Someone had wanted a source for examples; I found this looking up Arnold
Barnett in Google.com. He has other interesting examples.
>From http://209.58.177.220/articles/oct94/barnett.html
Arnold Barnett
The Odds of Execution
A powerful example of the first problem arose in 1987, when the
U.S. Supreme Court issued its controversial McClesky v. Kemp ruling
concerning racial discrimination in the imposition of the death
penalty. The Court was presented with an extensive study of Georgia death
sentencing, the main finding of which was explained by the New York Times
as follows: "Other things being as equal as statisticians can make them,
someone who killed a white person in Georgia was four times as likely to
receive a death sentence as someone who killed a black."
The Supreme Court understood the study the same way. Its majority opinion
noted that "even after taking account of 39 nonracial variables,
defendants charged with killing white victims were 4.3 times as likely to
receive a death sentence as defendants charged with killing blacks."
But the Supreme Court, the New York Times, and countless other newspapers
and commentators were laboring under a major misconception. In fact, the
statistical study in McClesky v. Kemp never reached the "factor of
four" conclusion so widely attributed to it. What the analyst did conclude
was that the odds of a death sentence in a white-victim case were 4.3
times the odds in a black-victim case. The difference between
"likelihood" and "odds" (defined as the likelihood that an event will
happen divided by the likelihood that it will not) might seem like a
semantic quibble, but it is of major importance in understanding the
results.
The likelihood, or probability, of drawing a diamond from a deck of cards,
for instance, is 1 in 4, or 0.25. The odds are, by definition, 0.25/0.75,
or 0.33. Now consider the likelihood of drawing any red card (heart or
diamond) from the deck. This probability is 0.5, which corresponds to an
odds ratio of 0.5/0.5, or 1.0. In other words, a doubling of probability
from 0.25 to 0.5 results in a tripling of the odds.
The death penalty analysis suffered from a similar, but much more serious,
distortion. Consider an extremely aggravated homicide, such as the torture
and killing of a kidnapped stranger by a prison escapee. Represent as PW
the probability that a guilty defendant would be sentenced to death if the
victim were white, and as PB the probability that the defendant would
receive the death sentence if the victim were black. Under the "4.3 times
as likely" interpretation of the study, the two values would be related by
the equation:
If, in this extreme killing, the probability of a death sentence is very
high, such that PW = 0.99 (that is, 99 percent), then it would follow that
PB = 0.99/4.3 = 0.23. In other words, even the hideous murder of a black
would be unlikely to evoke a death sentence. Such a disparity would
rightly be considered extremely troubling.
But under the "4.3 times the odds" rule that reflects the study's actual
findings, the discrepancy between PW and PB would be far less
alarming. This yields the equation:
If PW = 0.99, the odds ratio in a white-victim case is 0.99/0.01; in other
words, a death sentence is 99 times as likely as the alternative. But even
after being cut by a factor of 4.3, the odds ratio in the case of a black
victim would take the revised value of 99/4.3 = 23, meaning that the
perpetrator would be 23 times as likely as not to be sentenced to
death. That is:
Work out the algebra and you find that PB = 0.96. In other words, while a
death sentence is almost inevitable when the murder victim is white, it is
also so when the victim is black - a result that few readers of the "four
times as likely" statistic would infer. While not all Georgia killings are
so aggravated that PW = 0.99, the quoted study found that the heavy
majority of capital verdicts came up in circumstances when PW, and thus
PB, is very high.
None of this is to deny that there is some evidence of race-of-victim
disparity in sentencing. The point is that the improper interchange of two
apparently similar words greatly exaggerated the general understanding of
the degree of disparity. Blame for the confusion should presumably be
shared by the judges and the journalists who made the mistake and the
researchers who did too little to prevent it.
(Despite its uncritical acceptance of an overstated racial disparity, the
Supreme Court's McClesky v. Kemp decision upheld Georgia's death
penalty. The court concluded that a defendant must show race prejudice in
his or her own case to have the death sentence countermanded as
discriminatory.)
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