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    Plan détaillé Texte intégral I. Composite Indicators II. Approaches Linked to Factor Analysis III. Complete Disjunctive Coding IV. Qualitative Harmonic Analysis V. Optimal Matching VI. Choosing a Method Notes de bas de page

    The Statistical Analysis of Trajectories

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    Table des matières

    Chapter 2

    Measuring the Dissimilarity Between Trajectories

    Texte intégral I. Composite Indicators II. Approaches Linked to Factor Analysis III. Complete Disjunctive Coding IV. Qualitative Harmonic Analysis V. Optimal Matching 1. Description 2. Choosing the Costs 3. Some Alternatives VI. Choosing a Method Notes de bas de page

    Texte intégral

    1Although optimal matching (OM) is the most widespread approach to measuring the dissimilarity between trajectories in the social sciences, there are numerous techniques for doing so, each with its advantages and limitations. To make it easier to follow the presentation of different methodologies, we will use a purely fictitious example: the employment trajectories of a set of individuals ages 18–29. Their situation over this time has been observed every year. We, thus, have a series of 12 successive states for each person, chosen from the following list of three: in education or training (E), unemployed or inactive (I), and in activity or employment (A). We will begin with the example of a person we will call Calvin, who becomes unemployed at the end of their studies at age 20 and finds a job at age 23. Their trajectory can be represented as follows:

    Table 1. Example of an Individual Trajectory: Calvin

    I. Composite Indicators

    2One approach is to summarize trajectories using a set of simple indicators—of duration or number—describing the states and transitions of which they are composed (Charlot & Pottier, 1987; Studer, 2018). This strategy is often neglected, and yet it is both simple and valuable when we have an a priori idea of the aspects of the trajectories that we want to highlight and study.

    3Some example indicators:

    • the initial situation and the final situation

    • the modal state (i.e., the situation that the individual was in for the largest amount of time) or the modal transition (i.e., their most frequent transition between two states)

    • the mean total duration of each state (e.g., the average amount of time spent in unemployment)

    • the mean number of transitions over the course of trajectories, which distinguishes more stable trajectories from more chaotic ones

    • the mean proportion of the trajectory spent in the principal situation

    • the mean number of episodes in each state (e.g., the average number of periods of unemployment or the fact of having experienced a particular state at all)

    • the average time taken to access a particular state (e.g., the average amount of time that passes before an individual enters their first stable job)

    • the presence or number of occurrences of a given subsequence

    4There are also various indicators of the complexity of trajectories, such as longitudinal entropy (Gabadinho et al., 2009); turbulence (Elzinga & Liefbroer, 2007); and Gabadinho, Ritschard, Müller et al.’s (2011) complexity index. Longitudinal entropy is based on the Shannon entropy of the durations of the periods spent in the different states. It, thus, measures the occurrence and distribution of the durations of the different states in the trajectory. It is lowest when the entire trajectory is spent in the same state and highest when the trajectory spends identical amounts of time in all possible states. Here, the focus is on the prevalence of these situations. Transversal entropy—based on the variety of situations at a given time—does not take into account the frequency of transitions or the order of events.

    5Turbulence is calculated on the basis of the number of distinct subsequences that can be extracted from the trajectory and the variance of the durations of the episodes in the different states. The complexity index is calculated as the geometric mean of longitudinal entropy and the number of transitions in the sequence.

    6This is not an exhaustive list, and many more indicators could be devised and potentially combined, depending on the aims of the research. The table constructed on the basis of these variables can be used directly with a standard measure, like Euclidean distance, or after using a normalized principal components analysis (PCA) to eliminate statistical noise and focus in on the relevant information.

    II. Approaches Linked to Factor Analysis

    7One of the families of methods for constructing typologies of trajectories is based on the application of factor analysis techniques. It is mainly the fruit of the many studies on employment and training carried out by researchers at CEREQ (the Centre for Studies and Research on Qualifications).1 CEREQ carried out its first major survey on occupational pathways in 1980. Other longitudinal surveys followed over the subsequent decades, both requiring and fostering the development of specific methods to make use of these data in a context marked by the increasing complexity of labor market integration, which became progressively more difficult both to define and to identify (Fénelon et al., 1997). Entry into working life is no longer a simple change of state—an irreversible transition between an initial situation (being a student) and a final situation (a stable employment contract). It has gradually transformed into a complex process that takes place over a relatively long time and is often described as a “path.”

    8In practical terms, most of the work at CEREQ on career paths draws on data from (generally, month-by-month) calendars that accompany the survey questionnaires and that collect data on the individual’s activity status. While the nomenclature of the states recorded in the calendars varies between surveys, it always concentrates on one or more dimensions of the respondent’s situation with respect to paid employment (e.g., activity status, employment contract, working time, pay, occupation, socio-occupational category), at varying levels of precision.

