Score models

• We can generalise these linear models by applying scores other than the row and column numbers.
• We may feel there is a smaller distance between `strongly agree' and `agree' than between `agree' and `neutral', and score them thus:
  • 1 strongly agree
  • 2 agree
  • 4 neutral
  • 6 disagree
  • 7 strongly disagree

• Alternatively, we may have quantitative information about the categories.
  • Medical experiments may have several groups of subjects given different doses of the treatment.
  • Categories may refer to entities about which we have other information.
  • They may be constructed from continuous variables such as income and we can use mid-point values etc. .

• An example: pregnant mice are exposed to differing levels of a toxin and three outcomes are recorded: normal foetus, abnormal foetus, dead foetus:
  

  Dose	Outcome
  	Normal	Abnormal	Dead
  .00	281	1	15
  62.50	225	0	17
  125.0	283	7	22
  250.0	202	59	38
  500.0	9	132	144

• Independence gives $G^2$ of 911.6904 for 8 df, a very poor fit with these residuals:
  

  Dose	Outcome
  	Normal	Abnormal	Dead
  .00	10.50	-7.58	-5.95
  62.50	8.65	.	-4.34
  125.0	9.13	-6.72	-5.06
  250.0	-.90	3.30	-1.96
  500.0	-27.30	17.71	17.34

• There is obviously association between the toxin and bad outcomes, so let's fit a linear term for dose (1, 2, 3, 4, 5) in interaction with outcome. $G^2$ falls to 155.2920 for 5 df, a significant improvement.

  The residuals are smaller but still many are too big:
  

  Dose	Outcome
  	Normal	Abnormal	Dead
  .00	-8.92	1.93	8.66
  62.50	-3.76	.	3.76
  125.0	3.97	-1.78	-3.32
  250.0	6.94	1.08	-8.17
  500.0	-10.69	-.69	11.37

• But we have quantitative information on dose: what if we fit this as row scores: $G^2$ of 48.9363 for 5 df.
  

  Dose	Outcome
  	Normal	Abnormal	Dead
  .00	-3.15	-1.07	4.43
  62.50	-2.82	.	2.82
  125.0	1.66	-2.18	-.36
  250.0	3.08	2.90	-5.70
  500.0	-4.23	-2.80	6.69

• It may be that the effect of the toxin is non-linear in the dose: since the residuals for the 500mg/kg dose are so high, it may be that the effect increases faster than dose: we can use the square of dose to model this: $G^2$ of 37.1931 for 5 df, a somewhat better fit.

• Outcome can be considered ordinal as well (true ordinal rather than quantitative as dose). We can fit a variant of the linear-by-linear model, using the square of dose and the column index for outcome:

  compute assoc = outcome*dose*dose.
  
  genlog dosecat outcome with assoc
    /print=est/plot=none
    /design = dosecat outcome assoc.
  

• This doesn't fit well: $G^2$ is 288.4299 for 7 df. Compared with the less parsimonious model with the square of dose in interaction with outcome, it is not preferred. Looking back at the residuals from independence, this is not surprising:
  

  Dose	Outcome
  	Normal	Abnormal	Dead
  .00	10.50	-7.58	-5.95
  62.50	8.65	.	-4.34
  125.0	9.13	-6.72	-5.06
  250.0	-.90	3.30	-1.96
  500.0	-27.30	17.71	17.34