708:
565:
1030:
550:
1252:
703:{\displaystyle ~p_{\mathcal {}}=\sum _{\mathbf {y} \,:\;\operatorname {\mathbb {P} } \left(\mathbf {y} \right)\,\leq \,\operatorname {\mathbb {P} } \left(\mathbf {x} \right)_{0}}\operatorname {\mathbb {P} } \left(\mathbf {y} \right)~}
2002:
1668:
321:
2122:
903:
427:
161:
1807:
555:
The significance probability for the test is the probability of occurrence of the data set observed, or of a data set less likely than that observed, if the null hypothesis is true. Using an
867:
384:
224:
2443:
Smith, P.J.; Rae, D.S.; Manderscheid, R.W.; Manderscheid, S. (1981). "Approximating the moments and distribution of the likelihood ratio statistic for multinomial goodness of fit".
2174:
895:
419:
1889:
807:
2346:
2289:
1853:
1506:
1463:
1370:
1698:
1542:
765:
increase so it is probably only worth using exact tests for small samples. For larger samples, asymptotic approximations are accurate enough and easier to calculate.
2232:
1732:
1406:
1286:
is chosen to make the statistic asymptotically chi-squared distributed, for convenient comparison to a familiar statistic commonly used for the same application.)
1107:
1284:
1118:
2200:
1313:
1075:
763:
737:
62:
82:
1897:
1550:
231:
2016:
713:
where the sum ranges over all outcomes as likely as, or less likely than, that observed. In practice this becomes computationally onerous as
2483:
1025:{\displaystyle ~\operatorname {\mathbb {P} } \left(\mathbf {x} \right)_{A}=N!\;\prod _{i=1}^{k}{\frac {\;p_{i}^{x_{i}}\,}{x_{i}!}}~.}
545:{\displaystyle ~\operatorname {\mathbb {P} } \left(\mathbf {x} \right)_{0}=N!\,\prod _{i=1}^{k}{\frac {\pi _{i}^{x_{i}}}{x_{i}!}}~.}
87:
1740:
1469:(false positives). The difference between the moments of chi-squared and those of the test statistic are a function of
2493:
2373:
812:
2234:
degrees of freedom when the null hypothesis is true but does so from below, as it were, rather than from above as
326:
166:
1408:
degrees of freedom. However it has long been known (e.g. Lawley) that for finite sample sizes, the moments of
2008:
2389:
Lawley, D.N. (1956). "A general method of approximating to the distribution of likelihood ratio criteria".
2488:
2130:
1466:
872:
396:
1373:
28:
1858:
779:
2294:
2237:
1819:
1472:
1411:
1318:
773:
1676:
1511:
1247:{\displaystyle ~-2\ln()=-2\;\sum _{i=1}^{k}x_{i}\ln \left({\frac {\pi _{i}}{p_{i}}}\right)~.}
1110:
769:
2202:
under the null hypothesis. This statistic also converges to a chi-squared distribution with
2205:
1703:
1379:
1080:
2416:
Williams, D.A. (1976). "Improved
Likelihood Ratio Tests for Complete Contingency Tables".
