Gary Kleck responds to Tim Lambert

Gary Kleck writes:

my position that estimates of DGUs with a wounding are unstable is
correct. The prevalence of DGUs with a wounding in the Kleck-Gertz
(K-G) survey was 0.0011 (1.326% of U.S. adults had a DGU of some
kind in the previous year, and 8.3% of DGUs involved a wounding —
see pp. 184-185 of K-G article; 0.083 x 0.01326 = 0.00110058).
Assuming simple random sampling, the 95% confidence interval
estimate of the national annual prevalence of DGUs with a wounding
would be 0.0011 +/- 1.96((.0011 x .9989)/4,977)) = 0.0011 +/- 0.0009,
or 0.000179-0.00202.

This is not the correct way to do the calculation. Kleck has
effectively assumed that there were only 0.0011*4977=5.5 cases with
woundings in the sample. If there were only that many, then the
confidence interval would indeed be as large as Kleck claims. (Well,
almost — the formula that Kleck uses doesn’t give a very
approximation to the confidence interval when the number of cases is
as small as 5.)

However, on pages 173-4 Kleck reports that there were 17 sample cases
with woundings, 8.3% of the DGUs. This lets us work out a 95%
confidence interval: 8.3% +/- 1.96 x sqrt((.083 x .917)/205)
= 8.3% +/- 3.8% or 4.5% – 12.1% of DGUs involve a wounding.

Multiplying these proportions times the U.S.
population age 18+ yields 34,438-388,673 DGUs with woundings per
year, a considerably wider range than Lambert’s miscalculations
implied.

Multiplying the percentages by Kleck’s preferred estimate of 2.5M DGUs
yields 110,000 – 300,000 DGUs with woundings per year. Allowing for
the sampling error in Kleck’s 2.5M DGU estimate and rounding things
off to avoid spurious precision in the confidence interval gives the
100,000-300,000 interval that I gave earlier.

Further, the assumption of simple random sampling tends
to understate the magnitude of sampling error for a survey that
actually used a more complex sampling design, which means that
the estimate of DGUs with a wounding is actually even more
unstable (i.e. has an even wider interval estimate) than even
these calculations imply.

The confidence intervals given on pages 166-167 assume simple random
sampling. Kleck and Gertz do not mention in their paper that because
of their sampling design they have understated the sampling error of
their DGU estimates.

In sum, my position that the Kleck-Gertz survey sample, however
adequate for estimating total DGUs, was too small for estimating
specific subtypes of DGUs such as those with a wounding (see
K-G rebuttal of David Hemenway, J. Crim. Law & Criminology,
Summer 1997,p. 1453), was correct, and Lambert is wrong.

Kleck is mistaken. Just to double check: the relative uncertainity of
an estimate of a proportion is roughly proportional to the inverse of
the square root of the number of positive cases. This implies that
the relative uncertainty of the wounding estimate (based on 17
positive cases) is about twice that of the 2.5M DGU estimate (based on
66 positive cases).

I have no idea where Lambert came up with 100,000-300,000.

I hope I have explained the derivation to everyone’s satisfaction.

However, even if it had been correct, an interval estimate
as wide as 100,000-300,000, with the upper limit three times
as large as the lower limit, is itself quite imprecise.

It is however precise enough to observe that it is not all consistent
with the estimate of 7700-18,500 that appears on page 164 of
“Targetting Guns”. It seems likely that most of the people who
reported a wounding in Kleck’s survey, did not, in fact, wound a
criminal. This could be because they were honestly mistaken, or
embellishing a real DGU or making the whole thing up.

While you would expect that some could make an honest mistake about
the matter, it seems unlikely that 90% would, especially since that
implies that when DG users fire shots at a criminal they only get a
hit a mere 5% of the time.