International Scholarly Research Notices

Research Article

A Computational Study Assessing Maximum Likelihood and Noniterative Methods for Estimating the Linear-by-Linear Parameter for Ordinal Log-Linear Models

Table 1

Mean ML and 𝐿 π‘œ 𝑔 𝑁 𝐼 estimates of πœ™ for three different sized contingency tables with varying various sample sizes [iterations] give the median iterations taken for Newton’s method to converge to four decimal places and the range of iterations observed.
Sample SizeEstimation 3 Γ— 3 4 Γ— 5 5 Γ— 5
𝑁 Method πœ™ = 0 . 8 % Ξ” πœ™ = 0 . 5 % Ξ” πœ™ = 0 . 3 % Ξ”
10 L o g N I   
MLE
[Iterations]		 1.04281
0.93413
[ 4 1 , 1 - 8 4 6 ] 		30.4
16.8 		0.64944
0.64554
[ 7 7 ; 3 - 4 5 0 ] 		29.9
29.1 		0.44185
0.42392
[ 4 5 ; 4 - 2 2 5 ] 		47.3
41.3
50 L o g N I   
MLE
[Iterations]		 0.86133
0.81255
[ 3 4 ; 1 - 3 4 8 ] 		7.7
1.6 		0.57181
0.51608
[ 4 4 ; 1 4 - 1 3 1 ] 		14.4
3.2 		0.35003
0.31199
[ 2 4 ; 1 1 - 5 9 ] 		16.7
4.0
100 L o g N I   
MLE
[Iterations]		 0.82731
0.80296
[ 3 4 ; 1 1 - 2 7 8 ] 		3.4
0.4 		0.53740
0.50386
[ 4 3 ; 1 8 - 1 4 0 ] 		7.5
0.8 		0.32444
0.30406
[ 2 3 ; 7 - 5 2 ] 		8.1
1.4
200 L o g N I   
MLE
[Iterations]		 0.81199
0.80078
[ 3 4 ; 2 1 - 1 3 9 ] 		1.5
0.1 		0.52043
0.50151
[ 4 3 ; 1 9 - 1 0 1 ] 		4.1
0.3 		0.30965
0.30096
[ 2 3 ; 1 2 - 4 2 ] 		3.2
0.3
500 L o g N I   
MLE
[Iterations]		 0.80535
0.80057
[ 3 4 ; 2 1 - 8 5 ] 		0.7
0.1 		0.50774
0.50044
[ 4 3 ; 1 8 - 1 0 4 ] 		1.5
0.1 		0.30326
0.30033
[ 2 3 ; 1 3 - 4 3 ] 		1.1
0.1
1000 L o g N I   
MLE
[Iterations]		 0.80081
0.80009
[ 3 4 ; 2 1 - 9 1 ] 		0.1
<0.05		 0.50250
0.50007
[ 4 2 ; 2 2 - 8 5 ] 		0.5
<0.05		 0.30128
0.30011
[ 2 3 ; 1 2 - 4 1 ] 		0.4
<0.05
2500 L o g N I   
MLE
[Iterations]		 0.80031
0.79999
[ 3 4 ; 2 1 - 7 9 ] 		<0.05
<0.01		 0.50102
0.50003
[ 4 3 ; 1 9 - 1 1 2 ] 		0.2
<0.01		 0.30022
0.29995
[ 2 3 ; 1 4 - 4 4 ] 		0.07
<0.05
5000 L o g N I   
MLE
[Iterations]		 0.80004
0.80000
[ 3 4 ; 2 1 - 7 6 ] 		<0.01
<0.01		 0.50047
0.49999
[ 4 3 ; 1 9 - 9 3 ] 		0.1
<0.01		 0.30007
0.29999
[ 2 3 ; 1 4 - 5 0 ] 		<0.05
<0.01
10000 L o g N I   
MLE
[Iterations]		 0.80003
0.79996
[ 3 4 ; 2 2 - 8 1 ] 		<0.01
<0.01		 0.50008
0.49997
[ 4 3 ; 2 1 - 9 0 ] 		<0.05
<0.01		 0.30003
0.29999
[ 2 3 ; 1 3 - 4 9 ] 		0.01
<0.01
50000 L o g N I   
MLE
[Iterations]		 0.79998
0.79998
[ 3 4 ; 2 1 - 6 4 ] 		<0.01
<0.01		 0.50004
0.49998
[ 4 2 ; 2 1 - 1 1 4 ] 		<0.01
<0.01		 0.30000
0.29999
[ 2 3 ; 1 2 - 4 7 ] 		0
<0.01
100000 L o g N I   
MLE
[Iterations]		 0.80000
0.80000
[ 3 4 ; 2 0 - 7 9 ] 		0
0		 0.50000
0.49997
[ 4 3 ; 2 0 - 8 6 ] 		0
<0.01		 0.30000
0.29999
[ 2 3 ; 1 3 - 3 9 ] 		0
<0.01