1260:
8:
2179:
1292:
1038:
742:
716:
41:
2460:
67:
2442:
2369:
1997:{\displaystyle ~q_{2}=1+{\frac {\,k+1\,}{\,6N\,}}+{\frac {\;k^{2}\,}{\;6N^{2}\,}}~.}
1663:{\displaystyle ~q_{1}=1+{\frac {\;\sum _{i=1}^{k}\pi _{i}^{-1}\,-\,1\;}{6N(k-1)}}~.}
2452:
2425:
2398:
32:
20:
316:{\displaystyle ~H_{0}:{\boldsymbol {\pi }}=(\pi _{1},\pi _{2},\ldots ,\pi _{k})~,}
2117:{\displaystyle ~\chi ^{2}=\sum _{i=1}^{k}{\frac {\;(x_{i}-E_{i})^{2}\,}{E_{i}}}~}
387:
24:
2402:
590:
2477:
2429:
1465:
are greater than those of chi-squared, thus inflating the probability of
2464:
2436:
556:
1816:. derived a dividing factor which matches the first moment as far as
1673:
In the special case where the null hypothesis is that all the values
2456:
1734:(i.e. it stipulates a uniform distribution), this simplifies to
1508:
Williams showed that the first moment can be matched as far as
2324:
2267:
1441:
1348:
1148:
1053:
587:
584:
581:
64:
items each of which has been observed to fall into one of
2366:
Goodness-of-Fit
Statistics for Discrete Multivariate Data
2291:
does, so may be preferable to the uncorrected version of
156:{\displaystyle ~\mathbf {x} =(x_{1},x_{2},\dots ,x_{k})~}
1544:
if the test statistic is divided by a factor given by
2297:
2240:
2208:
2182:
2133:
2019:
1900:
1861:
1822:
1743:
1706:
1679:
1553:
1514:
1475:
1414:
1382:
1321:
1295:
1263:
1121:
1083:
1041:
906:
875:
815:
782:
745:
719:
568:
430:
399:
329:
234:
169:
163:
as the observed numbers of items in each cell. Hence
90:
70:
44:
1802:{\displaystyle ~q_{1}=1+{\frac {\,k+1\,}{\,6N\,}}~.}
869:
The exact probability of the observed configuration
393:
The exact probability of the observed configuration
578:
2451:(375). American Statistical Association: 737–740.
2340:
2283:
2226:
2194:
2168:
2116:
1996:
1883:
1847:
1801:
1726:
1692:
1662:
1536:
1500:
1457:
1400:
1364:
1307:
1278:
1246:
1101:
1069:
1024:
889:
861:
801:
757:
731:
702:
544:
413:
378:
315:
218:
155:
76:
56:
2475:
2357:
2007:The null hypothesis can also be tested by using
2445:Journal of the American Statistical Association
1077:between these two probabilities, multiplied by
1035:The natural logarithm of the likelihood ratio,
862:{\displaystyle ~p_{i}={\frac {\;x_{i}\,}{N}}~.}
809:is replaced by its maximum likelihood estimate
2415:
897:under the alternative hypothesis is given by
2409:
2388:
2176:is the expected number of cases in category
379:{\displaystyle ~\sum _{i=1}^{k}\pi _{i}=1~.}
2363:
2382:
2060:
1970:
1956:
1627:
1579:
1168:
974:
949:
835:
613:
2097:
1984:
1967:
1946:
1939:
1936:
1926:
1789:
1782:
1779:
1769:
1623:
1619:
997:
912:
846:
676:
644:
641:
637:
616:
609:
473:
436:
386:These are the parameter values under the
219:{\displaystyle ~\sum _{i=1}^{k}x_{i}=N~.}
1289:If the null hypothesis is true, then as
776:can be defined under which each value
252:
31:equal specified values; it is used for
2476:
2364:Read, T.R.C.; Cressie, N.A.C. (1988).
421:under the null hypothesis is given by
228:Next, defining a vector of parameters
84:categories. It is possible to define
768:One of these approximations is the
13:
2321:
2264:
1438:
1345:
1145:
1050:
14:
2505:
2484:Categorical variable interactions
2368:. New York, NY: Springer-Verlag.
