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THE UNIVERSITY OF CHICAGO

THE ECONOMICS OF NUTRITION, BODY BUILD, AND HEALTH: WAALER SURFACES AND PHYSICAL HUMAN CAPITAL

A DISSERTATION SUBMITTED TO

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CONTENTS List of Tables List of Figures Acknowledgements Chapter I Physical Human Capital 2 Waaler Surfaces 2.1 Waaler Surfaces 2 .2 000 0 0 2.000004

2.1.1 Definition of Waaler Surface and Waaler Curves

2.1.2 The Waaler Surface as a Graphical Analytical Tool

2.2 Empirical Properties of WaalerSurfaces

2.3 Stability across Age among NorwegianMen 2.4 WaalerIndex .2.2 0 0.- 0.2 000 3 Sex Discrimination 3.1 Summary StAiSHCS Ặ Q Q TQ Q Q HQ HQ v2 3.2 Waaler Curves and Surfaces for Norwegian Women 3.3 SexDIiscriminaion - Q Q Q Q Q Q Q Q QQ Q y2 3-4 Sensitivity AnalySiS Q Q Q Q Q Q Q Q Quà va 3.4.1 Sensitivity with Respect to Dispersion

3.4.2 Sensitivity with RespecttoLevels

3-5 Equivalent Variations .0000.000004 3.6 Concluding Remarks .2 0 000,4 4 Body Build as Rational Choice 4.1 The Waaler Surface as an ObjectiveFuncion

4.2 Household Production of Health and EngelsLaw ili

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iv

5 Long-term Optimization and Efficiency of Human Secular Growth 85

6 The Health of the Elderly, 1990-2035 98

6.1 ÏIntroduction Q Q Q Q HQ HQ vn s2 99

6.2 Waaler SurfacesinMorbidity IOI

6.3 Forecasting the Health oftheElderly 108

6.3.1 EmpiricalEvidence 108

6.3.2 Forecasts ofElderly Health 123

6.4 Implicatons for Health-CareCosts 129

7 Concluding Remarks 134 7.1 Summary 020.000 cee eee ee es 134 7.2 Future Research 2 0.0.2 00000002 ee eae 136 Appendix A Description of Norwegian Data 140 B_ Derivation of Waaler Surfaces 145 BI Degree ofPolynomial 145

B2 OLSandLogit Ặ Q Q Q Q HQ Q.2 156 B.3 Smoothing vs Precision: AnExample 160

B.4 Sensitivity with Respect to Risk Level and Body Build Distribution 162

B.4.I Senstivity with RespecttoRiskLevel 163

B.4.2 Sensitivity with Respect to Height and BMI Distribution 166

C Waaler Surfaces for Norwegian Adults by Age and Sex 170

D Simulation of Risk Levels 185

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2.1 2.2 2.3 3.1 3.2 3.3 3.4 3.5 4.1 6.1 6.2 6.4 6.5 A.I A.2 A.3 B.1 LIST OF TABLES Eighteen-Year Mortality by Height and Weight for Norwegian Men Aged “má ee 20

Relative Risk of 18-Year Mortality by Height and Weight among Norwegian Men Aged 45-59 2.2.2 20.0000 eee ee ee ee ee 28 Relative Risk of 18-Year Mortality by Height and BMI among Norwegian

Men Aged 45-59 1.22 eee ee ee 30

Summary Statistics by Age and Sex for Norwegian Sample 2 38 Comparison of Sex Mortality Differentials by Age between Indians and Nor-

wegians "Ha ee ee 56

Mean Anthropometric Measurements of Indian Adults, 1988~1990 58 Sensitivity of Risk Levels with Respect to Height and BMI Dispersion 62 Sensitivity of Risk Levels with Respect to Height and BMI Means 64 Equations for Predicting Basal Metabolic Rate from Body Weight 73 Comparison of the Prevalence of Chronic Conditions Among Union Army Veterans in 1910, Veterans in 1983 (reporting whether they ever had specific chronic conditions), and Veterans in NHIS 1985-1988 (reporting whether they had specific chronic conditions during the preceding 12 months), Aged 65 and Above, Percentages , 109 Projected Changes in Height (in centimeters) and BMI by Age Group, 1990—

