Contour Maps of Demographic Surfaces

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W O R K I N G P A P E R

CONTOUR W S OF

DEXOGRAPHIC

SURFACES

James W . Vaupel B r a d l e y A. Gambill Anatoli I. Yashin Alan J . B e r n s t e i n

!lay

19

85 WF-85-33

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I n t e r n a t i o n a l I n s t i t u t e for Applied Systems Analysis

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NOT FOR QUOTATIOK WITHOUT PERMISSION OF THE AUTHORS

CONTOUR MAPS OF DEMOGRAPHIC SURFACES

James W. Vaupel Bradley A. Gambiil Anatoii I . Yashin Alan J . Bernstein

About the authors James W . Vaupel and Anatoli I . Yashin are r e s e a r c h s c h o l a r s and Bradley A. Gambill and Alan J. Bernstein are r e s e a r c h assistants in t h e Population Program, led by Nathan Keyfitz, a t t h e International Institute f o r Applied Systems Analysis in Laxenburg, Austria. Vaupel is also Professor of Public Affairs and Planning a t t h e University of Minnesota, USA; Yashin is a senior r e s e a r c h e r a t t h e Institute f o r Control Sciences in Moscow, USSR; and Gambill and Bernstein are students a t D u ~ e University, USA.

Please adriresc correspondence to: James W . Vaupel, International Institute f o r Applied Systems Analysis, Laxenburg, A-2361 Austria.

Working Papers are interim r e p o r t s on work of t h e International Institute f o r Applied Systems Analysis and have received only lim- ited review. Views o r opinions expressed herein do not neces- sarily r e p r e s e n t those of t h e Institute o r of its National Member Organizations.

INTERNATIONAL IKSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria

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CONTOUR MAPS OF DEMOGRAPHIC SURFACES

James W . Vaupel, Bradley A . Gambill, Anato!i I. Yashin and Alan J . Bernstein

INTRODUCTION

Contour maps, which are widely used in depicting s p a t i a l p a t t e r n s , c a n b e r e a d i l y a d a p t e d t o r e p r e s e n t a n y s u r f a c e t h a t i s defined o v e r two dimensions. In p a r t i c u l a r , a n a r r a y of demographic d a t a c a n often b e p i c t u r e d in a n intelligibie a n d g r a p h i c a l l y s t r i k i n g way by a c o n t o u r map. The d a t a might p e r t a i n t o population l e v e l s o r t o rates of f e r t i l i t y , m a r r i a g e , d i v o r c e , migration, morbidity, o r mortality. Most o f t e n t h e d a t a are s t r u c - tured. by a g e and time-e.g., age-specific mortality rates o v e r time-but in some c a s e s o t h e r dimensions might b e used, such as l i f e e x p e c t a n c y o r c r u d e b i r t h r a t e . Contour maps permit visualization of demographic s u r - f a c e s a n d o f f e r a panoramic view impossible t o o b t a i n from t h e usual g r a p h s of l e v e l s o r rates at s e l e c t e d a g e s o v e r time o r a t s e l e c t e d times o v e r a g e . F u r t h e r m o r e , a c o n t o u r map is often s u p e r i o r t o a three-dimensional p e r - s p e c t i v e plot ir! providing a c l e a r , y e t r i c h r e p r e s e n t a t i o n of a demographic s u r f a c e ; i t i s usually difficult on a three-dimensional p l o t t o d i s c e r n t h e e x a c t position of t h e s u r f a c e a b o v e t h e a g e and time dimensions and t h r e e -

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dimensional plots become confusing if made t o o d e t a i l e d , especially when displayed on a moderately-priced monitor o r p r i n t e r . Contour maps a r e p a r t i c u i a r l y e f f e c t i v e in highlighting p a t t e r n s in t h e i n t e r a c t i o n of a g e , p e r i o d , and c o h o r t e f f e c t s .

Contour maps h a v e been used only occasionally by d e m o g r a p h e r s , p e r h a p s b e c a u s e of t h e computational e f f o r t r e q u i r e d o r b e c a u s e of t h e lack of detailed d a t a o v e r long s t r e t c h e s of a g e and time. In t h e i r influen- t i a l study of c h a n g e s in d e a t h r a t e s o v e r time, Kermack, McKendrick, and McKinlay (1934) superimpose on t h r e e of t h e i r t a b l e s some rough lines t h a t a r e , in e f f e c t , c o n t o u r s of r e l a t i v e mortality. The p i o n e e r i n g study by Dela- p o r t e (1941) includes a set of c o n t o u r maps t h a t summarize mortality pat- t e r n s in s e v e r a l E u r o p e a n c o u n t r i e s ; Federici (1955) d i r e c t s a t t e n t i o n t o D e i a p o r t e ' s c o n t o u r maps in h e r s u r v e y of demographic methods. R e c e n t a d v a n c e s in c o m p u t e r s , including t h e development of powerful micro- c o m p u t e r s , as well as t h e collection a n d publication of e x t e n s i v e a r r a y s of demographic s t a t i s t i c s f o r single y e a r s of a g e a n d single y e a r s of time (e.g., Natale a n d Bernassola (1973), Vallin (1973), H e u s e r (1976) and Veys (1983)), should lead t o g r e a t e r u s e of demographic c o n t o u r maps in t h e f u t u r e .

This p a p e r p r e s e n t s a bouquet of c o n t o u r maps t o s u g g e s t t h e b r o a d p o t e n t i a l of t h e i r use in demographic s t u d i e s . E v e r y p i c t u r e p r e s e n t e d could s e r v e as t h e b a s i s f o r a thousand words o r more of explanation and analysis, b u t h e r e w e merely s e r v e up t h e maps as i l l u s t r a t i o n s of t h e method. F o r a n example of how such maps c a n b e used in demographic analysis, s e e t h e s t u d y by Caselli, Vaupel, and Yashin of Italian mortality (1985).

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L E L S . SHADES. ArYD GRIDS ILLUSTRATED BY ITALIAN MALE MORTALITY F i g u r e l a d i s p l a y s t h e c o n t o u r s of mortality for Italian males from a g e 0 t o 79 and f o r y e a r s 1 8 7 0 t o 1 9 7 9 . The map i s b a s e d on mortaiity rates, q , f o r single y e a r s of a g e a n d time assembled by N a t a l e and B e r n a s s o l a (1973) and Caselli (forthcoming). Data are d i s c r e t e , b u t a s u r f a c e i s continuous:

t h e s u r f a c e q (x , y ) c a n b e defined by l i n e a r l y i n t e r p o i a t i n g between adja- c e n t d a t a points. The v a l u e s of q (x , y ) give t h e h e i g h t of t h e mortality s u r - f a c e o v e r a g e x and time y

.

The l i n e s on a c o n t o u r map c o n n e c t a d j a c e n t points t h a t a r e of equal h e i g h t ; t h e s e l i n e s are sometimes called level l i n e s o r isograms. In F i g u r e l a , o n e of t h e l e v e l l i n e s r e p r e s e n t s a mortality r a t e of a b o u t 11 p e r c e n t : t h e line starts in 1 8 7 0 at a g e 3 5 a n d e n d s in 1 9 7 9 at a g e 5 6 , indicating t h a t 56-year-old Italian mer! in r e c e n t y e a r s f a c e d t h e same c h a n c e of mortality t h a t 35-year-olds f a c e d a b o u t a c e n t u r y a g o .

An i m p o r t a n t c o n s i d e r a t i o n when designing a c o n t o u r map i s how many d i f f e r e n t l e v e l s t o u s e . The c o m p u t e r p r o g r a m t h a t we employed t o draw t h e maps, which was developed by Gambill u n d e r Vaupel's d i r e c t i o n at Duke University and a t t h e I n t e r n a t i o n a l I n s t i t u t e for Applied Systems Analysis, allows l i n e s t o b e drawn a t up t o 15 i e v e l s , s e p a r a t i n g t h e s u r f a c e i n t o 1 6 t i e r s . Use of fewer l i n e s s a c r i f i c e s d e t a i l , w h e r e a s u s e of m o r e Lines t e n d s t o make t h e map i e s s inteiligibie: 15 l e v e l s i s a r e z s o n a b l e compromise, although u s e of 1 0 o r 2 0 l e v e l s might b e c o n s i d e r e d . D e l a p o r t e d r a w s lines a t 1 9 ! 2 0 , o r 2 1 l e v e l s on h i s v a r i o u s maps of E u r o p e a n m o r t a l i t y ; a number of t h e f i g u r e s in t h i s p a p e r , including F i g u r e s 1 5 a n d 1 6 , u s e fewer t h a n 1 5 levels. F i g u r e l b p r e s e n t s t h e c o n t o u r s of Italian male mortality using 1 0 leve!s r a t h e r t h a n t h e 1 5 l e v e l s used in F i g u r e 1.

