2736:
36:
1276:
2311:
1787:
2031:
2461:
965:
for each age determination. As regards weighting, one can either weight all of the measured ages equally, or weight them by the proportion of the sample that they represent. For example, if two thirds of the sample was used for the first measurement and one third for the second and final measurement,
3090:
M. J. Streule, R. J. Phillips, M. P. Searle, D. J. Waters and M. S. A. Horstwood 2009. Evolution and chronology of the
Pangong Metamorphic Complex adjacent to themodelling and U-Pb geochronology Karakoram Fault, Ladakh: constraints from thermobarometry, metamorphic modelling and U-Pb geochronology.
873:
MSWD > 1 if the observed scatter exceeds that predicted by the analytical uncertainties. In this case, the data are said to be "overdispersed". This situation is the rule rather than the exception in (U-Th)/He geochronology, indicating an incomplete understanding of the isotope system. Several
1052:
2039:
2454:
1554:
1794:
1549:
1401:
2731:{\displaystyle {\text{MSWD}}_{w}={\frac {\sum _{i=1}^{N}w_{i}}{{\big (}\sum _{i=1}^{N}w_{i}{\big )}^{2}-\sum _{i=1}^{N}w_{i}^{2}}}\cdot \sum _{i=1}^{N}{\frac {w_{i}(x_{i}-{\overline {x}}^{*})^{2}}{(\sigma _{x_{i}})^{2}}}.}
306:
3080:
Lance P. Black, Sandra L. Kamo, Charlotte M. Allen, John N. Aleinikoff, Donald W. Davis, Russell J. Korsch, Chris
Foudoulis 2003. TEMORA 1: a new zircon standard for Phanerozoic U–Pb geochronology. Chemical Geology 200,
869:
MSWD < 1 if the observed scatter is less than that predicted by the analytical uncertainties. In this case, the data are said to be "underdispersed", indicating that the analytical uncertainties were overestimated.
2318:
461:
1040:
540:
1271:{\displaystyle \sigma ^{2}={\frac {\sum _{i=1}^{N}(x_{i}-{\overline {x}})^{2}}{N}}{\text{ and }}s^{2}={\frac {N}{N-1}}\cdot \sigma ^{2}={\frac {1}{N-1}}\cdot \sum _{i=1}^{N}(x_{i}-{\overline {x}})^{2}.}
205:
2306:{\displaystyle s^{2}={\frac {\sum _{i=1}^{N}w_{i}x_{i}^{2}\cdot \sum _{i=1}^{N}w_{i}-{\big (}\sum _{i=1}^{N}w_{i}x_{i}{\big )}^{2}}{{\big (}\sum _{i=1}^{N}w_{i}{\big )}^{2}-\sum _{i=1}^{N}w_{i}^{2}}}.}
1408:
840:, the MSWD is a measure of goodness of fit that takes into account the relative importance of both the internal and external reproducibility, with most common usage in isotopic dating.
586:
2749:, the reduced chi-squared statistic is called the outfit mean-square statistic, and the information-weighted reduced chi-squared statistic is called the infit mean-square statistic.
1782:{\displaystyle \sigma ^{2}={\frac {\sum _{i=1}^{N}w_{i}x_{i}^{2}\cdot \sum _{i=1}^{N}w_{i}-{\big (}\sum _{i=1}^{N}w_{i}x_{i}{\big )}^{2}}{{\big (}\sum _{i=1}^{N}w_{i}{\big )}^{2}}}.}
1286:
807:
717:
679:
2026:{\displaystyle s^{2}={\frac {\sum _{i=1}^{N}w_{i}}{{\big (}\sum _{i=1}^{N}w_{i}{\big )}^{2}-\sum _{i=1}^{N}w_{i}^{2}}}\cdot {\sum _{i=1}^{N}w_{i}(x_{i}-{\overline {x}}^{*})^{2}}.}
749:
625:
963:
341:
2915:
381:
54:
214:
929:
902:
3155:
769:
401:
3037:
3025:
3015:
972:
489:
3150:
1283:
When individual determinations of an age are not of equal significance, it is better to use a weighted mean to obtain an "average" age, as follows:
874:
reasons have been proposed to explain the overdispersion of (U-Th)/He data, including unevenly distributed U-Th distributions and radiation damage.