2055 ee ee ee 124

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D.1 Comparison of Actual and Simulated Height-Weight Samples D.2 Comparison of Actual and Simulated Risk Levels

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2.1 2.2 2.3 2.4 2.5 2.6 2.7 3.1 3.2 3-3 3-4 3-5 3.6 3-7 3.8 4.1 5.1 5.2 5.3 6.1 6.2 6.3 6.4 LIST OF FIGURES Relative 18-Year Mortality Risk by Height and by Bm1 for Norwegian Men Aged 45-59 6 ee ee ee 9

Relative 18-Year Mortality Risk for Norwegian Men Aged 45-59 10

Iso-Risk Contour Map of Relative 18-Year Mortality Surface for Norwegian Men Aged 45-59 2.2.2 2 eee ee ee ee ee 12 Graphically Derived Optimal Weight Curve 14

Mathematically Derived Optimal WeightCurve 15

[so-BMI Curves in WaalerSpace L7 Example of a Waaler Diagram, Showing the Relative Risk of 18-Year Mor- tality among Norwegian Men Aged 4s-40 18

Mean Height by Age and Sex Among Norwegians 39

Mean Weight by Age and Sex Among Norwegians 42

Mean BMI by Age and Sex Among Norwegians 43

Comparison of Waaler Space for Men and Women 45

Waaler Curves in Height for Norwegian Women, Selected Ages 47

Waaler Curves in BMI for Norwegian Adults, Selected Ages 2 48

Comparison of Waaler Surfaces, Norwegian Adults Aged 45~59 50

Comparison of Waaler Surfaces for Men and Women with Similar Mortality Rates 2 AT ee ee 52 Iso-cost Map in WaalerSpace .Ố 74

Efficient Region of Body Build for Health Production on a Waaler Surface in Mortality for Norwegian Men Aged 45-59 89

Mean Height and Weight of 140 Adult Male Populations in 990 go Mean Height and Weight of French Men at Five Different Dates gI The Relationship between Height and Risk in Several Populations 102

The Relationship between BMI and Prospective Risk Among Norwegian Males Aged so-ó4 at Risk between 1963 and !979 103

Comparison of Waaler Surfaces, NHIs and Norwegian Men Aged 45-64 107

Trends in Health of Men Aged 65 and Above, NHIS 1982-1992 113

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Vill

6.5 Trends in Health of Men Aged 65-84, NHIS 1982-1992 II5

6.6 Trends in Health of Men Aged 85 and Above, NHIS 1982-1992 116

6.7 Trends in Mean Height and BMI, NHIS Men 1982~1992 118

6.8 Trends in Mean Height and BMI by Age Group, NHIS Males 1982-1992 [19

6.9 Mean Heights of White, Native-Born Males by Birth Cohort, I7IO-1970 [2I 6.10 Changes in Optimum and Mean BMI over Time, NHIS Males Aged 45-64 During the PeriodsShown 128

B.t Waaler Surface for Norwegian Men Aged 50~54, 2nd-degree Polynomial 149 B.2 Waaler Surface for Norwegian Men Aged 50~54, 3rd-degree Polynomial 1 50 B.3 Waaler Surface for Norwegian Men Aged 50~54, 4th-degree Polynomial 1 5I B.4 Waaler Surface for Norwegian Men Aged 50~54, 5th-degree Polynomial 152

B.5 Waaler Surface for Norwegian Men Aged 50-54, 6th-degree Polynomial 1 53 B.6 Waaler Surface for Norwegian Men Aged 50—54, 7th-degree Polynomial 1 54 B.7 Waaler Surface for Norwegian Men Aged 50-54, 8th-degree Polynomial 155

