Linear and log-multiplicative models

• $\ell$EM fits many models, described in a 100-page manual available from the website. Among these models are some for linear effects, and for log-multiplicative models. (See sections 2.3 and 11.5.4 of the manual.)

• To fit a model with a single linear effect (as on page ) we use the ass1 subcommand on the mod command:

  man 2
  dim 4 5
  lab V O
  mod {V,O,ass1(V,O,3a)}
  dum 4 5
  dat [123     383     538     703     230    
       214     492     732    1172     613    
        39     121     237     302     177    
        10      28      47      82      60]
  

  Output from this procedure is in vote-op.lem.out in the datasets directory.

• This is using the EVOTE by EOPFAMF table in unit 5. Fitting this is equivalent to the model above . The fit is reproduced exactly, though the parameterisation is different. In particular the parameters for the EVOTE*ELIN term are set to sum to zero, rather than having a reference category.

• The ass1(V,O,3a) phrase means V is the row variable, O the column variable, and the 3a means to fit an association with a linear row-effect. 4a will give a column effect.

• The linear-by-linear model is given by a 2a term. Using the AOPFAMC*EOPFAMC table used above, we get the same fit and the same linear-by-linear $\beta$ of 0.6260, though with the other parameters having different values.

  man 2
  dim 5 5 
  lab A E
  mod {A,E,ass1(A,E,2a)}
  dum 5 5
  dat [22      63      47      29      4       
       39      393     549     186     11      
       40      362     2285    752     46      
       18      137     801     825     81      
       3       21      70      127     51]
  

• A log-multiplicative row and column model is like a linear-by-linear model where the row and column indices are replaced by estimated scale values. For this table such a model is fitted with a model statement like this:
  mod A,E,ass2(A,E,5a)

• The results look like this:

   type 2 association (row=A column=E)
    association    5.3907
    row           -0.5531  -0.4008  -0.0231   0.3214   0.6555
    adj row       -1.2841  -0.9305  -0.0535   0.7463   1.5219
    column        -0.5059  -0.4335  -0.0373   0.2913   0.6855
    adj column    -1.1747  -1.0065  -0.0866   0.6763   1.5915
  

  The $\phi$ is 5.3907, and two sets of the log-multiplicative scales are given: all sum to zero, and the first (row and column, not adj) sum to one when squared.

• Log-multiplicative models with only a row scale are fitted by using e.g., ass2(A,E,3a), or ass2(A,E,4a) for column only.

• $\ell$EM fits also these models on tables of higher dimension, where for instance the R*C association can differ across a grouping variable.

Loglinear Analysis Unit 9