A restricted maximum likelihood estimator for truncated height samples

Economics and Human Biology 2 (2004) 5–19

A restricted maximum likelihood estimator for truncated height samples Brian A’Hearn Department of Economics, Franklin & Marshall College, Box 3003, Lancaster, PA 17604-3003, USA Received 4 December 2003; received in revised form 4 December 2003; accepted 4 December 2003

Abstract A restricted maximum likelihood (ML) estimator is presented and evaluated for use with truncated height samples. In the common situation of a small sample truncated at a point not far below the mean, the ordinary ML estimator suffers from high sampling variability. The restricted estimator imposes an a priori value on the standard deviation and freely estimates the mean, exploiting the known empirical stability of the former to obtain less variable estimates of the latter. Simulation results validate the conjecture that restricted ML behaves like restricted ordinary least squares (OLS), whose properties are well established on theoretical grounds. Both estimators display smaller sampling variability when constrained, whether the restrictions are correct or not. The bias induced by incorrect restrictions sets up a decision problem involving a bias–precision tradeoff, which can be evaluated using the mean squared error (MSE) criterion. Simulated MSEs suggest that restricted ML estimation offers important advantages when samples are small and truncation points are high, so long as the true standard deviation is within roughly 0.5 cm of the chosen value. © 2003 Elsevier B.V. All rights reserved. JEL classification: C4; I1; N0 Keywords: Truncated-normal distributions; Maximum likelihood estimation; Mean squared error; Anthropometrics; Height estimation

1. Introduction Human height is a widely used indicator of biological living standards in many different settings, such as underdeveloped economies, historical contexts, and circumstances in which economic indicators are either unreliable or completely lacking, as for the children of Soweto (Cameron, 2003), for example. Physical stature is positively correlated with net E-mail address: [email protected] (B. A’Hearn). 1570-677X/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.ehb.2003.12.003

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nutrition—the balance between the quantity and quality of nutrient intake and the demands on those resources by the human organism for growth, metabolic maintenance, work, and for resistance to diseases. Of course, individual heights depend more on genetic potential than nutrition, but at the population level environmental factors play a very substantial role in determining mean height at a given age (Bogin, 1999). Statistical analysis of height data is facilitated considerably by the approximately normal distribution of genetic potential within a population—a regularity so consistent that it may be considered a biological law. The height of a randomly selected individual can therefore be modeled as the sum of a population mean that depends systematically on factors affecting average net nutrition, and a random deviation that is normally distributed. The estimators considered in this paper all proceed from this premise. A fact that is perhaps less widely appreciated is that the dispersion of heights around their mean is relatively constant; the standard deviation of heights varies over a range of only about 1 cm, while mean heights can easily vary by more than 15 cm across populations or within a population over time (Cole, 2000, 2003; Komlos and Baur, 2003).1 This paper explores the significance of this regularity for estimating mean heights in historical samples.

2. The problem of truncation Historical height data drawn from military records are often truncated from below. Most armies imposed a minimum height requirement, and prior to the introduction of universal conscription, records typically contain measurements only for recruits who were actually enlisted.2 The requirement was often not strictly enforced, and varied across units or over time, leaving the distribution with a deficient but not clearly-truncated left tail, a problem known as shortfall (Watchter and Trussell, 1982). The estimators considered here effectively discard the deficient portion of the sample, reducing the problem again to one of truncation. Correct identification of the point at which shortfall begins is frequently an important practical problem. The several procedures employed for this purpose have been surveyed by Komlos (2003b) and their sampling performance evaluated through simulation studies (Heintel, 1996a,b; Baten and Heintel, 1995). In what follows it is assumed that the truncation point has been correctly identified. In the presence of truncation, the ordinary least squares (OLS) estimator is biased, as illustrated in Fig. 1. The graph depicts a sample of (y∗ , x), where yi∗ = xi + εi , ε ∼ N(0, 1), is a latent variable indicated by open circles. Also plotted are the corresponding values of the truncated variable yi = yi∗ if yi∗ > 0, yi unobserved otherwise, which are indicated by crosses. The dashed line in Fig. 1 depicts the OLS regression of y on x in this sample; it clearly underestimates the true slope of the population regression function E(yi∗ |xi ) indicated by the solid line through the origin. This bias toward zero is typical of OLS slope estimates in truncated samples (Ruud, 2000, p. 805). The bias is caused by the behavior of the random disturbance term ε, which is the vertical distance between y∗ and the 1 More specifically, the standard deviation of heights appears to range between 6 and 7 cm among modern male populations, between 5.3 and 6.5 cm among females (Cole, 2000, p. 402). 2 Truncation from above is also observed on occasion, but will not be dealt with here.