    9CEREQ research most often uses factor analysis techniques, from the French current of multidimensional exploratory statistics (see, for example, Bry, 1995, 1996; Le Roux & Rouanet, 2004; Lebart et al., 1997), adapted and applied to longitudinal data.

    10There are multiple variants of this application of factor analysis techniques to trajectories. The differences lie mainly in data coding and dissimilarity measures (Grelet, 2002).

    III. Complete Disjunctive Coding

    11One method is to recode the calendar of the trajectory in complete disjunctive form. In this type of coding, a binary variable is associated with each state for each time unit in the trajectory. In our example, for each year, we create three variables corresponding to each of the possible states. They take a value of 1 when the individual was in that state in that year and a value of 0 otherwise. We, thus, have 12 x 3 = 36 dichotomous variables for each individual. Calvin’s trajectory is summarized as follows:

    Table 2. Complete Disjunctive Table of Calvin’s Trajectory

    12This can be viewed as a frequency table and submitted to correspondence analysis. The measure used in this case is the chi-square distance. But the same table can also be submitted to a nonnormalized PCA, which uses the standard Euclidean distance metric. This method is occasionally referred to as the LIRHE2  method (Béduwé, 2001; Béduwé et al., 1995; Espinasse, 1993), after the laboratory at which some of the first studies using it were produced. With correspondence analysis, and thus with the chi-square metric, the distance between two individual trajectories is weighted by the inverse of the frequency of the variables. This means situations that are uncommon in a given year contribute more to the calculation of the distance between two trajectories than do common ones. In other words, more importance is given to rare states. Euclidean distance, in contrast, is simply based on the number of discrepancies between the trajectories, and the contribution of the states at different time points is equal.3 In our example (Table 3), the trajectories of Calvin and Hobbes differ at ages 20, 22, and 23; there are, thus, three discrepancies between them.

    Table 3. The Occupational Trajectories of Calvin and Hobbes

    13Using the combination of complete disjunctive coding with factor analysis to measure the dissimilarity between trajectories, thus, emphasizes the contemporaneity of identical situations—whether they occur in immediate succession or at widely separated moments. In this approach, it is the duration of the time spent simultaneously in the same state that defines the similarity between individuals’ trajectories. The notion of simultaneity implies taking into account not only the occurrence of situations but also their timing. The nature and order of the transitions, however, are not integrated into the analysis.

    14Note that the use of factor analysis is not indispensable. Clustering can be performed directly on the basis of Euclidean4 or chi-square distances. However, the use of factor analysis is strongly recommended. This allows the analysis to be focused on the dimensions that capture most of the information (or inertia) contributed by the different variables—eliminating residual dimensions, which are too weakly correlated with the sequences and, as a result, lack robustness (these can be thought of as “noise” dimensions). This remark also applies to the qualitative harmonic analysis presented in the next subsection.

    15Moreover, factor analysis is based on correlations (or covariances) between variables. In the case of trajectories encoded in the form of complete disjunctive tables, these variables correspond to an individual being in a given state in a given year. Individual trajectories are often composed of only a limited number of changes of state. It follows that an individual being in a given state at time t is likely correlated with their being in the same state at time t+1. This implies that factor analysis limits the emphasis on the simultaneous occurrence of identical situations. In other words, two trajectories that are identical but slightly offset in time (e.g., both feature a transition from education to employment but with a difference of one year in the timing of the change) will nonetheless be considered relatively similar.

    IV. Qualitative Harmonic Analysis

    16An alternative to complete disjunctive coding is to describe trajectories by summarizing the series of events that feature in them, a method that is sometimes called qualitative harmonic analysis (QHA). To do so, we divide the period studied into intervals and then, for each of these, measure the time spent in each state. Variables counting the transitions from one state to another can also be used. We, then, apply correspondence analysis (chi-square distance).

    17Harmonic analysis is a branch of mathematics that has long been widely applied in the physical sciences and biology. Its use in the social sciences is more recent, beginning in the 1970s (Deville, 1974, 1977). The aim was to incorporate duration into the explanation of social phenomena through data on individual histories:

    18Faced with such rich data, the statistician experiences a certain perplexity. Increasingly complex tables become uninterpretable without the aid of “automatic” methods of analysis. They thus seek to define a method of analysis that allows them to extract what is essential from the data at their disposal. Here, the word “essential” takes on a precise, quantifiable meaning, which is linked to the method that they employ. (Deville, 1977)

    19Deville, then, adapted this technique for use in the exploratory statistical analysis of complex trajectories, under the label of QHA (Deville, 1982; Deville & Saporta, 1980). For decades, the method was rarely used (Degenne et al., 1996; Grelet, 2002). In the end, it would not be until the revival of the collection of event history data that practical applications were developed. This occurred first with Latin American data (Barbary, 1997; Barbary & Pinzon Sarmiento, 1998; Dureau et al., 1994) and then, more recently, with French data—notably, with the Biographies et entourage survey, which provides data on family (Robette & Thibault, 2006), geographical (Robette et al., 2012), and occupational trajectories (Robette & Thibault, 2008).