Which s p e c i f i c e l e v a t i o n s t h e c o n t o u r lines should c o n n e c t i s a second.

i m p o r t a n t design decision. On mortality s u r f a c e s , w h e r e mortality r a t e s might a p p r o a c h a minimum of t h e o r d e r of magnitude of 0.0001 and a max-

imum of 1, u s e of equally s p a c e d lines-say a t 0.01, 0.02, a n d s o on up t o 0.15--results in a map w h e r e t h e c o n t o u r s a r e clumped t o g e t h e r a t t h e youngest and o l d e s t a g e s , with a l a r g e l y empty e x p a n s e in-between. F i g u r e l c i l l u s t r a t e s t h i s f o r Italian male mortality. The map i s far m o r e informa- t i v e when t h e l i n e s a r e s p r e a d o u t at c o n s t a n t multipies--e.g., e a c h line

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r e p r e s e n t i n g a level 5 0 p e r c e n t h i g h e r t h a n t h e p r e v i o u s line, as in F i g u r e l a . Alternatively, a convenient s c a l e c a n b e used: D e l a p o r t e p l a c e s his lines a t levels of mortality of 1, 2 , 3, ..., 9 , 1 0 , 1 2 , 15, 20, 3 0 , 5 0 , 1 0 0 , 1 5 0 , 200, 250, 300, 3 5 0 , and 400 p e r thousand, and in s e v e r a l f i g u r e s in t h i s p a p e r , including F i g u r e s 5 and 33, level lines are s e l e c t i v e l y p l a c e d at convenient levels.

D e m o g r a p h e r s o f t e n work with t r a n s f o r m a t i o n s s u c h as t h e log o r logit, s o i t might seem r e a s o n a b l e to t r a n s f o r m t h e s u r f a c e q ( z , y ) i n t o t h e s u r - f a c e o f , s a y , log q ( z , y ) and t h e n t o draw !eve! l i n e s at equa! iritervals o n t h e t r a n s f o r m e d s u r f a c e . If t h e t r a n s f o r m a t i o n i s monotonic, l i k e t h e log or logit t r a n s f o r m a t i o n , a n i d e n t i c a l c o n t o u r map c a n b e drawn by s p a c i n g t h e level lines at a p p r o p r i a t e l y unequal i n t e r v a l s on t h e o r i g i n a l s u r f a c e . In t h e c a s e of l o g a r i t h m s , t h e !eve1 lines should b e at multiples of e a c h o t h e r r a t h e r t h a n being equally s p a c e d . Thus, t h e map in F i g u r e l a c a n a l s o b e i n t e r p r e t e d a s depicting log mortality r a t e s .

An innovation in t h e c o m p u t e r p r o g r a m we used i s t h e shading of r e g i o n s a c c o r d i n g t o t h e h e i g h t of t h e s u r f a c e . The shading v a r i e s from l i g h t t o d a r k as t h e s u r f a c e s r i s e from low t o high l e v e l s of mortality. Such shading, which i s time-consuming t o d o by hand b u t e a s y with t h e h e l p of a computer-, makes t h e o v e r a l l p a t t e r n of a mortality s u r f a c e more immedi- ate!y c o m p r e h e n s i b l e , e s p e c i a l l y if t h e map i s viewed at a d i s t a n c e . A t t h e same time, t h e d e t a i l s of small p e a k s and p i t s and of t h e twists and t u r n s of t h e c o n t o u r s lines are s t i l l t h e r e t o b e s c r u t i n i z e d at c l o s e r a n g e . L i t e r a - t u r e , c r i t i c s n o t e , c a n b e p r o f i t a b l y r e a d at d i f f e r e n t l e v e l s of u n d e r s t a n d - ing; w e s u g g e s t t h e r e a d e r t r y viewing F i g u r e l a a n d p e r h a p s some of t h e o t h e r f i g u r e s in t h i s p a p e r at l e v e l s of 5 m e t e r s a n d 2 5 c e n t i m e t e r s .

Sometimes i t i s useful t o d r a w a g r i d on a c o n t o u r map s o t h a t t h e c o o r - d i n a t e s of v a r i o u s p o i n t s c a n b e conveniently l o c a t e d . In F i g u r e I d t h e map in F i g u r e l a i s r e d r a w n with a superimposed g r i d e v e r y twenty y e a r s of time a n d a g e . The g r i d d e t r a c t s a b i t from t h e underlying p a t t e r n - - t h a t i s t h e p r i c e of adding additional information. Grids are included in some of t h e maps p r e s e n t e d l a t e r in t h i s p a p e r .

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To see g e n e r a l t r e n d s i t may b e helpful t o s u p p r e s s t h e c o n t o u r lines in a map of a population s u r f a c e . In Figure l e t h e map in Figure l a is r e d r a w n with shading but without lines. A!ternatively, o n e could draw a traditiona!

c o n t o u r map with lines but without shading. Figure I f displays s u c h a map f o r Italian male mortality. The lines in Figure I f a r e not labelled but t h e y could b e .

SMOOTHED MAPS

It is useful t o t a k e a c l o s e look a t t h e small black blemishes isolated from c o n t o u r l i n e s on a c o n t o u r map, b e c a u s e t h e s e d a r k s p o t s indicate outliers--very localized p e a k s o r pits--that might b e d u e t o e r r o n e o u s d a t a values. Consider, f o r i n s t a n c e , t h e black s p o t in Figure l a at a b o u t a g e 5 4 a n d y e a r 1878: i t t u r n e d o u t t h a t t h i s blemish w a s indeed p r o d u c e d by a n e r r o r made in t r a n s c r i b i n g t h e Italian mortality d a t a t o a computer t a p e . (The e r r o r w a s c o r r e c t e d , b u t w e l e f t t h e s p o t as a n illustration.) On t h e o t h e r hand, t h e mark at a b o u t a g e 20 in 1962 r e p r e s e n t s a point where t h e mortality s u r f a c e b a r e l y c r o s s e s a c o n t o u r level, l i k e t h e t o p of a sea mount t h a t a p p e a r s a s a small island just rising a b o v e t h e l e v e l of t h e s u r r o u n d i n g o c e a n .

In addition t o t h e s e blemishes, some rough black blots are s m e a r e d a c r o s s l e v e l lines in Figure l a . These r e p r e s e n t v i r t u a l p l a t e a u s where t h e morta!ity s u r f a c e is r e p e a t e d l y c r o s s i n g and r e c r o s s i n g a l e v e l line. To eliminate t h i s kind of noise a n d t o s u p p r e s s t h e d e t a i l s of locai fluctuations s o t h a t global p a t t e r n s c a n b e more c l e a r l y p e r c e i v e d , i t may b e useful t o smooth a s u r f a c e . D e i a p o r t e p r e s e n t e d both raw and smoothed c o n t o u r maps of mortality rates in v a r i o u s E u r o p e a n c o u n t r i e s : on h i s "adjusted"

maps, D e l a p o r t e drew smooth c o n t o u r lines b a s e d on h i s feeling f o r t h e d a t a . We used a mechanistic, computer algorithm t o p r o d u c e t h e smoothed map shown in Figure I g . In t h i s map t h e height of t h e s u r f a c e a t a g e z in y e a r y w a s r e p i a c e d by t h e a v e r a g e of t h e 25 h e i g h t s in t h e 5 by 5 s q u a r e of points from z -2 t o z +2 a n d from y -2 t o y +2. On t h e e d g e s of t h e map, where a full 5 by 5 a r r a y of d a t a points i s not a v a i l a b l e , t h e smoothing p r o - c e d u r e a v e r a g e s t h e a v a i l a b l e d a t a .