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Fig. 1. OLS bias in a truncated sample.

population regression line. In the truncated sample, ε does not have a mean of zero (being instead positive), is correlated with x (being larger, the smaller is x), and has a variance that is not constant (being lower for small x). Three of the standard OLS assumptions are thus violated. Data on quantitative variables that determine height, such as income, number of disease attacks, or protein consumption, are seldom available at the individual level. Typically, analysis focuses on estimating differences in means across groups whose net nutrition can be expected to have differed, for example, birth cohorts, birth regions, or occupations. In this case too, truncation creates problems for estimation of the population parameters of interest. Fig. 2 illustrates the effect of truncation at 0 of a continuum of normal random variables with a standard deviation of 1 and means varying from −2 to 2 along the horizontal axis. The curve indicates the truncated mean as a function of the population mean µ, E(yi |µ), while the line through the origin has a slope of 45◦ , indicating points where population and truncated mean would coincide. When the population mean is well above the truncation point τ (to repeat, zero in Fig. 2), the distortion induced is relatively minor. The truncated mean function asymptotically approaches the population mean (the 45◦ line) on the right of Fig. 1 and is quite close for values of µ, two standard deviations or more above τ. As the mean falls closer to and eventually below τ, the bias grows; with truncation at zero, the truncated mean can never be negative, so it asymptotically approaches zero instead on the left of Fig. 1. OLS applied to a sample truncated from below is clearly biased upward as an estimator of the population mean.

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Fig. 2. The truncated mean function.

3. The truncated-normal density The truncated-normal probability density function (p.d.f.) describes conditional probabilities: the probability that y falls within some range of values, conditional on y∗ exceeding the truncation point τ, so that y is observed. Suppose y∗ is given by yi∗ = µ + εi , ε ∼ N(0, σ 2 ), and y by yi = yi∗ if yi∗ > τ, yi not observed otherwise. Conditional on being observed, the relative probabilities of y falling in ranges of values above the truncation point are still described by (1/σ)φ((y − µ)/σ), where φ is the standard normal p.d.f. The probability of y being observed is the probability that yi∗ > τ: Pr(yi∗ ) > τ = Pr(ε > τ − µ) = 1 − Φ((τ − µ)/σ), where Φ is the standard normal cumulative distribution function. From the definition of conditional probability, it follows that the truncated-normal p.d.f. is: f(yi ) = (1/σ)φ((yi − µ)/σ)/(1 − Φ((τ − µ)/σ)) for yi > τ, and f(yi ) = 0 otherwise. A visual intuition for this specification is provided in Fig. 3, which gives the p.d.f. for a standard normal random variable and for the same variable truncated at zero. The missing probability mass from the lower tail of the normal distribution is reallocated such that the area under the truncated p.d.f. is one. The relative probabilities of different outcomes in the range (0, ∞) are unchanged. 4. Estimators for truncated-normal samples The truncated-normal p.d.f. is the basis of the maximum likelihood estimator (MLE), which chooses those parameter values that maximize the average of the logarithm of f(yi ) = (1/σ)φ((yi −µ)/σ)/(1−Φ((τ −µ)/σ)) for the sample data. The truncated-normal MLE has

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Fig. 3. Truncated and standard normal densities.

a unique maximum, is consistent, asymptotically efficient and normally distributed (Ruud, 2000, pp. 801, 805; Cohen, 1991, pp. 5, 17; Amemiya, 1973).3 Early work in anthropometric history typically favored other methods. In an influential article, Wachter and Trussell (1982) presented an alternative procedure which they argued was more robust to anomalies common in historical height data, in particular deficient but not clearly-truncated sampling. Their quantile bend estimator (QBE) can be interpreted as fitting a normal curve adjusted for lower tail deficiency to the sample histogram, and adopting the mean and standard deviation of this artificial distribution as estimates of the population parameters. The QBE proved to have several disadvantages, however, including cumbersome application and unsuitability for multivariate analysis (Komlos, 1989). Komlos and Kim (1990) further criticized QBE-estimated time trends as displaying excessive short-term variability: fluctuations that were implausible, given biological limits to the short run variability in the physical stature of a population. Subsequent simulation studies by Heintel (1996a,b, 1997a,b) confirmed that the QBE is not efficient (suffers higher sampling variability) relative to alternative estimators, and its popularity has waned.4