    20Constructing the data table for QHA involves a series of steps. Once the study period has been defined (between ages 18 and 29, in our example), it must be divided into intervals for analysis. If the number of intervals is equal to the length of the trajectory, we have a complete disjunctive coding, which means that there is no added value in using QHA. On the other hand, if we use too few intervals, a valuable part of the available information will be lost. Thus, a compromise is needed.

    21Another trade-off concerns the lengths of the different intervals. There is no requirement that they be equal. Certain periods in life—most often in youth—tend to be marked by many events in a relatively short time, while other periods are much more stable.5 Choosing shorter intervals for these times of greater change allows them to be given more emphasis in constructing a typology of trajectories. In our example, if we decide to divide the period studied into three intervals, one option is to make them each 4 years long. The three intervals will then be from age 18 to 21, from 22 to 25, and from 26 to 29. But if we believe that more of the events (transitions from one state to another) likely occur at the beginning of the trajectory, we can emphasize these early years by dividing it up into unequal intervals: from age 18 to 20, from 21 to 23, and from 24 to 29. This gives us two successive periods of 3 years followed by a period of 6 years.

    22Next, for each individual, we calculate the proportion of each interval that they spent in each of the possible states. This gives us a number of variables equal to the number of intervals multiplied by the number of states. In our example, with three states and the study period divided into three equal intervals, the total number of variables is 3 x 3 = 9. For Calvin, they would take the following values:

    Table 4. Coding of Calvin’s Trajectory for QHA

    23Other authors propose adding another series of active variables to the data representing transitions (Degenne et al., 1995). In our example, the number of possible types of transitions is equal to the square of the number of states,6 3² = 9, and the total number of transitions over the course of the trajectory is one less than the number of years in the period: here, 12 – 1 = 11. We, then, count the number of transitions of each type, which would yield the following:

    Table 5. Transition Variables for Calvin’s Trajectory

    24Transitions can also be defined to include only changes of state (Robette & Thibault, 2006). The number of possible transitions here is then equal to

    25(number of states) × (number of states – 1). Here, 3 × 2 = 6.

    26In this case, we count the examples of each type of transition (in the sense of a change of state) and divide that number by the total number of transitions. Calvin's trajectory includes two changes of state: one from E to I and the other from I to A. The values of the associated transition variables are as follows:

    Table 6. State Change Variables for Calvin’s Trajectory

    27Transition variables do not radically alter the results obtained by performing basic QHA on state variables alone but they often enable us to obtain a typology of trajectories with more homogeneous clusters (Robette & Thibault, 2006). Furthermore, data tables constructed in this way integrate three crucial dimensions of trajectories into the analysis (Degenne et al., 1995): the duration and timing of states, as with complete disjunctive coding, but also the transitions, as there is a process only if there is a change of state. However, the sequence of changes of state—viewed as a whole—is not taken into account in this type of analysis.

    28Compared with methods using complete disjunctive coding, division into intervals decreases the sensitivity of the analysis to precise simultaneity in a common state. In other words, QHA will treat two sequences of events that are identical but slightly shifted in time with respect to each other as more similar than the previous variants. In addition, the possibility of dividing the study period into intervals of unequal sizes gives the method a certain flexibility, allowing us to focus more on the parts of the trajectory that we expect to include the most interesting information.

    V. Optimal Matching

    29Another family of measures of the dissimilarity between trajectories is based on the notion of sequences. A great variety of studies in the social sciences take an interest in sequences of events or situations. They may investigate the events that make up the life courses of individuals (e.g., in terms of education, work, family, residential mobility), changes in policy, law, or culture, etc. In practical terms, a sequence is defined as an ordered list of elements, which can be of any kind (e.g., events, numbers). Sequence analysis is a body of techniques for processing this type of data. Studies using sequence analysis mainly attempt to answer three questions (MacIndoe & Abbott, 2004):

    1. Are there any patterns or typical sequences in a given set?

    2. If there are, what produces them or determines the form that they take?

    3. What are the consequences of these typical sequences?

    30Sequential data can be analyzed in many ways. What distinguishes sequence analysis is the fact that it takes the sequence as a whole—not a series of separate points or observations—as its unit of analysis. Here, in contrast to event history analysis or time series, the sequence is not looked at as the product of a stochastic process, generated step by step, but as a unit of analysis in its own right (Abbott & Tsay, 2000).7 There are different sequence analysis methods, which have been applied in various fields within the social sciences, such as psychology, archaeology, linguistics, political science, and sociology (Abbott, 1995). One of these methods—OM—is discussed, used, and disseminated much more widely than the others.

