Estimating linear-by-linear association has long been an important topic in the analysis of contingency tables. For ordinal variables, log-linear models may be used to detect the strength and magnitude of the association between such variables, and iterative procedures are traditionally used. Recently, studies have shown, by way of example, three non-iterative techniques can be used to quickly and accurately estimate the parameter. This paper provides a computational study of these procedures, and the results show that they are extremely accurate when compared with estimates obtained using Newton’s unidimensional method.
During the 1970s and 1980s parameter estimation procedures for ordinal log-linear and ordinal association models gained considerable attention in the literature. For example, Haberman [
A different approach to estimating parameters from ordinal log-linear (or association) models is to consider the direct, iterative-free, estimation procedure originally proposed by Beh and Davy [
By performing a computational study, this paper will explore the accuracy and reliability of Newton’s method and four noniterative procedures on randomly generated contingency tables with a fixed association parameter. Such a study will be made by first briefly describing the ordinal log-linear model (Section
Suppose we consider an
see Beh and Davy [
The relationship between the two-ordered categorical variables may be determined by estimating the parameter
Therefore, if equidistant scores are chosen to reflect the ordered structure of the variables so that
If one considers the log-linear model (
There are a number of iterative procedures that may be used for estimating
Suppose that each of the cell frequencies of a contingency table
The computational study considered in this paper evaluates four non-iterative procedures for estimating
An alternative approach to approximating the natural logarithm present in the
which takes into account that
may also be considered. By adopting these two approximations of the logarithm function, two estimates of
Beh and Farver [
This standard error provides an indication of the maximum standard error that one would obtain for a contingency table with sample size
In order to study the accuracy and reliability of Newton’s method and the four non-iterative procedures described above, we consider a simple procedure for generating a two-way contingency table of dimension
respectively, ensuring that
respectively. For this computational study, natural row scores,
where
It is apparent that the
The computational study performed in this paper is based on random contingency tables (using the procedure described in Section
Tables
Mean estimate (and standard error) of
Size of contingency table | ||||||||||
True | 2 × 2 | 2 × 3 | 2 × 4 | 2 × 5 | 3 × 3 | 3 × 4 | 3 × 5 | 4 × 4 | 4 × 5 | 5 × 5 |
