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    Plan

    Plan détaillé Texte intégral I. Data II. Choosing a Population III. Choosing an Observation Period IV. Choosing States V. Choosing a Dissimilarity Measure and Coding VI. Choosing a Clustering Method VII. Choosing the Number of Clusters in the Typology VIII. Why a Typology? IX. Frequently Asked Questions Notes de bas de page

    The Statistical Analysis of Trajectories

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    Chapter 1

    Creating a Typology of Trajectories

    Texte intégral I. Data II. Choosing a Population III. Choosing an Observation Period IV. Choosing States V. Choosing a Dissimilarity Measure and Coding VI. Choosing a Clustering Method VII. Choosing the Number of Clusters in the Typology VIII. Why a Typology? IX. Frequently Asked Questions 1. What do I do when there is information missing? 2. Can I analyze trajectories of different lengths? 3. To weight or not to weight? 4. Can I build a typology with big data? 5. Can I analyze quantitative sequences? Notes de bas de page

    Texte intégral

    1The trajectories of individuals are often diverse and the potential number of distinct trajectories within a population is very large: For example, with three states and 12 years of observation, the number of trajectories would be 312 or 531,441. While, in practice, many individuals—particularly those with the most stable situations—tend to go through a similar trajectory, the task of grouping similar individuals together simply by observing the data can quickly prove insurmountable. In principle, we might want to create a typology a priori, choosing a set of criteria to group individuals together. For example, we might want to form a class of individuals who have never experienced a given situation or one with life courses that include a long period in a given state (Degenne et al., 1994). But the variability of trajectories can quickly pose problems: Some atypical trajectories may not fulfill any of our chosen criteria, while others fall into multiple categories. Another possibility is to center the analysis on stable trajectories (individuals who remain in the same state from the beginning to the end of the period) or those with few transitions; however, this implies leaving aside a part of the study population. Thus, we have no choice but to consider more sophisticated statistical tools. This means constructing typologies based on a data set.

    2If you want to study relatively few trajectories or trajectories over only a few timepoints, you may want to put this book down and content yourself with simple statistical treatments (two- or three-way tables) or with constructing your categories by hand. It is important not to confound the technical complexity of different methods with their scientific validity or ability to provide compelling evidence.

    I. Data

    3Typologies are especially suitable for situations in which we have survey data that allow us to (at least partially) reconstruct the respondents’ trajectories. In some cases, only a few life events (e.g., first marriage, first child) are recorded, and we will analyze simplified trajectories. But many longitudinal surveys record individual trajectories in detail, with data on many transitions, representing a diversity of situations. These surveys can be retrospective—as in the case of biographical surveys, such as those on urban integration in Africa (Antoine et al., 2012) and the perceived health of patients with HIV in Thailand (Lelièvre & Le Cœur, 2010)—or prospective, like the French (Étude longitudinale française depuis l’enfance [ELFE]) child cohort study or the Panel Study of Income Dynamics. Trajectories can also be reconstructed on the basis of other sources, such as administrative records (Fréchon & Robette, 2013) and prosopography (Dubois & François, 2013; Lemercier, 2005).

    Figure 1. Distribution of Perceived Health Status Over the Health Trajectories of People Infected with HIV in Thailand

    Interpretation: Perceived health trajectories are shown from 5 years before the beginning of antiretroviral treatment to 3 years later, with the beginning of treatment as Year 0. At the scale of the entire study population, the proportion of people reporting poor or very poor health increased during the years preceding treatment and decreased rapidly afterward.

    Source: LIWA (Living with Antiretrovirals) survey.

    II. Choosing a Population

    4The first step in constructing a typology of trajectories is to circumscribe the population we want to study. While the choice of how to do this depends above all on the research question, one technical issue in particular demands attention: the risk—when simultaneously analyzing multiple subpopulations with markedly differentiated types of trajectories—of masking the specific regularities of the trajectories of each subpopulation in the results. Imagine, for example, that we want to analyze the timing of family formation (e.g., couple formation, birth of children) to identify trajectories characterized by early or late transitions. If women in the study population experience these transitions earlier than men, then analyzing the two together will blur the distinction between early and late trajectories. In this case, it would be better to analyze data from women and men separately.