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Instead of smoothing by a v e r a g i n g o v e r a 5 by 5 s q u a r e , a l a r g e r ( o r s m a l l e r ) s q u a r e might b e used. In Figure l h we smoothed Italian male mor- t a l i t y on a n 11 by 11 s q u a r e . Global p a t t e r n s in t h i s map a r e somewhat c l e a r e r t h a n in F i g u r e l g but some i n t e r e s t i n g l o c a l d e t a i l is lost and e f f e c t s t h a t a r e c o n c e n t r a t e d in time o r a g e , s u c h as infant mortality a n d mortality d u r i n g t h e 1 9 1 8 Spanish influenza epidemic, are s m e a r e d o u t .

A v a r i e t y of a l t e r n a t i v e smoothing p r o c e d u r e s might b e used, including p r o c e d u r e s t h a t r e p l a c e points by a weighted a v e r a g e of a d j a c e n t points, t h e weights diminishing with d i s t a n c e . Figure l i p r e s e n t s a map of Italian male moi-tality smoothed by a n algorithm in which t h e weights given t o t h e points in a 5 by 5 s q u a r e w e r e p r o p o r t i o n a l to:

Thus t h e points in t h e c o r n e r s of t h e s q u a r e were given weights of 1/256, w h e r e a s t h e point in t h e c e n t e r r e c e i v e d a weight of 36/256. The t h e o r e t i - c a l a d v a n t a g e s of s u c h weighted smoothing algorithms ( s e e Tukey (1977) f o r a n i n t r o d u c t o r y discussion) h a v e t o b e balanced a g a i n s t t h e c o n c e p t u a l sim- plicity a n d computational convenience of t h e kind of s t r a i g h t f o r w a r d a v e r a g i n g il!ustratec! in F i g u r e s l g and h . Note t h a t in F i g u r e l i t h e s u r f a c e is r e d u c e d by two y e a r s along e a c h e d g e b e c a u s e t h e smoothing p r o c e d u r e used r e q u i r e s a full 5 by 5 a r r a y of d a t a ; specia! modification of t h e p r o - c e d u r e could b e made, analogous t o t h e modification of t h e smoothing p r o - c e d u r e used t o p r o d u c e F i g u r e s l g a n d h , s o t h a t a weighted smoothing could handle d a t a points up t o t h e e d g e s of t h e s u r f a c e .

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CLOSE-UPS

-4s discussed by Casel!i, Vaupe! and Yashin (1985), t h e p a t t e r n s of male mortality in Italy from a g e s 1 0 t o 49 f o r y e a r s 1 9 1 0 t o 1 9 6 9 r e v e a l some i n t e r e s t i n g c o h o r t e f f e c t s . Figures 2 a and 2b p r e s e n t c o n t o u r maps of t h i s r e s t r i c t e d a g e and time arez: t h e maps c a n b e c o n s i d e r e d a n enlargement o r ciose-up of a s e c t i o n of t h e map in Figure l a . Thus c o n t o u r maps c a n b e used both t o display a l a r g e d a t a a r r a y a n d a l s o t o f o c u s in on s e l e c t e d p o r - tions of t h e a r r a y . Note t h a t in Figure 2 a t h e c o n t o u r s a r e drawn at dif- f e r e n t , more n a r r o w l y s p a c e d l e v e l s t n a n t h e c o n t o u r s in Figure l a , but in 2b t h e y a r e drawn at t h e same i n t e r v a l s as in l a : p a r t of t h e a d v a n t a g e of a close-up is t h a t if t h e h e i g h t of t h e s u r f a c e v a r i e s l e s s in t h e r e s t r i c t e d r e g i o n being s c r u t i n i z e d , t h e n t h e levei lines c a n b e l o c a t e d at c l o s e r i n t e r - v a l s t o r e v e a l more l o c a l d e t a i l .

M A P S FROM INTERPOLATED DATA

The mortality rates f o r Italian males used in F i g u r e s 1 a n d 2 are avail- a b l e by singie y e a r of a g e a n d single y e a r of time. Frequently demogra- p h e r s h a v e t o work with l e s s finely-spaced d a t a ; mortality rates, for i n s t a n c e , may b e a v a i l a b l e e v e r y d e c a d e o r s o , by five-year a g e c l a s s e s . Figure 3 displays t h e evolution of Italian male mortality based on d a t a published in P r e s t o n , Keyfitz, and Schoen (1972). Data sets from t h i s s o u r c e w e r e a v a i i a b l e f o r 1 8 8 1 , 1 8 9 1 , 1 9 0 1 , 1910, 1921, 1931, 1960, and 1964.

Death rates w e r e given f o r five-year a g e c a t e g o r i e s from a g e 5 up t o a g e 8 0 , as well as f o r a g e z e r o and t h e f o u r - y e a r c a t e g o r y from a g e 1 to 5. W e c o n v e r t e d t h e n - y e a r d e a t h rates into single-year d e a t h rates using a stan- d a r d method ( d e s c r i b e d in Vaupel, Manton, a n d S t a l l a r d 1979) and t h e n used simple l i n e a r i n t e r p o l a t i o n between t h e a v a i l a b l e d a t a points o v e r time t o estimate t h e height of t h e mortality s u r f a c e at i n t e r m e d i a t e points in time.

Comparison of F i g u r e s l a and 3 r e v e a l s t h e d i f f e r e n c e i t makes t o work with detailed d a t a as opposed t o i n t e r p o l a t e d d a t a .

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The longest time s e r i e s of mortality r a t e s a r e a v a i l a b l e f o r Sweden: we used. d a t a , from Keyfitz a n d Flieger (1968) f o r 1 7 7 8 t o 1882 a n d from v a r i - ous editions of t h e Swedish S t a t i s t i c a l Yearbook f o r 1881 t h r o u g h 1 9 8 1 , which w a s a v a i l a b l e f o r t h e most p a r t f o r five y e a r p e r i o d s and f o r f i v e y e a r a g e c a t e g o r i e s b e f o r e 1880 b u t by singie y e a r of a g e t h e r e a f t e r . Fig- u r e 4 shows t h e evolution of Swedish female mortality f r o m 1 7 7 8 t o 1 9 8 1 based on i n t e r p o l a t i o n s (Vaupel, Manton, and S t a l l a r d 1979) we made using t h i s d a t a .

AGE-PERIOD. AGE-COHORT. AND COHORT-PERIOD

MAPS

O F

U S .

FEMALE

FERTILITY

Figure 5 displays t h e c o n t o u r s of U.S. b i r t h rates from 1917 t o 1980 f o r womer! from a g e 1 4 t o 49; t h e f i g u r e is based on d a t a compiled by H e u s e r

(1976; 1984). In t h e d a r k c e n t e r of t h e baby boom, f o r women a r o u n d a g e 23 a r o u n d 1960, fully a q u a r t e r of women g a v e b i r t h e a c h y e a r . The concen- t r a t i o n of high b i r t h rates among women in t h e i r e a r l y and mid twenties a n d t h e c y c l e s of high and low b i r t h rates t h a t underly E a s t e r l i n ' s t h e o r y are s t r i k i n g i y r e v e a i e d on t h e map.

Figure 5 i s a standard. map in which c u r r e n t y e a r r u n s along t h e h o r - izontal a x i s a n d a g e r u n s u p t h e v e r t i c a l a x i s . O t h e r c o o r d i n a t e s h e l p r e v e a l c o h o r t e f f e c t s . In p a r t i c u l a r , b e c a u s e t h e e y e c a n follow v e r t i c a l a n d horizontal lines more easily t h a n diagonals, i t may b e useful t o twist a c o n t o u r map s o t h a t y e a r of b i r t h , r a t h e r t h a n c u r r e n t y e a r , r u n s along t h e h o r i z o n t a l a x i s . Figure 6 i l l u s t r a t e s t h i s a p p r o a c h . Alternatively, as shown in Figure 7 , y e a r of b i r t h may r u n along t h e h o r i z o n t a l a x i s a n d c u r r e n t y e a r along t h e v e r t i c a l a x i s . We only used f i v e c o n t o u r lines on Figure 7 b e c a u s e t h e lines w e r e o t h e r w i s e t o o closely s p a c e d t o b e intelligible: e v e n with five lines t h e r e a r e two black s p l o t c h e s w h e r e f e r t i l i t y rates a r e i n c r e a s i n g s o r a p i d l y t h a t t h e c o n t o u r lines r u n t o g e t h e r .

