is the square of the correct, conditional one.
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2628:"R documentation Conditional logistic regression"
2491:that maximizes the conditional log likelihood.
210:{\displaystyle X_{i\ell }\in \mathbb {R} ^{p}}
8:
2711:: CS1 maint: multiple names: authors list (
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2514:It is also available in python through the
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2013:first observations being the cases, is
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2397:{\displaystyle {\mathcal {C}}_{k}^{m}}
710:{\displaystyle {\boldsymbol {\beta }}}
622:{\displaystyle {\boldsymbol {\beta }}}
125:{\displaystyle Y_{i\ell }\in \{0,1\}}
7:
2518:package starting with version 0.14.
132:the label (e.g. case status) of the
34:. Its main field of application is
2577:10.1093/oxfordjournals.aje.a112623
2404:is the set of all subsets of size
2328:
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1929:
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717:) converge to their true values.
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310:
14:
2598:Breslow, N.E.; Day, N.E. (1980).
695:for each stratum and the vector
26:that allows one to account for
20:Conditional logistic regression
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1:
454:maximum likelihood estimation
432:is the constant term for the
2534:Cochran-Mantel-Haenszel test
42:. It was devised in 1978 by
2461:{\displaystyle \{1,...,m\}}
688:{\displaystyle \alpha _{i}}
570:{\displaystyle \alpha _{i}}
425:{\displaystyle \alpha _{i}}
2749:
652:{\displaystyle X_{i\ell }}
600:{\displaystyle Y_{i\ell }}
503:{\displaystyle X_{i\ell }}
66:Observational studies use
74:as a way to control for
2528:Paired difference test
2507:package. It is in the
2485:
2484:{\displaystyle \beta }
2462:
2418:
2398:
2359:
2233:
2175:
2007:
1987:
1964:
790:Conditional likelihood
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735:
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689:
653:
623:
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571:
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396:
211:
166:
152:th observation of the
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126:
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2419:
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2213:
2155:
2008:
1988:
1965:
770:
768:{\displaystyle N/2+p}
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712:
690:
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624:
602:
572:
545:
543:{\displaystyle \ell }
525:
505:
475:
473:{\displaystyle \ell }
447:
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212:
167:
147:
145:{\displaystyle \ell }
127:
36:observational studies
16:Statistical technique
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2428:
2408:
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2020:
1997:
1977:
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745:
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611:
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436:
409:
224:
176:
156:
136:
88:
2733:Logistic regression
2393:
2293:
82:Logistic regression
52:Katherine Halvorsen
24:logistic regression
22:is an extension of
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2414:
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2003:
1983:
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1958:
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619:
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162:
142:
122:
38:and in particular
2417:{\displaystyle k}
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2264:
2089:
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2006:{\displaystyle k}
1986:{\displaystyle m}
1951:
1823:
1815:
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1695:
1627:
1559:
1443:
1375:
1252:
1244:
734:{\displaystyle N}
523:{\displaystyle i}
445:{\displaystyle i}
390:
165:{\displaystyle i}
2740:
2717:
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2710:
2702:
2666:
2660:
2659:
2657:
2655:
2645:
2639:
2638:
2636:
2634:
2626:Lumley, Thomas.
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2110:
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2012:
2010:
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1992:
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56:Ross L. Prentice
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2720:
2703:
2683:10.2307/2530253
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2370:
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2111:
2088: for
2067:
2052: for
2031:
2018:
2017:
1995:
1994:
1975:
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1957:
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1934:
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332:
315:
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290:
280:
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235:
222:
221:
195:
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174:
173:
172:th stratum and
154:
153:
134:
133:
91:
86:
85:
64:
17:
12:
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5:
2746:
2744:
2736:
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2718:
2677:(3): 623–630.
2661:
2640:
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2590:
2571:(4): 299–307.