3 It is also not limited in application by regularity conditions. Use of the ML method for truncated-normal variables dates back to a 1931 article by Fisher, building on earlier work by Pearson using the method of moments (Cohen, 1991, pp. 2–3). Practical implementation of the MLE relied on the use of standardized tables to convert truncated sample moments into population parameter estimates (Cohen, 1991, Chapter 2). 4 Heintel has shown that the truncated ML estimator offers superior mean squared error performance (Heintel, 1996a,b) and that the KK method is more reliable in estimating the sign of time trends. See, in addition, Komlos (1985, 1989) and Heintel et al. (1998) for empirical investigations confirming the unreliability of the QBE.

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Komlos and Kim (KK) proposed an estimator they argued was both simpler and more robust: the use of sample means after appropriate truncation of the data, later generalized in Komlos (1989) to OLS for multivariate analysis. 5 Truncated OLS produces biased estimates of the population parameters, of course, but is unbiased in estimating the truncated population mean. And since the truncated population mean is a monotonic function of the overall population mean (Fig. 2), the sign of differences in truncated sample means across groups or over time is a reliable (unbiased) estimator of the sign of the true population differences. The KK method’s advantages of simplicity and reduced sensitivity to departures from normality were real (Heintel, 1996a,b, Table 2), but offset in part by several disadvantages. To infer the magnitude of differences in population means from differences in truncated sample means, it was necessary to reintroduce the normality assumption, along with an assumption about the population standard deviation, which was not estimated directly (Komlos and Kim, 1990, p. 118). (The KK method with this conversion procedure turns out to be identical to the restricted MLE described below).6 Nor did the KK method yield the variance–covariance matrix of estimates, useful for hypothesis testing. Recently, as use of the QBE has declined in favor of the MLE (Twarog, 1997; Heintel et al., 1998; A’Hearn, 2003). In addition to possessing the desirable asymptotic properties described previously, the MLE is flexible, accommodating complexities such as truncation points that change over time, for example; it directly estimates the population standard deviation; it provides the variance–covariance matrix of estimates; it is robust to the presence of rounding in the data;7 and it is easy to employ since the incorporation of commands for truncated-normal ML estimation into software packages such as STATA and EViews. Still, the performance of the MLE in practical applications, in which it sometimes yields implausibly wide estimated fluctuations in heights over short periods of times or across relatively homogeneous groups, motivates research into further refinements and alternatives.8 The remainder of the paper considers such an extension. 5 Komlos and Kim originally advocated applying the highest known truncation point used by any region, military unit, or time period. In this way, the artificial impact of potentially different shortfall patterns in the sub-samples is avoided by “equalizing” the bias over the complete sample see (Heintel and Baten, 1998, footnote 17, for an interesting study of artificial correlations if one fails to equalize the bias). Subsequently, the KK method has been used with other truncation rules, such as Heintel’s (1996a,b) truncation point estimator. This procedure first smoothes the sample histogram using a kernel density estimator, then identifies τˆ as the point where the estimated density’s slope is maximal (i.e., where its first difference is greatest). 6 Simulation results show that converting the truncated sample mean to a population mean estimate assuming a population standard deviation of 6.86 cm and a normal distribution (as in Komlos, 2003a) yields essentially identical estimates to the restricted MLE discussed in this paper. 7 Heintel (1997b) shows that rounding poses few difficulties. Heaping, or the clustering of observations around particular values such as even integers, may be more of a problem; see Komlos (1999) for a preliminary investigation. Note that the truncation point used must also be adjusted for rounding. If measurements were to the nearest 0.5 cm, for example, the truncation point in Eq. (6) should be set at τˆ − 0.25 cm. 8 In the cited papers by Twarog and A’Hearn, height estimates fluctuated by as much as 2–3 cm across 5-year birth cohorts, a figure as large as cumulative changes expected over about two decades of improving (deteriorating) living standards. In the absence of famine, wars, or comparable events and their aftermaths, such dramatic and temporary fluctuations in height of a population are implausible.