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ALTERNATIVE GRAPHIC DISPLAYS OF U.S. FEXALJ3 FERTILITY

The most commonly used method f o r displaying demographic rates o v e r a g e a n d time is t o plot t h e rates o v e r time f o r s e l e c t e d a g e s o r o v e r a g e f o r s e l e c t e d times. In F i g u r e 8, f o r i n s t a n c e , U.S. b i r t h rates a r e g r a p h e d o v e r time at a g e s 18, 23, 2 8 , 33, 38, a n d 43 a n d in F i g u r e 9 t h e b i r t h rates are g r a p h e d from a g e 1 4 t o 49 f o r y e a r s 1 9 2 0 , 1930, 1 9 4 0 , 1 9 5 0 , 1 9 6 0 , 1970, a n d 1980. Comparison of F i g u r e s 8 a n d 9 with t h e c o n t o u r maps p r e s e n t e d in Figures 5, 6, a n d 7 r e v e a l s some of t h e s t r e n g t h s a n d weaknesses of t h e s e a l t e r n a t i v e g r a p h i c displays.

Figures 1 0 a n d 11 show two p l o t s of t h e U.S. b i r t h rate d a t a drawn from a three-dimensional p e r s p e c t i v e . Figure 1 0 , l i k e a l l t h e c o n t o u r maps in t h i s p a p e r , w a s p r o d u c e d using a n o r d i n a r y p r i n t e r ; Figure 11 w a s drawn using a c o m p u t e r p l o t t e r . The three-dimensional p l o t s a c r i f i c e s some of t h e r i c h n e s s of d e t a i l t h a t i s c l e a r l y p o r t r a y e d on t h e c o r r e s p o n d i n g c o n t o u r maps; f u r t h e r m o r e , i t i s difficult o n t h e three-dimensional plot t o r e l a t e a point on t h e s u r f a c e t o t h e e x a c t a g e and y e a r underlying t h e point.

What g r a p h i c method should b e used t o display a demographical s u r - f a c e ? Clearly, e a c h method h a s i t s s t r e n g t h s a n d weaknesses, with c o n t o u r maps having s p e c i a l a d v a n t a g e s in some c i r c u m s t a n c e s . D e m o g r a p h e r s should c o n s i d e r adding c o n t o u r maps t o t h e i r toolkit of g r a p h i c a l t e c h - niques.

RELATIW SURFACES OF SWEDISH POPULATION.

ITALIAN

MORTALITY.

AND

U.S.

FERTILITY

To d e p i c t t h e c h a n g e t h a t h a s o c c u r r e d o v e r time in age-specific ferti!ity rates, mortality rates, population l e v e l s , o r o t h e r demographic s t a t i s - t i c s , i t i s useful t o draw c o n t o u r maps of r e l a t i v e s u r f a c e s o n which t h e value of t h e s t a t i s t i c at e a c h point i s c a l c u l a t e d r e l a t i v e t o value of t h e s t a t i s t i c in some b a s e y e a r . Consider, f o r example, Figures 1 2 a n d 13. Fig- u r e 1 2 displays Swedish population levels from b i r t h t o a g e 7 9 from y e a r s 1780 t o 1 9 6 3 , b a s e d o n i n t e r p o l a t i o n s of d a t a in Keyfitz a n d F l i e g e r (1971);

Figure 1 3 p r e s e n t s t h e s e population leveis r e l a t i v e t o t h e population l e v e l at t h e v a r i o u s a g e s in t h e f i r s t y e a r , 1780.

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Two o t h e r a p p l i c a t i o n s of t h i s a p p r o a c h a r e shown in F i g u r e s 1 4 and 1 5 . F i g u r e 1 4 d i s p l a y s age-specific mortality rates f o r Italian males r e l a - t i v e t o t h e i r l e v e l s in 1 9 2 5 , a y e a r roughly halfway t h r o u g h t h e p e r i o d studied. F i g u r e 1 5 p r e s e ~ t s age-specific b i r t h r a t e s f o r U.S. females r e l a t i v e t o t h e i r level in t h e final y e a r , 1980.

Instead of dividing a demographic a r r a y by t h e age-specific s t a t i s t i c s f o r a p a r t i c u l a r y e a r , t h e a r r a y could b e divided by t h e period-specific s t a t i s t i c s f o r a p e r t i c l ~ l a r a g e . F o r example, F i g u r e 1 6 shows I t a l i a n male mortalit?. r a t e s a t v a r i o u s a g e s r e l a t i v e t o t n e infant mortality r a t e in t h e o p p r o p r i a t e y e a r .

3emogro;lhic s t e t i s t i c s couid a l s o b e e x p r e s s e d r e l a t i v e t o some com- p o s i t e age-specific o r period-specific m e a s u r e . F i g u r e s 1 7 znc2 1 8 p r o v i d e two exsm;lles. To proc'uce F i g r e 1 7 . Selgiurr. oge-specific fexe!e population i e v e l s ( f r o n V q 7 s 1983) w e r e divided by t h e t o t a i Selgium female popuiatior, in e a c h y e e r . Thus, t h e x a p gives c o n t o u r s of t h e age-distribution of t h e popuiation, i . e . , t h e p e r c e n t a g e of t h e population in e a c h y e a r t h a t a r e at v a r i o u s a g e s . The black b l o t c h e s and s t r e a k s are d u e t o r a p i d l y changing o r f l u c t u z t i n g population l e v e l s , causing s e v e r a l c o n t o u r l i n e s t o rur.

t o g e t h e r . F i g u r e 1 9 , which i s b a s e d on U . S . f e r t i l i t y ciatz, i s similar in n a t u r e e x c e p t t h a t t h e c o n t o u r s p e r t a i n t o cumulative l e v e l s up t h r o u g h a g e 49 r e l a t i v e t o t h e t o t a l !eve! o v e r oll a g e s . The map c a n b e i n t e r p r e t e d as showing t h e p r o p o r t i o n of a l l b i r t h s in a given y e a r t h a t o c c u r r e d t o women of some a g e o r less--in a s y n t h e t i c population in which t h e r e w e r e equal numbers of women a t e a c h a g e .

Finally, i t may sometimes b e useful t o examine c o n t o u r maps b a s e d on s t a t i s t i c s r e l a t i v e t o a c o h o r t - s p e c i f i c m e a s u r e r a t h e r t h a n e i t h e r a n a g e - s p e c i f i c c r g e r i o d - s p e c i f i c m e a s u r e . C o ~ s i d e r , f o r i n s t a n c e , F i g u r e 1 9 , which i s sir.iiar t o F i g u r e 1 8 e x c e p t t h a t cumulative f e r t i l i t y i s compute2

-

^*^{e k L A}^,-hivet o t o t a l c o h o r t f e r t i l i t y u;: t h r o u g h a g e 39.

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SMALL MULTIPLES

To c o m p a r e global p a t t e r n s among s e v e r a l population s u r f a c e s i t may b e useful t o s h r i n k c o n t o u r maps down in size a n d p r e s e n t s e v e r a l of them on t h e same page: Tufte (1983) c a l l s t h i s t h e "smal! multiples" a p p r o a c h . Figure 20 p r e s e n t s maps of U.S. b i r t h r a t e s at v a r i o u s p a r i t i e s : in Figure 20a, f i r s t - b i r t h rates (i.e., t h e p r o p o r t i o n of c h i l d l e s s women of some a g e in some y e a r who h a v e t h e i r f i r s t child) are displayed; in Figure 20b, second- b i r t h rates a r e displayed, a n d s o on. In e a c h of t h e small multiples, t h e same c o n t o u r l e v e l s a r e used.

Figure 2 1 p r e s e n t s a n o t h e r i l l u s t r a t i o n of t h e use of small multiples, t h i s time t o c o m p a r e mortality r a t e s from a g e 1 0 t o 49 f o r y e a r s 1910 t o 1 9 6 5 f o r I t a l i a n , F r e n c h , a n d Belgian males and females. The e f f e c t s of t h e F i r s t a n d Second World Wars a r e most prominent in t h i s p e r i o d and f o r t h e s e a g e s .