2169:{\displaystyle ~E_{i}=N\pi _{i}~}
1855:For the case of equal values of
926:
880:
689:
658:
629:
605:
450:
404:
95:
1315:increases, the distribution of
2332:
2329:
2316:
2313:
2275:
2272:
2259:
2256:
2088:
2061:
1648:
1636:
1449:
1446:
1433:
1430:
1356:
1353:
1340:
1337:
1156:
1153:
1140:
1137:
1109:is then the statistic for the
1058:
1045:
890:{\displaystyle ~\mathbf {x} ~}
414:{\displaystyle ~\mathbf {x} ~}
304:
259:
147:
102:
1:
2351:
7:
1884:{\displaystyle ~\pi _{i}~,}
38:Beginning with a sample of
10:
2510:
2009:Pearson's chi-squared test
802:{\displaystyle ~\pi _{i}~}
2403:10.1093/biomet/43.3-4.295
2341:{\displaystyle ~-2\ln()~}
2284:{\displaystyle ~-2\ln()~}
1848:{\displaystyle ~N^{-3}~.}
1501:{\displaystyle ~N^{-1}~.}
1458:{\displaystyle ~-2\ln()~}
1365:{\displaystyle ~-2\ln()~}
27:that the parameters of a
2494:Nonparametric statistics
1693:{\displaystyle \pi _{i}}
1537:{\displaystyle ~N^{-2}~}
559:, this is calculated as
29:multinomial distribution
2430:10.1093/biomet/63.1.33
2342:
2285:
2228:
2196:
2170:
2118:
2056:
1998:
1885:
1849:
1803:
1728:
1694:
1664:
1600:
1538:
1502:
1459:
1402:
1366:
1309:
1280:
1248:
1189:
1103:
1071:
1026:
970:
891:
863:
803:
774:alternative hypothesis
759:
733:
704:
546:
494:
415:
380:
353:
317:
220:
193:
157:
78:
58:
2343:
2286:
2229:
2227:{\displaystyle ~k-1~}
2197:
2171:
2119:
2036:
1999:
1886:
1850:
1804:
1729:
1727:{\displaystyle ~1/k~}
1695:
1665:
1580:
1539:
1503:
1460:
1403:
1401:{\displaystyle ~k-1~}
1372:converges to that of
1367:
1310:
1281:
1249:
1169:
1111:likelihood ratio test
1104:
1102:{\displaystyle ~-2~,}
1072:
1027:
950:
892:
864:
804:
760:
734:
705:
547:
474:
416:
381:
333:
318:
221:
173:
158:
79:
59:
2295:
2238:
2206:
2180:
2131:
2017:
1898:
1859:
1820:
1812:Subsequently, Smith
1741:
1704:
1677:
1551:
1512:
1473:
1412:
1380:
1319:
1293:
1279:{\displaystyle ~-2~}
1261:
1119:
1081:
1039:
904:
873:
813:
780:
743:
717:
566:
428:
397:
327:
232:
167:
88:
68:
42:
2348:for small samples.
2195:{\displaystyle ~i~}
1618:
1308:{\displaystyle ~N~}
1070:{\displaystyle ~~,}
996:
758:{\displaystyle ~N~}
732:{\displaystyle ~k~}
518:
57:{\displaystyle ~N~}
2338:
2281:
2224:
2192:
2166:
2114:
1994:
1881:
1845:
1799:
1724:
1690:
1660:
1601:
1534:
1498:
1467:type I errors
1455:
1398:
1362:
1305:
1276:
1244:
1099:
1067:
1022:
975:
887:
859:
799:
755:
729:
700:
673:
542:
497:
411:
376:
313:
216:
153:
74:
54:
2489:Statistical tests
2337:
2300:
2280:
2243:
2223:
2211:
2191:
2185:
2165:
2136:
2113:
2109:
2022:
1990:
1986:
1948:
1903:
1877:
1864:
1841:
1825:
1795:
1791:
1746:
1723:
1709:
1656:
1652:
1556:
1533:
1517:
1494:
1478:
1454:
1417:
1397:
1385:
1361:
1324:
1304:
1298:
1275:
1266:
1240:
1232:
1124:
1095:
1086:
1063:
1044:
1018:
1014:
909:
886:
878:
855:
851:
818:
798:
785:
754:
748:
728:
722:
699:
599:
571:
538:
534:
433:
410:
402:
372:
332:
309:
237:
212:
172:
152:
93:
77:{\displaystyle k}
53:
47:
2501:
2469:
2468:
2440:
2434:
2433:
2413:
2407:
2406:
2386:
2380:
2379:
2361:
2347:
2345:
2344:
2339:
2335:
2328:
2327:
2298:
2290:
2288:
2287:
2282:
2278:
2271:
2270:
2241:
2233:
2231:
2230:
2225:
2221:
2209:
2201:
2199:
2198:
2193:
2189:
2183:
2175:
2173:
2172:
2167:
2163:
2162:
2161:
2146:
2145:
2134:
2123:
2121:
2120:
2115:
2111:
2110:
2108:
2107:
2098:
2096:
2095:
2086:
2085:
2073:
2072:
2058:
2055:
2050:
2032:
2031:
2020:
2003:
2001:
2000:
1995:
1988:
1987:
1985:
1983:
1982:
1968:
1966:
1965:
1954:
1949:
1947:
1937:
1924:
1913:
1912:
1901:
1891:this factor is
1890:
1888:
1887:
1882:
1875:
1874:
1873:
1862:
1854:
1852:
1851:
1846:
1839:
1838:
1837:
1823:
1808:
1806:
1805:
1800:
1793:
1792:
1790:
1780:
1767:
1756:
1755:
1744:
1733:
1731:
1730:
1725:
1721:
1717:
1707:
1699:
1697:
1696:
1691:
1689:
1688:
1669:
1667:
1666:
1661:
1654:
1653:
1651:
1628:
1617:
1609:
1599:
1594:
1577:
1566:
1565:
1554:
1543:
1541:
1540:
1535:
1531:
1530:
1529:
1515:
1507:
1505:
1504:
1499:
1492:
1491:
1490:
1476:
1464:
1462:
1461:
1456:
1452:
1445:
1444:
1415:
1407:
1405:
1404:
1399:
1395:
1383:
1371:
1369:
1368:
1363:
1359:
1352:
1351:
1322:
1314:
1312:
1311:
1306:
1302:
1296:
1285:
1283:
1282:
1277:
1273:
1264:
1253:
1251:
1250:
1245:
1238:
1237:
1233:
1231:
1230:
1221:
1220:
1211:
1199:
1198:
1188:
1183:
1152:
1151:
1122:
1108:
1106:
1105:
1100:
1093:
1084:
1076:
1074:
1073:
1068:
1061:
1057:
1056:
1042:
1031:
1029:
1028:
1023:
1016:
1015:
1013:
1009:
1008:
998:
995:
994:
993:
983:
972:
969:
964:
939:
938:
933:
929:
916:
915:
907:
896:
894:
893:
888:
884:
883:
876:
868:
866:
865:
860:
853:
852:
847:
845:
844:
833:
828:
827:
816:
808:
806:
805:
800:
796:
795:
794:
783:
770:likelihood ratio
764:
762:
761:
756:
752:
746:
738:
736:
735:
730:
726:
720:
709:
707:
706:
701:
697:
696:
692:
680:
679:
672:
671:
670:
665:
661:
648:
647:
636:
632:
620:
619:
608:
595:
594:
593:
569:
551:
549:
548:
543:
536:
535:
533:
529:
528:
517:
516:
515:
505:
496:
493:
488:
463:
462:
457:
453:
440:
439:
431:
420:
418:
417:
412:
408:
407:
400:
385:
383:
382:
377:
370:
363:
362:
352:
347:
330:
322:
320:
319:
314:
307:
303:
302:
284:
283:
271:
270:
255:
247:
246:
235:
225:
223:
222:
217:
210:
203:
202:
192:
187:
170:
162:
160:
159:
154:
150:
146:
145:
127:
126:
114:
113:
98:
91:
83:
81:
80:
75:
63:
61:
60:
55:
51:
45:
33:categorical data
21:statistical test
17:Multinomial test
2509:
2508:
2504:
2503:
2502:
2500:
2499:
2498:
2474:
2473:
2472:
2457:10.2307/2287541
2441:
2437:
2414:
2410:
2387:
2383:
2376:
2362:
2358:
2354:
2320:
2319:
2296:
2293:
2292:
2263:
2262:
2239:
2236:
2235:
2207:
2204:
2203:
2181:
2178:
2177:
2157:
2153:
2141:
2137:
2132:
2129:
2128:
2103:
2099:
2091:
2087:
2081:
2077:
2068:
2064:
2059:
2057:
2051:
2040:
2027:
2023:
2018:
2015:
2014:
1978:
1974:
1969:
1961:
1957:
1955:
1953:
1938:
1925:
1923:
1908:
1904:
1899:
1896:
1895:
1869:
1865:
1860:
1857:
1856:
1830:
1826:
1821:
1818:
1817:
1781:
1768:
1766:
1751:
1747:
1742:
1739:
1738:
1713:
1705:
1702:
1701:
1684:
1680:
1678:
1675:
1674:
1629:
1610:
1605:
1595:
1584:
1578:
1576:
1561:
1557:
1552:
1549:
1548:
1522:
1518:
1513:
1510:
1509:
1483:
1479:
1474:
1471:
1470:
1437:
1436:
1413:
1410:
1409:
1381:
1378:
1377:
1344:
1343:
1320:
1317:
1316:
1294:
1291:
1290:
1262:
1259:
1258:
1226:
1222:
1216:
1212:
1210:
1206:
1194:
1190:
1184:
1173:
1144:
1143:
1120:
1117:
1116:
1082:
1079:
1078:
1049:
1048:
1040:
1037:
1036:
1004:
1000:
999:
989:
985:
984:
979:
973:
971:
965:
954:
934:
925:
921:
920:
911:
910:
905:
902:
901:
879:
874:
871:
870:
840:
836:
834:
832:
823:
819:
814:
811:
810:
790:
786:
781:
778:
777:
744:
741:
740:
718:
715:
714:
688:
684:
675:
674:
666:
657:
653:
652:
643:
642:
628:
624:
615:
614:
604:
603:
577:
576:
572:
567:
564:
563:
524:
520:
519:
511:
507:
506:
501:
495:
489:
478:
458:
449:
445:
444:
435:
434:
429:
426:
425:
403:
398:
395:
394:
388:null hypothesis
358:
354:
348:
337:
328:
325:
324:
298:
294:
279:
275:
266:
262:
251:
242:
238:
233:
230:
229:
198:
194:
188:
177:
168:
165:
164:
141:
137:
122:
118:
109:
105:
94:
89:
86:
85:
69:
66:
65:
43:
40:
39:
25:null hypothesis
12:
11:
5:
2507:
2497:
2496:
2491:
2486:
2471:
2470:
2435:
2408:
2381:
2374:
2355:
2353:
2350:
2334:
2331:
2326:
2323:
2318:
2315:
2312:
2309:
2306:
2303:
2277:
2274:
2269:
2266:
2261:
2258:
2255:
2252:
2249:
2246:
2220:
2217:
2214:
2188:
2160:
2156:
2152:
2149:
2144:
2140:
2125:
2124:
2106:
2102:
2094:
2090:
2084:
2080:
2076:
2071:
2067:
2063:
2054:
2049:
2046:
2043:
2039:
2035:
2030:
2026:
2005:
2004:
1993:
1981:
1977:
1973:
1964:
1960:
1952:
1945:
1942:
1935:
1932:
1929:
1922:
1919:
1916:
1911:
1907:
1880:
1872:
1868:
1844:
1836:
1833:
1829:
1810:
1809:
1798:
1788:
1785:
1778:
1775:
1772:
1765:
1762:
1759:
1754:
1750:
1720:
1716:
1712:
1687:
1683:
1671:
1670:
1659:
1650:
1647:
1644:
1641:
1638:
1635:
1632:
1626:
1622:
1616:
1613:
1608:
1604:
1598:
1593:
1590:
1587:
1583:
1575:
1572:
1569:
1564:
1560:
1528:
1525:
1521:
1497:
1489:
1486:
1482:
1451:
1448:
1443:
1440:
1435:
1432:
1429:
1426:
1423:
1420:
1394:
1391:
1388:
1358:
1355:
1350:
1347:
1342:
1339:
1336:
1333:
1330:
1327:
1301:
1272:
1269:
1255:
1254:
1243:
1236:
1229:
1225:
1219:
1215:
1209:
1205:
1202:
1197:
1193:
1187:
1182:
1179:
1176:
1172:
1167:
1164:
1161:
1158:
1155:
1150:
1147:
1142:
1139:
1136:
1133:
1130:
1127:
1098:
1092:
1089:
1066:
1060:
1055:
1052:
1047:
1033:
1032:
1021:
1012:
1007:
1003:
992:
988:
982:
978:
968:
963:
960:
957:
953:
948:
945:
942:
937:
932:
928:
924:
919:
914:
882:
858:
850:
843:
839:
831:
826:
822:
793:
789:
751:
725:
711:
710:
695:
691:
687:
683:
678:
669:
664:
660:
656:
651:
646:
640:
635:
631:
627:
623:
618:
612:
607:
602:
598:
592:
589:
586:
583:
580:
575:
553:
552:
541:
532:
527:
523:
514:
510:
504:
500:
492:
487:
484:
481:
477:
472:
469:
466:
461:
456:
452:
448:
443:
438:
406:
375:
369:
366:
361:
357:
351:
346:
343:
340:
336:
312:
306:
301:
297:
293:
290:
287:
282:
278:
274:
269:
265:
261:
258:
254:
250:
245:
241:
215:
209:
206:
201:
197:
191:
186:
183:
180:
176:
149:
144:
140:
136:
133:
130:
125:
121:
117:
112:
108:
104:
101:
97:
73:
50:
9:
6:
4:
3:
2:
2506:
2495:
2492:
2490:
2487:
2485:
2482:
2481:
2479:
2466:
2462:
2458:
2454:
2450:
2446:
2439:
2431:
2427:
2423:
2419:
2412:
2404:
2400:
2396:
2392:
2385:
2377:
2375:0-387-96682-X
2371:
2367:
2360:
2356:
2349:
2310:
2307:
2304:
2301:
2253:
2250:
2247:
2244:
2218:
2215:
2212:
2186:
2158:
2154:
2150:
2147:
2142:
2138:
2104:
2100:
2092:
2082:
2078:
2074:
2069:
2065:
2052:
2047:
2044:
2041:
2037:
2033:
2028:
2024:
2013:
2012:
2011:
2010:
1991:
1979:
1975:
1971:
1962:
1958:
1950:
1943:
1940:
1933:
1930:
1927:
1920:
1917:
1914:
1909:
1905:
1894:
1893:
1892:
1878:
1870:
1866:
1842:
1834:
1831:
1827:
1815:
1796:
1786:
1783:
1776:
1773:
1770:
1763:
1760:
1757:
1752:
1748:
1737:
1736:
1735:
1718:
1714:
1710:
1700:are equal to
1685:
1681:
1657:
1645:
1642:
1639:
1633:
1630:
1624:
1620:
1614:
1611:
1606:
1602:
1596:
1591:
1588:
1585:
1581:
1573:
1570:
1567:
1562:
1558:
1547:
1546:
1545:
1526:
1523:
1519:
1495:
1487:
1484:
1480:
1468:
1427:
1424:
1421:
1418:
1392:
1389:
1386:
1375:
1334:
1331:
1328:
1325:
1299:
1287:
1270:
1267:
1241:
1234:
1227:
1223:
1217:
1213:
1207:
1203:
1200:
1195:
1191:
1185:
1180:
1177:
1174:
1170:
1165:
1162:
1159:
1134:
1131:
1128:
1125:
1115:
1114:
1113:
1112:
1096:
1090:
1087:
1064:
1019:
1010:
1005:
1001:
990:
986:
980:
976:
966:
961:
958:
955:
951:
946:
943:
940:
935:
930:
922:
917:
900:
899:
898:
856:
848:
841:
837:
829:
824:
820:
791:
787:
775:
771:
766:
749:
723:
693:
685:
681:
667:
662:
654:
649:
638:
633:
625:
621:
610:
600:
596:
573:
562:
561:
560:
558:
539:
530:
525:
521:
512:
508:
502:
498:
490:
485:
482:
479:
475:
470:
467:
464:
459:
454:
446:
441:
424:
423:
422:
391:
389:
373:
367:
364:
359:
355:
349:
344:
341:
338:
334:
310:
299:
295:
291:
288:
285:
280:
276:
272:
267:
263:
256:
248:
243:
239:
226:
213:
207:
204:
199:
195:
189:
184:
181:
178:
174:
142:
138:
134:
131:
128:
123:
119:
115:
110:
106:
99:
71:
48:
36:
34:
30:
26:
22:
18:
2448:
2444:
2438:
2421:
2417:
2411:
2394:
2390:
2384:
2365:
2359:
2126:
2006:
1813:
1811:
1672:
1288:
1257:(The factor
1256:
1034:
767:
712:
554:
392:
227:
37:
16:
15:
2397:: 295–303.
1374:chi-squared
2478:Categories
2418:Biometrika
2391:Biometrika
2352:References
557:exact test
2424:: 33–37.