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Table 1 Mean squared error of the restricted MLE τ

σ

n = 250

n = 500

n = 1000

Bias2

Variance

MSE

Bias2

Variance

MSE

Bias2

Variance

MSE

167

6.00 6.50 6.86 7.00 7.50 8.00

6.105 1.023 0.000 0.082 2.226 6.759

0.620 0.692 0.591 0.613 0.571 0.555

6.725 1.714 0.594 0.695 2.797 7.314

6.180 0.952 0.000 0.123 2.323 6.918

0.383 0.301 0.314 0.300 0.279 0.271

6.563 1.253 0.315 0.423 2.602 7.190

6.122 0.932 0.000 0.112 2.374 6.855

0.187 0.153 0.157 0.151 0.135 0.135

6.309 1.085 0.158 0.263 2.509 6.990

165

6.00 6.50 6.86 7.00 7.50 8.00

4.350 0.720 0.000 0.096 1.731 4.967

0.574 0.534 0.526 0.536 0.537 0.489

4.923 1.254 0.528 0.632 2.268 5.456

4.334 0.725 0.000 0.103 1.530 5.038

0.282 0.259 0.254 0.261 0.278 0.251

4.616 0.984 0.255 0.364 1.808 5.289

4.293 0.669 0.000 0.083 1.759 5.086

0.148 0.128 0.131 0.126 0.123 0.108

4.440 0.796 0.131 0.209 1.882 5.194

163

6.00 6.50 6.86 7.00 7.50 8.00

2.857 0.498 0.000 0.040 1.305 3.648

0.436 0.449 0.428 0.483 0.408 0.405

3.293 0.947 0.428 0.523 1.713 4.053

2.821 0.462 0.000 0.052 1.305 3.699

0.222 0.213 0.218 0.249 0.206 0.197

3.043 0.674 0.218 0.301 1.511 3.896

2.823 0.462 0.000 0.059 1.194 3.651

0.105 0.114 0.113 0.116 0.107 0.105

2.928 0.575 0.113 0.175 1.301 3.757

161

6.00 6.50 6.86 7.00 7.50 8.00

1.830 0.281 0.000 0.027 0.768 2.528

0.353 0.383 0.405 0.352 0.406 0.398

2.183 0.664 0.406 0.379 1.174 2.926

1.750 0.304 0.000 0.037 0.678 2.541

0.166 0.193 0.199 0.189 0.188 0.186

1.916 0.497 0.199 0.226 0.866 2.727

1.760 0.282 0.000 0.045 0.844 2.553

0.091 0.092 0.098 0.093 0.094 0.093

1.851 0.374 0.098 0.138 0.938 2.646

159

6.00 6.50 6.86 7.00 7.50 8.00

1.057 0.184 0.000 0.020 0.565 1.657

0.286 0.325 0.322 0.317 0.320 0.348

1.343 0.509 0.322 0.337 0.885 2.005

1.062 0.240 0.000 0.022 0.542 1.712

0.157 0.162 0.171 0.157 0.166 0.161

1.219 0.402 0.171 0.180 0.708 1.873

1.017 0.172 0.000 0.025 0.520 1.682

0.074 0.076 0.085 0.079 0.088 0.084

1.091 0.247 0.085 0.104 0.607 1.767

Note: Each combination reported here is the mean of 2000 simulations.

5. Restricted ML estimation Accurate estimation of population parameters is increasingly difficult, the higher the truncation point. While the asymptotic unbiasedness and efficiency of the MLE do not depend on either τ or µ, its variance increases steadily with the degree of truncation. This pattern is difficult to establish theoretically in the absence of an analytic expression for the truncated-normal MLE, but has been demonstrated repeatedly in simulation studies (Heintel, 1996a,b; Wachter and Trussell, 1982; and the results given below). It is intuitively clear that once τ ≥ µ, so that the sample mode coincides with the truncation point, it becomes difficult to distinguish between alternative possible distributions underlying the observed sample. Only the relative curvature of the sample histogram—something easily distorted by small