155:
3004:
719:
indicates that the fit has not fully captured the data (or that the error variance has been underestimated). In principle, a value of
3060:
3052:
2895:
2870:
2816:
2789:
72:
2449:{\displaystyle {\text{MSWD}}_{u}={\frac {1}{N-1}}\cdot \sum _{i=1}^{N}{\frac {(x_{i}-{\overline {x}})^{2}}{\sigma _{x_{i}}^{2}}}.}
544:
3111:, Jon Woodhead 2002. Improving isochron calculations with robust statistics and the bootstrap. Chemical Geology 185, 191–204.
2924:
2768:
Wendt, I., and Carl, C., 1991, The statistical distribution of the mean squared weighted deviation, Chemical
Geology, 275–285.
149:
3071:
McDougall, I. and
Harrison, T. M. 1988. Geochronology and Thermochronology by the Ar/Ar Method. Oxford University Press.
1045:
When each measured value can be assumed to have the same weighting, or significance, the biased and unbiased (or "
877:
Often the geochronologist will determine a series of age measurements on a single sample, with the measured value
771:
indicates that the extent of the match between observations and estimates is in accord with the error variance. A
476:
1544:{\displaystyle \sigma ^{2}={\frac {\sum _{i=1}^{N}w_{i}(x_{i}-{\overline {x}}^{*})^{2}}{\sum _{i=1}^{N}w_{i}}},}
817:
3003:
Measurements and Their
Uncertainties: A Practical Guide to Modern Error Analysis, By Ifan Hughes, Thomas Hase
866:
age) space, or if the compositional data fit a bivariate normal distribution in -space (for the central age).
2917:
How Bad is Good? A Critical Look at the
Fitting of Reflectivity Models using the Reduced Chi-Square Statistic
2458:
By analogy, the weighted mean square of the weighted deviations (weighted MSWD) can be computed as follows:
589:
145:
774:
684:
646:
2315:
The unweighted mean square of the weighted deviations (unweighted MSWD) can then be computed, as follows:
483:
395:
208:
133:
114:
813:" the data: either the model is improperly fitting noise, or the error variance has been overestimated.
722:
598:
934:
314:
2036:
The unbiased weighted estimator of the sample variance can also be computed on the fly as follows:
847:
3047:
Dickin, A. P. 1995. Radiogenic
Isotope Geology. Cambridge University Press, Cambridge, UK, 1995,
1046:
1042:
but this value can be misleading, unless each determination of the age is of equal significance.
628:
471:
is the weight matrix, the inverse of the input (diagonal) covariance matrix of observations. If
3056:
3048:
2891:
2866:
2812:
2806:
2785:
2779:
1396:{\displaystyle {\overline {x}}^{*}={\frac {\sum _{i=1}^{N}w_{i}x_{i}}{\sum _{i=1}^{N}w_{i}}}.}
354:
17:
3092:
907:
880:
855:
106:
94:
2831:
863:
754:
3144:
1049:" and "population" respectively) estimators of the variance are computed as follows:
837:
2964:
1791:
The unbiased weighted estimator of the sample variance can be computed as follows:
3108:
2746:
2033:
Again, the corresponding standard deviation is the square root of the variance.
810:
301:{\displaystyle \chi ^{2}=\sum _{i}{\frac {(O_{i}-C_{i})^{2}}{\sigma _{i}^{2}}}}
3096:
86:
3036:
Computational
Methods in Physics and Engineering, By Samuel Shaw Ming Wong
309:
3122:
3014:
Dealing with
Uncertainties: A Guide to Error Analysis, By Manfred Drosg
639:
As a general rule, when the variance of the measurement error is known
966:
then one might weight the first measurement twice that of the second.
816:
When the variance of the measurement error is only partially known,
456:{\displaystyle \chi _{\nu }^{2}={\frac {r^{\mathrm {T} }Wr}{\nu }},}
3026:
Practical
Statistics for Astronomers, By J. V. Wall, C. R. Jenkins
1035:{\displaystyle {\overline {x}}={\frac {\sum _{i=1}^{N}x_{i}}{N}},}
535:{\displaystyle \chi _{\nu }^{2}={\frac {\mathrm {RSS} }{\nu }},}
3123:"What do Infit and Outfit, Mean-square and Standardized mean?"
29:
1405:
The biased weighted estimator of variance can be shown to be
2808:
Parameter Estimation and Hypothesis Testing in Linear Models
818:
the reduced chi-squared may serve as a correction estimated
2991:
Data Reduction and Error Analysis for the Physical Sciences
1280:
The standard deviation is the square root of the variance.