    1. Description

    31The use of OM was mainly developed in molecular biology to analyze proteins and DNA sequences. The objective, in that case, was to search large databases for sequences that are similar to a particular sequence, such as one linked to a given protein. Algorithms to perform this task first appeared in the early 1970s, then proliferated into the 1980s (Sankoff & Kruskall, 1983). The first use of an OM algorithm in the social sciences was in a study on sequences of figures in English folk dances, which aimed to use these data to characterize models of solidarity in rural England in the 19th century (Abbott & Forrest, 1986). Since then, numerous studies have made use of this method, most often to study occupational trajectories (Abbott & Hrycak, 1990; Blair-Loy, 1999; Halpin & Chan, 1998; Robette & Thibault, 2008; Stovel et al., 1996) and school-to-work transitions (Brzinsky-Fay, 2007; McVicar & Anyadike-Danes, 2002; Scherer, 2001). But it has been used to explore a wide variety of topics, such as the transition to retirement (Han & Moen, 1999) or adulthood (Aassve et al., 2007; Robette, 2010), activist careers (Blanchard, 2010), daily activity patterns (Lesnard & de Saint Pol, 2009; Wilson, 1998;), and residential trajectories (Stovel & Bolan, 2004). It has also been applied to more unexpected themes, such as lynchings in the United States (Stovel, 2001), the rhetorical structure of articles in sociology journals (Abbott & Barman, 1997), the content of school textbooks (Levitt & Nass, 1989), and business networks (Stark & Vedres, 2006).

    32In practice, OM constitutes only one of the steps in sequence analysis. Its basic principle is to measure the dissimilarity between each pair of sequences in a given sample.8 The resulting dissimilarity matrix then serves as the starting point for the second step in the analysis. Most often, this second step is the construction of a typology of sequences, based on identifying groups of similar sequences using techniques such as hierarchical clustering.9 This answers the first question in sequence analysis: Are there typical sequences in the data set? The typology can then be used as a dependent or independent variable for further analyses, which constitute the third and final step of sequential analysis. It is aimed at answering the following question: What are the causes and consequences of the existence of these typical sequences? Like most other sequence analysis methods, then, OM is considered a descriptive method.

    33OM algorithms define a calculable measure of the distance between sequences. The general idea is to assess the dissimilarity between any two sequences by transforming one into the other using a set of elementary operations.10 The three elementary operations are insertion (an element is placed into the sequence), deletion (an element is removed from the sequence), and substitution (one element is switched for another). There are numerous ways to transform one sequence into another using these operations. In the simplest version, the distance between two sequences is equal to the smallest number of operations needed to transform one into the other. This is known as the Levenshtein I distance (Levenshtein, 1966).

    34As an illustration, let us return to the example of the trajectories of Calvin and Hobbes, as represented in Table 3. That representation already encodes the trajectories as sequences. The two can be matched using the three operations in various ways. One possibility is to delete a year of study (E) at the beginning of Hobbes’s sequence, insert a year of activity (A) at the end, and substitute inactivity (I) for activity (A) between the two phases of inactivity. This takes three operations. Another possibility is to substitute I for E at age 20, I for A at age 22, and A for I at age 23. Here again, this takes three operations. The two alternatives are, thus, equivalent in terms of the distance between Calvin and Hobbes’s sequences.

    35In this simple version of OM, we treat the three elementary operations as equally important in defining the dissimilarity between trajectories. But we can also associate a specific cost with each operation. The cost of a series of operations is equal to the sum of the costs of each operation. The distance between two sequences is then defined as the minimum cost needed to transform one into the other. Various dynamic algorithms have been devised to calculate the minimum costs (Sankoff & Kruskal, 1983). The most widely used in the social sciences is that of Needleman-Wunsch (1970).11

    2. Choosing the Costs

    36The choice of transformation costs—the costs associated with the elementary operations—is an important step in OM. In the words of Stovel et al. (1996, p. 394), it “haunts all optimal matching analyses.” The researcher’s freedom to set these costs is what gives the method its flexibility and makes it adaptable to different objects of study (Lesnard & de Saint Pol, 2009). In practice, insertion and deletion are considered one and the same operation, insofar as in matching a pair of sequences, deleting an element in one sequence is equivalent to inserting an element in the other.12 This operation is known as indel, a contraction of insertion and deletion. Indel operations emphasize the order of events, bringing together sections in different sequences that are identical but that occur at different times. Inserting or deleting an element changes the temporal structure of a sequence, “warping time.” Substitution, in contrast, preserves temporal structure: It compares situations located at the same point in each sequence but changes the unfolding events themselves (Lesnard & de Saint Pol, 2009).

