Elliot,
It appears to me that Arnold Barnett is guilty
of a serious misuse of statistical argument.
I don't think readers are apt to be misled by the media
reports; there is a very LOW rate of capital punishment
in the US, so the likelihoods are (indeed) essentially
the same as Odds Ratios. (There's a 10-year
minimum stay on Death Row, and fewer than 1000
people awaiting execution, so that means they are
adding fewer than 100 per year. Ten thousand murders per
year but less than 1% get Death.) There are other comments
inserted, below.
On 16 Mar 2001 02:48:42 GMT, Elliot Cramer <[EMAIL PROTECTED]>
wrote: [note: certain equations are referred to, but are shown by my
newsreader as a gap each time. ]
> 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.
- yeah, I don't think any readers were doing that equation, so they
never get so far as concluding THAT.
When I saw mention of these data a few years ago, my first tendency
was to doubt the "what-if." P[death sentence] = 0.99? not
generally.... Rates of executions are low, as I said earlier.
But the background statistics did classify cases, so that there could
be some high marginal rates.
- So what would make the articles perfect? The articles should have
used Odds ratios; or emphasized that the 4.3 likelihood applied,
strictly, for low overall rates; or mentioned that the 4.3 applied
to 'getting off' at the other extreme.
>
> 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.
- Now, the author is asserting that 1% versus 4% is far different
from 99% versus 96%. Statisticians should be leery of that.
Yes, there are occasions when they differ: 1 versus 4 is an important
difference if you multiply the fractions times costs or benefits.
But I don't sense the relevance, when moving a fraction between
categories of 'life in prison' and 'death'.
Steve Simon posted a few weeks ago to one stats-group. He rather
likes the likelihood approach, and he was citing someone else who
does; whereas, I have posted several times about how foolish it seems
to me, both logically and mathematically, to model 'Likely' instead
of using Log-Odds.
> 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.
- the judges and journalists missed the word; they missed the math
that would have made the word important; so they ended up with the
right conclusion.
>
> (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.)
====
For what I have noticed, omitting the Odds ratio is more likely to be
abusive, than *using* it. For instance, 98% of whites will complete
certain training, and 92% of blacks, that is another 4:1 Odds ratio.
There is not much difference in terms of success-rate (or
money-invested for training); that is a big difference in failure
rate, which did seem to matter.
- I have seen that oversight in a newspaper report.
--
Rich Ulrich, [EMAIL PROTECTED]
http://www.pitt.edu/~wpilib/index.html
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