    III. Choosing an Observation Period

    5Choosing an observation period means specifying both a beginning and an end. In the case of life courses, these are most often ages: for example, activity trajectories between ages 14 and 65. But it is perfectly possible to define the period as the span between two dates or events; to cover an event and the n years that follow it (e.g., individuals’ trajectory after leaving formal education) or precede it (e.g., individuals’ trajectory before retirement); or to “frame” an event (e.g., the perceived health trajectory of people living with HIV, from 5 years before the beginning of antiretroviral treatment until 3 years afterward; see Figure 1).

    IV. Choosing States

    6In general, it is better to choose a relatively limited space of possible situations—that is, to code trajectories on the basis of a limited number of states. Otherwise, the results may be less robust and harder to interpret. Some cases do, nonetheless, require a large number of states. This can be true, in particular, when we are dealing with multidimensional trajectories and we want to use states that combine various dimensions, such as marital status, number of children, and activity status (see Chapter 5). In this context, some studies have found that once the difficulty of interpreting the results has been overcome, they are relatively robust (Robette, 2010).

    V. Choosing a Dissimilarity Measure and Coding

    7Another important step in creating a typology is the choice of a measure of the dissimilarity (or distance) between trajectories. Any of the methods described in Chapter 2 can be used to do this. This can involve its own trade-offs, as we will see with optimal matching (OM).

    8These decisions are not neutral. In making them, we must take into account the specificities of the different methods, the data, and the research objectives. No single method can be considered “better” than another in general without considering their use in a particular context. One of the benefits of the methods presented in this book (and in this regard, of OM, in particular) is precisely the need to make our choices explicit and, thus, to think through their meaning from a theoretical perspective (Lesnard, 2010).

    9The choices of how to code trajectories and of a dissimilarity measure are linked. In most cases, we will have one variable per element of the trajectory: For example, a family history based on the annual observation of individuals’ marital status over 35 years will be coded using 35 variables, with the nth variable representing their marital status in the nth year of observation. However, other types of coding are also presented in Chapter 2 and, in certain cases, we may want to devise one that is specific to our research question.1

    VI. Choosing a Clustering Method

    10Once we have obtained a matrix of distances between trajectories using our chosen dissimilarity measure, the last step in constructing a typology is based on an automatic clustering procedure. This aims to divide the population into a limited number of relatively homogeneous and distinct groups, and to identify a set of “typical” trajectories. There are many clustering methods but, for the most part, they belong to one of two families: hierarchical clustering and partitioning.2

    11Among hierarchical clustering algorithms, divisive approaches start with the entire population and progressively break it down into smaller groups, step by step. Agglomerative hierarchical clustering (AHC), in contrast, iteratively groups together the most similar individuals according to a predefined measure of resemblance (the aggregation criterion). There are many types of aggregation criteria (or linkage rules) to choose from: for example, minimum, average, or maximum dissimilarity (termed single, average, or complete linkage, respectively); the centroid method. The most widely used in the social sciences is Ward’s criterion, which, at each step, minimizes the heterogeneity of the trajectories within each cluster (intraclass inertia), which is equivalent to maximizing heterogeneity between clusters (interclass inertia). It is known for producing relatively homogeneous and compact clusters (Nakache & Confais, 2004). Moreover, analyses have shown that the flexible WPGMA (weighted pair group method using arithmetic average) and flexible UPGMA (unweighted pair group method using arithmetic averages) criteria are particularly suitable for use with empirical data that include noise or outliers (Belbin et al., 1992; Lesnard, 2010; Milligan, 1981).