B.8 Comparison of Waaler Surfaces Derived by OLs and Logit 159

B.9 Optimum BMI Estimates for U.S Men Aged 45-64 from Two Different Model Specifications 2.2 2 Q Q Q Quy 161 B.10 Influence of Risk Level on the Shape of Waaler Curves 164

B.11 Three Hypothetical Height Distributions with Different Means 166

B.12 Two Height Distributions with Different Variances 168

C.I Waaler Surface for Norwegian Men Aged 4Q-44 172

C.2 Waaler Surface for Norwegian Men Aged 45-49 173

C.6 Waaler Surface for Norwegian Men Aged 65-69 .22 177

C.7 Waaler Surface for Norwegian Women Aged 40-44 178

C.g9 Waaler Surface for Norwegian Women Aged 50-54 180

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ACKNOWLEDGEMENTS

I thank the members of my thesis committee for their guidance throughout the course of writing this dissertation Robert Fogel, for first introducing this topic to me, teaching me how to do empirical research, and for always finding ways to extract the most out of my abil- ities Gary Becker, for his constant reminders to focus my attention on the economics and for enriching my thinking on economic behavior Robert Willis, for his incisive comments that clarified basic concepts that I missed or did not understand very well I feel very fortu- nate to have had such great teachers, and in the case of Bob Fogel, a great mentor as well The final product here certainly exceeds by far anything that I would have thought myself capable of writing

I would also like to acknowledge the generous help [ received from friends and col- leagues Dora Costa often provided prompt and very practical feedback Sven Wilson’s comments were no less helpful Insights that I gleaned from early discussions with Kyung- soo Choi have proved to be invaluable in finding my way later through difficult empirical issues I owe particular thanks to Chulhee Lee His constant willingness to be a patient and very thorough sounding board for new and only half-formed ideas has earned my respect both for his scholarship and for his gentle personality

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Thanks go also to several individuals and institutions who have made this research possible The National Health Survey, Oslo, Norway, collected the Norwegian data on body height and weight, which was matched subsequently to the data on mortality collected by the Central Bureau of Statistics, Oslo, Norway, by Mr Ernst Risan at the National Health Survey I would also like to express my appreciation for the efforts by Drs Kjell Bjartveit and Hans Th Waaler in making this data available Financial support from the National In-

stitute of Aging (AGI0120), the National Science Foundation (SBRQI14981), and the Center

for Population Economics, the University of Chicago, is gratefully acknowledged

My late father, my mother, and my sister have always given me their wholehearted support and encouragement To this list I would like to add my undergraduate advisors, Drs Soon Cho and Un-Chan Chung, who believed enough in me to give me a second chance

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CHAPTER 1

PHYSICAL HUMAN CAPITAL

Few of us would deny that health is an important factor in almost any discussion of human behavior In itself it is a direct determinant of one’s well-being, or utility, which is in some sense the ultimate subject matter of economics Health and longevity also have direct bearing on other matters that are of interest to economists, including productivity of the labor force, consumption and saving patterns over the life cycle, investment in human capital, and economic growth

Indeed health has been, and continues to be, identified and discussed as an important topic by economists.' Yet it is hard to avoid the impression that analytical research in health has lagged behind other areas in economics This is particularly noticeable in the human capital literature, a field that studies changes in the “quality” of the population Early pro-

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ponents of human capital theory (Schultz 1961, Becker 1964) did not forget to count as an important component of human capital the physical aspects of labor force quality, such as its health, the strength or energy level it brings to production, and its longevity, which collec- tively we may call physical human capital However, as Schultz (1970, 1992) summarizes, the focus of human capital research has predominantly been on the intellectual aspect of labor quality,* such as education, skills, knowledge, and technology Besides an intellectual demand for symmetry and completeness in complementing intellectual human capital theory with a theory of physical human capital, there is also a practical need to be able to understand the interactions between the biological and economic processes that underlie both the great improvements in health and longevity achieved during this century among devel- oped countries, and also the unexpected rapid changes in morbidity and mortality patterns currently observed among their elderly citizens

