Deltoid

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.