    12Hierarchical clustering methods—whether agglomerative or divisive—produce a clustering tree, known as a dendrogram. Each level in a dendrogram corresponds to a partition of the entire set of individuals in the study population. We, as users, decide how many clusters to include in the typology and, in doing so, we may make use of statistical indicators (see Section VII on choosing the number of clusters).

    13Partitioning methods include the k-means method and its variants (dynamic clustering, k-medoids). This method begins with a user-defined number of center points (or centroids), with each individual integrated into the cluster around the closest one. This operation is then repeated multiple times, with the centers of gravity of the clusters in a given partition taken as the new centroids for each new iteration. The process stops when a stable partition is obtained. The final number of clusters is dictated by the initial, user-defined number of centroids. One of the advantages of methods in this family is speed: They are much faster to compute than AHC, which can be useful with large numbers of observations. However, the results depend on the (arbitrary or random) choice of initial centroids. Choosing the number of clusters a priori also limits the exploration of the data. With hierarchical methods, in contrast, we can choose the number on the basis of statistical or other criteria and analyze the results at different levels of partition.

    14The most widely used hierarchical partitioning method is the k-means algorithm; however, it has the drawback of being sensitive to atypical observations, or outliers. A preferable method is the k-medoids algorithm: In this approach, it is not the center of gravity that is used at each step, but a medoid—an observation that is actually part of the data set and whose distance from the other observations in the cluster is the smallest. This approach is more robust to outliers.

    15In practice, these methods are often used as a complement to hierarchical clustering methods:

    1. Beforehand: to simplify the data when a large number of observations makes computing time with hierarchical methods alone too large. We might first partition the data into 50 clusters using k-medoids, before using AHC starting from these 50 clusters to explore a more parsimonious partition. This is known as “mixed clustering” and has the advantage of allowing the use of large data sets while enjoying the advantages of AHC (e.g., nested partitions, dendrogram for understanding and identifying clusters).

    2. Afterward: to make the clusters (a little) more homogeneous after the user has chosen the number on the basis of hierarchical clustering. This is known as consolidating the partition. The idea is to perform k-means (or k-medoids) clustering, taking the centers of the clusters identified using AHC as the initial centroids or medoids.

    16One possibility—inspired by k-means/medoids methods—is to define some “typical” trajectories (or templates) a priori and then group each individual trajectory together with whichever it is most similar to. This approach is of particular interest when we have strong hypotheses about the regularities to be found in the trajectories we are studying (Elzinga & Liefbroer, 2007) or when we have a large number of observations (because it requires the calculation of a significantly smaller number of dissimilarities).

    17Another approach, known as monothetic (or property-based) clustering, requires all observations in each cluster to have at least one shared property. This property is generally obtained by dividing a cluster on the basis of a so-called split variable. The main advantage of this approach is the ease with which the resulting clusters can be interpreted. Each is defined by a specific combination of characteristics, which constitutes a necessary and sufficient condition for membership. Additional observations can, thus, readily be assigned to a cluster. Unlike the approaches described above, monothetic classification does not take a distance matrix as an input. When it is used to analyze trajectories, then a certain number of indicators describing them must be predefined, such as the amount of time spent in the different states, commonly encountered transitions, or the number of episodes in the different states (Studer, 2018). The DIVCLUS-T (divisive clustering tree) algorithm is an example of a monothetic divisive hierarchical clustering method (Chavent et al., 2007).3

    18Most clustering methods assign each observation to just one cluster. This can be questionable, particularly for observations that fall close to the boundaries between clusters, whose assignment to one or the other can come down to very small differences in methodology or parameters. Fuzzy clustering—with algorithms such as Fuzzy C-means and FANNY—defines each observation’s “degree of membership” in each cluster, indicating its proximity to the other members. With three clusters, A, B, and C, an observation’s degree of membership may, for example, be 0.1 for Cluster A, 0.6 for Cluster B, and 0.3 for Cluster C (the sum being equal to 1). This type of method may be helpful if we want to “return to the data” (study particular observations), study the boundaries between clusters, or characterize clusters in terms of their “core” (i.e., most central observations).