0.00 | −0.00068 (0.2014) | 0.00020 (0.1160) | 0.00235 (0.0829) | 0.00042 (0.0583) | −0.00023 (0.0611) | −0.00029 (0.0488) | −0.00007 (0.0346) | 0.00008 (0.0312) | 0.00004 (0.0239) | 0.00005 (0.0178) |
0.01 | 0.01617 (0.2765) | 0.00740 (0.1237) | 0.01503 (0.0792) | 0.00868 (0.0631) | 0.01040 (0.0631) | 0.01016 (0.0446) | 0.01012 (0.0344) | 0.01020 (0.0307) | 0.01002 (0.0241) | 0.01000 (0.0179) |
0.05 | 0.05454 (0.2188) | 0.05134 (0.1194) | 0.05046 (0.0824) | 0.04911 (0.0621) | 0.04964 (0.0615) | 0.05009 (0.0424) | 0.05021 (0.0338) | 0.04997 (0.0301) | 0.05001 (0.0236) | 0.05009 (0.0183) |
0.10 | 0.10278 (0.2165) | 0.09998 (0.1089) | 0.10087 (0.0775) | 0.10061 (0.0608) | 0.09994 (0.0690) | 0.09990 (0.0436) | 0.09980 (0.0341) | 0.10006 (0.0301) | 0.10024 (0.0234) | 0.10001 (0.0181) |
0.20 | 0.20473 (0.2048) | 0.20097 (0.1149) | 0.19999 (0.0770) | 0.20010 (0.0580) | 0.19969 (0.0605) | 0.19970 (0.0431) | 0.20021 (0.0317) | 0.19995 (0.0290) | 0.20008 (0.0223) | 0.19995 (0.0165) |
0.30 | 0.30392 (0.2016) | 0.30622 (0.1130) | 0.30312 (0.0770) | 0.30040 (0.0558) | 0.30019 (0.0616) | 0.30022 (0.0404) | 0.30008 (0.0314) | 0.30007 (0.0278) | 0.29995 (0.0202) | 0.30001 (0.0149) |
0.40 | 0.40491 (0.2295) | 0.39656 (0.1170) | 0.40184 (0.0771) | 0.40082 (0.0629) | 0.40079 (0.0649) | 0.40080 (0.0412) | 0.40030 (0.0295) | 0.40001 (0.0263) | 0.40015 (0.0187) | 0.40036 (0.0129) |
0.50 | 0.50762 (0.2353) | 0.50533 (0.1164) | 0.50139 (0.0869) | 0.50326 (0.0564) | 0.49981 (0.0591) | 0.49788 (0.0422) | 0.50018 (0.0284) | 0.50030 (0.0249) | 0.50022 (0.0167) | 0.50025 (0.0113) |
0.60 | 0.61482 (0.2229) | 0.60065 (0.1102) | 0.60149 (0.0751) | 0.60076 (0.0561) | 0.59980 (0.0577) | 0.59945 (0.0373) | 0.59976 (0.0272) | 0.60012 (0.0230) | 0.59999 (0.0156) | 0.60015 (0.0096) |
0.70 | 0.71150 (0.2131) | 0.70272 (0.1175) | 0.69969 (0.0690) | 0.70033 (0.0524) | 0.69953 (0.0547) | 0.70031 (0.0359) | 0.70049 (0.0250) | 0.70026 (0.0209) | 0.70015 (0.0139) | 0.69996 (0.0078) |
0.80 | 0.80482 (0.2773) | 0.80215 (0.1081) | 0.79924 (0.0785) | 0.79978 (0.0485) | 0.79985 (0.0524) | 0.80013 (0.0345) | 0.80002 (0.0226) | 0.80009 (0.0197) | 0.79995 (0.0117) | 0.80038 (0.0066) |
0.90 | 0.89570 (0.2133) | 0.89081 (0.1033) | 0.90338 (0.0687) | 0.90103 (0.0513) | 0.89998 (0.0519) | 0.90018 (0.0314) | 0.90058 (0.0214) | 0.89980 (0.0174) | 0.90041 (0.0102) | 0.90025 (0.0055) |
1.00 | 1.01856 (0.2358) | 0.99917 (0.1074) | 0.99251 (0.0804) | 1.00302 (0.0504) | 0.99954 (0.0517) | 0.99992 (0.0301) | 1.00014 (0.0188) | 0.99983 (0.0167) | 0.99987 (0.0091) | 0.99659 (0.0045) |
1.10 | 1.10279 (0.2169) | 1.10185 (0.1187) | 1.10167 (0.0646) | 1.10041 (0.0472) | 1.10064 (0.0475) | 1.10020 (0.0282) | 1.09998 (0.0174) | 1.10030 (0.0146) | 1.10028 (0.0076) | 1.09676 (0.0035) |
1.20 | 1.20637 (0.2047) | 1.20018 (0.1057) | 1.20161 (0.0626) | 1.20078 (0.0430) | 1.20001 (0.0444) | 1.20077 (0.0260) | 1.19968 (0.0160) | 1.19960 (0.0132) | 1.19975 (0.0066) | 1.19886 (0.0030) |