The need to introduce and integrate biological evidence into economic analysis, rather than simply adapt existing human capital theory to deal with an exogenous process of health trends or analyze some abstract endogenous “health stock”, is illustrated by the debate on quality of life in the 1980s Upon discovering that mortality rates among the old (age 6s and over) and particularly among the old-old (age 85 and over) had started to decline unexpect- edly, some thoughtful analysts suggested that the increase in lifespan was largely owed to better but costly medical technology that kept alive “marginal survivors” who would pre-

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viously have died sooner after contracting a disease or developing a chronic condition.3 This viewpoint implicitly assumes that increased lifespan comes mainly from technological advances in medical intervention and focuses our attention on a tradeoff between quantity (length of life) and quality Although such effects undoubtedly do exist, a growing body of evidence since then suggests that gains are being made in both quantity and quality: the elderly in the U.S not only live longer, but are also healthier (Manton et al 1993, Fogel et al 1993, Waidmann et al 1995) Such evidence in turn raises the question of whether factors other than medical intervention could possibly have a significant influence on secular trends in health and longevity Thus it appears that to be able to understand this unexpected trend in morbidity and mortality among the old and the old-old and project its future course, as well as consider the implications for a number of economic issues, one requires a synthesis of biological evidence and economic analysis

Some of the difficulty in formulating an analytical framework for physical human capital lies in identifying suitable measures of both health and its determinants Ideally, the measures should be unambiguously quantifiable and comparable across populations, both in space and time We also require that the relationship between health and its determinants be stable, be in a form that is exploitable by economic analysis, and also be capable of explaining a significant part of the differences in health among populations both present and past In short, we need some stylized biological evidence around which an analytical frame-

work can be built

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CHAPTER 2 WAALER SURFACES

The material presented in this chapter is an attempt to contribute to the use of anthropometric measures in economics and other disciplines’ by identifying and clarifying certain empirical regularities in the body build-health relationship and formulating that relationship in a conceptual framework that is amenable to the tools of economic analysis After investigating height and weight separately as predictors of mortality during the 1960s and 1970s among Norwegian men and women, Waaler (1984) suggests that it would be possible to use height and weight jointly to produce a height-weight-mortality surface I follow this lead and show how such a surface, which I will call a Waaler surface, can be estimated and how to interpret the results using graphical methods that are familiar to economists

The first section following this introduction lays out the basic tool of analysis I introduce the concept of the Waaler surface and explain how to represent such a surface on a diagram that will allow us to probe the properties of the body build-health relationship in

'Steckel (1995) reviews the literature on height in economic history Also sce Floud et al (1990) and Fogel (1993) Behrman and Deolalikar (1988) survey research on health and nutrition in developing countries, including a brief summary on the use of anthropometric measures For the use of height and weight in the biomedical literature, see Tanner (1978), Eveleth and Tanner (1990), and Kushner (1995) James and Ralph

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detail This is followed by a section on the empirical properties of the Waaler surface, using as an example a Waaler surface in mortality from a sample of Norwegian men I show that the height-weight-risk relationship can be characterized as a convex function, which means that height-weight-health is concave The next two sections introduce some concepts, or themes, that we will be repeatedly exploring throughout the subsequent chapters: stability of the Waaler surface across populations and time, and the Waaler index, which will be used to measure the effectiveness of the Waaler surface in explaining differences in health between populations

2.1 Waaler Surfaces

2.1.1 Definition of Waaler Surface and Waaler Curves

Since I intend to rely heavily on graphical methods of analysis, it is helpful to think of the body build-health relationship as a continuous surface in three-dimensional space— height, weight, and risk—with risk on the vertical axis Each point on the surface represents a height-weight pair and the degree or severity of a risk event predicted by that pair The risk measured by such a surface will typically be some measure, or index, of health For example, the risk may be the mortality rates associated with different heights and weights for a certain age-sex group of a population, or it could be some index of general health, or even the odds of contracting a specific disease or chronic condition within some specified period