    19Also noteworthy is the development of neural clustering methods associated with Kohonen maps (Cottrell & Ponthieux, 2002; Delaunay & Lelièvre, 2006; Giret & Rousset, 2007). These “self-organizing maps” provide an interesting visualization of the proximity between clusters, similar to projections from factor analysis.

    20An approach adapted from latent class analysis has also been used to obtain typologies of life courses (Barban & Billari, 2012; Han et al., 2017; MacMillan & Eliason, 2003). However, this is a parametric method and, as such, requires stronger assumptions than the other techniques described in this book.

    21Finally, while there is a diversity of clustering methods and it is advisable to try several to ensure the robustness of the results obtained, it is worth noting that, to date, the combination of AHC with Ward's aggregation criterion is the most widely used and successful approach to producing typologies of trajectories.

    VII. Choosing the Number of Clusters in the Typology

    22The question of the number of clusters produced with automated clustering methods has been the focus of recurring critiques. Because this choice is left to the researcher’s discretion, some consider it arbitrary, arguing that the method lacks robustness as a result. This critique results from a misunderstanding of the nature of these methods and, often, of a causal and inferential perspective on quantitative analysis. Clustering is descriptive and nonparametric. Its goal is not to precisely measure a phenomenon or quantify the effect of one characteristic on another but to identify regularities while keeping the assumptions made about the data to a minimum. It is in this flexibility that its value and analytical power reside. The idea of determining the “true” or “best” number of clusters in a typology on the basis of statistical criteria—independently of the research question—makes little sense: “Classifications so produced can never be true or false, or even probable or improbable; they can only be profitable or unprofitable” (Williams & Lance, 1965, p. 160). The creation of a taxonomy in the social sciences should be guided by theoretical foundations, the heuristic value of the results, and a trade-off between the parsimony of the partition and the homogeneity of the clusters.

    23In practice, observing the typologies at different levels of partition is strongly recommended.4 It is often helpful to explore the nature and homogeneity of a cluster by looking at the subclusters that constitute it. Too many clusters and the results will be difficult to interpret and describe; too few and there is a risk that their internal heterogeneity will be too great, making it difficult to identify typical trajectories. Choosing a typology, thus, implies a trade-off, with the main criterion being that the finally selected typology should both be coherent and shed light on the research questions at hand.5

    24That said, statistical indicators can, nonetheless, prove useful—particularly for guidance in the first steps in analyzing the results. In other words, because the number of partitions that can potentially result from clustering is large, an indicator can help direct us to the first one to look at, before exploring typologies with higher or lower numbers of clusters. There are many such indicators (e.g., inertia gains, Calinski-Harabasz criterion, Hartigan criterion6), most often based on the comparison of inter- and intraclass variances. The general aim is to minimize the variations within each cluster and maximize the difference between clusters.7

    VIII. Why a Typology?

    25A typology of trajectories summarizes a diverse corpus of data. But what use is such a summary in an empirical analysis?

    26First of all, a typology is useful if we lack a priori knowledge on the forms of trajectories that we are likely to find in the corpus. In reality, this rarely happens, unless we are lucky enough to be studying a topic that has never before been investigated in the social sciences. It is not uncommon, however, for surprising types of trajectories to emerge, even if there are relatively few of them. In this sort of situation, the typology can unveil unexpected facts and, thereby, open up new avenues for research.

    27Next, a typology is useful when we want to assess how common different types of trajectories are. But caution is needed: A desire for overly precise quantification can lead to overinterpretation. While the regularities identified in a typology may be robust, the number of observations in each cluster is relatively sensitive to our methodological choices, particularly at the automatic clustering stage.

    28Finally, a typology of trajectories is valuable as a step in our analysis—a summary of the data that allows us to take our investigation further, for example, by comparing trajectories across different subpopulations or time periods.

