Program LOGLIN performs generalised log-linear modelling of categorical data. It can fit any of the log-linear models available to standard packages such as GLIM, SAS, BDMP or SPSS, including models with structural zeros (as in PROC CATMOD). In addition, it can fit models for missing data and/or unobserved data. Although it can fit the more general latent variable models described by Haberman (1980), Goodman (1981) or Hagenaars (1990a, 1990b), these can be cumbersome and slow to converge (David Rindskopf was very helpful in pointing out how to fit these in the present log-linear framework).
LOGLIN can be used for:
The general framework underlying these models is summarised by Espeland (1986), and Espeland & Hui (1987), and is originally due to Thompson & Baker (1981). An observed contingency table y, which will be treated as a vector, is modelled as arising from an underlying complete table z, where observed count y(j) is the sum of a number of elements of z, such that each z(i) contributes to no more than one y(j). Therefore one can write y=F'z, where F is made up of orthogonal columns of ones and zeros.
We then specify a loglinear model for z, so that log(E(z))=X'b, where X is a design matrix, and b a vector of loglinear parameters. The loglinear model for z and thus y, can be fitted using two methods, both of which are available in LOGLIN. The first was presented as AS207 by Michael Haber (1984) and combines an iterative proportional fitting algorithm for b and z, with an EM fitting for y, z and b. The second is a Fisher scoring approach, presented in Espeland (1986).
Each iteration of the Fisher scoring algorithm is
b(t+1) = b(t) + I-1 (PX')' (m - F(F'F)-1 y) ,
where,
b(t) is the estimate of b for the tth iteration,
m = exp(X'b) ,
P = F (F' diag(m) F)-1 F' diag(m) ,
and
I = (PX')' diag(m) (PX').
The default option provided by the program is to use the EM algorithm to provide starting values for the scoring algorithm, thus gaining a modest improvement in speed. However, each method can be called in isolation. The EM algorithm needs to call the scoring algorithm to get the information matrix for the loglinear parameters in any case. In the case of missing data, one is usually interested in collapsing the complete table to give expected counts for subtables, and often summary measures for these subtables. Standard errors of collapsed counts, and measures can be calculated using the covariance matrix for the loglinear parameters of the complete table using the delta method.
As an alternative, LOGLIN allows (nonparametric) bootstrap estimates of standard errors to be obtained. These are currently only for Poisson models, and will differ if sampling is constrained - eg product-multinomial - for incomplete tables. Espeland (1985) discusses approaches for this and other situations. Bootstrap percentiles for the model LR chi-square are also produced.
The program reads commands from standard input, and writes to standard output. The commands are made up of the following key words and data (note that the parser usually reads only the first two to four characters of a keyword, and will usually read a long form key word as well eg bootstrap|boot|bs):
! simplest table data 4 31 109 17 122 ! intercept row and col, odds ratio mo 4 4 1 1 1 1 1 0 1 0 1 1 0 0 1 0 0 0 ! labels for loglinear terms la intercept row col oddsr ! fit saturated model, and reverses the order of parameter printing se 4 4 3 2 1 or 1 2 3 4 bs 200
! DZ 2x2 table then MZ 2x2 table data 8 12 12 10 1335 5 12 24 1506 mo 8 6 1 2 1 2 2 2 1 1 1 0.5 1 0.5 1 1 1 0.5 1 0.5 1 0 1 0 0 0 1 2 0 2 0 0 1 1 0 0.5 0 0 1 1 0 0.5 0 0 1 0 0 0 0 0 !-------------------- ! 1 2 3 4 5 6 ! i a z a1 a a1 ! a2 z a2 ! z la i a z aa az aaz
! ! Adjust cross-reported asthma in singles using data from complete pairs ! cells 8 1 1 2 2 3 4 5 6 ! ! One 2x1 tables and one 2x2 table giving sens and spec ! data 6 116 540 451 91 168 2075 model 8 5 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 1 0 1 1 0 1 1 0 0 1 1 1 0 1 0 1 1 1 0 0 1 1 1 1 0 ! i L T A AT la i L T A AT conv 0.001 cl 4 1 2 1 2 pr 1 2 bs 200 ou
! ! Look at Mark Jenkins' asthma data - Brit Med J 1994;309:90-3. ! compare delta estimator of SE for stratified sample to that in LOGLIN ! cells 8 1 2 3 4 5 5 6 6 ! ! 2x2 table for the sampled probands (A+,A- in C+, then C-). ! One 2x1 table for unsampled subjects, giving therefore the sampling fraction. ! data 6 414 327 127 626 608 6240 model 8 6 1 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1 0 1 1 0 1 1 1 0 0 0 0 1 1 0 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 ! ! i S C A SC CA ! ! S=sampled; C=childhood asthma; A=adult asthma ! la i S C A SC CA conv 0.001 cl 8 1 2 1 2 1 2 1 2 pr 1 2 bs 500 ou