1.30 | 1.32157 (0.2372) | 1.30192 (0.1100) | 1.30332 (0.0551) | 1.30048 (0.0396) | 1.30092 (0.0452) | 1.30039 (0.0241) | 1.29982 (0.0143) | 1.30007 (0.0121) | 1.30016 (0.0057) | 1.28541 (0.0023) |
1.40 | 1.41818 (0.2433) | 1.40873 (0.0898) | 1.40110 (0.0564) | 1.39902 (0.0395) | 1.40079 (0.0423) | 1.40040 (0.0224) | 1.40003 (0.0132) | 1.40001 (0.0106) | 1.39921 (0.0051) | 1.38124 (0.0021) |
1.50 | 1.49777 (0.1907) | 1.50079 (0.0939) | 1.50105 (0.0531) | 1.50184 (0.0384) | 1.49974 (0.0427) | 1.49900 (0.0223) | 1.50036 (0.0118) | 1.50045 (0.0091) | 1.49371 (0.0045) | 1.47004 (0.0017) |
Mean estimate (and standard error) of
Size of contingency table | ||||||||||
True | 2 × 2 | 2 × 3 | 2 × 4 | 2 × 5 | 3 × 3 | 3 × 4 | 3 × 5 | 4 × 4 | 4 × 5 | 5 × 5 |
0.00 | −0.00068 (0.2014) | −0.00026 (0.1161) | 0.00350 (0.0828) | 0.00019 (0.0583) | −0.00023 (0.0611) | −0.00030 (0.0448) | −0.00003 (0.0346) | 0.00002 (0.0312) | 0.00006 (0.0239) | 0.00005 (0.0178) |
0.01 | 0.01618 (0.2760) | 0.00562 (0.1233) | 0.01509 (0.0783) | 0.00810 (0.0629) | 0.01028 (0.0631) | 0.01024 (0.0446) | 0.01012 (0.0344) | 0.01021 (0.0307) | 0.01005 (0.0241) | 0.01001 (0.0179) |
0.05 | 0.05454 (0.2187) | 0.05164 (0.1193) | 0.05092 (0.0824) | 0.04926 (0.0620) | 0.04962 (0.0615) | 0.05012 (0.0423) | 0.05022 (0.0338) | 0.05003 (0.0300) | 0.05000 (0.0236) | 0.05016 (0.0182) |
0.10 | 0.10278 (0.2163) | 0.10001 (0.1088) | 0.10092 (0.0773) | 0.10061 (0.0607) | 0.10010 (0.0689) | 0.09997 (0.0434) | 0.09989 (0.0339) | 0.10008 (0.0299) | 0.10034 (0.0231) | 0.10018 (0.0177) |
0.20 | 0.20474 (0.2045) | 0.20054 (0.1145) | 0.20012 (0.0765) | 0.20043 (0.0574) | 0.19977 (0.0600) | 0.19992 (0.0424) | 0.20054 (0.0308) | 0.19996 (0.0280) | 0.20032 (0.0212) | 0.20024 (0.0151) |
0.30 | 0.30392 (0.2007) | 0.30788 (0.1103) | 0.30350 (0.0750) | 0.30083 (0.0545) | 0.30022 (0.0604) | 0.30042 (0.0389) | 0.30045 (0.0296) | 0.30035 (0.0259) | 0.30004 (0.0179) | 0.30088 (0.0124) |
0.40 | 0.40491 (0.2281) | 0.39618 (0.1155) | 0.40358 (0.0748) | 0.40355 (0.0598) | 0.40126 (0.0629) | 0.40262 (0.0386) | 0.40080 (0.0265) | 0.40078 (0.0232) | 0.40167 (0.0151) | 0.40312 (0.0092) |
0.50 | 0.50762 (0.2308) | 0.50795 (0.1130) | 0.49748 (0.0845) | 0.50464 (0.0518) | 0.50083 (0.0561) | 0.49881 (0.0385) | 0.50105 (0.0240) | 0.50318 (0.0205) | 0.50290 (0.0122) | 0.50552 (0.0069) |
0.60 | 0.61482 (0.2188) | 0.60143 (0.1069) | 0.60722 (0.0700) | 0.60415 (0.0506) | 0.59982 (0.0535) | 0.59997 (0.0324) | 0.60070 (0.0218) | 0.60368 (0.0175) | 0.60521 (0.0102) | 0.61267 (0.0048) |
0.70 | 0.71150 (0.2063) | 0.70642 (0.1128) | 0.70040 (0.0639) | 0.70262 (0.0463) | 0.70007 (0.0493) | 0.70457 (0.0298) | 0.70614 (0.0185) | 0.70520 (0.0145) | 0.70790 (0.0079) | 0.71631 (0.0029) |
0.80 | 0.80483 (0.2665) | 0.80608 (0.1018) | 0.80105 (0.0717) | 0.79954 (0.0419) | 0.74372 (0.0469) | 0.80350 (0.0275) | 0.80484 (0.0154) | 0.80512 (0.0124) | 0.81298 (0.0055) | 0.82603 (0.0018) |
