Death penalty example

• In practical  a well known data set is/was modelled:

  Victim's	Defendent's	Death Penalty		%age yes
  race	race	Yes	No
  White	White	53	414	11.3
  	Black	11	37	22.9
  Black	White	0	16	0.0
  	Black	4	139	2.9
  Total	White	53	430	11.0
  	Black	15	176	7.9

  (Source: Agresti p. 54ff.)

• A $2\times2\times2$ table of defendent's race, victim's race and sentence for US murder cases.

• Careful inspection of the ``% yes'' column tells us much of the story:
  • On the whole, black defendents are less likely than white to be sentenced to death.
  • However, the percentages conditional on the victim's race are quite different: for black or white victims taken separately, black defendents have a higher rate of receiving the death penalty.

• This is another case of conditional and marginal association diverging, and the explanation lies in other two 2-way associations:
  • Black killers have more black victims and white killers have more white victims;
  • Killers of white victims are more likely to receive the death penalty.

• This emerges directly in the modelling:
  1. Adding drace by verdict does not improve on the 3-way independence model ($G^2$ falls from 402.84 to 401.30, for 1 df: p = 0.22) (very different from 2-D table!)
  2. Adding the vrace by verdict term does improve ($G^2$ falls to 385.96)
  3. So does adding vrace by drace (dramatically), and the model with vrace by drace and vrace by verdict approaches a good fit (p=0.07).
  4. However, if we now add the drace by verdict term it improves the fit even more, raising the fit to p=0.5.
• The parameter estimates are informative:

  Constant                   3.96685      (0.137378)
  Penalty(No)                2.05946      (0.145845)
  VicRace(B)                -5.66961      (0.645704)
  DefRace(B)                -1.55253      (0.326164)
  Penalty(No).DefRace(B)   -0.867797      (0.367062)
  VicRace(B).DefRace(B)      4.59497      (0.313523)
  Penalty(No).VicRace(B)     2.40444      (0.600437)
  

  • Two very large positive terms linking white victims to the death penalty and linking defendent's and victim's race
  • One smaller, though significant, negative term indicating black defendents are more likely to receive the death penalty, after controlling for the confounding effect of the other two relationships.

• Careful inspection of the table would have told the same story but the modelling has two advantages:
  • a framework for formal hypothesis testing
  • a tool valid for tables too large to analyse by hand.