*The National Health Interview Survey asks survey respondents to rate their own health from five cate- gories: excellent, very good, good, fair, or poor An example of a Waaler-surface risk might be the odds of

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of time It is also possible to think of a surface that shows the odds of some behavior or outcome that is significantly affected by morbidity or mortality, such as, e.g., a surface showing the risk by height and weight of labor-force-nonparticipation among older men I will call such height-weight-risk surfaces Waaler surfaces

In order to facilitate comparisons of Waaler surfaces from different populations, or even different types of risk from the same population, risk levels in Waaler surfaces are typically given in relative risks, rather than absolute, or crude, risks Therefore the predicted risk for any height-weight pair on a Waaler surface gives the risk level relative to the mean risk over all heights and weights of the population from which the surface was derived

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The Waaler surface may be thought of as a three-dimensional extension of the two-dimensional risk-by-height or risk-by-weight tables widely used in medical literature, examples of which are shown in Figure 2.1 Alternatively, the risk-by-height and risk-by-weight tables can also be derived from Waaler surfaces if the height-weight distribution is known:

RH(H) = / R(H.z)ƒ(H.z) dz, and

oo

RWW) = / R(y, W) f(y W) dy,

oo

where RH(-) and RW(-) give the relative risk by height and weight, respectively Thus risk- by-height and risk-by-weight tables can be thought of as marginal curves derived from a

Waaler surface; therefore I will call them Waaler curves

2.1.2 The Waaler Surface as a Graphical Analytical Tool

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FIGURE 2.1

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II

the Waaler surface as explained above, it is not surprising that such properties of the surface are essentially the same as those observed in the Waaler curves in Figure 2.1

However, more detailed analysis of the relationship between body build and risk is possible with a Waaler surface than with Waaler curves For instance, Waaler curves cannot show how the relationship between risk and BMI} changes over different heights Neither is it possible to say whether or not the excess risk of a shorter height will dominate the excess risk of, say, overweight when the mortality risk of a short but fit person is compared to that of a taller but very overweight person Such questions can be answered using a Waaler surface, especially when it is drawn as a contour map on two-dimensional space showing height and weight (let us call this Waaler space), rather than as a three-dimensional diagram of the type shown in Figure 2.2 Figure 2.3 presents a contour map, which is a two-dimensional representation of the surface shown in Figure 2.2 Each contour on this diagram is an iso- mortality-risk curve that gives the locus of height-weight pairs that yield a constant predicted risk level.4 Since risk is a “bad”, unlike utility or production surfaces commonly encoun- tered in economic analysis, the contour map shows a surface that is convex to the origin—the “inner” contours represent lower risk levels

An advantage of the Waaler contour map is the ease with which a precise optimal-weight- for-height curve can be derived using graphical techniques already familiar to economists To find the weight that minimizes risk at a given height, one simply draws a horizontal line

3A measure of relative weight See definition below

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Height

(m)

FIGURE 2.3

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13 (representing constant height) on the diagram to find the contour whose nadir touches that line The point where the contour and the horizontal line touch (and are tangent to each other) gives the weight that minimizes risk at that height Any contour above that line (and the lower risk represented by that contour) is not attainable at that height, and therefore a person of that height cannot lower his risk any further than the level represented by the tan- gential contour Thus the optimal weight-for-height curve must pass through each of the nadirs of the contours, and the curve can be graphically derived by simply connecting the lowest points of the contours.5

If the risk function for a Waaler surface is known, mathematical derivation of the optimum weight curve is also straightforward Given a risk function, R(H W), the optimum weight curve is given implicitly by the points that satisfy the first-order condition for mini- mizing risk with respect to weight:

_ ðR(H,W)

0

aw

Figures 2.4 and 2.5 show examples of optimum weight curves derived by both methods Because weight is positively correlated to height, it is often convenient to use a different measure of weight that conveys more accurately the degree of obesity One such measure of weight-for-height commonly used in the medical and economics literature is BMI (Body Mass Index), which is defined as body weight in kilograms divided by the square of body