    37There are, thus, two categories of costs: substitution costs and indel costs. In practice, substitution costs are generally chosen first, with the indel cost coming after, depending on the importance we want to give to the order of elements in the sequences. One option is for substitution costs to be the same regardless of the specific elements that are being swapped. In this case, the cost of substituting A for E is the same as that of substituting I for E. But they can also be specific to each pair of elements; in this case, we have to define a substitution cost matrix. This matrix must be symmetrical: When we are matching two sequences, AEA and AIA, replacing E with I in the first sequence is equivalent to doing the opposite in the second.13 In our example, a substitution cost matrix could take the following form (Table 7):

    Table 7. Example Substitution Cost Matrix

    38The substitution costs can be determined in various ways. Generally speaking, the aim is to adapt these costs to our data and hypotheses. Many studies adopt differentiated substitution costs on the basis of considerations that are specific to the object of study: the more similar the elements, the lower the substitution cost. This strategy is most often built on a hierarchical structure of elements, which may be either preexisting or specifically constructed for the analysis. For example, in studies of occupational trajectories, substitution costs may be based on the relative positions of different occupations in an overall hierarchy of socio-occupational categories (Blair-Loy, 1999; Halpin & Chan, 1998; Scherer, 2001; Solis & Billari, 2002; Stovel et al., 1996). Another type of solution is to let the data drive the determination of substitution costs, deriving them from the transition rates between specific elements (Rohwer & Pötter, 2005). The cost of substituting one element for another is, then, higher where the probability of a transition from one to the other is lower (Aassve et al., 2007; Han & Moen, 1999; Pollock et al., 2002; Robette & Thibault, 2008). Finally, more complex strategies are also possible, for example, by combining a hierarchy of elements with transition rates (Abbott & Hrycak, 1990; Stovel & Bolan, 2004).

    39The choice of the indel cost (relative to substitution costs) is important as well. Some researchers prefer to make substitution and indel costs the same, arguing that theoretical justifications for other approaches are lacking (Dijkstra & Taris, 1995). This approach represents the simple version of the OM algorithm presented above (Levenshtein I distance; see Table 8).

    Table 8. Substitution and Insertion/Deletion Costs with Hamming and Levenshtein Distances

    40In the first applications of OM, indel costs tended to be set at quite high levels. But with sequences of equal length, if the indel costs are higher than the maximum substitution cost multiplied by half of the length of the sequences, insertion and deletion operations are never used. We are, then, dealing with the Hamming distance (Hamming, 1950), which is based on the simultaneity of identical elements. Here, the dissimilarity between two sequences is defined as equivalent to the number of substitutions needed to match them—that is, in terms of trajectories, the number of time units where the two are in different states.14 When lengths of the sequences are different, an indel cost that exceeds this threshold results in insertion-deletion operations being used only to compensate for the length difference. When, as often, one of the objectives of the analysis is to identify portions of sequences that are identical but that can occur with variable timing, the indel cost should be much lower (MacIndoe & Abbott, 2004). On the other hand, if the indel cost is lower than or equal to half of the minimum substitution cost, then only indel operations will be used. In this case, the dissimilarity between two sequences corresponds to the length of their longest common subsequence, also called the Levenshtein II distance. In the end, choosing the costs amounts to “positioning the cursor” between the two limit cases of the Hamming and Levenshtein II distances, depending on whether we are focusing more on the timing of situations or the order in which they unfold within different trajectories. If there is just one substitution cost, setting the indel cost at three-quarters of its value is an intermediate solution.

    41It is worth noting that a number of studies have shown that variations in the choice of costs do not lead to major differences in results (Chan, 1995; Levitt & Nass, 1989; McVicar & Anyadike-Danes, 2002).

