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  • Working Without a Typology
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    Plan

    Plan détaillé Texte intégral I. Graphical Representations II. Distances from a Reference Sequence III. Intergroup/Intragroup Distances IV. Composite Indicators V. Analysis of Variance VI. Implicative Statistics VII. Sequence Analysis and Event History Analysis Notes de bas de page

    The Statistical Analysis of Trajectories

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    Table des matières

    Chapter 6

    Working Without a Typology

    Texte intégral I. Graphical Representations II. Distances from a Reference Sequence III. Intergroup/Intragroup Distances IV. Composite Indicators V. Analysis of Variance VI. Implicative Statistics VII. Sequence Analysis and Event History Analysis Notes de bas de page

    Texte intégral

    1While this book is mostly focused on typologies of trajectories, doing descriptive analysis does not necessarily require constructing a typology. Other approaches are possible, either as complements or as alternatives.

    I. Graphical Representations

    2All of the graphical representations presented in Chapter 3 are also suited to comparing subpopulations (e.g., women vs. men).

    II. Distances from a Reference Sequence

    3Some studies focus on the distance between individual sequences and a reference sequence (Kogan, 2004; Munoz-Bullon & Malo, 2003; Scherer, 2001). The reference trajectory may be the most frequent one or may be a theoretically constructed “normal” trajectory. The aim, in this case, is to try and explain the difference between a given trajectory and the norm. For example, what factors play a role in deviations from the model of a continuous, full-time career?

    4Let us return to the application from Chapter 4. We can decide to define the reference trajectory as composed of 4 years of schooling (between ages 14 and 17), followed by 33 years of full-time employment (between ages 18 and 50), which corresponds to the largest group of trajectories in the sample. For each individual, we calculate the dissimilarity between their trajectory and the reference trajectory.1 Then, we examine the distribution of deviations from the reference trajectory as a function of gender and number of children (Figure 6). We can see here that women’s trajectories deviate more from the continuous, full-time employment trajectory than do men’s. Looking in more detail, this gendered differentiation is present only among individuals with children and grows with the number of children. The deviation from the reference trajectory is particularly large among women with three or more children.

    Figure 6. Distribution of Deviations from the Reference Trajectory by Gender and Number of Children

    Coverage: A sample of 500 individuals.

    Source: Biographies et entourage survey (INED, 2001).

    5One of the limitations of this approach is that two sequences can be equidistant from the reference sequence and yet be very different from each other. The distance measure alone tells us nothing about the nature of the difference between two trajectories. In our example, it does not tell us whether, on average, the trajectories of women with three or more children deviate from a full-time trajectory by way of part-time work, odd jobs, or economic inactivity.

    III. Intergroup/Intragroup Distances

    6The dissimilarities between trajectories—calculated using one of the measures described in Chapter 2—can be analyzed in their own right without first clustering or classifying the trajectories. For example, we might compare the distribution of dissimilarity between trajectories within (or between) different subpopulations (e.g., by gender, birth cohort). Are women’s trajectories more diverse than men’s? Do we observe a “destandardization” of life courses with successive cohorts (Elzinga & Liefbroer, 2007; Robette, 2010)? Are the life courses of skilled manual workers more similar to those of individuals in intermediate occupations or to those of unskilled manual workers?

    7We could, for example, calculate the mean distance between men’s and women’s trajectories to see whether gender is associated with marked differences in trajectories; however, this value in itself is difficult to interpret, as we lack points of comparison to decide how high it is. Liao and Fasang (2021) proposed an adaptation of the Bayesian information criterion (BIC) and the likelihood-ratio test for just this case. Here, the difference in BIC is very large for gender (32.6), indicating a substantial difference between the occupational trajectories of women and men.

    8The information contained in the distance matrix can be summarized using multidimensional scaling, a form of factor analysis that is adapted to dealing with distance matrixes. As with principal components analysis (PCA), correspondence analysis, or multiple correspondence analysis, for example, here the data are represented in the form of a scatter plot on a plane defined by the factorial axes. We can then project supplementary variables (like the characteristics of individuals—or even the clusters in a typology of trajectories) and observe how close or far apart the categories are.

    9In Figure 7, on the first (horizontal) axis, Cluster 1 (full-time employment trajectories) stands out against the other clusters. This contrast is linked to gendered social relations, insofar as women are markedly skewed to the left and men to the right (on the side of full-time employment).

    10On the second (vertical) axis, the opposition is less clear, but types of trajectories that contain inactivity seem to fall more in the lower part of the graph and the others higher up.

    Figure 7. Multidimensional Scaling of Employment Trajectories

    Coverage: A sample of 500 individuals.