0.90 | 0.89570 (0.2076) | 0.90016 (0.0967) | 0.90264 (0.0605) | 0.90583 (0.0421) | 0.90236 (0.0442) | 0.90229 (0.0234) | 0.91092 (0.0135) | 0.90512 (0.0100) | 0.91617 (0.0039) | |
1.00 | 1.01857 (0.2247) | 0.99812 (0.0995) | 0.99674 (0.0698) | 1.01016 (0.0394) | 1.00154 (0.0429) | 1.00645 (0.0209) | 1.01148 (0.0105) | 1.01483 (0.0085) | 1.02763 (0.0028) | |
1.10 | 1.10280 (0.2085) | 1.09827 (0.1085) | 1.10866 (0.0533) | 1.10855 (0.0361) | 1.10441 (0.0376) | 1.11082 (0.0186) | 1.11704 (0.0088) | 1.12472 (0.0063) | ||
1.20 | 1.20638 (0.1958) | 1.20485 (0.0944) | 1.21104 (0.0507) | 1.20885 (0.0319) | 1.20120 (0.0338) | 1.21299 (0.0157) | 1.21861 (0.0070) | 1.22115 (0.0049) | ||
1.30 | 1.32158 (0.2227) | 1.30226 (0.0970) | 1.31646 (0.0494) | 1.30883 (0.0286) | 1.30870 (0.0334) | 1.31479 (0.0137) | 1.32367 (0.0053) | 1.32639 (0.0039) | ||
1.40 | 1.41819 (0.2224) | 1.40275 (0.0774) | 1.40607 (0.0435) | 1.41200 (0.0270) | 1.41515 (0.0299) | 1.41974 (0.0115) | ||||
1.50 | 1.49778 (0.1785) | 1.50663 (0.0795) | 1.50531 (0.0402) | 1.51515 (0.0250) | 1.50567 (0.0290) | 1.50459 (0.0108) |
Mean estimate (and standard error) of
Size of contingency table | ||||||||||
True | 2 × 2 | 2 × 3 | 2 × 4 | 2 × 5 | 3 × 3 | 3 × 4 | 3 × 5 | 4 × 4 | 4 × 5 | 5 × 5 |
0.00 | −0.00068 (0.2014) | −0.00008 (0.1160) | 0.00299 (0.0829) | 0.00032 (0.0583) | −0.00024 (0.0611) | −0.00030 (0.0448) | −0.00004 (0.0346) | 0.00005 (0.0312) | 0.00006 (0.0239) | 0.00005 (0.0178) |
0.01 | 0.01610 (0.2759) | 0.00633 (0.1234) | 0.01424 (0.0785) | 0.00836 (0.0630) | 0.01029 (0.0631) | 0.01020 (0.0446) | 0.01011 (0.0344) | 0.01021 (0.0307) | 0.01004 (0.0241) | 0.01001 (0.0179) |
0.05 | 0.05446 (0.2187) | 0.05152 (0.1193) | 0.05075 (0.0824) | 0.04922 (0.0620) | 0.04962 (0.0615) | 0.05008 (0.0423) | 0.05016 (0.0338) | 0.04996 (0.0300) | 0.04992 (0.0236) | 0.05000 (0.0182) |
0.10 | 0.10231 (0.2154) | 0.09998 (0.1099) | 0.10078 (0.0773) | 0.10046 (0.0607) | 0.09993 (0.0689) | 0.09975 (0.0434) | 0.09954 (0.0339) | 0.09969 (0.0299) | 0.09964 (0.0231) | 0.09911 (0.0177) |
0.20 | 0.20451 (0.2045) | 0.20025 (0.1145) | 0.19952 (0.0765) | 0.19932 (0.0574) | 0.19893 (0.0600) | 0.19818 (0.0424) | 0.19766 (0.0308) | 0.19676 (0.0281) | 0.19492 (0.0213) | 0.19101 (0.0153) |
0.30 | 0.30337 (0.2007) | 0.30434 (0.1110) | 0.29977 (0.0753) | 0.29721 (0.0546) | 0.29743 (0.0604) | 0.29481 (0.0390) | 0.29092 (0.0297) | 0.28925 (0.0261) | 0.28149 (0.0182) | 0.26891 (0.0129) |
0.40 | 0.40352 (0.2275) | 0.39385 (0.1155) | 0.39791 (0.0749) | 0.39275 (0.0602) | 0.39458 (0.0629) | 0.38791 (0.0388) | 0.37732 (0.0269) | 0.37233 (0.0236) | 0.35294 (0.0159) | 0.32652 (0.0103) |
0.50 | 0.50350 (0.2310) | 0.49954 (0.1134) | 0.48823 (0.0846) | 0.48801 (0.0526) | 0.48693 (0.0563) | 0.47029 (0.0389) | 0.45542 (0.0248) | 0.44396 (0.0214) | 0.40896 (0.0135) | 0.36225 (0.0086) |