    3. Some Alternatives

    42Alongside the many studies putting the useful properties of OM techniques to work, there have also been a number of critiques (Robette, 2011). These critiques are often based on comparing OM with parametric approaches, whose widespread applications have demonstrated their value, but which tend to direct researchers’ attention to particular questions. The two are, in fact, suited to the pursuit of different objectives—they are not competing, but complementary approaches. When our focus is on the humble but indispensable task of exploring and describing complex longitudinal data, most of the problems that arise with OM are entirely tolerable (Halpin, 2003). In any case, these critiques have stimulated the development of a “second wave” of sequence analysis (Aisenbrey & Fasang, 2010) and new measures of the dissimilarity between sequences.15

    43Some of the alternatives focus on ways of defining the costs of the elementary operations. Laurent Lesnard (2010) proposed an approach that links substitution costs to the timing of events, termed the dynamic Hamming distance (DHD). In this approach, a separate substitution cost matrix is calculated for each position—in temporal terms, each time step—in the sequence, on the basis of the probability of the different state transitions at that position. For a given pair of sequences, this yields distances for each position, which are then summed to obtain the overall measure of the distance between the two sequences. This technique has the particularity of not using indel operations. It can, thus, only be used to deal with sequences of identical length. It has proven to be particularly suited to the analysis of time use, to identify regularities in the timing of everyday activities (Lesnard, 2010). Halpin (2010) proposes a modified version of the OM algorithm that weights elementary operations by the inverse of the length of episodes (Studer and Ritschard [2016] called it spell length-sensitive optimal matching). The indel cost can also be varied in accordance with the similarity between the inserted or deleted element and the neighboring elements, as measured by the cost of substituting between them (localized optimal matching; see Hollister, 2009). Finally, the costs can be determined using an iterative optimization procedure (cost optimization method; see Gauthier et al., 2009).

    44Studer and Ritschard (2014) introduced optimal matching of spell sequences, or OMspell for short. Here, sequences are recoded to take into account the duration of episodes (or spells) by extending the alphabet.16 At each step, the encoding of the state includes the time elapsed since the start of the episode. In our example, Calvin’s sequence would, thus, be coded as E1-E2-I1-I2-I3-A1-A2-A3-A4-A5-A6-A7. This approach increases the size of the alphabet and the number of costs to be defined for the elementary operations (substitution and indel) but we can reformulate them to limit the number of parameters.

    45Biemann (2011) suggests recoding sequences in terms of categories of transitions before using OM. Calvin’s sequence (see Table 1) would then be EE-EI-II-II-IA-AA-AA-AA-AA-AA-AA. This approach considerably increases the size of the alphabet (taking us from 3 to 32 = 9 possible states in our example). But this problem can be avoided through a suitable reformulation of the costs, with a new measure known as optimal matching of sequences of transitions, or OMstran (see Studer & Ritschard, 2016).

    46Cees Elzinga, a strong critic of OM, has developed various measures that differ markedly from it. The foremost among them is based on the number of matching subsequences (NMS) in two sequences that are being compared (see Elzinga 2003, 2007). Elzinga and colleagues generalized it with the subsequence vector representation-based measure SVRspell (see Elzinga and Studer, 2015). SVRspell introduces weights based on the length of subsequences and, thus, the duration of episodes.

    VI. Choosing a Method

    47As we have seen, there are many dissimilarity measures to choose from, making the question of the robustness of our chosen method a crucial one. Do different techniques produce convergent results? What regularities are identified by what metrics?

    48Although comparing methods is not a central issue for most authors,17 a number of articles have examined the implications of different alternatives, detailing their methodological protocols. Some have chosen OM and tested various cost parameters. For example, using data from a study on the spread of Morris dancing in rural England, Forrest and Abbott (1990) introduce variation in substitution costs and find the behavior of the method to be robust. In another article, Abbott and Hrycak (1990, p. 176) conclude that “there is substantial robustness with respect to substitution costs.” Chan (1995, p. 477) tests three substitution cost matrixes on career data, noting “an impressive common core,” and adding that “the variation...follows interpretable patterns.” Similarly, in a study on school-to-work transitions, Anyadike-Danes and McVicar (2010) carry out a set of sensitivity analyses, varying the substitution and indel costs. They find that “relatively well-defined careers show up whether the cost matrix is designed to pick them out or not” (p. 495) and that the nature of the groups identified with different cost matrixes is similar, although there can be differences in their size and membership (see also McVicar and Anyadike-Danes, 2002).