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16

height in meters The primary advantage of BMI as a measure of relative weight-for-height is that it is known to be only very weakly correlated to height.° Conversion from height and weight into BMI units can be done using a map of iso-BMI curves in height-weight space, such as that shown in Figure 2.6.7 It is important to note that, whereas the iso-risk contours and the optimum-weight curve show relationships that are empirically observed and hence not necessarily invariant, the iso-BMI curves are entirely definitional, and not behavioral Indeed, the iso-BMI contours may be interpreted as a mapping of a different axis onto height- weight space

While it is possible to examine the properties of a Waaler surface using only a contour map of iso-risk curves, it is often easier to identify key features of the surface and interpret their meaning in terms of BMI units when the three different types of curves (iso-risk contours, optimum-weight curve, and iso-BMI curves) are superimposed upon each other in a single diagram Figure 2.7 shows an example of such a diagram

Strictly speaking, a Waaler surface is the height-weight-risk surface itself, of which the contour map is but one of the many possible ways of representing it However, a diagram that shows the contour levels of the surface together with the optimum-weight curve and the isO-BMI curves is clearly the most useful way of showing the surface, since that particular format allows for the greatest level of detail Thus the term “Waaler surface” is implicitly understood to mean such a diagram as an analytical tool

®This property of BM holds in cross-sectional analysis of a population [In time-series or cross-country data, BMI and height tend to be positively correlated See Figures 5.2 and 5.3 in Chapter 6 for examples

7From the definition of BMI, BMI = kg/m’, the equation for an iso-BMI curve for BMI = k can be derived

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FIGURE 2.7

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19 2.2 Empirical Properties of Waaler Surfaces

Let us derive a Waaler surface from actual data and see what are the notable empirical properties Table 2.1 gives the 18-year mortality rate by height and weight for a large sample of Norwegian men aged 45-59 at the time of measurement.’ Each cell in this table shows the 18-year relative mortality rate,? 18-year (absolute) mortality rate, and the cell size

The table shows that relative mortality risk drops as height increases This property holds both in the overall for all weights (the leftmost column in the table) and for specific weight ranges (for each column) As for the relationship between relative mortality risk and weight, it can be seen from the rows of the table that risk is very high at low weights, decreases as weight increases, and rises again beyond a certain weight level, but not as steeply as in the lower-weight region The relationship can be characterized as an asymmetric U- shape Again, such a relationship holds whether one looks at overall weight (the bottom- most row) or weight at specific heights These properties are interesting mainly because they agree with and thus confirm previous studies which investigated the hei ght-mortality and weight-mortality relationship separately However, they fail to provide new insight

Let us move on to Waaler-surface analysis Applying a smoothing procedure’® to the data used to derive Table 2.1 yields a surface estimate, which we have already seen in Fig- ure 2.7 The tradeoffs between height and weight in terms of mortality risk are easier to

®The data set is described in Appendix A

9The cell-specific mortality rate divided by the (weighted) mean rate over all cells, 0.246 By definition, the (weighted) mean of the relative risk thus calculated has to be equal to one

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2I

examine in that diagram By definition, any two points on the same iso-risk contour on a Waaler surface represent height-weight combinations that yield the same predicted risk Thus as one moves along an iso-risk contour, the height-weight tradeoff is such that predicted risk remains constant In Figure 2.7, this means that height does not matter as much as weight in regions where BMI is very low or very high In those regions, the iso-risk contours run more or less parallel to (as opposed to an angle that cuts directly across) the iso-BMI curves Therefore the change in height, holding BMI constant along an iso-BMI curve, required to move to a different iso-risk curve is comparatively larger than the change in BMI to move to a different iso-risk curve when height is held constant Conversely, weight does not matter as much as height at medium-range BMI levels around the optimum weight curve

The behavior just described of the slope of the iso-risk contours at different regions in Waaler space is a necessary consequence of having iso-risk contours that are U-shaped that decrease in risk level as height increases Since the iso-risk contours cannot intersect,'' U- shaped curves must be nested The iso-risk contour map therefore consists of concentric U-shaped curves Figure 2.7 shows that the U-shape iso-risk contours are slanted to the right