Figure 22 dispiays r e l a t i v e mortality r a t e s f o r males a n d f o r females in England a n d Wales, Sweden, and Italy from a g e 5 t o 8 0 f o r y e a r s 1870 t o 1978. In e a c h case, t h e mortality rate f o r a given a g e i s r e l a t i v e t o t h e r a t e at t h a t a g e in 1870. Thus t h e maps p r o v i d e a p i c t u r e of t h e p a t t e r n of p r o g r e s s made in r e d u c i n g mortality rates s i n c e 1870. The maps are analagous t o t h e t a b l e s with r o u g h c o n t o u r lines in Kermack, McKendrick, a n d McKinlay (1933); P r e s t o n a n d van d e r Walle (1978) a n d Coale a n d K i s k e r (1985) p r e s e n t similar t a b l e s . These a n a l y s t s a s c r i b e t h e diagonal c o n t o u r s in t h e i r t a b l e s t o c o h o r t e f f e c t s . The maps in Figure 22 p r o v i d e a r i c h e r , more detailed p i c t u r e of t h e v a r i o u s local a n d global p a t t e r n s in t h e c h a n g e s in mortality rates, in v e r t i c a l , horizontal, a n d diagonal d i r e c t i o n s .

The maps in F i g u r e 22 f o r t h e t h r e e c o u n t r i e s w e r e p r o d u c e d using dif- f e r e n t kinds of d a t a . As noted e a r l i e r , mortality rates f o r Italy were avail- a b l e f o r single y e a r s of time a n d a g e . F o r Sweden, t h e rates w e r e g e n e r a l l y a v a i l a b l e f o r f i v e y e a r p e r i o d s ; b e f o r e 1880 t h e rates w e r e f o r f i v e y e a r a g e c a t e g o r i e s a n d a f t e r w a r d s f o r single y e a r s of a g e . Finally, f o r England and Wales, t h e rates were a v a i l a b l e f o r f i v e y e a r a g e c a t e g o r i e s f o r single y e a r s of time a b o u t o n c e p e r d e c a d e . D i f f e r e n c e s in t h e smoothness of t h e maps, especially f o r England a n d Wales compared with I t a l y , a r e p r o b a b l y

l a r g e l y a t t r i b u t a b i e t o t h e s e d i f f e r e n c e s in d a t a r i c h n e s s .

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In t h e a n a l y s e s of Kermack, McKendrick, and McKinlay, P r e s t o n and van d e r Walle, a n d Coale and K i s k e r , mortality rates were t a k e n r e l a t i v e t o a p e r i o d e a r l i e r t h a n 1 8 7 0 , t h e underlying assumption being t h a t at a n e a r l y enough p e r i o d t h e r e would h a v e b e e n no s y s t e m a t i c p a t t e r n of p r o g r e s s a g a i n s t mortality. F i g u r e 2 3 shows mortality r a t e s for Swedish males and females r e l a t i v e to t h e a v e r a g e l e v e l s at e a c h a g e in t h e p e r i o d from 1 7 7 8 t o 1799. The f i g u r e r e v e a l s t h e fluctuating p a t t e r n of mortality b e f o r e t h e middle of t h e n i n e t e e n t h c e n t u r y a n d t h e g e n e r a l p a t t e r n of p r o g r e s s a g a i n s t mortality s u b s e q u e n t l y . The p a t t e r n i s c l e a r l y more complex t h a n a p u r e c o h o r t - e f f e c t model would s u g g e s t .

A d i r e c t way of c o n s i d e r i n g t h e h y p o t h e s i s t h a t "a c o h o r t c a r r i e s i t s mortality l e v e l with it" i s to examine mortality s u r f a c e s t h a t a r e c a l c u l a t e d r e l a t i v e t o a c o h o r t ' s mortality levels. In Figure 2 4 , for i n s t a n c e , in e a c h of t h e s i x s u r f a c e s shown t h e mortality r a t e at e a c h a g e and y e a r was divided by t h e mortality r a t e at t h a t a g e for t h e c o h o r t b o r n in 1870. The p a t t e r n t h a t e m e r g e s shows some s t r o n g diagonals, b u t i t i s a p p a r e n t t h a t t h e r e a r e a!so i m p o r t a n t e f f e c t s in h o r i z o n t a l and v e r t i c a l d i r e c t i o n s . I n t e r p r e t a t i o n of t h e s e and similar s u r f a c e s should a l s o b e t e m p e r e d by r e a l i z a t i o n t h a t diagonal p a t t e r n s c a n e m e r g e not only as a r e s u l t of c o h o r t e f f e c t s b u t a l s o as t h e r e s u l t of t h e i n t e r a c t i o n of p e r i o d a n d a g e e f f e c t s .

RATIO

SURFACES

Instead of using small multiples, a n o t h e r a p p r o a c h to comparing two or more d e m o g r a p h i c s u r f a c e s i s to compute some new s u r f a c e s t h a t r e p r e s e n t at e v e r y point e i t h e r t h e d i f f e r e n c e or t h e r a t i o of t h e h e i g h t of o n e of t h e o r i g i n a l s u r f a c e s to a n o t h e r . F i g u r e 2 5 a , for i n s t a n c e , shows t h e r a t i o of male to female mortality rates in I t a l y , smoothed on a 5 by 5 g r i d . To highlight t h e a g e s a n d p e r i o d s when Italian male and female mortality rates w e r e roughly e q u a l , F i g u r e 25b p r e s e n t s a modified v e r s i o n of t h i s map in which only t h r e e c o n t o u r s l i n e s a r e d r a w n , for e q u a l male a n d female d e a t h rates and for l e v e l s t e n p e r c e n t a b o v e and below equality.

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Two f a r t h e r r a t i o s u r f a c e s a r e p r e s e n t in F i g u r e s 26 and 27. F i g u r e 26 dis;~!ays t h e r z t i o of l t e l i e n mele mortzlity t o F r e n c h me!e mortality from a g e TO t o 7 3 f o r y e a r s 1 9 3 3 t o 1950. F i g u r e 27 gives t h e d i f f e r e n c e between f i r s t end s e c o n d b i r t h r a t e s in t h e U.S.

I n addition t o maps of d e a t h and b i r t h r a t e s and of population l e v e l s , c o n t o u r maps c a n b e drawn b a s e d on a n y o t h e r kin? of d a t a t h a t i s s t r u c - t u r e d along t w o dimezsions. F i g u r e s 29 ar!d 2 9 s u g g e s t two possibilities. Fig- u r e 28 displays t h e r a t i o of males t o females in Belgium, b a s e d on Veys' (1993) 6 a t e . And F i g u r e 29 shows age-specific m a r r i a g e r a t e s for Italian females, based or, pre!irr.inary, unpublished d a t a supplied to us by t h e D e p e r t m e ~ t of Demographic: S c i e ~ c e at t h e University of Rome.

L F E

T A K E STATISTICS

FOR

BELGIAN FQdALES

Life t a b l e s o f t e n p r o v i d e s t a t i s t i c s by a g e and ^o\7ertime on population s i z e , number of d e a t h s , d e a t h rates, p e r i o d s u r v i v o r s h i p , p e r i o d life e x p e c - t a n c y , a n a sometimes c o h o r t s u r v i v o r s h i p . All six of t h e s e s t a t i s t i c s a r e a v a i l a b l e , for example, in Veys' (1983) compilatior. of Belgian l i f e t a b l e s from a g e 0 t o 99 f o r y e a r s 1 8 9 2 t o 1 9 7 7 . F i g u r e s 30 t h r o u g h 35 use Veys' d a t a f o r Belgiar! females t o i l l u s t r a t e t h e d i f f e r e n t kinds of c o n t o u r map p a t t e r n s p r o d u c e d by d i f f e r e n t kinds of life t a b l e s t a t i s t i c s . The d i f f e r e n c e s in t h e p a t t e r n s a r e q u i t e s t r i k i n g ; n o t e in p a r t i c u l a r t h e d i f f e r e n c e between t h e p e r i o c and c o h o r t p a t t e r n s of s u r v i v o r s h i p .

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U.S. FENALE MORTALRY RATES FROM 1900 TO 2050

F a b e r (1982) published l i f e t a b l e s f o r U.S. males and females by single y e a r of a g e from b i r t h t o 119 f o r e v e r y t e n t h y e a r from 1900 t h r o u g h 2050.

The mortality rates at advanced a g e s and after 1980 a r e b a s e d on e x t r a p o -

!ations. Figure 36 displays t h e s u r f a c e of annual d e a t h s rates (i.e., of q ) f o r U.S. females a n d P i g u r e 37 displays t h e s u r f a c e of t h e f o r c e of mortality.

