Estimating Historical Heights: CommentAuthor(s): David RothmanSource: Journal of the American Statistical Association, Vol. 77, No. 378 (Jun., 1982), p. 298Published by: American Statistical AssociationStable URL: http://www.jstor.org/stable/2287235 .

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Comment DAVID ROTHMAN*

Wachter and Trussell's estimates from London Marine Society records of boys' heights indicate some kind of step in ages 13-16 around 1820. A statistical test should be made of two alternative mechanisms for that jump:

1. Improvements in nutrition were participated in by a wide range of age groups, leading to a nearly simulta- neous and equal jump in average heights shortly there- after by all groups 13-16.

2. Improvements in prenatal or neonatal nutrition about 1805 benefited a specific cohort first, leading to a jump in age 13 for 1818 or 1819, a nearly equal jump in age 14 one year later, etc.

Actually, there is at least one other model providing re- sults intermediate between the first two:

3. Improvements in general nutrition around 1818 tended to advance the onset of puberty, leading to a large jump for those of age 13. Those of age 14 would see less of this jump, and those of age 15 and 16 would see even

* David Rothman is Senior Staff Engineer, Agbabian Associates, El Segundo, California 90245.

smaller jumps (most of these would have reached puberty already without the nutritional effect).

Model 3 is not too likely because the steady-state in- creases for all age groups appear to be similar in magnitude.

I would like to see a more complete exploration of convenient four-parameter models before resorting to the procedures of the authors. Their statement, "the pro- cesses are of the kind that simple functional forms, ex- ponentials and so forth, are going to fail to describe," is not supported by any evidence. A two-parameter normal density of heights, f1, and a two-parameter density of truncation levels, f2, would lead to the likelihood function

?(x) - f l (x) f2(Y) dy/(1 - F, (y))

where F1 is the cdf for f 1. In this way, we can estimate f1 and f2 simultaneously, leading to information about the selection process (treated as a nuisance by the authors).

? Journal of the American Statistical Association June 1982, Volume 77, Number 378

Applications Section

Comment NANCY BRANDON TUMA*

In "Estimating Historical Heights," Wachter and Trussell propose, explicate, and illustrate two new esti- mators for the mean and variance of a normal distribution when observed data are subject to a variety of problems. Most of these problems not only occur often but have been discussed on many occasions. However, I have not previously seen a statistical treatment of one problem that is quite common in data I have analyzed. The stan- dard problems considered by Wachter and Trussell are discretization (due to rounding), heaping (due to num- ber preferences of the original data recorders), and random errors (due to mistakes in transcription, etc.).

* Nancy Brandon Tuma is Professor, Department of Sociology, Stan- ford University, Stanford, CA 94305.

The "new" problem addressed in this paper is "short- fall -undercounts in the tail of a distribution that be- come proportionately smaller (and eventually vanish) near the center of the distribution. Although shortfall may occur in either tail, Wachter and Trussell concentrate on the case of shortfall in the lower tail, the problem that plagues their data and most other data familiar to me. Unlike truncation, shortfall is gradual so that an analyst cannot readily tell just where the uncontaminated region of the empirical distribution begins. With shortfall, unlike censoring, an analyst does not know the fraction of cases lacking the desired information.

? Journal of the American Statistical Association June 1982, Volume 77, Number 378

Applications Section

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