200:{\displaystyle \chi _{\nu }^{2}={\frac {\chi ^{2}}{\nu }},}
398:, the definition is often written in matrix notation as
50:
2464:
2321:
2042:
1797:
1557:
1411:
1289:
1055:
975:
937:
910:
883:
777:
757:
725:
687:
649:
601:
547:
492:
404:
357:
317:
217:
158:
207:where the chi-squared is a weighted sum of squared
45:
may be too technical for most readers to understand
2730:
2448:
2305:
2025:
1781:
1543:
1395:
1270:
1034:
957:
923:
896:
801:
763:
743:
711:
673:
619:
580:
534:
455:
375:
335:
300:
199:
969:The arithmetic mean of the age determinations is
134:Ordinary least squares § Reduced chi-squared
3091:Journal of the Geological Society 166, 919–932
2923:, University California, Davis, archived from
2811:. Springer Berlin Heidelberg. Section 3.2.5.
2557:
2518:
2247:
2208:
2193:
2144:
1888:
1849:
1762:
1723:
1708:
1659:
8:
2984:
2982:
595:When the fit is just an ordinary mean, then
2784:. Wellesley-Cambridge Press. p. 301.
581:{\displaystyle \mathrm {RSS} =\sum r^{2},}
2863:Standard Mathematical Tables and Formulae
2716:
2704:
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2063:
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2041:
2013:
2003:
1993:
1983:
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1949:
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1932:
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1054:
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1007:
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989:
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888:
882:
787:
782:
776:
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735:
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724:
697:
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686:
659:
654:
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611:
606:
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569:
548:
546:
513:
511:
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497:
491:
431:
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409:
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327:
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251:
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235:
222:
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183:
177:
168:
163:
157:
73:Learn how and when to remove this message
57:, without removing the technical details.
2952:, University Science Books, p. 268
2764:
2762:
2758:
2914:Laub, Charlie; Kuhl, Tonya L. (n.d.),
2909:
2907:
2834:Practical Regression and Anova using R
387:minus the number of fitted parameters
2865:. Chapman&Hall/CRC. p. 626.
802:{\displaystyle \chi _{\nu }^{2}<1}
712:{\displaystyle \chi _{\nu }^{2}>1}
674:{\displaystyle \chi _{\nu }^{2}\gg 1}
55:make it understandable to non-experts
7:
3156:Statistical deviation and dispersion
2778:Strang, Gilbert; Borre, Kae (1997).
383:, equals the number of observations
2846:Kenney, J.; Keeping, E. S. (1963).
555:
552:
549:
520:
517:
514:
432:
25:
2950:An introduction to error analysis
2781:Linear algebra, geodesy, and GPS
744:{\displaystyle \chi _{\nu }^{2}}
620:{\displaystyle \chi _{\nu }^{2}}
486:, the definition simplifies to:
467:is the vector of residuals, and
126:standard error of the regression
34:
3151:Geochronological dating methods
958:{\displaystyle \sigma _{x_{i}}}
846:MSWD = 1 if the age data fit a
336:{\displaystyle \sigma _{i}^{2}}
99:mean squared weighted deviation
3127:Rasch Measurement Transactions
2890:. Princeton University Press.
2745:In data analysis based on the
2713:
2692:
2681:
2647:
2409:
2382:
2010:
1976:
1493:
1459:
1256:
1229:
1120:
1093:
848:univariate normal distribution
681:indicates a poor model fit. A
271:
244:
130:standard error of the equation
18:Mean square weighted deviation
1:
2989:Bevington, Philip R. (1969),
809:indicates that the model is "
97:testing. It is also known as
2948:Taylor, John Robert (1997),
2850:. van Nostrand. p. 187.
2669:
2403:
1998:
1481:
1296:
1250:
1114:
981:
91:reduced chi-square statistic
588:where the numerator is the
3172:
2805:Koch, Karl-Rudolf (2013).
120:Its square root is called
3097:10.1144/0016-76492008-117
2848:Mathematics of Statistics
1551:which can be computed as
477:generalized least squares
351:. The degree of freedom,
122:regression standard error
2966:Chi-Square Curve Fitting
931:and an associated error
376:{\displaystyle \nu =n-m}
3121:Linacre, J. M. (2002).
2993:, New York: McGraw-Hill
2963:Kirkman, T. W. (n.d.),
2886:Hayashi, Fumio (2000).
2861:Zwillinger, D. (1995).