The two s u r f a c e s are similar e x c e p t at advanced a g e s when mortality is v e r y high. Note t h a t t h e maps g r a p h i c a l l y r e v e a l P a b e r ' s underlying assumptior. t h a t p r o g r e s s a g a i n s t mortality wi!l slow down in t h e f u t u r e . The n a t u r e of t h i s assumption is r e v e a l e d even more c l e a r l y in Figure 38, which displays t h e f o r c e of morta!ity o v e r a g e re!ative t o t h e f o r c e of mortality in 1980.

MODEL LIFE T A B X S

Demographers f r e q u e n t l y make use of mode! life t a b l e s , especia!!~

t h o s e developed by Coale and Demeny (1983). In t h e Coale and Demeny t a b l e s , d e a t h rates in v a r i o u s a g e c a t e g o r i e s a r e given by t h e life e x p e c - t a n c y of t h e population, f o r males and f o r females, and f o r f o u r d i f f e r e n t kinds of h y p o t h e t i c a l populations, labelled E a s t , West, N o r t h , a n d South.

Figure 39 p r e s e n t s c o n t o u r maps f o r females of t h e s e f o u r t y p e s . Note t h a t t h e horizontal a x i s gives life e x p e c t a n c y r a t h e r t h a n time. Thus Figure 39 i l l u s t r a t e s n o t o ~ l y t h e u s e of smal! multiples t o p o r t r a y s y n t h e t i c d a t a , b u t a l s o t h e u s e of a v a r i a b l e o t h e r t h a n a g e o r time as o n e of t h e dimensions on a c o n t o u r map.

Another a p p r o a c h t o c o n s t r u c t i o n of model life t a b l e s w a s developed by Brass (1971). In t h i s a p p r o a c h , a s t a n d a r d t r a j e c t o r y of s u r v i v o r s h i p p r o - p o r t i o n s , p ( ~ ) w h e r e z s t a n d s f o r a g e , i s modified by p a r a m e t e r s a and b t o p r o d u c e a l t e r n a t i v e t r a j e c t o r i e s , p '(z), s u c h t h a t

w h e r e t h e logit function is given by

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Given a t r a j e c t o r y of s u r v i v o r s h i p p r o p o r t i o n s , a t r a j e c t o r y of f o r c e s of morta!ity (i.e., d e a t h r a t e s ) c a n b e r e a d i l y c a l c u l a t e d . F i g u r e 40 i!!ustrates hotll t h e values of t h e p a r a m e t e r s a and b a f f e c t t h e r e s u l t i n g age-specific f o r c e of mortality: t h e p a r a m e t e r a r u n s along t h e h o r i z o n t a l a x i s of e a c h map and t h e f i v e maps r e p r e s e n t d i f f e r e n t va!ues of b .

MkPPIMG RESIDUALS TO SHOW GOODNESS OF FIT

How well d o e s a mode! f i t some empirica! d a t a ? If t h e d a t a a r e defined o v e r t w o dimensions, t h e n a c o n t o u r map c a n b e used to display t h e r e s i d u - a l s , i . e . , t h e d i f f e r e n c e s between t h e a c t u a l values a n d t h e va!ues p r e d i c t e d by t h e mode!. By scrutinizing t h e p a t t e r n of t h e r e s i d u a l s , a n a n a l y s t may glean some c l u e s as to how to improve t h e mode!. (Tukey (1977) a n d Mos- t e l l e r and Tukey (1977) p r o v i d e c l e a r discussions of t h e use of r e s i d u a l s in d a t a analysis a n d model building.) A s a n i l l u s t r a t i o n of t h i s g e n e r a l method, F i g u r e 4 1 shows how well a modified form of B r a s s ' model f i t s Italian female mortality d a t a . The modification made involves t h e u s e of 1 9 2 5 Italian female mortality rates as t h e s t a n d a r d r a t h e r t h a n B r a s s ' o r i g i n a l s t a n d a r d . F i g u r e 41a d i s p l a y s t h e a c t u a l values of Italian female m o r t a l i t y r a t e s . Fig- u r e 41b displays t h e v a l u e s estimated by t h e modified B r a s s model. F i g u r e 41c disp!ays t h e s u r f a c e of r e s i d u a l s , i.e., t h e s u r f a c e of d equal to ^qminus q ' , w h e r e q i s t h e o b s e r v e d mortality rate and q ' i s t h e m o r t a l i t y r a t e p r e d i c t e d by t h e modified B r a s s model. D i f f e r e n t values of a a n d b in t h e mode! w e r e c h o s e n for e a c h y e a r of time. The values used for a a n d b are displayed in t h e g r a p h in F i g u r e 41d.

A s a f i r s t s t e p in e x p l o r a t o r y d a t a analysis a n d model building, i t may b e useful to r e m o v e a g e a n d p e r i o d e f f e c t s ( o r , m o r e g e n e r a l l y , t h e main e f f e c t s along t h e x a n d y dimensions) f r o m t h e d a t a a n d t h e n look a t t h e r e s i d u a l s . The r e l a t i v e s u r f a c e s shown in F i g u r e s 13 t h r o u g h 18 c a n b e i n t e r p r e t e d as residua! s u r f a c e s f o r which e i t h e r a p e r i o d e f f e c t o r a n a g e e f f e c t h a s b e e n removed. F i g u r e 42a displays a s u r f a c e of r e s i d u a l s calcu- l a t e d by removing both a p e r i o d a n d a n a g e e f f e c t : t h e o r i g i n a l s u r f a c e , whicn p r e s e n t s U . S . f e r t i l i t y rates, w a s shown in F i g u r e 5; f r o m t h i s s u r f a c e t h e a v e r a g e ferti!ity r a t e at e a c h a g e and t h e a v e r a g e f e r t i l i t y in e a c h y e a r

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w a s divided o u t and t h e n , t o normalize t h e resu!ting v a l u e s , t h e v a l u e s w e r e multip!ied by t h e o v e r a l l a v e r a g e f e r t i l i t y r a t e . The v a l u e s of t h e s e a v e r - a g e rates a r e shown in F i g u r e s 42b and 42c. The c o n t o u r line at t h e level of o n e in F i g u r e 42a shows when ferti!ity rates car! b e exact!y e x p l a i n e d by a simple multiplicative mode! of a g e and p e r i o d a v e r a g e s ; t h e d a r k a n d light areas on t h e map show when ferti!ity rates are h i g h e r o r lower t h a n t h e rates p r e d i c t e d by t h e simple age-period model.

KAPS OF TI3EOL2,ETICAL DEMOGRAPHIC MODELS

To u n d e r s t a n d a c t u a l population phenomenon, d e m o g r a p h e r s o f t e n ana!yze simp!ified, t h e o r e t i c a l mode!s t h a t c a p t u r e some a s p e c t of r e a l i t y (Keyfitz 1977). C o n t o u r s maps c a n b e used t o show how some v a r i a b l e of i n t e r e s t in s u c h mode!s r e s p o n d s t o c h a n g e s in two of t h e p a r a m e t e r s . Fig- u r e s 4 3 and 44 p r o v i d e s u c h i l l u s t r a t i o n s . S u p p o s e mortality rates follow t h e female mode! West s c h e d u l e of Coale a n d Demeny. F u r t h e r s u p p o s e t h a t a population i s s t a b l e and is g o v e r n e d by t h e s e mortality t r a j e c t o r i e s (which c a n b e classified by t h e single mortality m e a s u r e e o ) a n d by some growth r a t e r

.

What p r o p o r t i o n of t h e population will b e a b o v e a g e 65? The c o n t o u r map in F i g u r e 4 3 d i s p l a y s t h e a n s w e r , f o r v a r i o u s va!ues of e o a n d b .

A s a s e c o n d example, s u p p o s e t h a t t h e f o r c e of mortality at a n y a g e i s given by a Gompertz c u r v e s u c h t h a t u ( z )

=

a e

'=.

How will life e x p e c t a n c y at b i r t h c h a n g e as a a n d b v a r y ? F i g u r e 44 g r a p h i c a l l y reso!ves t h i s ques- tion.

