## Abstract

The second chapter specifies how a Lexis diagram is constructed and shows that cohorts are depicted on the 45^{∘} line. It briefly discusses the so-called identification problem of standard methods of age-, period-, and cohort analysis and explains how those effects look like in the Lexis diagram. The chapter concludes with a brief history of the depiction of population dynamics in three dimensions.

### Keywords

- Lexis Diagram
- Effect Looks
- Lex Plane
- Lexical Surface
- International Institute For Applied Systems Analysis (IIASA)

*These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.*

Download chapter PDF

Any dynamics in vital events such as births and deaths involve change over calendar time, age, and/or cohort. The so-called Lexis diagram represents the ideal canvas to illustrate such dynamics. The Lexis Diagram as we use it today consists of a Cartesian coordinate system where calendar time (“period”) is depicted on the *x*-axis and age on the *y*-axis (see Fig. 2.1 on page 6).^{Footnote 1} We added horizontal and vertical reference lines to facilitate orientation.

Birth cohorts move in such a diagram along the 45^{∘} line since a person is 1 year later 1 year older. Expressed differently: The current age of a person can be calculated if we subtract the birth date from the current calendar date. We used the example of three eminent demographers of the twentieth century in Fig. 2.1 to illustrate this relationship: William Brass, Ansley Coale, and Nathan Keyfitz. To be able to follow the cohorts on the 45^{∘} line, we made sure in Fig. 2.1—as well as in all other figures in this monograph—that the aspect ratio maps the length of one calendar year to exactly one age year.

Of course, we are not restricted to depict individuals on the Lexis plane. The standard approach is, indeed, to use population level data. It is obvious that we can not draw lines for every individual in that case. Colors are used instead to indicate the same value for the chosen statistic. While most figures in the remaining chapters show (smoothed) age-specific mortality or its time derivative, we opted to illustrate the basic approach of Lexis surface maps by depicting the population size of the United States for women and men combined from 1900 until 2010 for ages 0–110 in Fig. 2.2. Thus, we have 111 × 111 = 12, 321 individual datapoints. They are less than the 20,000 mentioned by Keyfitz in Vaupel et al. (1985a) but considerably more than the median number of entries in data matrices for statistical graphics found by Tufte (2003) in various scientific and non-scientific publications. Tufte—who was described as the “da Vinci of Data” by The New York Times (Deborah 1998)—states in a related book (Tufte 2001, p. 166): “Data graphics should often be based on large rather than small data matrices and have a high rather low data density. More information is better than less information, especially when the marginal costs of handling and interpreting additional information are low, as they are for most graphics.”

In our Lexis maps we employed a color scheme reminiscent of geographic maps where green colors indicate lower values and brown colors are used for high “altitudes”. Analogously to standard maps, we added contour lines to emphasize areas of equal elevation, which translates to the same number of people in our figure.

Depicting mortality, fertility or other population characteristics in the Lexis diagram provides a useful framework to analyze data for the presence of age-, period-, and cohort- (“APC”) effects. The major problem of standard statistical approaches (e.g., regression analysis) in this area is the so-called *Identification Problem*, which refers to the perfect correlation of age plus cohort equaling period. Various methods have been introduced (e.g., constraining the parameters in a regression setting) but “there is no magic solution” (Wilmoth 2006, p. 235).^{Footnote 2} With our surface maps, we suggest instead a graphical approach that can be used for questions such as “[w]hether mortality improvements takes place by cohorts or by periods” (Keyfitz in Vaupel et al. 1985a, p. ix).