0.60 | 0.60742 (0.2165) | 0.59252 (0.1070) | 0.58724 (0.0709) | 0.57402 (0.0516) | 0.57580 (0.0539) | 0.55214 (0.0331) | 0.52330 (0.0231) | 0.50859 (0.0189) | 0.45332 (0.0121) | |
0.70 | 0.70298 (0.2073) | 0.69001 (0.1127) | 0.67442 (0.0644) | 0.65328 (0.0473) | 0.66050 (0.0499) | 0.62525 (0.0311) | 0.57783 (0.0204) | 0.55366 (0.0166) | ||
0.80 | 0.79400 (0.2685) | 0.78320 (0.1027) | 0.75942 (0.0726) | 0.73482 (0.0431) | 0.74372 (0.0469) | 0.68976 (0.0291) | 0.62194 (0.0178) | 0.59053 (0.0151) | ||
0.90 | 0.88478 (0.2076) | 0.86906 (0.0972) | 0.84284 (0.0619) | 0.80984 (0.0446) | 0.81907 (0.0456) | 0.74370 (0.0256) | ||||
1.00 | 0.99208 (0.2242) | 0.95982 (0.1003) | 0.91896 (0.0711) | 0.87180 (0.0428) | 0.88914 (0.0447) | 0.78265 (0.0240) | ||||
1.10 | 1.07792 (0.2070) | 1.04284 (0.1102) | 0.99425 (0.0563) | 0.93162 (0.0397) | 0.94721 (0.0399) | |||||
1.20 | 1.17866 (0.1918) | 1.12713 (0.0961) | 1.06491 (0.0539) | 0.98924 (0.0355) | 1.00068 (0.0366) | |||||
1.30 | 1.27633 (0.2260) | 1.20889 (0.0981) | 1.12928 (0.0542) | 1.05000 (0.0322) | 1.05912 (0.0370) | |||||
1.40 | 1.37037 (0.2244) | 1.29294 (0.0806) | 1.19599 (0.0470) | 1.07872 (0.0317) | 1.09808 (0.0340) | |||||
1.50 | 1.44834 (0.1785) | 1.37143 (0.0823) | 1.24810 (0.0439) |
Mean estimate (and standard error) of
Size of contingency table | ||||||||||
True | 2 × 2 | 2 × 3 | 2 × 4 | 2 × 5 | 3 × 3 | 3 × 4 | 3 × 5 | 4 × 4 | 4 × 5 | 5 × 5 |
0.00 | −0.00068 (0.2014) | −0.00020 (0.1161) | 0.00331 (0.0828) | 0.00026 (0.0583) | −0.00023 (0.0611) | −0.00030 (0.0448) | −0.00003 (0.0346) | 0.00003 (0.0312) | 0.00006 (0.0239) | 0.00005 (0.0178) |
0.01 | 0.01617 (0.2760) | 0.00587 (0.1233) | 0.01482 (0.0783) | 0.00818 (0.0629) | 0.01028 (0.0631) | 0.01022 (0.0446) | 0.01012 (0.0344) | 0.01021 (0.0307) | 0.01005 (0.0241) | 0.01001 (0.0179) |
0.05 | 0.05454 (0.2187) | 0.05161 (0.1193) | 0.05075 (0.0824) | 0.04925 (0.0620) | 0.04962 (0.0615) | 0.05011 (0.0423) | 0.05021 (0.0338) | 0.05002 (0.0300) | 0.05000 (0.0236) | 0.05014 (0.0182) |
0.10 | 0.10271 (0.2162) | 0.10001 (0.1088) | 0.10089 (0.0773) | 0.10061 (0.0607) | 0.10007 (0.0689) | 0.09996 (0.0434) | 0.09987 (0.0339) | 0.10007 (0.0299) | 0.10029 (0.0231) | 0.10012 (0.0177) |
0.20 | 0.20473 (0.2045) | 0.20054 (0.1145) | 0.20012 (0.0765) | 0.20036 (0.0574) | 0.19974 (0.0600) | 0.19982 (0.0424) | 0.20031 (0.0308) | 0.19976 (0.0280) | 0.19977 (0.0212) | 0.19881 (0.0152) |
0.30 | 0.30391 (0.2007) | 0.30672 (0.1106) | 0.30252 (0.0752) | 0.30051 (0.0545) | 0.30013 (0.0604) | 0.30002 (0.0389) | 0.29933 (0.0296) | 0.29895 (0.0259) | 0.29652 (0.0179) | 0.29131 (0.0125) |
0.40 | 0.40486 (0.2281) | 0.39614 (0.1155) | 0.40305 (0.0748) | 0.40154 (0.0599) | 0.40072 (0.0629) | 0.40021 (0.0386) | 0.39647 (0.0266) | 0.39431 (0.0233) | 0.38530 (0.0154) | 0.36861 (0.0097) |
0.50 | 0.50682 (0.2309) | 0.50599 (0.1131) | 0.49682 (0.0845) | 0.50235 (0.0521) | 0.49920 (0.0561) | 0.49339 (0.0386) | 0.48907 (0.0242) | 0.48343 (0.0208) | 0.46052 (0.0128) | 0.42230 (0.0079) |
