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Table 1 Evaluation criteria for models for which MLEs were found with 2-parameter symmetrical protection curves fitted to illustrative datasets

From: Some extensions in continuous models for immunological correlates of protection

−2×log-likelihood (rank)       
Goodness-of-fit (rank)       
Coef. of Var. \( \widehat{\lambda} \) (rank) Protection curve/function
Dataset (cases of disease:subjects) Error function Logistic function Square root sigmoid Double exponential Arctangent function Absolute sigmoid function
German pertussis FHA IgG (44:1988) 375.441 (6) 0.1799 (1) 0.4365 (6) 375.247 (5) 0.1645 (3) 0.3247 (5) 374.949 (4) 0.1427 (4) 0.2234 (4) 374.393 (3) 0.1775 (2) 0.2072 (3) 374.384 (2) 0.1302 (6) 0.1708 (2) 373.629 (1) 0.1329 (5) 0.1552 (1)
German pertussis PT IgG (44:1987) 379.477 (6) 0.1891 (6) 2.1025 (6) 378.741 (5) 0.1906 (5) 0.5796 (5) 376.344 (3) 0.2837 (3) 0.2250 (3) 377.239 (4) 0.2622 (4) 0.2296 (4) 373.950 (2) 0.5100 (2) 0.1748 (2) 373.595 (1) 0.5233 (1) 0.1530 (1)
German pertussis PRN IgG (44:1992) 381.059 (2) 0.8700 (1) 0.4318 (6) 381.081 (4) 0.8619 (2) 0.3044 (5) 381.124 (6) 0.8263 (4) 0.1855 (4) 381.028 (1) 0.8560 (3) 0.1819 (3) 381.080 (3) 0.8141 (5) 0.1719 (2) 381.087 (5) 0.7829 (6) 0.1514 (1)
German pertussis FIM IgG (44:1986) 376.847 (6) 0.8273 (1) 0.5638 (6) 376.683 (5) 0.8039 (3) 0.4063 (5) 376.553 (3) 0.7381 (4) 0.2784 (4) 376.080 (1) 0.8222 (2) 0.2771 (3) 376.640 (4) 0.7028 (5) 0.2288 (2) 376.518 (2) 0.6791 (6) 0.1936 (1)
German pertussis FHA IgA (44:1932) 418.378 (4) 0.7157 (4) 0.1317 (1) - 417.617 (3) 0.7216 (3) 0.1490 (4) - 417.604 (2) 0.7217 (1) 0.1490 (3) 417.603 (1) 0.7217 (2) 0.1490 (2)
German pertussis PT IgA (44:1933) 418.782 (5) 0.5103 (1) 0.2860 (4) 418.784 (6) 0.5099 (2) 0.3207 (5) 418.101 (3) 0.4777 (6) 0.1490 (3) 418.751 (4) 0.5042 (3) 0.3621 (6) 418.076 (2) 0.4781 (5) 0.1490 (2) 418.075 (1) 0.4781 (4) 0.1490 (1)
German pertussis PRN IgA (44:1968) - - 406.932 (3) 0.2489 (3) 0.1490 (3) - 406.479 (2) 0.2528 (2) 0.1489 (2) 406.432 (1) 0.2532 (1) 0.1488 (1)
German pertussis FIM IgA (44:1994) 410.398 (6) 0.0107 (5) 0.8372 (6) 410.361 (5) 0.0113 (3) 0.7657 (5) 410.349 (4) 0.0118 (2) 0.5205 (4) 409.896 (3) 0.0109 (4) 0.2282 (3) 408.899 (2) 0.0063 (6) 0.1776 (2) 407.798 (1) 0.0196 (1) 0.1721 (1)
Piedra RSV/A (34:175) - - 159.291 (3) 0.6080 (4) 0.9513 (4) 159.477 (4) 0.6156 (3) 0.4410 (3) 158.612 (2) 0.6694 (2) 0.2956 (2) 157.895 (1) 0.7361 (1) 0.2119 (1)
Piedra RSV/B (34:175) - - 154.415 (2) 0.6934 (3) 0.7328 (3) - 154.395 (1) 0.7439 (2) 0.3734 (2) 154.553 (3) 0.7591 (1) 0.2643 (1)
White/varicella (79:3459) 643.035 (4) 0.9848 (5) 0.2887 (6) 642.145 (3) 0.9987 (3) 0.2297 (5) 641.683 (1) 1.0000 (1) 0.1802 (3) 641.765 (2) 1.0000 (2) 0.2000 (4) 643.060 (5) 0.9934 (4) 0.1403 (2) 643.507 (6) 0.9841 (6) 0.1314 (1)
Swedish pertussis FHA IgG (92:209) - - 284.404 (1) 0.0100 (1) 0.0777 (1) - - -
Swedish pertussis PT IgG (92:209) 270.365 (6) 0.9223 (6) 0.1577 (4) 270.205 (5) 0.9274 (5) 0.1587 (5) 269.832 (4) 0.9379 (4) 0.1483 (3) 269.757 (3) 0.9407 (3) 0.1938 (6) 269.374 (2) 0.9529 (2) 0.1341 (2) 268.871 (1) 0.9754 (1) 0.1216 (1)
Swedish pertussis PRN IgG (92:209) 247.593 (5) 0.7605 (6) 0.0807 (4) 247.621 (6) 0.7700 (5) 0.0813 (5) 247.571 (4) 0.8123 (1) 0.0813 (6) 247.304 (3) 0.7736 (4) 0.0791 (3) 246.989 (2) 0.7884 (2) 0.0772 (2) 246.071 (1) 0.7863 (3) 0.0763 (1)
Swedish pertussis FIM IgG (92:209) 251.075 (6) 0.2775 (6) 0.3644 (6) 250.822 (5) 0.2921 (5) 0.2708 (5) 250.161 (4) 0.3250 (4) 0.1697 (4) 249.765 (3) 0.3530 (3) 0.1470 (3) 248.902 (2) 0.3691 (2) 0.1103 (2) 247.731 (1) 0.4765 (1) 0.0941 (1)
Black Nicolay HAI (22:777) 181.035 (6) 0.5029 (6) 0.2875 (6) 180.649 (5) 0.5241 (4) 0.2661 (5) 179.402 (3) 0.5209 (5) 0.2267 (3) 179.675 (4) 0.5886 (2) 0.2329 (4) 177.247 (2) 0.5479 (3) 0.2122 (2) 176.743 (1) 0.6813 (1) 0.2120 (1)