2565:Am J Epidemiol
2550:
2549:
2547:
2544:
2543:
2542:
2531:
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2520:
2496:
2495:Implementation
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2077:
2074:
2070:
2066:
2063:
2060:
2057:
2049:
2046:
2041:
2038:
2034:
2030:
2026:
2002:
1982:
1971:
1970:
1955:
1949:
1944:
1941:
1937:
1931:
1926:
1921:
1918:
1915:
1912:
1909:
1904:
1901:
1897:
1891:
1886:
1881:
1878:
1875:
1870:
1865:
1862:
1858:
1852:
1847:
1842:
1839:
1836:
1830:
1827:
1825:
1820:
1819:
1810:
1805:
1802:
1798:
1792:
1787:
1782:
1777:
1773:
1769:
1766:
1763:
1760:
1757:
1752:
1747:
1744:
1740:
1734:
1729:
1724:
1719:
1715:
1711:
1708:
1705:
1699:
1693:
1688:
1685:
1681:
1675:
1670:
1665:
1660:
1656:
1652:
1649:
1646:
1643:
1640:
1636:
1631:
1625:
1620:
1617:
1613:
1607:
1602:
1597:
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1581:
1578:
1575:
1572:
1568:
1563:
1557:
1552:
1549:
1545:
1539:
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1529:
1524:
1520:
1516:
1513:
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1507:
1504:
1499:
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1487:
1481:
1476:
1471:
1466:
1462:
1458:
1455:
1452:
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1436:
1433:
1429:
1423:
1418:
1413:
1408:
1404:
1400:
1397:
1394:
1391:
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1379:
1373:
1368:
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1361:
1355:
1350:
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1340:
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1332:
1329:
1326:
1323:
1320:
1315:
1310:
1307:
1303:
1297:
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1287:
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1274:
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1268:
1259:
1256:
1254:
1249:
1248:
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1234:
1230:
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1213:
1210:
1206:
1202:
1198:
1194:
1189:
1186:
1182:
1177:
1173:
1170:
1165:
1162:
1158:
1154:
1150:
1146:
1143:
1138:
1135:
1131:
1126:
1122:
1119:
1114:
1111:
1107:
1103:
1099:
1095:
1090:
1087:
1083:
1078:
1074:
1071:
1066:
1063:
1059:
1055:
1051:
1045:
1040:
1037:
1033:
1028:
1024:
1021:
1016:
1013:
1009:
1005:
1001:
997:
992:
989:
985:
980:
976:
973:
968:
965:
961:
957:
953:
946:
943:
941:
939:
936:
933:
930:
925:
922:
918:
914:
909:
906:
902:
898:
893:
890:
886:
882:
877:
874:
870:
865:
861:
858:
853:
850:
846:
842:
839:
836:
831:
828:
824:
820:
816:
812:
810:
791:
788:
764:
761:
758:
754:
750:
741:datapoints is
730:
705:
682:
678:
664:
661:
646:
643:
639:
617:
594:
591:
587:
564:
560:
539:
519:
497:
494:
490:
469:
441:
419:
415:
403:
402:
388:
383:
380:
376:
370:
365:
360:
355:
351:
347:
344:
341:
338:
335:
330:
325:
322:
318:
312:
307:
302:
297:
293:
289:
286:
283:
277:
274:
269:
266:
262:
257:
253:
250:
245:
242:
238:
234:
230:
204:
199:
194:
189:
186:
182:
161:
141:
121:
118:
115:
112:
109:
106:
101:
98:
94:
68:stratification
63:
60:
44:Norman Breslow
28:stratification
15:
13:
10:
9:
6:
4:
3:
2:
2745:
2734:
2731:
2730:
2728:
2714:
2708:
2700:
2696:
2692:
2688:
2684:
2680:
2676:
2672:
2665:
2662:
2650:
2644:
2641:
2629:
2622:
2619:
2608:on 2016-12-26
2607:
2603:
2602:
2594:
2591:
2586:
2582:
2578:
2574:
2570:
2566:
2562:
2555:
2552:
2545:
2540:
2535:
2532:
2529:
2526:
2525:
2522:Related tests
2521:
2519:
2512:
2494:
2492:
2478:
2469:
2452:
2449:
2446:
2443:
2440:
2437:
2434:
2411:
2389:
2384:
2352:
2341:
2338:
2334:
2316:
2313:
2310:
2306:
2299:
2296:
2289:
2284:
2272:
2269:
2265:
2254:
2251:
2247:
2229:
2224:
2221:
2218:
2214:
2207:
2204:
2198:
2192:
2189:
2184:
2181:
2177:
2171:
2166:
2163:
2160:
2156:
2152:
2147:
2144:
2140:
2136:
2133:
2130:
2127:
2124:
2119:
2116:
2112:
2103:
2100:
2097:
2094:
2091:
2083:
2080:
2075:
2072:
2068:
2064:
2061:
2058:
2055:
2047:
2044:
2039:
2036:
2032:
2016:
2015:
2014:
2000:
1980:
1953:
1942:
1939:
1935:
1916:
1913:
1910:
1902:
1899:
1895:
1876:
1873:
1863:
1860:
1856:
1837:
1834:
1828:
1826:
1803:
1800:
1796:
1780:
1775:
1771:
1764:
1761:
1758:
1755:
1745:
1742:
1738:
1722:
1717:
1713:
1706:
1703:
1697:
1686:
1683:
1679:
1663:
1658:
1654:
1647:
1644:
1641:
1638:
1634:
1629:
1618:
1615:
1611:
1595:
1590:
1586:
1579:
1576:
1573:
1570:
1566:
1561:
1550:
1547:
1543:
1527:
1522:
1518:
1511:
1508:
1505:
1502:
1492:
1489:
1485:
1469:
1464:
1460:
1453:
1450:
1434:
1431:
1427:
1411:
1406:
1402:
1395:
1392:
1389:
1386:
1382:
1377:
1366:
1363:
1359:
1343:
1338:
1334:
1327:
1324:
1321:
1318:
1308:
1305:
1301:
1285:
1280:
1276:
1269:
1266:
1257:
1255:
1235:
1232:
1228:
1219:
1216:
1211:
1208:
1204:
1187:
1184:
1180:
1171:
1168:
1163:
1160:
1156:
1144:
1136:
1133:
1129:
1120:
1117:
1112:
1109:
1105:
1088:
1085:
1081:
1072:
1069:
1064:
1061:
1057:
1038:
1035:
1031:
1022:
1019:
1014:
1011:
1007:
990:
987:
983:
974:
971:
966:
963:
959:
944:
942:
931:
928:
923:
920:
916:
912:
907:
904:
900:
896:
891:
888:
884:
880:
875:
872:
868:
859:
856:
851:
848:
844:
840:
837:
834:
829:
826:
822:
801:
800:
799:
796:
789:
787:
785:
784:related tests
780:
778:
762:
759:
756:
752:
748:
728:
718:
680:
676:
662:
660:
644:
641:
637:
592:
589:
585:
562:
558:
537:
517:
495:
492:
488:
467:
457:
455:
439:
417:
413:
381:
378:
374:
358:
353:
349:
342:
339:
336:
333:
323:
320:
316:
300:
295:
291:
284:
281:
275:
267:
264:
260:
251:
248:
243:
240:
236:
220:
219:
218:
202:
192:
187:
184:
180:
159:
139:
116:
113:
110:
104:
99:
96:
92:
83:
79:
77:
73:
69:
61:
59:
57:
53:
49:
45:
41:
37:
33:
29:
25:
21:
2707:cite journal
2674:
2670:
2664:
2652:. Retrieved
2643:
2631:. Retrieved
2621:
2610:. Retrieved
2606:the original
2600:
2593:
2568:
2564:
2554:
2513:
2498:
2470:
2367:
1972:
797:
793:
781:
719:
666:
550:. The value
458:
404:
80:
65:
48:Nicholas Day
40:epidemiology
19:
18:
2633:November 3,
2516:statsmodels
2424:of the set
1993:, with the
530:in stratum
76:confounding
2671:Biometrics
2612:2016-11-04
2539:score test
777:odds ratio
663:Motivation
62:Background
2654:March 25,
2479:β
2329:⊤
2324:β
2314:∈
2307:∑
2300:
2273:∈
2266:∑
2242:⊤
2237:β
2215:∑
2208:
2157:∑
2101:≤
2059:≤
1930:⊤
1925:β
1917:
1890:⊤
1885:β
1877:
1851:⊤
1846:β
1838:
1791:⊤
1786:β
1772:α
1765:
1733:⊤
1728:β
1714:α
1707:
1698:×
1674:⊤
1669:β
1655:α
1648:
1606:⊤
1601:β
1587:α
1580:
1562:×
1538:⊤
1533:β
1519:α
1512:
1480:⊤
1475:β
1461:α
1454:
1422:⊤
1417:β
1403:α
1396:
1378:×
1354:⊤
1349:β
1335:α
1328:
1296:⊤
1291:β
1277:α
1270:
704:β
677:α
645:ℓ
616:β
593:ℓ
559:α
538:ℓ
496:ℓ
468:ℓ
414:α
382:ℓ
369:⊤
364:β
350:α
343:
324:ℓ
311:⊤
306:β
292:α
285:
268:ℓ
244:ℓ
193:∈
188:ℓ
140:ℓ
105:∈
100:ℓ
2727:Category
2509:survival
2505:survival
72:matching
32:matching
2691:2530253
2503:in the
2699:497345
2697:
2689:
2585:727199
2583:
2501:clogit
2368:where
1822:
1251:
405:where
2687:JSTOR
2546:Notes
2713:link
2695:PMID
2656:2023
2635:2016
2581:PMID
2095:<
30:and
2679:doi
2573:doi
2569:108
2297:exp
2205:exp
1914:exp
1874:exp
1835:exp
1762:exp
1704:exp
1645:exp
1577:exp
1509:exp
1451:exp
1393:exp
1325:exp
1267:exp
340:exp
282:exp
78:.
70:or
2729::
2709:}}
2705:{{
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