0.60 | 0.61277 (0.2177) | 0.60066 (0.1069) | 0.60309 (0.0704) | 0.59827 (0.0509) | 0.59648 (0.0540) | 0.58824 (0.0326) | 0.57337 (0.0223) | 0.56490 (0.0181) | 0.52303 (0.0112) | |
0.70 | 0.70991 (0.2067) | 0.70403 (0.1126) | 0.69708 (0.0640) | 0.69114 (0.0466) | 0.69288 (0.0494) | 0.67813 (0.0302) | 0.64850 (0.0193) | 0.63058 (0.0155) | 0.56907 (0.0094) | |
0.80 | 0.80340 (0.2670) | 0.80302 (0.1021) | 0.79317 (0.0719) | 0.78409 (0.0422) | 0.78794 (0.0462) | 0.76014 (0.0281) | 0.71204 (0.0166) | 0.68449 (0.0139) | ||
0.90 | 0.89526 (0.2076) | 0.89665 (0.0968) | 0.88885 (0.0610) | 0.87568 (0.0431) | 0.87881 (0.0447) | 0.83495 (0.0243) | 0.76251 (0.0152) | 0.73009 (0.0118) | ||
1.00 | 1.01160 (0.2247) | 0.99328 (0.0996) | 0.97792 (0.0701) | 0.95673 (0.0410) | 0.96469 (0.0435) | 0.89309 (0.0225) | ||||
1.10 | 1.09777 (0.2076) | 1.08835 (0.1090) | 1.07277 (0.0546) | 1.03428 (0.0379) | 1.04616 (0.0384) | 0.94742 (0.0205) | ||||
1.20 | 1.20275 (0.1938) | 1.18780 (0.0948) | 1.16055 (0.0520) | 1.10968 (0.0337) | 1.11730 (0.0350) | 1.00053 (0.0181) | ||||
1.30 | 1.31224 (0.2236) | 1.28023 (0.0971) | 1.24276 (0.0517) | 1.18733 (0.0305) | 1.19350 (0.0351) | |||||
1.40 | 1.41026 (0.2230) | 1.37640 (0.0786) | 1.32384 (0.0451) | 1.23586 (0.0297) | 1.25046 (0.0320) | |||||
1.50 | 1.49244 (0.1784) | 1.46870 (0.0806) | 1.39468 (0.0420) | 1.29823 (0.0282) | 1.31951 (0.0314) |
Mean estimate (and standard error) of
Size of contingency table | ||||||||||
True | 2 × 2 | 2 × 3 | 2 × 4 | 2 × 5 | 3 × 3 | 3 × 4 | 3 × 5 | 4 × 4 | 4 × 5 | 5 × 5 |
0.00 | −0.00064 (0.2014) | 0.00017 (0.0060) | 0.00229 (0.0829) | 0.00042 (0.0583) | −0.00023 (0.0611) | −0.00029 (0.0448) | −0.00007 (0.0346) | 0.00008 (0.0312) | 0.00004 (0.0239) | 0.00005 (0.0178) |
0.01 | 0.01578 (0.2745) | 0.00740 (0.1236) | 0.01351 (0.0787) | 0.00868 (0.0630) | 0.01038 (0.0631) | 0.01016 (0.0446) | 0.01012 (0.0344) | 0.01020 (0.0307) | 0.01002 (0.0241) | 0.01000 (0.0179) |
0.05 | 0.05537 (0.2182) | 0.05137 (0.1193) | 0.05045 (0.0823) | 0.04906 (0.0620) | 0.04959 (0.0615) | 0.05004 (0.0423) | 0.05008 (0.0338) | 0.04987 (0.0300) | 0.04982 (0.0236) | 0.04983 (0.0182) |
0.10 | 0.10060 (0.2125) | 0.09994 (0.1087) | 0.10065 (0.0773) | 0.10031 (0.0606) | 0.09990 (0.0688) | 0.09950 (0.0434) | 0.09900 (0.0339) | 0.09925 (0.0299) | 0.09871 (0.0231) | 0.09771 (0.0177) |
0.20 | 0.20554 (0.2037) | 0.20065 (0.1143) | 0.19853 (0.0765) | 0.19771 (0.0574) | 0.19736 (0.0600) | 0.19566 (0.0424) | 0.19334 (0.0308) | 0.19195 (0.0280) | 0.18773 (0.0214) | 0.18065 (0.0154) |
0.30 | 0.30442 (0.1998) | 0.30099 (0.1113) | 0.29682 (0.0754) | 0.29088 (0.0546) | 0.29373 (0.0604) | 0.28762 (0.0390) | 0.27933 (0.0299) | 0.27595 (0.0262) | 0.26320 (0.0184) | 0.24319 (0.0132) |
0.40 | 0.39883 (0.2243) | 0.38977 (0.1152) | 0.38999 (0.0749) | 0.38243 (0.0602) | 0.38611 (0.0628) | 0.37154 (0.0389) | 0.35374 (0.0272) | 0.34521 (0.0240) | 0.31750 (0.0164) | 0.28401 (0.0109) |