Figure 2.3 gives an overview how age-, period-, and cohort effects would ideally look like on the Lexis surface. The same color indicates the same value in the variable of interest (e.g., death rates). The left panel represents “pure” age effects. That means that the only variation in the variable of interest takes place across the age dimension, regardless of calendar year or cohort. The panel in the middle denotes “pure” period effects, i.e., the same values are measured at all ages but they differ along the calendar time/period dimension (“Year”). Finally, the panel on the right illustrates how a surface map would like if (birth) cohorts alone were driving the development in the variable of interest. The same color along the 45^{∘} line shows that each cohort has their own characteristic value of the variable of interest, which does not change throughout their life course. Obviously, those are idealized and simplified representations. We expect to find rather interactions of these three forces than such “pure” effects. Furthermore, we should acknowledge the biggest drawback of our method: In contrast to other methods of APC analysis, our visual approach does not attribute any numerical value to each of those effects. Hence, one can neither compare various effects with each other nor is it possible to conduct significance tests that are typical of regression analyses and other standard methods in statistics.

We are not the first to illustrate demographic phenomena in three dimensions, i.e., *either* on the Lexis plane using colors to indicate the third dimension *or* by wireframe plots. An interesting overview of the history of such “Frequency Surfaces and Isofrequency Lines” is given in Caselli and Vallin (2006). They cite the example of Luigi Perozzo’s depiction of the change in the Swedish age pyramid in 1880, based on a diagram by Gerard Van Den Berg (1860), as one of those earliest examples. We have reproduced Perozzo’s diagram in Fig. 2.4. About 60 years later, Pierre Delaporte used such wireframes to depict French mortality (1938) and contour lines for European mortality (1942).

An explicit case of using such plots to separate age-, period-, and cohort-effects from each other can be found in Thomas Pullum’s article on US fertility published in 1980. A few years later, the population program at the International Institute for Applied Systems Analysis (IIASA) in Laxenburg in Austria turned out to be an incubator for advancing the display of population dynamics on the Lexis plane in the 1980s. Vaupel, Yashin, Caselli, and others introduced colored/shaded contour maps to depict, for example, population size, mortality, or birth rates (e.g., Vaupel et al. 1985a,b, 1987; Caselli et al. 1985; Gambill and Vaupel 1985). The “democratization” effort described in the introductory chapter was also mirrored in the late 1990s for Lexis surfaces: Kirill Andreev developed not only the user-friendly software Lexis to analyze demographic trends in Denmark and other highly developed countries (Vaupel et al. 1997; Andreev 2002). He also shared it freely with anyone interested.^{Footnote 3} Despite being a milestone for the creation of Lexis surface maps, almost no one is using it anymore. The aforementioned specialized languages such as Matlab (Mathworks 2017) or R (R Development Core Team 2015) have become the favorite tools nowadays along with Python (van Rossum 1995). With the exception of the reproduction of Perozzo’s plot all figures in this monograph were created with R as we will explain in Sect.3.2 and in the appendix, starting on page 161.

## Notes

- 1.
It should be noted that the Lexis diagram can be considered to represent an example of “Stigler’s law of eponymy” that states “No scientific discovery is named after its original discoverer.” Please see Vandeschrick (2001) for a discussion about the problem of calling the diagram used in this book a “Lexis diagram”.

- 2.
Please refer to this article also for a systematic overview of APC models used in demographic research.

- 3.
While writing his Master’s thesis, the first author of this monograph received the Lexis software from Kirill Andreev simply via email in early 2000.

## References

Andreev, K. F. (2002).

*Evolution of the Danish population from 1835 to 2000*(Odense Monographs on Population Aging, Vol. 9), Odense: University Press of Southern Denmark.Caselli, G., & Vallin, J. (2006). Frequency surfaces and isofrequency lines. In G. Caselli, J. Vallin, & G. Wunsch (Eds.),

*Demography. Analysis and synthesis*(Vol. I, Chap. 7, pp. 69–77). Amsterdam: Elsevier.Caselli, G., Vaupel, J. W., & Yashin, A. I. (1985). Mortality in Italy: Contours of a century of evolution.

*Genus, 41*(1–2), 39–55.Deborah, S. (1998).

*The da Vinci of Data*. The New York Times, 30 March 1998.Delaporte, P. (1938). Évolution de la mortalité française depuis un siècle.