2311:
2302:−
2254:
2245:−
2216:−
2155:π
2075:−
2038:∑
2025:χ
1867:π
1832:−
1682:π
1643:−
1621:−
1612:−
1603:π
1582:∑
1524:−
1485:−
1428:
1419:−
1390:−
1335:
1326:−
1268:−
1214:π
1204:
1171:∑
1163:−
1135:
1126:−
1088:−
952:∏
918:
788:π
682:
650:
639:≤
622:
601:∑
499:π
476:∏
442:
356:π
335:∑
296:π
289:…
277:π
264:π
253:π
175:∑
132:…
2465:2287541
323:where:
23:of the
19:is the
2463:
2372:
2336:
2299:
2279:
2242:
2222:
2210:
2190:
2184:
2164:
2135:
2127:where
2112:
2021:
1989:
1902:
1876:
1863:
1840:
1824:
1794:
1745:
1722:
1708:
1655:
1555:
1532:
1516:
1493:
1477:
1453:
1416:
1396:
1384:
1360:
1323:
1303:
1297:
1274:
1265:
1239:
1123:
1094:
1085:
1062:
1043:
1017:
908:
885:
877:
854:
817:
797:
784:
753:
747:
727:
721:
698:
570:
537:
432:
409:
401:
371:
331:
308:
236:
211:
171:
151:
92:
52:
46:
2461:JSTOR
1814:et al
1376:with
772:. An
2370:ISBN
739:and
2453:doi
2426:doi
2399:doi
2480::
2459:.
2449:76
2447:.
2422:63
2420:.
2395:43
2393:.
2308:ln
2251:ln
1425:ln
1332:ln
1201:ln
1132:ln
390:.
35:.
2467:.
2455::
2432:.
2428::
2405:.
2401::
2378:.
2333:)
2330:]
2325:R
2322:L
2317:[
2314:(
2305:2
2276:)
2273:]
2268:R
2265:L
2260:[
2257:(
2248:2
2219:1
2213:k
2187:i
2159:i
2151:N
2148:=
2143:i
2139:E
2105:i
2101:E
2093:2
2089:)
2083:i
2079:E
2070:i
2066:x
2062:(
2053:k
2048:1
2045:=
2042:i
2034:=
2029:2
1992:.
1980:2
1976:N
1972:6
1963:2
1959:k
1951:+
1944:N
1941:6
1934:1
1931:+
1928:k
1921:+
1918:1
1915:=
1910:2
1906:q
1879:,
1871:i
1843:.
1835:3
1828:N
1797:.
1787:N
1784:6
1777:1
1774:+
1771:k
1764:+
1761:1
1758:=
1753:1
1749:q
1719:k
1715:/
1711:1
1686:i
1658:.
1649:)
1646:1
1640:k
1637:(
1634:N
1631:6
1625:1
1615:1
1607:i
1597:k
1592:1
1589:=
1586:i
1574:+
1571:1
1568:=
1563:1
1559:q
1527:2
1520:N
1496:.
1488:1
1481:N
1450:)
1447:]
1442:R
1439:L
1434:[
1431:(
1422:2
1393:1
1387:k
1357:)
1354:]
1349:R
1346:L
1341:[
1338:(
1329:2
1300:N
1271:2
1242:.
1235:)
1228:i
1224:p
1218:i
1208:(
1196:i
1192:x
1186:k
1181:1
1178:=
1175:i
1166:2
1160:=
1157:)
1154:]
1149:R
1146:L
1141:[
1138:(
1129:2
1097:,
1091:2
1065:,
1059:]
1054:R
1051:L
1046:[
1020:.
1011:!
1006:i
1002:x
991:i
987:x
981:i
977:p
967:k
962:1
959:=
956:i
947:!
944:N
941:=
936:A
931:)
927:x
923:(
913:P
881:x
857:.
849:N
842:i
838:x
830:=
825:i
821:p
792:i
750:N
724:k
694:)
690:y
686:(
677:P
668:0
663:)
659:x
655:(
645:P
634:)
630:y
626:(
617:P
611::
606:y
597:=
591:]
588:g
585:i
582:s
579:[
574:p
540:.
531:!
526:i
522:x
513:i
509:x
503:i
491:k
486:1
483:=
480:i
471:!
468:N
465:=
460:0
455:)
451:x
447:(
437:P
405:x
374:.
368:1
365:=
360:i
350:k
345:1
342:=
339:i
311:,
305:)
300:k
292:,
286:,
281:2
273:,
268:1
260:(
257:=
249::
244:0
240:H
214:.
208:N
205:=
200:i
196:x
190:k
185:1
182:=
179:i
148:)
143:k
139:x
135:,
129:,
124:2
120:x
116:,
111:1
107:x
103:(
100:=
96:x
72:k
49:N
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