! ! Estimation AB0 frequencies Elandt-Johnson, 1971, p 401, Ex 14.1 ! A B AB 0 ! data 4 725 258 72 1073 ce 9 1 3 1 3 2 2 1 2 4 model 9 3 2 0 0 1 1 0 1 0 1 1 1 0 0 2 0 0 1 1 1 0 1 0 1 1 0 0 2 la A B 0
! ! Test for HWE ApoE Cauley et al 1993 across two age cohorts ! ! 2-2, 3-2, 4-2, 3-3, 4-3, 4-4 ! data 12 2 47 5 315 98 6 5 126 11 581 135 12 ce 18 1 2 3 2 4 5 3 5 6 7 8 9 8 10 11 9 11 12 ! ! e2 e3 e4 age ! model 18 7 2 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 1 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 2 0 0 0 0 2 0 0 1 2 0 0 1 1 0 1 1 1 0 1 0 1 1 1 0 1 1 1 0 1 1 1 0 0 2 0 1 0 2 0 0 1 1 1 0 1 1 1 0 1 1 1 0 1 0 1 1 1 0 1 1 0 0 2 1 0 0 2 ! !1 2 3 4 5 6 7 ! la e2 e3 e4 age e2*age e3*age e4*age ! ! se 4 Comparing LR for full model versus no interaction ! 1 2 3 4 model tests for gene frequencies conditional on ! HWE
! gametic (pair) frequency gamma two alleles A(ij), B(kl) ! allelic " alpha two gametes G1(ik), G2(jl) ! deviation from HWE phi ! intragametic allelic assoc epsilon ! intergametic allelic assoc delta ! ! ln g(ijkl) = mu + a(i) + a(j) + a(k) + a(l) + p(ij) + p(kl) ! + e(ik) + e(jl) + d(il) + d(jk) ! ! a(i) and a(j) are represented by a combined parameter in the model below, ! as is a(k) & a(l) and e(ik) and e(jl). ! epsilon and delta are confounded and cannot be simultaneously estimated. ! Locus B 3 alleles versus Locus H three alleles. data 36 2 2 1 7 3 3 6 11 10 18 30 15 6 9 12 22 45 45 14 19 11 31 23 19 31 66 37 110 93 72 37 57 15 53 43 22 cells 81 1 2 4 2 3 5 4 5 6 7 8 10 8 9 11 10 11 12 19 20 22 20 21 23 22 23 24 7 8 10 8 9 11 10 11 12 13 14 16 14 15 17 16 17 18 25 26 28 26 27 29 28 29 30 19 20 22 20 21 23 22 23 24 25 26 28 26 27 29 28 29 30 31 32 34 32 33 35 34 35 36 model 81 21 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 2 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 1 0 0 2 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 2 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 1 0 1 0 2 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 2 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0 1 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 1 0 0 2 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 1 2 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 2 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 2 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 2 0 2 0 1 0 0 0 1 0 0 0 2 0 0 0 2 0 0 0 1 2 0 1 1 1 0 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 2 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 2 0 1 1 1 0 0 0 0 1 0 0 1 0 1 0 1 0 1 0 1 2 0 0 2 1 0 0 0 0 0 0 1 0 0 2 0 0 0 2 0 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 1 1 2 0 0 0 1 0 1 0 0 0 1 1 0 0 1 1 0 0 1 1 1 1 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 1 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 2 0 0 1 0 0 0 0 1 0 0 1 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 2 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 1 0 1 0 2 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 2 0 0 1 0 0 1 0 0 0 1 1 0 0 1 1 0 0 1 1 1 1 1 0 1 0 0 0 0 1 0 0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 1 1 1 0 1 0 0 0 1 0 0 1 0 0 1 0 1 1 0 1 1 1 0 2 0 1 0 0 0 0 0 1 0 0 1 1 0 0 1 1 1 0 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 2 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 1 0 2 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 2 2 0 0 0 0 1 1 0 0 0 0 2 0 0 0 2 0 0 1 0 2 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 1 1 0 2 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 1 0 2 1 1 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 1 1 0 2 0 2 0 0 0 1 0 0 0 1 0 0 0 2 0 0 0 2 !------------------------------------------ ! 1 2 3 4 5 6 7 8 9101112131415161718192021 ! i a a a a p p p p p p p p e e e e d d d d ! ! Allelic association and deviation from HWE ! Since epsilon and delta terms are confounded - one set (delta's) is zeroed ! ie assume no intergametic association se 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 ! ! No allelic association - deviation from HWE ! se 13 ! 1 2 3 4 5 6 7 8 9 10 11 12 13 ! ! HWE; no allelic association ! se 5 ! 1 2 3 4 5 ou