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CONCLUSION

The f i g u r e s p r e s e n t e d in t h i s p a p e r suggest just a f e w of t h e numerous ways t h a t d e m o g r a p h e r s c a n u s e c o n t o u r maps t o c l e a r l y , efficiently, and simultaneously display both p e r s i s t e n t global a n d prominent l o c a l p a t t e r n s in population r a t e s o r levels o v e r two dimensions. In p a r t i c u l a r , c o n t o u r maps can strikingly r e v e a l t h e i n t e r a c t i o n between a g e , p e r i o d , and c o h o r t p a t t e r n s .

Even in c a s e s w h e r e demographic d a t a a l r e a d y h a s been c a r e f u l l y scrutinized by p e r c e p t i v e a n a l y s t s who h a v e u n c o v e r e d most of t h e avail- a b l e i n t e r e s t i n g p a t t e r n s , c o n t o u r maps may b e useful in highlighting t h e s e p a t t e r n s in a visually r e v e a l i n g manner. With c o n t o u r maps, what w a s b e f o r e understood now c a n b e s e e n . F u r t h e r m o r e , t h e maps, by giving demogra- p h e r s a new p e r s p e c t i v e on t h i s d a t a , may f o c u s a t t e n t i o n o n some neglected a s p e c t s a n d p a t t e r n s in e v e n thoroughly-studied d a t a .

Beyond efficient d e s c r i p t i o n , c o n t o u r maps c a n help d e m o g r a p h e r s with e x p l o r a t o r y d a t a analysis a n d with model building. S u r f a c e s c a n b e com- puted r e l a t i v e t o some p a r t of t h e s u r f a c e o r t o a n o t h e r s u r f a c e ; and dif- f e r e n t s u r f a c e s c a n b e p l a c e d n e x t t o e a c h o t h e r a n d compared. The p a t - t e r n s p r o d u c e d by a model c a n b e displayed as c a n t h e f i t of t h e model t o some empirical d a t a .

The r e s u l t i n g c o n t o u r maps c a n b e displayed n o t only as p r i n t e d o u t p u t but a l s o o n a c o m p u t e r monitor. The s h a d e s used in t h e maps p r e s e n t e d in t h i s p a p e r r a n g e from black t o light g r e y , but t h e maps c a n b e p r o d u c e d in glowing c o l o r s , on a c o l o r computer monitor o r using a c o l o r p r i n t e r . The e f f e c t s a r e d r a m a t i c , as i s t h e s p e e d with which a c o m p u t e r c a n draw a map.

A l a r g e computer i s not needed--as d e s c r i b e d in t h e Appendix, we h a v e used a n IBM P.C.

Tukey, in h i s lucid exposition of The Visual L)i-splay of Quantitative Information (1983), concludes t h a t g r a p h i c designs should give "visual a c c e s s t o t h e s u b t l e a n d difficult, t h a t is, t h e r e v e l a t i o n of t h e complex".

Demographic s u r f a c e s c a n b e p a r t i c u l a r l y complex. A mortality s u r f a c e , f o r example, might b e defined o v e r n e a r l y a c e n t u r y of a g e a n d more t h a n a c e n t u r y of time, comprising close t o t e n thousand d a t a points t h a t may v a r y o v e r f o u r o r d e r s of magnitude. Contour maps a r e a s t r i k i n g , e f f i c i e n t , and c l e a r means of giving d e m o g r a p h e r s visuai a c c e s s t o such s u r f a c e s .

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1910 1930 1950 YEAR

F i g u r e l a : I t a l i a n Male M o r t a l i t y R a t e s - w i t h c o n t o u r l i n e s from .000667 t o .195 a t m u l t i p l e s o f 1 . 5

From a g e 0 t o 79 and Year 1870 t o 1979

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1910 1930 1950 YEAR

Figure lb: Italian Male Mortality Rates - with contours from .000667 to .196 at multiples of 1.88

From Age 0 to 79 and Year 1870 to 1979

(22)

1910 1930 YEAR

Figure lc: Italian Male Mortality Rates - with contour lines from .O1 to .15 at even intervals

(23)

1910 1930 YEAR

Figure Id: Italian Male Mortality Rates - with contour lines from .000667 to .195 at multiples of 1.5, with a grid

(24)

YEAR

Figure le: Italian Male Mortality Rates - with contour lines from . 0 0 0 6 6 7 to .195 at multiples of 1.5, with contour lines drawn in white

(25)

YEAR

Figure lf: Italian Male Mortality Rates - with contour lines from .000667 to .I95 at multiples of 1.5, with no shading

(26)

1910 1930 1950 Year

Figure lg: Italian Male Mortality Rates - with contour lines from .000667 to .195 at multiples of 1.5, smoothed on a 5 by 5 sqaure

00100 .00225 .00507

.

^oil4 ^.0256 ^.0577 ^-130

(27)

1910 1930 1950 YEAR

F i g u r e l h : I t a l i a n Male M o r t a l i t y R a t e s - w i t h c o n t o u r l i n e s from -000667 t o .195 a t m u l t i p l e s of 1 . 5 , smoothed on an 11 by 11 s q u a r e

From Age 0 t o 79 and Year 1870 t o 1979

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YEAR

Figure li: Italian Male Mortality Rates - with contours from .000667 to.195 at multiples of 1.5, smoothed on a weighted 5 by 5 sqaure

(29)

YEAR

Figure 2a: Italian Male Mortality Rates - with contour lines from .000667 to -0263 at multiples of 1.3

(30)

YEAR

Figure 2b: Italian Male Mortality Rates - with contour lines from .000667 to .195 at multiples of 1.5

(31)

YEAR

F i g u r e 3: I t a l i a n Male M o r t a l i t y R a t e s ( i n t e r p o l a t e d d a t a ) ^-,w i t h c o n t o u r l i n e s from .000667 t o . 1 9 5 a t m u l t i p l e s o f 1 . 5

From Age 0 t o 79 a n d Year 1 8 8 1 t o 1963

(32)

1780 1800 1820 1840 1860 1880 1900 1920 1940 1960 1980 YEAR

Figure 4: Swedish Female Mortality Rates

-

with contour lines from .000667 to .195 at multiples of 1.5

(33)

1940 1960 YEAR

F i g u r e 5: U.S. B i r t h R a t e s - w i t h c o n t o u r l i n e s s e l e c t i v e l y p l a c e d from . 0 0 1 t o . 2 5 From A g e 1 4 t o 49 a n d Year 1914 t o 1980

(34)

YEAR OF BIRTH

Figure 6: U.S. Cohort Fertility - with contours selectively placed from .001 to .25 From Age 14 to 49 and Year of Birth 1868 to 1966

(35)

1890 1910 1930 YEAR OF BIRTH

F i g u r e 7: U . S . F e r t i l i t y by Year o f B i r t h and C u r r e n t Year - w i t h c o n t o u r s from . 0 5 t o .25 a t e v e n l y spaced i n t e r v a l s

From C u r r e n t Year 1917 t o 1980 and Year of B i r t h 1867 t o 1966

(36)

YEAR

F i g u r e 8: U . S . B i r t h R a t e s Over Time From 1917 t o 1980 a t Ages 18, 23, 28, 33, 38, 43

(37)

49 AGE

F i g u r e 9: U . S , : B i r t h R a t e s From A g e 14 t o 49 a t Y e a r s 1920, 1930; 1940, 1950, 1960, 1970, 1980

(38)

49 AGE

Figure 10: Three-dimensional perspective of U.S. Fertility Rates

(39)

1980 YEAR

Figure 11: Three-dimensional perspective of U.S. Fertility Rates -

by a computer plotter

'

*

From Fertility in America: Heterogeneity and the Effect of Birth Order, by William iiodges

(40)

YEAR

F i g u r e 1 2 : S w e d i s h P o p u l a t i o n - w i t h c o n t o u r s l i n e s p l a c e d e v e n l y frorK10,OOO ko 1 3 0 , 0 0 0

From Age 0 t o 79 a n d Year 1 7 8 0 t o 1 9 6 3

20 40 6 0 8 0 1 0 9 1 2 0

( i n t h o u s a n d s )

(41)

1780 1800 1820 1840 1860 1880 1900 1920 1940 1960 YEAR

Figure 13: Swedish Population Relative to Age Specific 1780 Levels - with contour lines from .9 to 6.3.7 at multiples of 1.15, smoothed on a 5 by 5 square From Age 0 to 79 and Year 1780 to 1963

(42)

1910 1930 1950 YEAR

Figure 14: Italian Male Mortality Rates Relative to Age Specific 1925 Levels - with contour lines from .3 to 2.12 at multiples of 1.15, with 5 by 5 smoothing From Age 0 to 79 and Year 1870 to 1979

(43)

. , , , . , . n , . # , , , n , , , , , , , > < , a , , , , , , , ,

. -

, , > , . ,

I , ,

^,^,

.