    49Other studies compare OM with other measures. Lesnard (2010) applies his own DHD technique to time use data. He finds that the results are quite similar to those with the original Hamming distance, whereas the Levenshtein II distance is slightly less sensitive to contemporaneity. Robette and Thibault (2008) study occupational trajectories using both OM and QHA. They find that the main groups identified using the two techniques are very similar and observe that OM is a little better at distinguishing stable trajectories from mobile ones, while QHA seems slightly more sensitive to the presence of rare states. Aisenbrey and Fasang (2010) round out the assessment of sequence analysis methods with a comparative overview of DHD, OM with substitution costs based on transition rates, and Elzinga's NMS (ignoring durations), applied to school-to-work transitions. The first two identify the same regularities, with small differences in size and sensitivity to timing variations. The results with NMS, on the other hand, “depart more radically” from those of the other two methods. It identifies various “internally homogeneous groups...and one extremely heterogeneous group...that comprises more than half of all cases” (p. 444). Finally, Grelet (2002) focuses on factor analysis techniques, applying PCA, correspondence analysis, and QHA to the study of occupational integration. The results are generally highly convergent, although correspondence analysis and QHA are more sensitive to rare situations than PCA.

    50Overall, most of these comparisons conclude that the identification of regularities is robust to the use of these different measures. In the words of Abbott and Hrycak (1990): “As is often the case, while care is needed, differences in minor analytic decisions are unlikely to drastically change results” (p. 164).

    51Another interesting result is that the variations observed in these empirical studies are in line with the statistical foundations of the different measures. For example, correspondence analysis and QHA use the chi-square distance, which weights dissimilarities by the inverse of the frequency of the states in the sequences—making it unsurprising that these methods are more sensitive to rare states. Or consider that the use of indel operations in OM moves subsequences backward or forward. This logically makes the Hamming distance (which does not use indel operations) less sensitive to subsequences and transitions than the Levenshtein II distance (which exclusively uses indel operations).

    52These comparison studies are nonetheless subject to a number of limitations. First, each examines only a handful of measures at a time; none attempts to encompass a wide range of methods. Second, they all use a single data set. It is, thus, difficult to assess whether the conclusions remain valid beyond the specific case that they examine. Finally, and most importantly, the comparisons are all based on examining typologies of sequences. And yet, as we saw above, constructing a typology of sequences involves a long series of methodological choices: not just a dissimilarity measure, but also the coding, automatic classification technique, parameterization, and number of clusters. With comparisons based on the end result, it is difficult to disentangle the influence of the different steps in the analysis.

    53However, Robette and Bry (2012) take a more systematic approach and their conclusions converge with the earlier findings. In the social sciences, the structures present in the data are marked enough that the principal regularities emerge with most of the available metrics. The differences arise at the margins: cluster sizes, clusters of rare cases, etc. Metrics linked to OM are more sensitive to the order of elements in sequences and, therefore, to the distinction between stability and mobility, while metrics linked to factor analysis are more sensitive to the timing of events.

    54Studer and Ritschard (2014; 2016) present an extremely detailed comparison, which can be used to choose a dissimilarity measure in accordance with the research question.18 To focus on the order of elements (sequencing), they recommend OMstran, OMspell, or SVRspell,19 rather than “standard” OM. For studies focused on the timing of events, they recommend the Hamming distance or the combination of complete disjunctive coding with either the chi-square or Euclidean distance. For analyses focused on duration at the scale of the whole sequence, they recommend a simplified representation of the sequence, such as the one used in QHA, with only a single interval. Finally, to study the duration of particular episodes, they recommend using OMspell, LCS, or “classical” OM.

    55Studer and Ritschard also advise against the use of certain metrics. DHD produces results very similar to the Hamming distance but it is more complicated; the same goes for OM with substitution costs based on transition rates versus OM with a single substitution cost.20 NMS lacks sensitivity to order, timing, and duration (and, thus, to all the temporal dimensions of trajectories). Localized OM and spell length-sensitive OM sometimes produce strange and counterintuitive results. The cost optimization method suffers from mathematical problems and can produce negative dissimilarities.

    56Finally, note that when we are analyzing a collection of trajectories, we do not always know whether we want to emphasize order, timing, or duration. In this case—and, indeed, in any case—trying out multiple metrics is advisable. The comparison can only be instructive, at least at the data exploration stage.

    Notes de bas de page

    1 A few examples can also be found in studies from the Netherlands (Martens, 1994; Van der Heijden, 1987; Van der Heijden et al., 1997).

    2 Laboratoire interdisciplinaire de recherche sur les ressources humaines and l’emploi [Interdisciplinary Laboratory for Research on Human Resources and Employment].

    3 For more on the mathematical formalization of these distances and their implications, see Grelet (2002).

    4 In this case, the method is equivalent to the Hamming distance, which is discussed above in the context of OM.

    5 Empirically, one of the ways to assess the concentration of events in different parts of the life course is to observe the age distribution of state changes.

    6 The notion of “transition” is understood here simply as the movement from the situation at time t to the situation at time t+1, whether the two are different or not.

    7 These observations, made by Andrew Abbott in the context of sequence analysis, in fact also apply to the analysis of trajectories using the factor analysis methods just presented.