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departures from normality due to random sampling or heaping—differentiates the upper one half of a distribution with µ = τ from, say, the upper one third of a distribution with a slightly lower mean and higher standard deviation. As the ML estimators of mean and standard deviation are not independent, prior information about one could be useful in more accurately estimating the other. It is not possible to make general statements about the performance of MLEs under parameter restrictions, but intuition can be drawn from results for least squares. The OLS estimator incorporating linear restrictions on the parameters (i.e., forcing linear combinations of the parameters to equal a particular value) can be shown to suffer from less sampling variability than its unconstrained version (Judge et al., 1988, pp. 235–240). This is true whether the restrictions are correct or not. In the latter case, the estimator is biased but the reduction in variance remains, creating a bias–precision tradeoff. It seems reasonable to conjecture that the restricted MLE behaves similarly in truncated-normal settings. In this case, the relative constancy of the standard deviation of heights remarked upon in the introduction could be exploited to more accurately estimate population mean heights. In practice, of course, σ is never known. And whenever an incorrect restriction is imposed some degree of bias is induced. The value of 6.86 cm has been suggested as plausible for males based on data for modern populations, but any such rule of thumb will never be exactly correct (Frisancho, 1990; Cole, 2000). The optimal choice of estimator thus depends on how

4

3 150

mse

2

155

160

1

0 6

τ

165 6.5

7

σ

7.5

8

Fig. 4. Mean squared error (cm2 ) of the restricted MLE (n = 500). Note: MSE values >4 not shown.

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we evaluate the tradeoff between bias and precision. A common criterion for balancing these risks is mean squared error: ˆ 2 + var(µ). ˆ MSE(µ) ˆ = E[(µ ˆ − µ)2 ] = bias(µ) If the restrictions imposed are close to true, the bias induced will be small, and the reduction in variance substantial. So the choice between restricted and unrestricted estimation depends on the researcher’s degree of confidence in the restrictions. The extent of the tradeoff between bias and variance is an empirical question, depending on sample size (n), τ and σ. We explore this tradeoff below using simulation methods and provide a practical guide to the circumstances in which the restricted MLE is an attractive alternative to unconstrained ML estimation. Table 2 Mean squared error of the unconstrained MLE τ

σ

n = 250 Bias2

Variance

n = 500 MSE

Bias2

Variance

n = 1000 MSE

Bias2

Variance

MSE

167

6.00 6.50 6.86 7.00 7.50 8.00

0.000 0.000 0.000 0.000 0.000 0.000

9.431 9.559 10.728 12.476 16.214 12.233

9.431 9.559 10.728 12.476 16.214 12.233

0.000 0.000 0.000 0.000 0.000 0.000

3.894 4.450 4.771 4.704 4.836 6.515

3.894 4.450 4.771 4.704 4.836 6.515

0.000 0.000 0.000 0.000 0.000 0.000

1.604 1.875 1.970 2.083 2.191 2.468

1.604 1.875 1.970 2.083 2.191 2.468

165

6.00 6.50 6.86 7.00 7.50 8.00

0.000 0.000 0.000 0.000 0.000 0.000

4.474 4.669 5.575 5.936 5.913 7.776

4.474 4.669 5.575 5.936 5.913 7.776

0.000 0.000 0.000 0.000 0.000 0.000

1.898 2.051 2.470 2.392 2.819 3.146

1.898 2.051 2.470 2.392 2.819 3.146

0.000 0.000 0.000 0.000 0.000 0.000

0.821 1.015 1.123 1.183 1.336 1.391

0.821 1.015 1.123 1.183 1.336 1.391

163

6.00 6.50 6.86 7.00 7.50 8.00

0.000 0.000 0.000 0.000 0.000 0.000

1.876 2.374 2.959 2.907 3.468 4.594

1.876 2.374 2.959 2.907 3.468 4.594

0.000 0.000 0.000 0.000 0.000 0.000

0.841 1.078 1.318 1.289 1.646 1.988

0.841 1.078 1.318 1.289 1.646 1.988

0.000 0.000 0.000 0.000 0.000 0.000

0.420 0.507 0.589 0.644 0.712 0.864

0.420 0.507 0.589 0.644 0.712 0.864

161

6.00 6.50 6.86 7.00 7.50 8.00

0.000 0.000 0.000 0.000 0.000 0.000

0.909 1.215 1.537 1.465 1.907 2.355

0.909 1.215 1.537 1.465 1.907 2.355

0.000 0.000 0.000 0.000 0.000 0.000

0.440 0.588 0.691 0.697 0.927 1.139

0.440 0.588 0.691 0.697 0.927 1.139

0.000 0.000 0.000 0.000 0.000 0.000

0.217 0.290 0.348 0.375 0.422 0.558

0.217 0.290 0.348 0.375 0.422 0.558

159

6.00 6.50 6.86 7.00 7.50 8.00

0.000 0.000 0.000 0.000 0.000 0.000

0.478 0.664 0.832 0.896 1.100 1.422

0.478 0.664 0.832 0.896 1.100 1.422

0.000 0.000 0.000 0.000 0.000 0.000

0.240 0.337 0.364 0.393 0.525 0.667

0.240 0.337 0.364 0.393 0.525 0.667

0.000 0.000 0.000 0.000 0.000 0.000

0.123 0.158 0.193 0.205 0.271 0.332

0.123 0.158 0.193 0.205 0.271 0.332

Note: Each combination reported here is the mean of 2000 simulations.