2832:Julian Faraway (2000),
590:residual sum of squares
111:variance of unit weight
93:is used extensively in
2732:
2633:
2591:
2543:
2503:
2450:
2378:
2307:
2281:
2233:
2169:
2128:
2079:
2027:
1965:
1922:
1874:
1834:
1783:
1748:
1684:
1643:
1594:
1545:
1524:
1448:
1397:
1376:
1333:
1272:
1228:
1092:
1036:
1012:
959:
925:
898:
803:
765:
745:
713:
675:
621:
582:
536:
484:ordinary least squares
475:is non-diagonal, then
457:
396:weighted least squares
377:
347:, and calculated data
337:
302:
201:
115:weighted least squares
2733:
2613:
2571:
2523:
2483:
2451:
2358:
2308:
2261:
2213:
2149:
2108:
2059:
2028:
1945:
1902:
1854:
1814:
1784:
1728:
1664:
1623:
1574:
1546:
1504:
1428:
1398:
1356:
1313:
1273:
1208:
1072:
1037:
992:
960:
926:
924:{\displaystyle w_{i}}
899:
897:{\displaystyle x_{i}}
804:
766:
746:
714:
676:
622:
583:
537:
458:
378:
338:
303:
202:
2462:
2319:
2040:
1795:
1555:
1409:
1287:
1053:
973:
935:
908:
881:
775:
755:
723:
685:
647:
599:
545:
490:
402:
355:
315:
215:
156:
2606:
2440:
2296:
2104:
1937:
1619:
904:having a weighting
792:
740:
702:
664:
616:
507:
419:
332:
295:
173:
2728:
2592:
2446:
2419:
2303:
2282:
2090:
2023:
1923:
1779:
1605:
1541:
1393:
1268:
1032:
955:
921:
894:
799:
778:
761:
741:
726:
709:
688:
671:
650:
629:standard deviation
627:equals the sample
617:
602:
578:
532:
493:
453:
405:
373:
333:
318:
298:
281:
240:
197:
159:
113:in the context of
2930:on 6 October 2016
2723:
2672:
2608:
2469:
2441:
2406:
2353:
2326:
2298:
2001:
1939:
1774:
1536:
1484:
1388:
1299:
1253:
1203:
1169:
1138:
1133:
1117:
1027:
984:
843:In general when:
764:{\displaystyle 1}
527:
448:
296:
231:
192:
150:degree of freedom
144:It is defined as
83:
82:
75:
16:(Redirected from
3163:
3135:
3134:
3118:
3112:
3105:
3099:
3088:
3082:
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3069:
3063:
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3017:
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2994:
2986:
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2911:
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701:
696:
680:
678:
677:
672:
663:
658:
626:
624:
623:
618:
615:
610:
587:
585:
584:
579:
574:
573:
558:
541:
539:
538:
533:
528:
523:
512:
506:
501:
462:
460:
459:
454:
449:
444:
437:
436:
435:
424:
418:
413:
382:
380:
379:
374:
342:
340:
339:
334:
331:
326:
307:
305:
304:
299:
297:
294:
289:
280:
279:
278:
269:
268:
256:
255:
242:
239:
227:
226:
206:
204:
203:
198:
193:
188:
187:
178:
172:
167:
78:
71:
67:
64:
58:
38:
37:
30:
21:
3171:
3170:
3166:
3165:
3164:
3162:
3161:
3160:
3141:
3140:
3139:
3138:
3120:
3119:
3115:
3106:
3102:
3089:
3085:
3079:
3075:
3070:
3066:
3046:
3042:
3035:
3031:
3024:
3020:
3013:
3009:
3002:
2998:
2988:
2987:
2980:
2971:
2969:
2962:
2961:
2957:
2947:
2946:
2942:
2933:
2931:
2927:
2920:
2913:
2912:
2905:
2898:
2885:
2884:
2880:
2873:
2860:
2859:
2855:
2845:
2844:
2840:
2830:
2826:
2819:
2804:
2803:
2799:
2792:
2777:
2776:
2772:
2767:
2760:
2755:
2743:
2712:
2700:
2695:
2691:
2680:
2663:
2650:
2637:
2636:
2554:
2544:
2515:
2504:
2482:
2465:
2460:
2459:
2424:
2408:
2385:
2381:
2342:
2322:
2317:
2316:
2244:
2234:
2205:
2190:
2180:
2170:
2129:
2080:
2058:
2043:
2038:
2037:
2009:
1992:
1979:
1966:
1885:
1875:
1846:
1835:
1813:
1798:
1793:
1792:
1759:
1749:
1720:
1705:
1695:
1685:
1644:
1595:
1573:
1558:
1553:
1552:
1525:
1503:
1492:
1475:
1462:
1449:
1427:
1412:
1407:
1406:
1377:
1355:
1344:
1334:
1312:
1290:
1285:
1284:
1255:
1232:
1192:
1174:
1158:
1140:
1137: and
1119:
1096:
1071:
1056:
1051:
1050:
1013:
991:
971:
970:
943:
938:
933:
932:
911:
906:
905:
884:
879:
878:
856:arithmetic mean
834:
829:
773:
772:
753:
752:
721:
720:
683:
682:
645:
644:
637:
597:
596:
565:
543:
542:
488:
487:
426:
425:
400:
399:
353:
352:
343:, observations
313:
312:
270:
260:
247:
243:
218:
213:
212:
179:
154:
153:
142:
107:isotopic dating
95:goodness of fit
79:
68:
62:
59:
51:help improve it
48:
39:
35:
28:
23:
22:
15:
12:
11:
5:
3169:
3167:
3159:
3158:
3153:
3143:
3142:
3137:
3136:
3113:
3107:Roger Powell,
3100:
3083:
3073:
3064:
3040:
3029:
3018:
3007:
2996:
2978:
2955:
2940:
2903:
2896:
2878:
2871:
2853:
2838:
2824:
2817:
2797:
2790:
2770:
2757:
2756:
2754:
2751:
2742:
2741:Rasch analysis
2739:
2727:
2719:
2715:
2707:
2703:
2698:
2694:
2687:
2683:
2677:
2671:
2668:
2662:
2657:
2653:
2649:
2644:
2640:
2631:
2626:
2623:
2620:
2616:
2612:
2604:
2599:
2595:
2589:
2584:
2581:
2578:
2574:
2570:
2565:
2559:
2551:
2547:
2541:
2536:
2533:
2530:
2526:
2520:
2511:
2507:
2501:
2496:
2493:
2490:
2486:
2479:
2474:
2445:
2438:
2431:
2427:
2422:
2415:
2411:
2405:
2402:
2397:
2392:
2388:
2384:
2376:
2371:
2368:
2365:
2361:
2357:
2351:
2348:
2345:
2341:
2336:
2331:
2302:
2294:
2289:
2285:
2279:
2274:
2271:
2268:
2264:
2260:
2255:
2249:
2241:
2237:
2231:
2226:
2223:
2220:
2216:
2210:
2201:
2195:
2187:
2183:
2177:
2173:
2167:
2162:
2159:
2156:
2152:
2146:
2141:
2136:
2132:
2126:
2121:
2118:
2115:
2111:
2107:
2102:
2097:
2093:
2087:
2083:
2077:
2072:
2069:
2066:
2062:
2055:
2050:
2046:
2022:
2016:
2012:
2006:
2000:
1997:
1991:
1986:
1982:
1978:
1973:
1969:
1963:
1958:
1955:
1952:
1948:
1943:
1935:
1930:
1926:
1920:
1915:
1912:
1909:
1905:
1901:
1896:
1890:
1882:
1878:
1872:
1867:
1864:
1861:
1857:
1851:
1842:
1838:
1832:
1827:
1824:
1821:
1817:
1810:
1805:
1801:
1778:
1770:
1764:
1756:
1752:
1746:
1741:
1738:
1735:
1731:
1725:
1716:
1710:
1702:
1698:
1692:
1688:
1682:
1677:
1674:
1671:
1667:
1661:
1656:
1651:
1647:
1641:
1636:
1633:
1630:
1626:
1622:
1617:
1612:
1608:
1602:
1598:
1592:
1587:
1584:
1581:
1577:
1570:
1565:
1561:
1540:
1532:
1528:
1522:
1517:
1514:
1511:
1507:
1499:
1495:
1489:
1483:
1480:
1474:
1469:
1465:
1461:
1456:
1452:
1446:
1441:
1438:
1435:
1431:
1424:
1419:
1415:
1392:
1384:
1380:
1374:
1369:
1366:
1363:
1359:
1351:
1347:
1341:
1337:
1331:
1326:
1323:
1320:
1316:
1309:
1304:
1298:
1295:
1267:
1262:
1258:
1252:
1249:
1244:
1239:
1235:
1231:
1226:
1221:
1218:
1215:
1211:
1207:
1201:
1198:
1195:
1191:
1186:
1181:
1177:
1173:
1167:
1164:
1161:
1157:
1152:
1147:
1143:
1132:
1126:
1122:
1116:
1113:
1108:
1103:
1099:
1095:
1090:
1085:
1082:
1079:
1075:
1068:
1063:
1059:
1031:
1026:
1020:
1016:
1010:
1005:
1002:
999:
995:
988:
983:
980:
950:
946:
941:
918:
914:
891:
887:
864:geometric mean
833:
830:
828:
825:
798:
795:
790:
785:
781:
760:
738:
733:
729:
708:
705:
700:
695:
691:
670:
667:
662:
657:
653:
636:
633:
614:
609:
605:
577:
572:
568:
564:
561:
557:
554:
551:
531:
526:
522:
519:
516:
510:
505:
500:
496:
452:
447:
443:
440:
434:
429:
422:
417:
412:
408:
372:
369:
366:
363:
360:
330:
325:
321:
293:
288:
284:
277:
273:
267:
263:
259:
254:
250:
246:
238:
234:
230:
225:
221:
196:
191:
186:
182:
176:
171:
166:
162:
141:
138:
81:
80:
42:
40:
33:
27:Test statistic
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3168:
3157:
3154:
3152:
3149:
3148:
3146:
3132:
3128:
3124:
3117:
3114:
3110:
3104:
3101:
3098:
3094:
3087:
3084:
3077:
3074:
3068:
3065:
3062:
3061:0-521-59891-5
3058:
3054:
3053:0-521-43151-4
3050:
3044:
3041:
3038:
3033:
3030:
3027:
3022:
3019:
3016:
3011:
3008:
3005:
3000:
2997:
2992:
2985:
2983:
2979:
2968:
2967:
2959:
2956:
2951:
2944:
2941:
2926:
2919:
2918:
2910:
2908:
2904:
2899:
2897:0-691-01018-8
2893:
2889:
2882:
2879:
2874:
2872:0-8493-2479-3
2868:
2864:
2857:
2854:
2849:
2842:
2839:
2836:
2835:
2828:
2825:
2820:
2818:9783662039762
2814:
2810:
2809:
2801:
2798:
2793:
2791:9780961408862
2787:
2783:
2782:
2774:
2771:
2765:
2763:
2759:
2752:
2750:
2748:
2740:
2738:
2725:
2717:
2705:
2701:
2696:
2685:
2675:
2666:
2660:
2655:
2651:
2642:
2638:
2629:
2624:
2621:
2618:
2614:
2610:
2602:
2597:
2593:
2587:
2582:
2579:
2576:
2572:
2568:
2563:
2549:
2545:
2539:
2534:
2531:
2528:
2524:
2509:
2505:
2499:
2494:
2491:
2488:
2484:
2477:
2472:
2456:
2443:
2436:
2429:
2425:
2420:
2413:
2400:
2395:
2390:
2386:
2374:
2369:
2366:
2363:
2359:
2355:
2349:
2346:
2343:
2339:
2334:
2329:
2313:
2300:
2292:
2287:
2283:
2277:
2272:
2269:
2266:
2262:
2258:
2253:
2239:
2235:
2229:
2224:
2221:
2218:
2214:
2199:
2185:
2181:
2175:
2171:
2165:
2160:
2157:
2154:
2150:
2139:
2134:
2130:
2124:
2119:
2116:
2113:
2109:
2105:
2100:
2095:
2091:
2085:
2081:
2075:
2070:
2067:
2064:
2060:
2053:
2048:
2044:
2034:
2020:
2014:
2004:
1995:
1989:
1984:
1980:
1971:
1967:
1961:
1956:
1953:
1950:
1946:
1941:
1933:
1928:
1924:
1918:
1913:
1910:
1907:
1903:
1899:
1894:
1880:
1876:
1870:
1865:
1862:
1859:
1855:
1840:
1836:
1830:
1825:
1822:
1819:
1815:
1808:
1803:
1799:
1789:
1776:
1768:
1754:
1750:
1744:
1739:
1736:
1733:
1729:
1714:
1700:
1696:
1690:
1686:
1680:
1675:
1672:
1669:
1665:
1654:
1649:
1645:
1639:
1634:
1631:
1628:
1624:
1620:
1615:
1610:
1606:
1600:
1596:
1590:
1585:
1582:
1579:
1575:
1568:
1563:
1559:
1538:
1530:
1526:
1520:
1515:
1512:
1509:
1505:
1497:
1487:
1478:
1472:
1467:
1463:
1454:
1450:
1444:
1439:
1436:
1433:
1429:
1422:
1417:
1413:
1403:
1390:
1382:
1378:
1372:
1367:
1364:
1361:
1357:
1349:
1345:
1339:
1335:
1329:
1324:
1321:
1318:
1314:
1307:
1302:
1293:
1281:
1278:
1265:
1260:
1247:
1242:
1237:
1233:
1224:
1219:
1216:
1213:
1209:
1205:
1199:
1196:
1193:
1189:
1184:
1179:
1175:
1171:
1165:
1162:
1159:
1155:
1150:
1145:
1141:
1130:
1124:
1111:
1106:
1101:
1097:
1088:
1083:
1080:
1077:
1073:
1066:
1061:
1057:
1048:
1043:
1029:
1024:
1018:
1014:
1008:
1003:
1000:
997:
993:
986:
978:
967:
948:
944:
939:
916:
912:
889:
885:
875:
871:
867:
865:
861:
857:
853:
849:
844:
841:
839:
838:geochronology
832:Geochronology
831:
826:
824:
822:
821:
814:
812:
796:
793:
788:
783:
779:
758:
736:
731:
727:
706:
703:
698:
693:
689:
668:
665:
660:
655:
651:
642:
634:
632:
630:
612:
607:
603:
593:
591:
575:
570:
566:
562:
559:
529:
524:
508:
503:
498:
494:
485:
480:
478:
474:
470:
466:
450:
445:
441:
438:
427:
420:
415:
410:
406:
397:
392:
390:
386:
370:
367:
364:
361:
358:
350:
346:
328:
323:
319:
311:
308:with inputs:
291:
286:
282:
275:
265:
261:
257:
252:
248:
236:
232:
228:
223:
219:
210:
194:
189:
184:
180:
174:
169:
164:
160:
151:
147:
139:
137:
135:
131:
127:
123:
118:
116:
112:
108:
104:
100:
96:
92:
88:
77:
74:
66:
56:
52:
46:
43:This article
41:
32:
31:
19:
3130:
3126:
3116:
3103:
3086:
3076:
3067:
3043:
3032:
3021:
3010:
2999:
2990:
2970:, retrieved
2965:
2958:
2949:
2943:
2932:, retrieved
2925:the original
2916:
2888:Econometrics
2887:
2881:
2862:
2856:
2847:
2841:
2833:
2827:
2807:
2800:
2780:
2773:
2744:
2457:
2314:
2035:
1790:
1404:
1282:
1279:
1044:
968:
876:
872:
868:
859:
858:age) or log(
851:
845:
842:
835:
827:Applications
820:a posteriori
819:
815:
640:
638:
594:
481:
472:
468:
464:
393:
388:
384:
348:
344:
143:
129:
125:
121:
119:
110:
102:
98:
90:
84:
69:
60:
44:
3109:Janet Hergt
2747:Rasch model
862:) (for the
811:overfitting
3145:Categories
2753:References
635:Discussion
209:deviations
146:chi-square
140:Definition
87:statistics
63:April 2021
3133:(2): 878.
2697:σ
2676:∗
2670:¯
2661:−
2615:∑
2611:⋅
2573:∑
2569:−
2525:∑
2485:∑
2421:σ
2404:¯
2396:−
2360:∑
2356:⋅
2347:−
2263:∑
2259:−
2215:∑
2151:∑
2140:−
2110:∑
2106:⋅
2061:∑
2005:∗
1999:¯
1990:−
1947:∑
1942:⋅
1904:∑
1900:−
1856:∑
1816:∑
1730:∑
1666:∑
1655:−
1625:∑
1621:⋅
1576:∑
1560:σ
1506:∑
1488:∗
1482:¯
1473:−
1430:∑
1414:σ
1358:∑
1315:∑
1303:∗
1297:¯
1251:¯
1243:−
1210:∑
1206:⋅
1197:−
1176:σ
1172:⋅
1163:−
1115:¯
1107:−
1074:∑
1058:σ
994:∑
982:¯
940:σ
854:(for the
784:ν
780:χ
732:ν
728:χ
694:ν
690:χ
666:≫
656:ν
652:χ
608:ν
604:χ
563:∑
525:ν
499:ν
495:χ
479:applies.
446:ν
411:ν
407:χ
368:−
359:ν
320:σ
283:σ
258:−
233:∑
220:χ
190:ν
181:χ
165:ν
161:χ
3081:155–170.
751:around
641:a priori
310:variance
592:(RSS).
49:Please
3059:
3051:
2972:30 May
2934:30 May
2894:
2869:
2815:
2788:
1047:sample
463:where
89:, the
2928:(PDF)
2921:(PDF)
132:(see
128:, or
105:) in
3057:ISBN
3049:ISBN
2974:2015
2936:2015
2892:ISBN
2867:ISBN
2813:ISBN
2786:ISBN
2468:MSWD
2325:MSWD
794:<
704:>
643:, a
148:per
109:and
103:MSWD
3093:doi
850:in
836:In
482:In
394:In
85:In
53:to
3147::
3131:16
3129:.