^{, ,}

.

^{, ,}

.

^,^,^,^,

.

^,

.

^{, ,}, # , , . , ,

. . .

^,

.

^,, ~ , ~ , : , i . j . ' , i : , : , : , ~ , i ^, , , a $ ,

i'

^,'^,'^,'

i

^{.', ,',},f s,~,m,~~;,:,~,:,~,~,~,333333~~,8~~~'~~,'~~i

I

I I 1 1 I

1920 1940 1960

YEAR

Figure 15: U.S. Birth Rates Relative to Age Specific 1980 Levels

-

with contour lines from .8 to 5.66 at multiples of 1.15

(44)

1910 1930 1950 YEAR

F i g u r e 16: I t a l i a n Male M o r t a l i t y R a t e s R e l a t i v e t o I n f a n t M o r t a l i t y - w i t h c o n t o u r l i n e s from . 0 0 1 t o 1.11 a t m u l t i p l e s o f 1 . 6 5 , 5 by 5 smoothing From Age 0 t o 79 and Year 1870 t o 1979

(45)

YEAR

F i g u r e 1 7 : A g e - d i s t r i b u t i o n o f B e l g i a n Female P o p u l a t i o n - w i t h c o n t o u r l i n e s s e l e c t i v e l y p l a c e d from .00005 t o .027

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YEAR

F i g u r e 18: Cumulative U . S . F e r t i l i t y R a t e s - w i t h c o n t o u r l i n e s s e l e c t i v e l y p l a c e d from .1 t o -999

(47)

YEAR OF BIRTH

F i g u r e 19: C u m u l a t i v e U . S . C o h o r t F e r t i l i t y - w i t h c o n t o u r l i n e s s e l e c t i v e l y p l a c e d from .1 t o .999

From Age 1 4 t o 39 and Year o f B i r t h 1903 t o 1 9 4 1

(48)

F i g u r e s 2 0 a - f : U.S. B i r t h - R a t e s a t P a r i t i e s 1 t h r o u g h 6

-

w i t h c o n t o u r l i n e s s e l e c t i v e l y p l a c e d from .0001 t o -1

(49)

F l g u r e s 21a-f: M o r t a l i t y R a t e Comparlson: Male M o r t a l i t y i n t h e L e f t Column, Female M o r t a l i t y i n t h e R l g h t Column, and I t a l i a n , B e l g i a n , and F r e n c h M o r t a i i t y From Top t o Bottom - w l t h c o n t o u r s from .000667 t o .195

(50)

Figure 22: Mortality Rates Relative to 1870 Age Specific Levels: Male Mortality in the first column, female mortality in the second column and Italian, Swedish, and Zngland and Wales from top to bottom - ^{w ~ t h}contours from

.1 to 1.28 at multiples of 1.2, smoothed on a 5 by 5 square From Age 5 to 79 and Year 1870 to 1978

(51)

F i g u r e 23: Swedish M o r t a l i t y R e l a t i v e t o Age S p e c i f i c M o r t a l i t y from 1780 t o 1799

-

w i t h c o n t o u r l i n e s from . 0 5 t o 1 . 1 3 7 , smoothed on a 5 by 5 s q u a r e From Age 0 t o 79 and Year 1780 t o 1981

(52)

Figure 24: Mortality Rates Relative to 1870 Cohort Levels.: Male Mortality in the first column, female mortality in the second column and Italian, Swedish, and England and Wales from top to bottom - with contours from .1 to 1.28 at multiples of 1.2 ,smoothed on a 5 by 5 square From Age 5 to 79 and Year 1870 to 1978

(53)

1910 1930 YEAR

Figure 25a: Italian Female Mortality Rates Divided by Italian Male Mortality Rates -

with contour lines from .51 to 1.95 at intervals of 10 percent change From Age 0 to 79 and Year 1870 to 1979, smoothed on a 5 by 5 square

(54)

1910 1930 1950 YEAR

Figure 25b: Italian Female Mortality Divided by Italian Male Mortality - with selected contour lines, smoothed on a 5 by 5 square

(55)

1920 1940 YEAR

Figure 26: French Male Mortality Divided by Italian Male Mortality - with contour lines from .51 to 1 . 9 5 a t multiples of 1.1 , smoothed on a 5 by 5 square From Age 10 to 70 and Year 1900 to 1960

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1940

YEAR

F i g u r e 27: U . S . F e r t i l i t y : P a r i t y One Minus P a r i t y Two - w i t h c o n t o u r l i n e s s e l e c t i v e l y p l a c e d from -.02 t o .07

(57)

1900 1920 1940 1960 YEAR

Figure 28: Belgian Female Population Divided by Belgian Male Population

-

^with

contour lines selectively placed from .85 to 2.01 5 by 5 smoothing From Age 0 to 99 and Year 1892 to 1977

(58)

YEAR

Figure 29: Age Specific Marriage Data for Italian Females - with contour lines from 1 0 to 15,000 at selected intervals

From Age 0 to 4 9 and Year 1952 t o 1981

(59)

1920 1940 YEAR

F i g u r e 30: B e l g i a n Female P o p u l a t i o n - w l t h c o n t o u r l i n e s from 1 0 , 0 0 0 t o 9 0 , 0 0 0 a t s e l e c t e d i n t e r v a l s

20 8

b

( i n t h o u s a n d s )

(60)

1920 1940 YEAR

F i g u r e 31: B e l g i a n Female D e a t h s - w i t h c o n t o u r l i n e s from 1 0 t o 1 5 , 0 0 0 From Age 0 t o 99 a n d Year 1892 t o 1977

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1920 1940 YEAR

Figure 32: Belgian Female Mortality Rates

-

with contour lines from .000667 to .195 at multiples of 1.5

(62)

1920 1940 YEAR

Figure 33: Belgian Female Period Survivorship - with contour lines selectively placed from .001 to .95

(63)

1920 1940 YEAR OF BIRTH

Figure 34: Belgian Female Cohort Survivorship

-

with contours selectively placed from .001 to .95

(64)

YEAR

Figure 35: Belgian Female Period Life Expectancy - with contours selectively placed from - 3 to 70

From Age 0 to 99 and Year 1892 to 1977

(65)

1920 1940 1960 1980 2000 2020 2040 YEAR

F i g u r e 36: P r o j e c t e d U . S . Female M o r t a l i t y Rates Based on Faber L i f e T a b l e s - w i t h c o n t o u r l i n e s from .000667 t o -195 a t m u l t i p l e s of 1 . 5

(66)

1900 1920 1940 1960 1980 2000 2020 2040 YEAR

Figure 37: Projected U.S. Force of Mortality Based on'Faber Life Tables - ^with

contour lines from .000667 to .195 at mdlclples of 1.5 From Age 0 to 99 and Year 1900 to 2050

(67)

1900 1920 1940 1960 1980 2000 2020 2040 YEAR

Figure 38: Projected Force of Mortality Rates Relative to 1980 Age Specific Levels -

with contour lines from .51 to 1.95 at multiples of 1.1, 5 by 5 smoothing From Age 0 to 99 and Year 1900 to 2050

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F l q u r e 39: C o a l e and Dsmeny Model D a t a , w i t h N o r t h , S o u t h , E a s t , and. West from L e f t t o F a q h t and Top t o Bottom and c o n t o u r s from .000667 t o .195

From Aqe 0 t o 99 on t h e V e r t i c l e Axis and L i f e E x p e c t a n c y 20 t o 8 0 a t i n t e r v a l s o f 2.5 on t h e H o r i z o n t a l A x i s

(69)

Figure 40: Brass's Model With Maps From Top to Bottom at Values of b=.6, .8, 1 , 1.2, 1.4 - and contour lines evenly spaced from . 5 to 7.5

From Age 0 to 99 on the Verticle Axls and Values of a From -1 to 1 at intervals of .1 on the Horizontal Axis