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4

3 150

mse

2

155

160

1

0 6

τ

165 6.5

7

7.5

σ

8

Fig. 5. Mean squared error (cm2 ) of the unconstrained MLE (n = 500). Note: MSE values >4 not shown.

6. MSE performance: simulation results The simulation results reported here are based on 2000 replications each for a range of n, σ, and τ.9 For the sake of direct applicability to practical research situations, the ranges for each parameter were chosen as representative of values likely to be encountered in historical height data: µ = 165 cm throughout; σ varies between 6 and 8 cm by increments of 0.5 cm (with special consideration of 6.86 cm); τ varies from 150 to 167 cm; n = 250, 500, and 1000. The restricted MLE imposes the restriction that σ = 6.86 cm in all cases, permitting assessment of the bias–precision tradeoff, while the unconstrained procedure estimates the parameter freely. Restricted ML results are summarized in Table 1, and illustrated in Fig. 4. They indicate that: (a) MSE and both of its constituent elements increase with increasing τ; (b) MSE is high if τ > µ; (c) if σ = 6.86 cm then E(µ ˆ = µ), with bias increasing very rapidly as the σ restriction error exceeds about 0.5 cm; and (d) the variance of the estimates decreases as n increases, while bias is roughly constant. Table 2 and Fig. 5 present the results for unconstrained ML estimation. They indicate that: (a) the bias of the MLE estimator is zero throughout, so that MSE is driven by variance alone; 9

The random heights generated for these simulations were continuous, i.e., not rounded, differing in this way from typical historical height samples.

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Table 3 Mean squared error: unconstrained − restricted MSE τ

σ

MSE difference n = 250

n = 500

n = 1000

167

6.00 6.50 6.86 7.00 7.50 8.00

2.706 7.845 10.134 11.781 13.417 4.919

−2.669 3.197 4.456 4.281 2.234 −0.675

−4.705 0.790 1.812 1.820 −0.318 −4.522

165

6.00 6.50 6.86 7.00 7.50 8.00

−0.449 3.415 5.047 5.304 3.645 2.320

−2.718 1.067 2.215 2.028 1.011 −2.143

−3.619 0.219 0.992 0.974 −0.546 −3.803

163

6.00 6.50 6.86 7.00 7.50 8.00

−1.417 1.427 2.531 2.384 1.755 0.541

−2.202 0.404 1.100 0.988 0.135 −1.908

−2.508 −0.068 0.476 0.469 −0.589 −2.893

161

6.00 6.50 6.86 7.00 7.50 8.00 6.00 6.50 6.86 7.00 7.50 8.00

−1.274 0.551 1.131 1.086 0.733 −0.571 −0.865 0.155 0.510 0.559 0.215 −0.583

−1.476 0.091 0.492 0.471 0.061 −1.588 −0.979 −0.065 0.193 0.213 −0.183 −1.206

−1.634 −0.084 0.250 0.237 −0.516 −2.088 −0.968 −0.089 0.108 0.101 −0.336 −1.435

159

Note: Positive numbers indicate superior MSE performance of the restricted estimator.

(b) MSE rises with τ, but much more rapidly than was the case for the restricted estimator (Table 1); MSE at τ = 167 is more than 10 times its value at τ = 159; (c) variance decreases with sample size, yielding a sharper decrease in MSE than for the restricted estimator since bias plays no role in its determination; and (d) MSE rises with the underlying population standard deviation σ.10 Clearly, for small n and τ ≥ µ, the unconstrained MLE is not reliable. At the extremes of τ = 167, σ = 8.0, and n = 250, MSE is 12 cm2 . This implies a root MSE of 3.5 cm, which is very large relative to the differences typically observed over