0.50 | 0.49941 (0.2300) | 0.48981 (0.1128) | 0.48156 (0.0833) | 0.46832 (0.0528) | 0.46943 (0.0564) | 0.44038 (0.0393) | 0.41736 (0.0253) | 0.39981 (0.0220) | 0.35530 (0.0142) | |
0.60 | 0.60336 (0.2113) | 0.58180 (0.1068) | 0.56500 (0.0712) | 0.54913 (0.0516) | 0.54854 (0.0541) | 0.50988 (0.0337) | 0.46819 (0.0238) | 0.44776 (0.0198) | ||
0.70 | 0.69972 (0.2058) | 0.66768 (0.1121) | 0.64706 (0.0645) | 0.60455 (0.0479) | 0.62207 (0.0503) | 0.56595 (0.0319) | 0.50408 (0.0214) | 0.47675 (0.0176) | ||
0.80 | 0.79011 (0.2568) | 0.75919 (0.1026) | 0.71197 (0.0728) | 0.67057 (0.0437) | 0.69012 (0.0476) | 0.61207 (0.0302) | ||||
0.90 | 0.86600 (0.2049) | 0.83271 (0.0969) | 0.78390 (0.0627) | 0.73735 (0.0455) | 0.75035 (0.0464) | 0.64749 (0.0269) | ||||
1.00 | 0.95778 (0.2192) | 0.92254 (0.0996) | 0.84033 (0.0739) | 0.77967 (0.0437) | 0.80391 (0.0458) | |||||
1.10 | 1.05198 (0.2011) | 0.99021 (0.1101) | 0.90467 (0.0571) | 0.81670 (0.0410) | 0.83903 (0.0413) | |||||
1.20 | 1.14028 (0.1859) | 1.05917 (0.0963) | 0.95848 (0.0551) | 0.85077 (0.0370) | 0.86977 (0.0383) | |||||
1.30 | 1.24366 (0.2000) | 1.12319 (0.0973) | 0.99862 (0.0553) | |||||||
1.40 | 1.34106 (0.2180) | 1.19429 (0.0788) | 1.04569 (0.0489) | |||||||
1.50 | 1.37638 (0.1752) | 1.25152 (0.0836) |
Median number of iterations needed for Newton’s unidimensional method to converge to four decimal places. In parenthesis is the range of iterations for the 200 simulated contingency tables.
Size of contingency table | ||||||||||
True | 2 × 2 | 2 × 3 | 2 × 4 | 2 × 5 | 3 × 3 | 3 × 4 | 3 × 5 | 4 × 4 | 4 × 5 | 5 × 5 |
0.00 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
(1–7) | (1–6) | (1–9) | (1–4) | (1–3) | (1–4) | (1–3) | (2-3) | (1–3) | (1–3) | |
0.01 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
(2–14) | (2–30) | (2–95) | (2–14) | (2–5) | (2-3) | (2-3) | (2-3) | (2-3) | (2-3) | |
0.05 | 4 | 4 | 4 | 4 | 5 | 4 | 4 | 4 | 5 | 5 |
(3–13) | (1–7) | (2–6) | (4–6) | (4–6) | (4-5) | (4-5) | (4-5) | (4-5) | (4–6) | |
0.10 | 5 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 7 |
(3–29) | (4–7) | (5–8) | (5–8) | (4–6) | (5–7) | (5–7) | (5–7) | (6–8) | (6–10) | |
0.20 | 7 | 7 | 7 | 8 | 8 | 8 | 9 | 10 | 11 | 13 |
(5–13) | (7–12) | (7–11) | (7–11) | (7–10) | (7–11) | (7–15) | (8–13) | (9–17) | (8–21) | |
0.30 | 9 | 9 | 9 | 10 | 10 | 12 | 13 | 14 | 17 | 23 |
(3–23) | (9–219) | (9–128) | (9–15) | (9–15) | (9–17) | (10–22) | (10–19) | (12–27) | (15–35) | |
0.40 | 11 | 11 | 12 | 13 | 13 | 16 | 19 | 21 | 28 | 39 |
(7–25) | (6–16) | (11–25) | (11–23) | (11–22) | (12–26) | (13–33) | (14–47) | (16–49) | (20–65) | |
0.50 | 13 | 14 | 15 | 16.5 | 17 | 21 | 27 | 30 | 44 | 68 |
(4–261) | (12–56) | (13–28) | (14–103) | (13–30) | (6–47) | (18–53) | (18–57) | (23–80) | (24–110) | |
0.60 | 15 | 16 | 18 | 21 | 21 | 28 | 37 | 42 | 66 | 113 |
(11–260) | (15–32) | (16–33) | (17–66) | (16–47) | (18–61) | (21–81) | (23–98) | (30–139) | (41–261) | |
