T o understand actual population phenomena, demographers often analyze simplified, theoretical models that capture some aspect of reality (Keyfitx, 1977).
Contour maps can be used to show how some variable of interest in such models responds to changes in two of the parameters. Figures 49 and
44
provide such illus- trations.Suppose mortality rates follow the female model West schedule of Coale and Demeny (1984). Further, suppose that a population is stable and is governed by these mortality trajectories (which can be classified by the single mortality measure i o , life expectancy a t birth) and by some growth rate r . What proportion of the instance, the situation described on the middle of the right margin of the map when a is 0.00050 and b is 0.10, yielding a life expectancy of about 48 years. Cutting the mortality level by a factor of ten, i.e., reducing a to 0.00005, would increase life expectancy to 70 years. Alternatively, holding a constant but slowing the rate of aging by reducing b from 0.10 to 0.06 would have the same effect. Clearly, the effect on life expectancy of a small proportional change in the rate of mortality increase is much greater than a similar change in the level of mortality.
10 30 50 70 Year
Figure 45. Interaction of age and period effects.
T h e maps in F i g u r e s 45, 46, and 47, which are also based on theoretical models, are designed t o illustrate the contour patterns produced by the interaction of age, period, and cohort effects. Each of the three maps can be interpreted as consisting of nine separate maps. T h e height of the surface in Figure
4 5
is simply A+
T. A , the age effect, increases from 0 t o 1 over t h e lowest third of the vertical axis, stays constant a t 1 over the middle third, and decreases from 1 t o 0 over the top third. Similarly, T , the time effect, increases from 0 to 1, stays a t I , and declines from 1 t o 0 along the horizontal axis. Thus, in the center block the height of the surface is 2, as indicated by the uniform black rectangle.T h e height of the surface in Figure 46 is A f T
+
C , where C is a cohort eRect t h a t , within each of the nine blocks of the map, varies linearly from -1 to 1.For the earliest cohort in each block, which only appears in t h e upper left corner, the value of C is 1 ; for the latest cohort, which only appears in the lower right corner, the value of C is 1. Note that for the cohort born a t the s t a r t of each block,
10 30 50 70 90 Year
Figure 46. Interaction of age, period, and positive cohort effects.
in t h e lower left corner, which runs diagonally u p t o the upper right corner, the value of C is zero. T h e block in t h e lower right corner of the overall m a p has a uni- form height of 1, produced by T declining from 1 t o 0, A increasing from 0 t o 1, a n d C increasing from -1 t o 1: the strong age, period, a n d cohort effects cancel out.
T h e surface in Figure 47 is similar, except t h a t the height is now given by A t T - C .
T h e maps provide a clear, cautionary warning t o demographers tempted t o ascribe diagonal patterns t o cohort effects, vertical patterns t o period effects, and horizontal patterns t o age effects. Note, for instance, t h a t in Figure 45, the period-age interactions produce strong diagonals a n d t h a t in Figure 46, the interac- tion of age a n d cohort effects with no period effect produces, in t h e lower central block, vertical lines, whereas the interaction of period a n d cohort effects with no age effect produces, in the right middle block, horizontal lines.
10 30 50 70 90 Year
Figure 47. Interaction of age, period, and negative cohort effects.
19. Conclusion
T h e figures presented in this paper suggest just a few of the numerous ways t h a t demographers can use contour maps t o clearly, efficiently, and simultaneously display both persistent global and prominent local patterns in population rates or levels over two dimensions. In particular, contour maps can strikingly reveal the interaction between age, period, and cohort patterns. By using small multiples or computer movies, demographers can use the maps t o gain access t o several dimen- sions.
Even in cases where some demographic d a t a already have been carefully scru- tinized by perceptive analysts who have uncovered most of the interesting patterns, contour maps may be useful in highlighting these patterns. With contour maps, what was before understood can now be seen. Furthermore, the maps, by giving demographers a new perspective on data, may focus attention on some neglected aspects and patterns in even the thoroughly studied data.
Beyond efficient description, contour maps can help demographers with exploratory d a t a analysis and with model building. Surfaces can be computed rela- tive t o some p a r t of the surface or t o another surface; a n d different surfaces can be ing, efficient, a n d clear means of giving demographers visual access t o such surfaces.
William Playfair (1801), the pioneer of graphical methods for presenting sta- tistical d a t a , argued t h a t with a good visual display "as much information may be
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About the Authors
J a m e s W . V a u p e l and A n a t o l i I. Y a s h i n are research scholars and B r a d l e y A . G a m - bill is a research assistant in the Population Program, led by Nathan Keyfitz, at the International Institute for Applied Systems Analysis (IIASA) in Laxenburg, Aus- tria. Vaupel is also Professor of Public Affairs and Planning a t the University of Minnesota, USA; Yashin is a senior researcher a t the Institute for Control Sciences in Moscow, USSR; a n d Gambill is a student a t Duke University, USA.