Another characteristic that Waaler surface analysis reveals is that the optimal BMI defined as the BMI that minimizes mortality risk for any given height, differs with height Fora person with a height of 160 cm, a BMI level around 25 minimizes mortality risk However, at 185 cm, the optimal BMI is close to 23 Although BMI rather than raw weight is often

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used in weight studies as a means for controlling for height, my results show that hei ght is still significant Optimal BMI decreases with height, and therefore weight recommendations based on BMI alone would be misleading

A stronger result of the inverse relationship between height and risk can be shown witha Waaler surface than with a Waaler curve in height only Returning to Figure 2.7, the lowest relative risk level that a group of short men at 169 cm can attain is 1.0 when all of them are at the optimum BMI level slightly below 25 Compare this to the risk level among taller men at 180 cm At that height, the relative risk level is 1.0 or lower for anyone with a BMI level between 21 and 27 This range is large enough that if the mean BMI among men 180 cm in height is on the optimum weight curve, roughly 70 percent!? of them would have a risk level equal to or less than 1.0, which is attainable by the 169 cm group only under the most favorable conditions

The weight-risk relationship in Waaler curves in weight, characterized as an asymmetric U-shape curve, can also be seen in the surface diagram At any height, the risk penalty for deviating from the optimum weight is higher when the deviation is negative (i.e., underweight) than positive (i.e., overweight) In Figure 2.7 this can be seen from the fact that the distance between the iso-mortality-risk contours are greater in the overweight region than in the underweight Hence at any height a greater variation in weight is required for the un-

'*With a standard deviation of about 3 for the BMI distribution, a range from 21 to 27 BMI would cover about

2 standard deviations, or one standard deviation in either direction around the mean If we use the standard normal distribution as a rough approximation (the BMI distribution is right-skewed), the interval within one

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23 derweight than the overweight to reach the next iso-mortality curve The implication is that the risk of being underweight exceeds the risk of being overweight

Another important point in the way weight affects risk is that as weight deviates in either direction from the optimum level, risk becomes increasingly sensitive to weight This can be seen in the Waaler surface from the increasingly shorter horizontal distance between the iso- risk contours as weight deviates from the optimum level This property can be characterized as convexity of the weight-risk relationship

A similar property holds for height At low height levels, the vertical distance between iso-risk contours is smaller than at taller heights Unlike weight, however, it is not clear whether there is an optimum height at which risk is minimized Table 2.1 suggests the exis- tence of an optimal height around 195 cm, beyond which risk levels will start rising again However, although the decline in risk evidently flattens out around 195 cm, the cell sizes in Table 2.1 for heights above 190 cm indicate that, even with such a large sample, whether or not risk starts rising again beyond that height cannot be conclusively determined because of small-cell problems

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2.3 Stability across Age among Norwegian Men

The most immediately noticeable finding thus far is probably the fact that optimum BMI varies with height It is also the finding that is the most directly applicable Many weight Studies implicitly assume that the influence of height is fully controlled for when weight is converted into BMI units Indeed most of the tables that show recommended weight levels for different heights simply take a single recommended BMI standard (usually the range 22— 25) that is applicable to all heights and translates it back into recommended height-specific raw weight levels in kilograms or pounds If optimal BMI levels vary by height, as we have shown, weight targets based on such tables would be misleading and could potentially have large practical consequences for health, especially among very tall or short people The analysis in the previous section also shows that it is somewhat safer to err on the side of being overweight

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25

Besides convexity, there are two additional issues that bear on the usefulness of Waaler surfaces in economic analysis The first of these is the issue of stability,'3 by which I mean verifying that the properties of the Waaler surface we have just identified can be generalized beyond the particular case we have examined The other issue, relevance, will be addressed later on using the Waaler index, introduced in the next section The balance of this section considers stability across age among men in the Norwegian sample."4