*Journal de la société de statistique de Paris, 79*, 181–206.Delaporte, P. (1942). Évolution de la mortalité en Europe depuis l’origine des statisques.

*Journal de la société de statistique de Paris, 83*, 183–203.Gambill, B. A., & Vaupel, J. W. (1985).

*The LEXIS program for creating shaded contour maps of demographic surfaces*. Tech. Rep. RR–85–94. International Institute for Applied Systems Analysis (IIASA), Laxenburg, A.Mathworks. (2017).

*Matlab*. Available at www.mathworks.com.Pullum, T. W. (1980). Separating age, period, and cohort effects in white US fertility, 1920–1970.

*Social Science Research, 9*(3), 225–244.R Development Core Team. (2015).

*R: A language and environment for statistical computing*. R Foundation for Statistical Computing, Vienna, Austria, http://www.R-project.org, ISBN:3-900051-07-0.van Rossum, G. (1995).

*Python reference manual*. Tech. rep., CWI (Centre for Mathematics and Computer Science), Amsterdam, The Netherlands.Tufte, E. R. (2001).

*The visual display of quantitative data*(2nd ed.). Cheshire: Graphics Press.Tufte, E. R. (2003).

*The cognitive style of PowerPoint*. Cheshire: Graphics Press.University of California, Berkeley (USA), and Max Planck Institute for Demographic Research, Rostock, (Germany). (2017).

*Human mortality database*. Available at http://www.mortality.org.Van Den Berg, G. J. (1860). Befolknings statistik in: Underdaniga berattelse for dren 1856–1860, ny folijd II, 3. Stockholm, Statistika Central-Byrans.

Vandeschrick, C. (2001). The Lexis diagram, a misnomer.

*Demographic Research, 4*(3), 97–124. DOI 10.4054/DemRes.2001.4.3, http://www.demographic-research.org/volumes/vol4/3/.Vaupel, J. W., Gambill, B. A., & Yashin, A. I. (1985a).

*Contour maps of population surfaces*. Tech. Rep. RR–85–47, International Institute for Applied Systems Analysis (IIASA), Laxenburg, A.Vaupel, J. W., Gambill, B. A., & Yashin, A. I., Bernstein, A. J. (1985b).

*Contour maps of demographic surfaces*. Tech. Rep. RR–85–33, International Institute for Applied Systems Analysis (IIASA), Laxenburg, A.Vaupel, J. W., Gambill, B. A., & Yashin, A. I. (1987).

*Thousands of data at a glance: Shaded contour maps of demographic surfaces*. Tech. Rep. RR–87–16, International Institute for Applied Systems Analysis (IIASA), Laxenburg, A.Vaupel, J. W., Zhenglian, W., Andreev, K. F., & Yashin, A. I. (1997).

*Population data at a glance: Shaded contour maps of demographic surfaces over age and time*(Odense Monographs on Population Aging, Vol. 4). Odense: University Press of Southern Denmark.Wilmoth, J. R. (2006). Age-period-cohort models in demography. In G. Caselli, J. Vallin, & G. Wunsch (Eds.),

*Demography. Analysis and synthesis*(Vol. I, Chap. 18, pp. 227–236). Elsevier, Amsterdam.

## Author information

### Authors and Affiliations

## Rights and permissions

This chapter is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.

## Copyright information

© 2018 The Author(s)

## About this chapter

### Cite this chapter

Rau, R., Bohk-Ewald, C., Muszyńska, M.M., Vaupel, J.W. (2018). The Lexis Diagram. In: Visualizing Mortality Dynamics in the Lexis Diagram. The Springer Series on Demographic Methods and Population Analysis, vol 44. Springer, Cham. https://doi.org/10.1007/978-3-319-64820-0_2

### Download citation

DOI: https://doi.org/10.1007/978-3-319-64820-0_2

Published:

Publisher Name: Springer, Cham

Print ISBN: 978-3-319-64818-7

Online ISBN: 978-3-319-64820-0

eBook Packages: Social SciencesSocial Sciences (R0)