The increase in MSE with σ is seen in the twisting of the surface in Fig. 5. This effect is pronounced for high values of τ. 10

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4

150

2

mse

155 0 160

-2 6

τ

165 6.5

7

σ

7.5

8

Fig. 6. Unconstrained minus restricted MSE (n = 500). Note: MSE differences <−2 not shown.

time or across groups in historical data. Note that MSE values are off the scale (>4) for τ ≥ 166 at the higher values of σ in Fig. 5. Comparison of the performance of the restricted and unconstrained estimators confirms the intuition that restricted ML exhibits the same behavior as restricted OLS. The restricted estimator has consistently lower variance, even when the σ = 6.86 cm restriction is incorrect, but is biased in this case. In some situations, it may be favorable to trade unbiasedness for precision. Table 3 and Fig. 6 map out those situations, reporting differences between unconstrained and restricted MSEs. (Positive numbers indicate superior performance of the restricted estimator.) It is clear from Fig. 6 that if the σ restriction is approximately true (within roughly 0.5 cm) and the truncation point exceeds about 160 cm (or a point about one standard deviation below the mean), the restricted estimator offers substantially better performance. Considering Table 3, the choice is clearest at the extremes (the upper left and lower right corners). At truncation points well above the mean in small samples, the restricted estimator offers dramatically better precision, far outweighing its bias. In contrast, at truncation points well below the mean in large samples, unconstrained estimation is generally preferred; it performs less well only in the immediate neighborhood of σ = 6.86 cm, and then only slightly. In historical datasets, truncation points are typically less than but close to the mean, while sample sizes are relatively small due to being drawn manually from archival records. Figs. 7a–c depict the MSE for each estimator in several typical situations: t = 163 cm,

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5.0

4.0 Unrestricted mse 3.0 (cm 2 )

2.0 1.0 Restricted

0.0 6

6.5

(a)

7 σ (cm)

7.5

8

4.0

3.0 mse (cm 2 )

Unrestricted 2.0

1.0 Restricted 0.0 6

6.5

(b)

7

σ (cm)

7.5

8

7.5

8

8.0 Unrestricted 6.0

mse (cm 2 )

4.0 Restricted 2.0

0 6

(c)

6.5

7

σ (cm)

Fig. 7. MSE of Restricted and Unconstrained MLE: (a) τ = 163 cm, n = 250; (b) τ = 163 cm, n = 500; (c) τ = 165 cm, n = 250.

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n = 250; t = 163 cm, n = 500; and t = 165 cm, n = 250. The curves are cross sections of the surfaces in Figs. 4 and 5. It is again evident that the optimal choice of estimator depends on the degree of confidence in the σ restriction. If one is confident that σ is within a half-centimeter or so of 6.86, the restricted MLE is clearly preferable. The reduction in risk can be substantial in small samples: on the order of about 2 cm2 when the restriction is approximately correct. That implies a reduction of about 1.4 cm in the expected error (i.e., root MSE). This is large, relative to likely true differences over time and across groups.

7. Conclusion Researchers in anthropometric history often face a difficult estimation problem in inferring the parameters of a normal distribution from a small sample, truncated from below at a point close to the population mean. The smaller the sample and the greater the degree of truncation, the greater is the variance of all estimators. The ML estimator, though possessing optimal asymptotic properties and several practical advantages over alternative methods, is no less vulnerable than the rest. The simulation studies summarized here show that the restricted ML estimator behaves similarly to the restricted OLS estimator, whose properties are well established. In particular, it exhibits smaller sampling variability than the unconstrained MLE. This suggests that the relative constancy of the standard deviation of heights across groups and over time can be exploited to improve the accuracy of estimates of the mean. The restricted OLS estimator has reduced sampling variability, regardless of whether the restrictions imposed are correct, but is biased in the latter case. The simulations show that the restricted ML estimator shares the same properties. With no restrictions ever likely to be exactly true in practice, this sets up a decision problem involving a bias–precision tradeoff. The tradeoff can be evaluated according to the minimum mean squared error criterion. Simulation results indicate that the restricted estimator’s precision is a decisive advantage if the truncation point is close to or above the population mean and with small sample sizes—situations typical of research with historical height data. Choice of estimation technique will depend on the degree of confidence in the restriction that the historical population’s standard deviation is near a particular value such as the modern figure of 6.86 cm. If that restriction is true to within roughly 0.5 cm, the restricted ML estimator is more reliable.11 Acknowledgements I thank John Komlos for encouraging me to pursue this project and for helpful discussions throughout its completion. I also thank Stephen Digaetano for cheerful and expert research assistance. 11 The restricted and unconstrained MLE can be used together, as in A’Hearn (2003), where the sample includes some periods in which the truncation point is well below sample modes and others where they coincide. Unconstrained estimation in the periods with a low truncation point establishes a standard deviation that is imposed in restricted estimation for the others.