    Source: Biographies et entourage survey (INED, 2001).

    IV. Composite Indicators

    11Indicators describing trajectories such as those listed in Chapter 2 (e.g., mean durations of states, complexity) can be directly used for all kinds of bivariate or multivariate statistical analyses, in combination, for example, with the characteristics of individuals. We can find, for example, that the mean time of economic inactivity in women’s trajectories is 7.7 years, versus 0.9 for men, or that the turbulence of employment trajectories seems to vary little across different cohorts.

    12In addition, these indicators and especially the codings of trajectories from approaches linked to factor analysis (see Chapter 2, Section II) can be subjected to PCA or correspondence analysis. The results are bound to be illuminating (Degenne et al., 1995), particularly in regard to the characteristics of the trajectories that have a structuring role in the study population. An additional step is to project individual characteristics (e.g., gender, level of education) into the same space as supplementary variables.

    13Finally, composite indicators can be used as explanatory variables in regression models, which can involve a feature selection step (Bolano & Studer, 2020).

    V. Analysis of Variance

    14If we want to investigate the extent to which trajectories vary between different subpopulations, there are techniques for the analysis of variance extended to the case where the “dependent variable” is not a variable but a distance matrix (Studer et al., 2011).

    15With a single explanatory variable, we can measure the proportion of the variation in dissimilarities that it explains (using a pseudo-R2), as well as the variation for each of the values of the variable (i.e., within each subpopulation). In our example, gender explains 7.4% of the variance in distances between employment trajectories, which is markedly higher among men than women (19.1 vs. 9.4).

    16These indicators can be detailed for each time position. The proportion of variance explained by gender is nearly 0 at the beginning of the trajectories, grows considerably between ages 18 and 30 (when it reaches 14%), and then decreases again to 5% at age 50 (Figure 8).

    Figure 8. Proportion of Variance Explained by Gender, Ages 14–50

    Coverage: A sample of 500 individuals.

    Source: Biographies et entourage survey (INED, 2001).

    17The variability in the early part of women’s and men’s trajectories is low, increasing through age 21 in both cases (Figure 9). The results then diverge, as the variance in women’s trajectories remains high up to age 50, whereas that of men’s trajectories drops rapidly after age 21 to very low levels between ages 30 and 50.

    Figure 9. Variability of Employment Trajectories Over Time, by Gender

    Coverage: A sample of 500 individuals.

    Source: Biographies et entourage survey (INED, 2001).

    18With multiple explanatory variables, we can obtain the proportion of variance explained by all combined and its decomposition into contributions from the individual variables. Here, year of birth, gender, level of education, and number of children together explain 16.2% of the variance in the dissimilarities between employment trajectories—7.7% for gender, 6.0% for level of education, 2.1% for number of children, and 0.3% for year of birth.

    19Finally, analysis of variance can be used to construct a decision tree (Studer et al., 2009). At the base of the decision tree, we choose the explanatory variable that explains the largest proportion of the variance in the dissimilarity matrix. Then, we divide the population into two subpopulations using that variable. The rest of the tree is constructed in iterative fashion: For each subpopulation, the variable that explains the most variance is selected and is further subdivided into two groups for each of the values of that variable. The tree can be “grown” until no further division is possible. Most often, though, we will want to “prune” the tree to keep its size interpretable, setting a threshold for the maximum number of divisions, the minimum number of observations in each of the final subpopulations, or a minimum degree of association between the distance matrix and the selected explanatory variable.

    20Figure 10 presents a decision tree constructed on the basis of dissimilarities between employment trajectories along with some explanatory variables: year of birth, gender, level of education, and number of children. Here, as above, we see that it is gender that explains the largest share of variance in dissimilarities. Among women, the most discriminating variable is the number of children—more specifically, whether or not they have three or more children. Inactivity and, to a lesser extent, part-time work are more prevalent in the trajectories of women with three or more children than in those of other women. Among men, in contrast, the most discriminating variable is level of education: Men with a secondary school diploma or higher enter the labor market later.

    Figure 10. Decision Tree for Employment Trajectories

    Coverage: A sample of 500 individuals.

    Source: Biographies et entourage survey (INED, 2001).

    21The advantages of this approach are the ease with which the graphical representation of the tree can be interpreted and its ability to deal with combinations of characteristics —in other words, interactions between explanatory variables.