In order to see if and how risk behaves differently from the Waaler surface in the previous section for 18-year mortality among Norwegian men aged 45-59, [ derived Waaler surfaces for men by 5-year age groups from the Norwegian data, also using 18-year mortality status as the dependent variable Not surprisingly, the Waaler surface loses predictive power beyond a certain age In the case of Norwegian men, surfaces for men 70 years of age or older had very little predictive power,'S which showed up as very flat surfaces with only a few iso-mortality-risk contour curves The surfaces for younger age groups also failed to show consistent behavior Although some parts of the Waaler surfaces for 5-year age groups from ages 15-19 up to ages 35~39 did show the right-slanted U-shaped iso-risk contours that imply convexity, overall the surfaces showed risk that behaved erratically without any regularity In contrast to age groups above 70, where Waaler surfaces could not be derived

"Besides providing a set of stylized facts upon which a theoretical framework of analysis can be built, stability indirectly addresses the issue of endogeneity Properties that are stable across a wide variety of factors such as income, race, culture, disease regime and health environment, and medical technology are not very likely to be the endogenous outcome of interactions within a system, and hence can be used as exogenous factors in building a theoretical model

‘4Later chapters will investigate other factors that can affect the shape of the Waaler surface, such as sex type of risk, and different populations

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26

because the mortality rates over an observation period of 18 years are too high, it appears that too few deaths are recorded at these younger ages for any regularity to emerge '®

The surfaces for 5-year age groups from 40-44 to 65-69 are shown in Appendix C Al- though there are variations in surface shape between age groups, the overall appearance is similar In particular, each surface shows that risk is a convex function of height and weight, with risk decreasing with height, and behaving in a U-shape relationship with weight The notable difference among the age groups is that the surface gets flatter as age increases, which can be seen from the wider distances between iso-mortality contours in the surfaces for older age groups.'?

While the similarities just noted among Waaler surfaces at different ages do hold broadly, there are also apparent differences or exceptions that cannot be dismissed so conveniently Let us take a moment to consider possible reasons for such irregularities and see if we can reconcile those irregularities with the hypothesis that the Waaler surface is relatively stable Earlier when discussing whether or not an optimum height exists, we saw from Table 2.1 that risk beyond very tall heights around 195 cm could not be estimated reliably because of sample size problems Generalizing this lesson, it is not difficult to see that the risk predicted by a Waaler surface will have different degrees of reliability in different regions of the surface Looking again at Table 2.1, estimates of risk in a height-weight cell with a large number of observations will be fairly reliable, whereas risk estimates in cells that fall into the tails of the

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27 height-weight distribution will have small-cell problems.'® In the extreme case, some parts of the Waaler surface will show risk predictions for height-weight cells that have no observations in the sample from which the surface was derived Clearly when that happens, the risk predicted for such a height-weight region by the Waaler surface should be understood to be a best-guess extrapolation from other, more densely populated regions, and appropriate caution must be exercised in using that prediction

Table 2.2 reproduces the risk by height and weight shown previously in Table 2.1, but in more detail in order to see how the sample distribution affects the reliability of risk estimates Instead of height and weight cells by 10cm and to kg, cells of § cm and 5 kg are used in this table For purposes of illustration, suppose we adopt the somewhat arbitrary criterion that cells of size 100 or larger are needed to get reliable risk estimates The marginal distribution at the leftmost column shows that a Waaler curve in height can be reliably estimated from 150 cm to 199 cm under this criterion Similarly, the bottom row shows that a Waaler curve in weight can be derived from 45 kg to 119 kg If we look at the full distribution however, rather than the marginals, it is clear that surface estimation is more sensitive to sample size For example, none of the cells in the rows for height 150-154 cm and 195-199 cm are large enough now to meet the 100-observation criterion The same is true for cells in the columns for weight 45-49 kg and 115-119 kg Therefore for surface estimation, if we apply the 100- observation criterion, the “necessary condition” for height and weight to yield reliable risk

'8The risk estimate for each cell is the mean risk among observations in the cell The tange of the confidence

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