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References A’Hearn, B., 2003. Anthropometric evidence on living standards in northern Italy, 1730–1860. J. Econ. Hist. 63, 351–381. Amemiya, T., 1973. Regression analysis when the dependent variable is truncated normal. Econometrica 41, 997–1016. Baten, J., Heintel, M., 1995. Zum Problem der Verteilungen mit Shortfall bei der Nutzung des Indikators Durchschnittliche Körpergröße. Hist. Soc. Res. 20, 135–154. Bogin, B., 1999. Patterns of Human Growth. Cambridge University Press, Cambridge. Cameron, N., 2003. Physical growth in a transitional economy: the aftermath of South Africa apartheid. Economics and Human Biology 1, 29–42. Cohen, C., 1997. Truncated and Censored Samples. Marcel Dekker, New York. Cole, T.J., 2000. Galton’s midparent height revisited. Ann. Hum. Biol. 27, 401–405. Cole, T.J., 2003. The secular trend in human physical growth: a biological view. Econ. Hum. Biol. 1, 161–168. Frisancho, R.A., 1990. Anthropometric Standards for the Assessment of Growth and Nutritional Status. University of Michigan Press, Ann Arbor. Heintel, M., 1996a. Historical height samples with shortfall: a computational approach. Hist. Comput. 8, 24–37. Heintel, M., 1996b. Estimating means and trends of normal samples with shortfall. Unpublished Working Paper, University of Munich. Heintel, M., 1997a. Height samples with shortfall. A persisting problem revisited. Unpublished Working Paper, University of Munich. Heintel, M., 1997b. Über einige statistische Probleme und Möglichkeiten der anthropometrischen Geschichte. In: Baten, J., Denzel, M. (Eds.), Wirtschaftsstruktur und Ernährungslage, 1770–1879. Scripta Mercaturae, St. Katharinen, pp. 108–151. Heintel, M., Baten, J., 1998. Smallpox and nutritional status in England, 1770–1873: On the difficulties of estimating historical heights. Economic History Review 51, 360–371. Heintel, M., Sandberg, L., Steckel, R., 1998. Swedish historical heights revisited. New estimation techniques and results. In: Komlos, J., Baten, J. (Eds.), The Biological Standard of Living in Comparative Perspective. Franz Steiner, Stuttgart, pp. 449–458. Judge, G., Hill, R.C., Griffiths, W., Lütkepohl, H., Lee, T.-C., 1988. Introduction to the Theory and Practice of Econometrics, 2nd ed. Wiley, New York. Komlos, J., 2003a. An anthropometric history of early-modern France. Eur. Rev. Econ. Hist. 7, 159–189. Komlos, J., 2003b. How to (and how not to) analyze deficient height samples. Unpublished Working Paper, University of Munich. Komlos, J., 1999. On the nature of the malthusian threat in the eighteenth century. Econ. Hist. Rev. 52, 730–748. Komlos, J., 1989. Nutrition and Economic Development in the Eighteenth Century Habsburg Monarchy. Princeton University Press, Princeton. Komlos, J., 1985. Stature and nutrition in the Habsburg monarchy: the standard of living and economic development in the eighteenth century. Am. Hist. Rev. 90, 1149–1161. Komlos, J., Baur, M., 2003. From the fattest to the tallest: the fate of the American population in the 20th century. Unpublished Working Paper, University of Munich. Komlos, J., Kim, J.H., 1990. Estimating trends in historical heights. Hist. Methods 23, 116–120. Ruud, P., 2000. An Introduction to Classical Econometric Theory. Oxford University Press, Oxford. Twarog, S., 1997. Heights and living standards in Germany, 1850–1939: the case of Württemberg. In: Steckel, R., Floud, R. (Eds.), Health and Welfare during Industrialization. University of Chicago Press, Chicago. Watchter, K., Trussell, J., 1982. Estimating historical heights. J. Am. Stat. Assoc. 77, 279–293.