0.70 | 17 | 19 | 23 | 28 | 27 | 37 | 53 | 64 | 101 | 211 |
(14–737) | (17–39) | (19–52) | (21–67) | (19–43) | (20–71) | (26–150) | (30–126) | (25–195) | (59–560) | |
0.80 | 20 | 24 | 28 | 34 | 34 | 49 | 75 | 89 | 170 | 359 |
(8–407) | (20–40) | (23–58) | (25–64) | (22–49) | (24–140) | (32–160) | (38–188) | (61–352) | (102–1026) | |
0.90 | 24 | 29 | 34 | 42 | 43 | 67 | 107.5 | 128.5 | 255 | 605 |
(9–39) | (24–46) | (27–716) | (29–114) | (26–101) | (32–116) | (39–207) | (58–332) | (62–653) | (147–2129) | |
1.00 | 28 | 32 | 42 | 55 | 53 | 90 | 145.5 | 172 | 395 | 1028 |
(16–987) | (27–83) | (17–101) | (37–286) | (31–122) | (42–281) | (67–349) | (66–480) | (99–1173) | (3–3243) | |
1.10 | 32 | 39 | 52 | 68.5 | 69 | 116 | 211 | 275 | 656.5 | 1875.5 |
(29–863) | (4–1983) | (38–187) | (48–285) | (39–181) | (45–383) | (57–580) | (68–590) | (151–2370) | (8–6656) | |
1.20 | 36 | 47 | 64 | 88 | 88 | 157.5 | 285 | 392 | 999 | 2725 |
(28–208) | (36–94) | (44–180) | (51–201) | (46–210) | (69–579) | (94–889) | (129–907) | (199–3120) | (589–12917) | |
1.30 | 42 | 55 | 77 | 109.5 | 109 | 207 | 420.5 | 556 | 1571.5 | 4730 |
(29–1239) | (33–267) | (53–275) | (58–265) | (52–222) | (69–441) | (135–1104) | (154–1606) | (260–4833) | (11–23803) | |
1.40 | 48 | 64 | 94 | 143.5 | 135.5 | 284 | 562 | 822.5 | 2249 | 6224 |
(23–2288) | (45–2781) | (58–262) | (78–372) | (63–431) | (88–557) | (129–1675) | (187–3370) | (24–7041) | (3–25352) | |
1.50 | 56 | 75 | 114 | 177 | 170 | 369.5 | 827 | 1233 | 3102.5 | 9769 |
(38–92) | (57–216) | (74–302) | (104–914) | (72–624) | (103–1071) | (189–2641) | (305–4608) | (22–11773) | (9–34759) |
Table
Graphical view of the performance of Newton’s unidimensional method for estimating
Table
Table
Graphical view of the performance of
Tables
Graphical view of the performance of
Graphical view of the performance of
Graphical view of the performance of
Two hundred (200) contingency tables of dimension 4 × 5 were randomly generated (using the algorithm described in Section
The
The three remaining non-iterative procedures also behave well, but provide poorer estimates of
Iterative procedures can be computationally very intensive. Certainly for those contingency tables that are of large dimension, or where
For small contingency tables (in terms of dimension and
Of the four non-iterative procedures considered in this paper,
The enhanced accuracy of
One can also observe the relative accuracy of the non-iterative methods. Since the right-hand side of (
Beh and Davy [
The computations performed here have focused on generating contingency tables with a sample size of approximately 1000. Further investigations need to be undertaken to study the behaviour of the non-iterative estimation procedures for varying sample sizes. Preliminary investigations reveal that, for contingency tables with a sample size exceeding about 200, the estimate