    8 For a set of N sequences, the total number of distances calculated is, thus,.

    9 One possibility would be to apply multidimensional scaling to the dissimilarity matrix to eliminate statistical “noise” in the manner of the factor analysis methods presented above (Section II). To my knowledge, this path has not been taken to date.

    10 To designate the transformation of one sequence into another, we may speak of matching or aligning a pair of sequences.

    11 This algorithm has been implemented in various software packages: TDA (Rohwer & Pötter, 2005), Stata (Brzinsky-Fay et al., 2006), and R (Gabadinho et al., 2009). For a more complete overview of software options, see the Appendix.

    12 For example, if we want to match two sequences, EIA and EA, deleting I from the first and inserting I between E and A in the second are equivalent.

    13 From a mathematical point of view, the symmetry of substitution costs ensures that the distance metric has the required properties: separation, symmetry, and the triangle inequality (MacIndoe & Abbott, 2004).

    14If substitution costs are constant. With varying substitution costs, this would be equivalent to the sum of the substitution costs.

    15 For a detailed analysis of the main dissimilarity measures, see Studer and Ritschard (2014, 2016).

    16i.e., the set of possible states in the corpus of sequences.

    17 Among the exceptions, see Aisenbrey and Fasang (2010), Grelet (2002), Robette and Thibault (2008), and Studer (2012).

    18 See Studer and Ritschard (2016, pp. 507–509) for more details, particularly on choosing parameters.

    19 These three metrics differ in terms of their sensitivity to small perturbations in the data.

    20 These results converge with those of Robette and Bry (2012).

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    1 A few examples can also be found in studies from the Netherlands (Martens, 1994; Van der Heijden, 1987; Van der Heijden et al., 1997).

    2 Laboratoire interdisciplinaire de recherche sur les ressources humaines and l’emploi [Interdisciplinary Laboratory for Research on Human Resources and Employment].

    3 For more on the mathematical formalization of these distances and their implications, see Grelet (2002).

    4 In this case, the method is equivalent to the Hamming distance, which is discussed above in the context of OM.

    5 Empirically, one of the ways to assess the concentration of events in different parts of the life course is to observe the age distribution of state changes.

    6 The notion of “transition” is understood here simply as the movement from the situation at time t to the situation at time t+1, whether the two are different or not.

    7 These observations, made by Andrew Abbott in the context of sequence analysis, in fact also apply to the analysis of trajectories using the factor analysis methods just presented.

    8 For a set of N sequences, the total number of distances calculated is, thus,.

    9 One possibility would be to apply multidimensional scaling to the dissimilarity matrix to eliminate statistical “noise” in the manner of the factor analysis methods presented above (Section II). To my knowledge, this path has not been taken to date.

    10 To designate the transformation of one sequence into another, we may speak of matching or aligning a pair of sequences.

    11 This algorithm has been implemented in various software packages: TDA (Rohwer & Pötter, 2005), Stata (Brzinsky-Fay et al., 2006), and R (Gabadinho et al., 2009). For a more complete overview of software options, see the Appendix.

    12 For example, if we want to match two sequences, EIA and EA, deleting I from the first and inserting I between E and A in the second are equivalent.

    13 From a mathematical point of view, the symmetry of substitution costs ensures that the distance metric has the required properties: separation, symmetry, and the triangle inequality (MacIndoe & Abbott, 2004).

    14If substitution costs are constant. With varying substitution costs, this would be equivalent to the sum of the substitution costs.

    15 For a detailed analysis of the main dissimilarity measures, see Studer and Ritschard (2014, 2016).

    16i.e., the set of possible states in the corpus of sequences.

    17 Among the exceptions, see Aisenbrey and Fasang (2010), Grelet (2002), Robette and Thibault (2008), and Studer (2012).

    18 See Studer and Ritschard (2016, pp. 507–509) for more details, particularly on choosing parameters.

    19 These three metrics differ in terms of their sensitivity to small perturbations in the data.

    20 These results converge with those of Robette and Bry (2012).

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    Robette, N. (2025). Measuring the Dissimilarity Between Trajectories. In The Statistical Analysis of Trajectories. Paris: Ined Éditions. https://doi.org/10.4000/153ia
    Robette, Nicolas. « Measuring the Dissimilarity Between Trajectories ». In The Statistical Analysis of Trajectories. Paris: Ined Éditions, 2025. doi:10.4000/153ia.
    Robette, Nicolas. « Measuring the Dissimilarity Between Trajectories ». The Statistical Analysis of Trajectories, Ined Éditions, 2025, https://doi.org/10.4000/153ia.

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    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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