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Visualisation of historical events using Lexis pencils
5. Further development - Extending the Lexis display into three dimensions
First, we note that the 2-D Lexis pencil display is rather a strange concept, with 3-D pencil objects being plotted in a 2-D co-ordinate system. An obvious extension is therefore to use a 3-D co-ordinate system to position the pencils in space, with the y-axis defined as before and representing time spent in the study, but with a base plane defined by the x and z-axes rather than a base x-axis. We define this to be a 3-D Lexis pencil display. This extension into three dimensions corresponds to the characteristics of many datasets. Often event history studies have more than one way of representing time; typical temporal variables will include the age of the individual and calendar date. If these are defined to be the x-axis and z-axis respectively, then the pencils can be anchored on the base plane according to age at first event and date of first event, and the pencils will slope at 45 degrees to both the x and z-axes. If the x-axis is defined to be 'age at first event' and the z-axis to be 'date of first event' then x and z are constant over time, and the pencils will instead be vertical.
If there is no obvious second temporal dimension, then any other continuous variable can be used to define the z-axis and to position the pencils. One special useful case is using a temporal variable such as age for the x-axis and the rank order of age for the z-axis. This will neatly space out the pencils along a curve which represents the cumulative distribution function of age when viewed perpendicular to the x-z base plane.
The 3-D Lexis pencil display is a complex diagram, and a single view of the 3-D world will not be sufficient to explore a dataset. We developed these ideas using AVS (Advanced Visual Systems, 1992) , a scientific visualisation system which provides powerful user interactivity for exploring a three-dimensional world, such as zooming, panning and fly-through, as well as the programming environment to construct the Lexis pencils. Using the facilities provided in the developed AVS module, the user has control over the assignment of variables to axes, the number of faces on each pencil and the variables assigned to them, the colour mapping of codes to colours, the width of the pencil face and the angle subtended by adjacent faces. Users can also rotate and spin the pencils around their own axes while keeping the viewpoint fixed, and can display identification information such as case number for any pencil by clicking on it.
Another useful facility is case selection. Subsets of cases may be defined either by selecting cases of the values of certain variables (which may or may not be variables used to construct the display) or by specifying a set of case numbers. This allows users to compare, for example, males and females in separate displays without the necessity of increasing pencil complexity by adding an additional pencil face. Associated with this is the idea of transparency, where a subset of pencils can be made transparent and thus less intrusive on the eye, drawing attention to the fully opaque pencils.
We can now provide further information about the dataset of 188 Kirkcaldy married couples. One of the questions of interest in this study was the relationship between female employment and male unemployment. Discussion in the literature suggested that the wives of males who were unemployed were less likely to work because of the workings of the benefit system at that time - it was not financially advantageous for the female to work. This hypothesis was investigated by Davies et al (1991) who fitted a random effects logistic model to the probability of the wife working in any month. After controlling for the age of the youngest child, the number of children in the household, calendar time and the ages of both the husband and wife, an effect of the duration of the husband's unemployment was indeed found, with unemployment longer than one year being a significant deflator on the probability of the wife working.
Using the pencil construction outlined earlier, we can examine the data graphically using a 3D lexis pencil display, to see if we can find any further insights into the data. The pencils are arranged on an x-z base plane which is defined by age of the wife at marriage and date of marriage. We present two displays. With 188 pencils, we examine subsets of the data, and use the zooming facility of AVS to examine a small number of histories in detail. The first display (Figure 4) looks at couples who married in 1967 and 1968. Recall that the central face of the pencil is the employment history of the woman, with dark blue indicating unemployment, and the bottom face of the pencil is the age of the youngest child, with green indicating no children in the family, and yellow indicating a child aged less than one. What is clear is that women are stopping work before the birth of their first child - a pregnancy effect on the probability of a wife working. This effect is not surprising, but was omitted from the statistical analysis. Other subsets of marriage years show the same effect - the choice of marriage period is not significant.
The second display (Figure 5) shows those couples who married in 1952 and 1953. This display has made of the pencils semi-transparent, in order to highlight two pencils at the front of the display. The leftmost highlighted pencil is untypical for this time period, and illustrates a woman returning to work soon after childbirth - this is more typical of patterns in the 1970s. The rightmost highlighted pencil is more typical, and represents a woman who worked for a short period after marrying but stopped work a long time before the birth of her first child, and then stayed unemployed. The semi-transparent pencils in the background follow the same pattern as this leftmost pencil. In all displyed pencils the husband is working solidly with no evidence of unemployment. The general pattern of women's employment in the 1950s is different from that in the late 1960s and 1970s, where women more typically return to work around five years after their family is complete. This analysis suggests that there may be strong marriage cohort effects in the data, and again, no such effect was controlled for in the original analysis.
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