    VI. Implicative Statistics

    22Studer and colleagues propose a way to investigate the different trajectories of multiple subpopulations using implicative statistics, by reconstituting the sequence of typical states for each one (Struffolino et al., 2016; Studer, 2012). In our example, military service is typical of the trajectories of men around age 20, while from age 25 full-time employment is typical (Figure 11). Economic inactivity is characteristic of women’s trajectories starting at age 18; this is also true of part-time employment, but in less marked fashion, beginning at age 30. In implicative statistics, a state A is considered typical of women if the rule “being a woman rather than a man implies A” is statistically significant, according to a measure defined on the basis of the number of counter-examples in the data set (Gras et al., 1996).

    Figure 11. Sequences of Typical States of Men and Women

    Coverage: A sample of 500 individuals.

    Source: Biographies et entourage survey (INED, 2001).

    VII. Sequence Analysis and Event History Analysis

    23As we have seen, sequence analysis and event history modeling are two very different approaches to the study of trajectories. The first is descriptive and treats trajectories as holistic units, while the second—which is more often used to test causal hypotheses inferentially—is centered on events.

    24The potential complementarity of the two is not lost on specialists in sequence analysis, who have been seeking methods that combine them. To date, three have been proposed. Because, to the best of my knowledge, they have thus far only been applied in research articles, I will present them briefly using the examples presented by the authors.

    25Sequence history analysis (SHA; see Rossignon et al., 2018) has been applied to study the determinants of home-leaving. Rossignon et al. seek, in particular, to estimate the effect that coresidence (e.g., with both parents, with one parent, with parents and siblings) has on departure from the parental home. Their first step is to reconstitute the trajectory of each individual’s coresidence from birth to time t, one step at a time, varying t from t0 (birth) to the moment of their departure from home (or to the end of observation, in the case of individuals who did not move out). They, thus, have multiple trajectories per individual and an overall set of trajectories with very heterogeneous lengths. Then, they construct a typology of these trajectories, recoded as sequences of episodes (disregarding the duration of the states, given the differences in the lengths of the trajectories), using optimal matching. The last step is to estimate an event history model with home-leaving as a dependent variable and some explanatory variables, including the typology of coresidence trajectories—a time-varying covariate.

    26The goal of the sequence analysis multistate model (SAMM; see Studer, Struffolino, & Fasang, 2018) procedure is to characterize the relationship between time-varying variables and trajectories. Studer and colleagues apply it to explore the association between the employment trajectories of women in East and West Germany and their time-varying family situation (being in a union or single, having children or not). Starting with data on employment trajectories, they first extract a series of subsequences with an identical 5-year duration for each individual, placing the beginning of the observational window at each change of state. This yields multiple subsequences per person (an average of 3.5) and, contrary to SHA, a set of sequences of identical length. Then, they construct a nine-cluster typology of employment subsequences. Finally, this typology becomes the dependent variable in a multistate model, which they estimate using a Cox model with random effects. The explanatory variables are the country and the period, along with the individual’s age, conjugal status, and parental status.

    27Studer, Liefbroer, and Mooyaart (2018) use competing trajectories analysis (CTA) to study the determinants of individuals’ pathways into adulthood. They code the transition into adulthood in terms of moving out of the parental home, being in a union, and having children. For each individual’s trajectory, they first extract the 5-year subsequence that immediately follows whichever of these events in the transition to adulthood occurs first. Each individual is, thus, characterized by a single subsequence. Then, they construct a typology of these subsequences, identifying seven clusters and interpreting them as the typical ways in which the process of family formation begins. Next, they examine the factors that influence which of these pathways individuals take, estimating an event history model with age at the start of the subsequence as a dependent variable. Finally, they investigate the factors influencing the trajectory itself by estimating a competing risk model with the type of subsequence as a dependent variable. The timing of the first event, the sequence of events, and the spacing of events are, thus, analytically separated.

    28These three approaches have points in common, particularly their use of the construction of “subtrajectories,” which are then used in event history models. With SHA, the typology is an explanatory variable, whereas with SAMM and CTA, it is a dependent variable.

    Notes de bas de page

    1 With optimal matching as a dissimilarity measure, a single substitution cost of 2 and an indel cost of 1.5, as in Chapter 4.

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    1 With optimal matching as a dissimilarity measure, a single substitution cost of 2 and an indel cost of 1.5, as in Chapter 4.

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    Robette, N. (2025). Working Without a Typology. In The Statistical Analysis of Trajectories. Paris: Ined Éditions. https://doi.org/10.4000/153ie
    Robette, Nicolas. « Working Without a Typology ». In The Statistical Analysis of Trajectories. Paris: Ined Éditions, 2025. doi:10.4000/153ie.
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    Robette, N. (2025). The Statistical Analysis of Trajectories (P. Reeve, Trad.). Paris: Ined Éditions. https://doi.org/10.4000/153ii
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