    IX. Frequently Asked Questions

    1. What do I do when there is information missing?

    29There is no general answer to this question. To date, there has been little research on the question of information missing from trajectories. It is possible, for example, to create an additional “missing value” state. If there are few nonresponses and they are not associated with a particular type of trajectory, this should have little effect on the results. But if there are many nonresponses, we risk ending up with clusters of individuals whose main point in common is the lack of information on their trajectory.

    30Imputing the missing value is a possible, if imperfect, option. Moreover, it poses specific problems in the case of longitudinal data. Trajectories are generally characterized by a certain stability and the standard imputation procedures tend to overestimate transition rates (Halpin, 2016). Brendan Halpin has developed a version of chained multiple imputation that fills the “gaps” in trajectories in a way that takes into account the nature of longitudinal data. The principle is to constrain the imputations to respect this longitudinality (Halpin, 2012, 2013).8

    31Another possibility is simply to delete the missing elements, which correspondingly shortens the sequences and can lead to biases.

    32Finally, as for most of the other choices, it is best to try out different options to compare the results and assess their robustness.

    2. Can I analyze trajectories of different lengths?

    33The length of the observation period may or may not be the same for all individuals. This remains a relatively problematic issue. First of all, the meaning of a difference in length depends on whether it is because of the nature of the underlying process or a feature of the data collection. For example, the variable length of the school-to-work transition in itself constitutes information about the life courses of individuals. We may want to take this into account in grouping together individual trajectories, considering young people with “short trajectories” relatively similar and relatively distinct from young people with “long trajectories” when we are constructing our typology.

    34The problem is entirely different when the variations in length are a consequence of how the data were collected, such as when they are right- or left-censored. For example, if a survey of people aged 30–60 reconstructs activity trajectories starting at age 14, their length will vary between 16 and 46 years for reasons external to the observed process. When we construct a typology, it will be difficult to distinguish what in the composition of the clusters is because of a “real” difference between trajectories from what is because of their varying lengths.

    35Moreover, not all dissimilarity measures allow differences in the length of trajectories to be taken into account. The Hamming distance—or one constructed using a complete disjunctive coding (see Chapter 2)—is based on simultaneity; without using coding tricks (adding “nonobserved” states), they are not compatible with variable lengths.

    36With certain dissimilarity measures, distances between trajectories can be normalized (Elzinga & Studer, 2019; Gabadinho, Ritschard, Müller et al., 2011). We can, for example, normalize the distance between two trajectories by dividing it by the length of the longest of the two (Abbott & Hrycak, 1990; Stovel et al., 1996), or by the theoretical maximum distance.

    37Finally, Stovel and Bolan (2004) propose the introduction of an indel cost that varies with sequence length. It is fixed when the sequences being compared are of equal length and equal to approximately a quarter of that fixed cost when the sequences are of different lengths.

    38Most importantly, we must ask what the meaning of similarity between trajectories of different lengths would be with respect to our research object. Should trajectories that consist of the same chain of events but take place over highly variable periods of time be considered similar?

    3. To weight or not to weight?

    39Weighting trajectories makes no difference at the stage of distance matrix construction. The degree of similarity between two trajectories is independent of their respective weights. But taking weights into account is preferable at the automatic classification stage and indispensable when analyzing the resulting typology (e.g., numbers of individuals in different clusters).

    4. Can I build a typology with big data?

    40If the sample of trajectories is very large, computing time can become a problem—in constructing the distance matrix but, also and above all, in the automatic classification step. There are “technical” strategies that can be used here: for example, choosing a clustering method that is less computationally costly (e.g., k-means, rather than AHC), using optimized software implementations9 or aggregating identical trajectories.

    41Another possibility is to take the template approach (see above), defining typical trajectories a priori and constituting groups of trajectories based on similarity with them. This limits the number of distances that need to be calculated10 and, most importantly, removes the need for an automatic classification step. If we do not want to define typical trajectories a priori, we can consider constructing a typology of trajectories on the basis of a subsample of limited size and, then, use the resulting clusters to define templates and apply the preceding strategy.