3125:.
3055:,
2981:^
2906:^
2761:^
823:.
631:.
391:.
211::
152::
136:)
124:,
117:.
3095::
2900:.
2875:.
2821:.
2794:.
2726:.
2718:2
2714:)
2706:i
2702:x
2693:(
2686:2
2682:)
2667:x
2656:i
2652:x
2648:(
2643:i
2639:w
2630:N
2625:1
2622:=
2619:i
2603:2
2598:i
2594:w
2588:N
2583:1
2580:=
2577:i
2564:2
2558:)
2550:i
2546:w
2540:N
2535:1
2532:=
2529:i
2519:(
2510:i
2506:w
2500:N
2495:1
2492:=
2489:i
2478:=
2473:w
2444:.
2437:2
2430:i
2426:x
2414:2
2410:)
2401:x
2391:i
2387:x
2383:(
2375:N
2370:1
2367:=
2364:i
2350:1
2344:N
2340:1
2335:=
2330:u
2301:.
2293:2
2288:i
2284:w
2278:N
2273:1
2270:=
2267:i
2254:2
2248:)
2240:i
2236:w
2230:N
2225:1
2222:=
2219:i
2209:(
2200:2
2194:)
2186:i
2182:x
2176:i
2172:w
2166:N
2161:1
2158:=
2155:i
2145:(
2135:i
2131:w
2125:N
2120:1
2117:=
2114:i
2101:2
2096:i
2092:x
2086:i
2082:w
2076:N
2071:1
2068:=
2065:i
2054:=
2049:2
2045:s
2021:.
2015:2
2011:)
1996:x
1985:i
1981:x
1977:(
1972:i
1968:w
1962:N
1957:1
1954:=
1951:i
1934:2
1929:i
1925:w
1919:N
1914:1
1911:=
1908:i
1895:2
1889:)
1881:i
1877:w
1871:N
1866:1
1863:=
1860:i
1850:(
1841:i
1837:w
1831:N
1826:1
1823:=
1820:i
1809:=
1804:2
1800:s
1777:.
1769:2
1763:)
1755:i
1751:w
1745:N
1740:1
1737:=
1734:i
1724:(
1715:2
1709:)
1701:i
1697:x
1691:i
1687:w
1681:N
1676:1
1673:=
1670:i
1660:(
1650:i
1646:w
1640:N
1635:1
1632:=
1629:i
1616:2
1611:i
1607:x
1601:i
1597:w
1591:N
1586:1
1583:=
1580:i
1569:=
1564:2
1539:,
1531:i
1527:w
1521:N
1516:1
1513:=
1510:i
1498:2
1494:)
1479:x
1468:i
1464:x
1460:(
1455:i
1451:w
1445:N
1440:1
1437:=
1434:i
1423:=
1418:2
1391:.
1383:i
1379:w
1373:N
1368:1
1365:=
1362:i
1350:i
1346:x
1340:i
1336:w
1330:N
1325:1
1322:=
1319:i
1308:=
1294:x
1266:.
1261:2
1257:)
1248:x
1238:i
1234:x
1230:(
1225:N
1220:1
1217:=
1214:i
1200:1
1194:N
1190:1
1185:=
1180:2
1166:1
1160:N
1156:N
1151:=
1146:2
1142:s
1131:N
1125:2
1121:)
1112:x
1102:i
1098:x
1094:(
1089:N
1084:1
1081:=
1078:i
1067:=
1062:2
1030:,
1025:N
1019:i
1015:x
1009:N
1004:1
1001:=
998:i
987:=
979:x
949:i
945:x
917:i
913:w
890:i
886:x
860:t
852:t
797:1
789:2
759:1
737:2
707:1
699:2
669:1
661:2
613:2
576:,
571:2
567:r
560:=
556:S
553:S
550:R
530:,
521:S
518:S
515:R
509:=
504:2
473:W
469:W
465:r
451:,
442:r
439:W
433:T
428:r
421:=
416:2
389:m
385:n
371:m
365:n
362:=
349:C
345:O
329:2
324:i
292:2
287:i
276:2
272:)
266:i
262:C
253:i
249:O
245:(
237:i
229:=
224:2
195:,
185:2
175:=
170:2
101:(
76:)
70:(
65:)
61:(
47:.
20:)
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