Models for ordered categories and trends (i)

• A lot of variables we work with have some ordinality.
  • Levels of education, or strength of opinion.
  • Some are clearly quantitative, as counts (number of children) or as grouped continuous variables (age groups, income groups).
  • Some have specific values of a variable attached to each category: school size where categories are different schools, dose where categories are experimental groups, etc.

• We can model these as nominal and get sensible results but it is more efficient to take account of the ordinality: we can make better fitting, simpler models and make stronger claims.

• There are several different approaches to imposing ordinality:
  • imposing a linear effect on a variable
  • imposing a set on non-linear scores
  • calculating a best-fit set of scores
  • imposing true ordinality: steps

• Imposing a linear effect simply says that one (or more) variables has an effect which increases proportionately from category to category.
  
  $$\log F_{ij} = \lambda_0 + \lambda_j + \beta i$$
  
  

• This model suggests that the effect of the row variable can be approximated by a linear term: There are $x$ in row 1, $x.e^\beta$ in row 2, $x.e^{2\beta}$ in row three and so on.

• If we try this on AOPFAMC by EOPFAMC like this:

  compute alin = aopfamc.
  genlog aopfamc eopfamc with alin
   /print=est/plot=none/design= eopfamc alin .
  

  we get $G^2$ of 6604.7571 for 19 df.
• This model is a constrained version of independence which says that the $\lambda_i$ set of parameters can be approximated by a linear effect. The fit of independence is 1167.1117 for 16, much better: the row distribution is not well approximated by a linear effect in this table.

• If we look at the parameter estimates for the $\lambda_i$s we see why a linear effect (of 0.0646) a poor fit:

     i       p e
  ----------------
     1     -.4948
     2     1.4743
     3     2.5569
     4     1.9288
     5      .0000
  

• Linear terms become more interesting at higher orders. Here they impose linearity on the association. If we are classifying social class by opinion on welfare, it may be plausible that for a particular category the association effect has a linear form: very unlikely to agree strongly, very likely to disagree strongly, with a smooth gradient in between. Each category may have a different gradient.

• Consider the EVOTE by EOPFAMF table (vote intention in 1995, by attitude on `Husband should earn, wife stay at home').

  EVOTE  Political party supported
  by  EOPFAMF  Husband should earn, wife stay at home
              EOPFAMF                                 Page 1 of 1
      Count  |Strongly Agree    Neithr a Disagree Strongly
             | agree            gree, di           disagre  Total
  EVOTE -----+--------+--------+--------+--------+--------+
          1  |   123  |   383  |   538  |   703  |   230  |  1977
  Consve     |        |        |        |        |        |  31.4
          2  |   214  |   492  |   732  |  1172  |   613  |  3223
  Labo       |        |        |        |        |        |  51.1
          3  |    39  |   121  |   237  |   302  |   177  |   876
  Lib b/SDP  |        |        |        |        |        |  13.9
          4  |    10  |    28  |    47  |    82  |    60  |   227
  Othe       |        |        |        |        |        |   3.6
             +--------+--------+--------+--------+--------+
      Column     386     1024     1554     2259     1080     6303
       Total     6.1     16.2     24.7     35.8     17.1    100.0
  

• Independence gives $G^2$ of 100.4906 for 12. If we fit a linear effect for opinion in association with vote, what happens?

• First, make a copy of the opinion variable. Then enter the copy as a covariate, and include it in the /design. By entering a copy as a covariate, SPSS treats the values 1, 2, 3, 4 and 5 as quantitative scores.

  compute elin = eopfamf.
  genlog evote eopfamf with elin
   /print=est/plot=none
   /design= eopfamf evote elin by evote.
  

• $G^2$ falls to 42.8971 for 9 df, a big improvement (but still not a well fitting model). The interaction uses up only 3 df because we fit parameters only for $4-1$ categories of vote.
  
  $$\log F_{ij} = \lambda_0 + \lambda_i + \lambda_j + \beta_ji$$
  
  

• The parameter estimates are worth looking at:

  [EVOTE = 1]*ELIN  -.3307  .0657
  [EVOTE = 2]*ELIN  -.1848  .0646
  [EVOTE = 3]*ELIN  -.1337  .0697
  [EVOTE = 4]*ELIN   .0000  .
  

• For conservatives, we see a decline as we move from agreement to disagreement that is larger than that for labour, with LibDems being even smaller, with the other category being zero.
• That is, after taking account of the margins, the association takes the form that Conservatives are least likely to agree, and `other' voters most, with the other parties in between.