    5. Can I analyze quantitative sequences?

    42Sometimes what changes over the course of a trajectory is not a categorical state but a quantity (e.g., salary). There have been few studies exploring typologies of trajectories in this type of context in social sciences such as sociology or demography.11 In econometrics, on the other hand, there is an abundant literature on time series.

    43One of the possibilities is simply to discretize the quantity to produce trajectories of the type dealt with in this book. If the trajectories are of the same length, then such classical distance measures as Euclidean or Manhattan distance can also be used.

    Notes de bas de page

    1 For example, Stovel (2001) recodes trajectories such that the state at a given moment integrates information about past events.

    2 For an overview of clustering techniques, see Kauffman and Rousseeuw (2009), for example.

    3 Billari and Piccarreta (2005, 2007) have developed comparable methods for trajectories composed of nonrepeatable events.

    4 This is easier to do with hierarchical methods than with partitioning methods.

    5 We might, for example, want a particular type of trajectory to emerge, even if it is rare, and bringing it out implies a relatively high number of clusters.

    6 For a review and an empirical test of these criteria, see Milligan and Cooper (1985) or Studer (2013).

    7 Note that Studer (2019) proposes a method for the validation of typologies of trajectories using permutation tests and bootstrapping.

    8 This approach is implemented in the Stata® module MICT (Halpin, 2016).

    9 In R, the fastcluster package provides an optimized implementation of AHC (Müllner, 2013).

    10 On the order of N x K instead of N2, with N being the number of trajectories and K being the number of clusters.

    11 For one of the few attempts to date, see Gaudin et al. (2006).

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    1 For example, Stovel (2001) recodes trajectories such that the state at a given moment integrates information about past events.

    2 For an overview of clustering techniques, see Kauffman and Rousseeuw (2009), for example.

    3 Billari and Piccarreta (2005, 2007) have developed comparable methods for trajectories composed of nonrepeatable events.

    4 This is easier to do with hierarchical methods than with partitioning methods.

    5 We might, for example, want a particular type of trajectory to emerge, even if it is rare, and bringing it out implies a relatively high number of clusters.

    6 For a review and an empirical test of these criteria, see Milligan and Cooper (1985) or Studer (2013).

    7 Note that Studer (2019) proposes a method for the validation of typologies of trajectories using permutation tests and bootstrapping.

    8 This approach is implemented in the Stata® module MICT (Halpin, 2016).

    9 In R, the fastcluster package provides an optimized implementation of AHC (Müllner, 2013).

    10 On the order of N x K instead of N2, with N being the number of trajectories and K being the number of clusters.

    11 For one of the few attempts to date, see Gaudin et al. (2006).

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    Référence numérique du chapitre

    Format

    Robette, N. (2025). Creating a Typology of Trajectories. In The Statistical Analysis of Trajectories. Paris: Ined Éditions. https://doi.org/10.4000/153i9
    Robette, Nicolas. « Creating a Typology of Trajectories ». In The Statistical Analysis of Trajectories. Paris: Ined Éditions, 2025. doi:10.4000/153i9.
    Robette, Nicolas. « Creating a Typology of Trajectories ». The Statistical Analysis of Trajectories, Ined Éditions, 2025, https://doi.org/10.4000/153i9.

    Référence numérique du livre

    Format

    Robette, N. (2025). The Statistical Analysis of Trajectories (P. Reeve, Trad.). Paris: Ined Éditions. https://doi.org/10.4000/153ii
    Robette, Nicolas. The Statistical Analysis of Trajectories. Traduit par Paul Reeve. Paris: Ined Éditions, 2025. doi:10.4000/153ii.
    Robette, Nicolas. The Statistical Analysis of Trajectories. Traduit par Paul Reeve, Ined Éditions, 2025, https://doi.org/10.4000/153ii.
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