! Fit teeth from Espeland et al 1986 cells 12 1 5 9 1 2 3 4 5 6 7 8 9 ! 3x3 table of rating of caries 3 point scale 2 observers data 9 1450 55 74 99 35 33 22 11 64 model 12 8 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 1 1 0 0 1 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0
! Fit Hochberg 1977 double sampling data cells 32 1 1 2 2 1 1 2 2 3 3 4 4 3 3 4 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ! 2x2 table of imprecise measures and 2x2x2x2 reliability data data 20 1196 13562 7151 58175 17 3 10 258 3 4 4 25 16 3 25 197 100 13 107 1014 ! ! model is AA*BB* + L (dummy study variable) ! so vars are intercept, A, A*, B, B*, L, A.A*, A.B, A.B*, A*.B, A*.B* ! B.B*, A.A*.B, A.A*.B*, A.B.B*, A*.B.B*, A.A*.B.B* model 32 17 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 0 1 0 1 1 0 1 1 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 0 0 0 1 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 0 0 1 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 0 0 0 1 1 1 0 0 0 1 0 1 1 0 1 1 0 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 1 0 1 0 1 0 1 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ! ! recover estimated A.B collapsed table for entire sample collapse 32 1 2 1 2 3 4 3 4 1 2 1 2 3 4 3 4 1 2 1 2 3 4 3 4 1 2 1 2 3 4 3 4 odds_ratio 1 2 3 4 ! ! get bootstrapped standard errors of mean values collapsed table bootstrap 150The edited output from example (4) is:
+---------------------------------+ | LOGLIN | | General Log-linear Modelling | | Using AS 207 (Haber, 1984) | +---------------------------------+ Written by David L Duffy 1992 QIMR Australia HP Fortran version Program LOGLIN run at 14:31:52 on 8-Apr-92 The following input lines were read: . . [as above] . Output: No. cells complete table= 32 No. cells observed table= 20 No. parameters estimated= 17 Convergence criterion = .100E-02 Fitting via Fisher score algorithm Mean observed cell size = 4094.00 Rank of design matrix = 17 Gibbs Chi-square = 6.49 P= .09 Pearson Chi-square = 6.18 P= .10 df = 3. Observed Table ------------------------- Observed Fitted F-T Deviate [ 1] 1196.00 1196.13 .00 . . [20] 1014.00 987.30 .85 Full Table ---- Fitted [ 1] 753.12 . [32] 987.30 Full Table ------------------------------- Parameter S.E. exp(Par) 95% Confidence Limits Term [ 1] 6.895 .028 987.297 933.776 1043.885 [ 2] -2.249 .103 .106 .086 .129 [ 3] -4.138 .240 .016 .010 .026 . [16] -2.794 .987 .061 .009 .423 [17] 3.991 1.349 54.099 3.846 761.043 Collapsed table ------------ [ 1] 3227.39 [ 2] 21071.03 [ 3] 10581.91 [ 4] 47002.67 -------------- OR .68 -------------- Bootstrap mean S.E. 95% CL----------- [ 1] 3198.97 376.18 2461.66 3936.28 [ 2] 21125.60 654.63 19842.53 22408.66 [ 3] 10601.53 596.20 9432.98 11770.08 [ 4] 46957.00 826.98 45336.11 48577.89 ------------------------------------------- logOR -.40 .16 -.72 -.09 OR .67 .49 .92 ------------------------------------------- No. of bootstrap samples= 150 Job completed in 153.0 seconds. 2.5 minutes.Espeland and Hui (1987) give their results for the same model. The overall model goodness-of-fit was G23=6.49. The standard errors are calculated using the delta method.
---------------------------------------------------------------------------- Precise Injury Precise belt use Fitted Estimate Standard Error ---------------------------------------------------------------------------- Yes Yes 3227.4 344.9 Yes No 21071.0 660.0 No Yes 10581.9 527.2 No No 47002.7 787.6 ---------------------------------------------------------------------------- Odds ratio from collapsed table 0.68 0.16 ----------------------------------------------------------------------------
! ! Example data from Lem manual ! data 16 59 56 14 36 7 15 4 23 75 162 22 115 8 68 22 123 ce 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ! ! X A B C D XA XB XC XD ! design 32 16 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 0 0 1 1 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 1 0 1 0 1 0 1 0 1 1 1 0 0 0 0 0 1 1 0 1 0 0 1 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 1 0 0 0 0 1 0 1 1 0 1 1 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 1 1 0 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 0 0 1 0 0 0 1 1 1 0 1 0 1 0 1 0 0 1 0 0 0 0 1 1 1 0 1 1 1 0 1 1 0 1 1 0 0 1 1 1 1 1 0 0 1 1 0 0 1 0 0 0 0 0 1 1 1 1 0 1 1 1 0 1 1 0 1 0 1 0 1 1 1 1 1 0 1 1 1 0 1 1 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 !-------------------------------- ! 1 2 3 4 5 6 7 8 910111213141516 !-------------------------------- !i X2A2B2C2D2X2X2X2X2A2A2A2B2B2C2 ! A2B2C2D2B2C2D2C2D2D2 ! ! la i X A B C D XA XB XC XD AB AC AD BC BD CD se 10 1 2 3 4 5 6 7 8 9 10 fi em conv 1e-6