Chapter 5
Multidimensional Sequences
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1In early applications of sequence analysis in the social sciences, it was difficult to simplify the successive aspects of the life courses of individuals to a single, limited set of states. Methodological adjustments were sometimes needed to take into account the diversity and complexity of individuals’ social situations. In other words, to study careers in detail as sequences, we must bring to bear multiple dimensions. For example, in their seminal article on the careers of musicians in Germany during the Baroque and Classical periods, Abbott and Hrycak (1990) combine position (e.g., vocalist or instrumentalist) and “sphere” (e.g., opera or church). In similar ways, Stovel et al. (1996) combine position and company size to create their variable for employment status; Blair-Loy (1999) combines socio-occupational category and company size; and Han and Moen (1999) use employment status, company, and occupation.
2Subsequent applications of sequence analysis have focused on life courses. The need to simultaneously address the different dimensions of an individual’s life, thus, became a major concern. Social scientists have explored the methodological repercussions of the interdependence of conjugal, parental, occupational, and housing careers (Courgeau & Lelièvre, 1989). Pollock (2007) introduced multiple-sequence analysis (MSA), which was then systematized by Gauthier et al. (2010) and renamed multichannel sequence analysis (MCSA).
I. Associations Between Dimensions
3Creating a typology of multidimensional sequences only makes sense if the different dimensions are statistically interlinked. Otherwise, the clustering procedure may produce a set of clusters with trajectories that are very homogeneous on one dimension and very heterogeneous on the others. In other words, we may be led to interpret the typology in terms of interdependence between dimensions when this is nothing but an illusion induced by our choice of methods.
4There are various ways to check that the dimensions are in fact statistically interrelated. The first is graphical. Consider a set of sequences with three dimensions: A, B, and C (which could be family, employment, and housing trajectories, for example). We can represent each dimension using an index plot, sorted on Dimension A on the basis of the results of a multidimensional scaling (Piccarreta & Lior, 2010). The order of the sequences in the index plot for Dimension A should seem to obey an interpretable logic. If that is also the case for the other dimensions, then we have reason to think that they are statistically associated with Dimension A.
5The other approaches are statistical (Piccarreta, 2017; Piccarreta & Elzinga, 2013). First of all, we can simply calculate the correlation between the distance matrixes for the different dimensions: A and B, B and C, A and C. The Mantel coefficient measures linear relationships, while rank correlation coefficients (Spearman’s or Kendall’s) evaluate monotonic relationships.
6We can take this a step further by submitting the matrix of the correlations between dimensions (here, a 3 × 3 matrix) to a principal components analysis (PCA). If the first axis explains a large part of the variance, then there is a linear relation between the dimensions. If, on the contrary, the subsequent axes explain a substantial part of the variance, then the results of the PCA can help us to better understand the structure of the relations between the dimensions.
7Another way to measure the association between different dimensions is joint sequence analysis. It is based on the correlations between the distance matrixes of the different dimensions (dA, d B, and dC) and the distance matrix of the multidimensional sequences (dABC). Taking the square of each correlation (e.g., between dA and dABC) gives us a measure of the proportion of the dissimilarities of the dimension (dA) that is explained by the dissimilarities between the multidimensional sequences (dABC). On the basis of these correlations, we can calculate the total proportion of the dissimilarities of the different dimensions that is explained by dABC. The value of this measure can range from 0 to 1, with 1 corresponding to a perfect linear relationship. It is high if the dimensions are related and if that relation can be efficiently summarized by the calculation of dissimilarities between multidimensional sequences.
8Piccarreta (2017) also suggests using Cronbach’s alpha to measure the association between the different possible combinations of dimensions: here, between A and B; A and C; B and C; and A, B, and C. High values indicate strong associations. Another option is to measure, for a given dimension, the correlation between its distance matrix and the matrix of distances between sequences that combine all the other dimensions but not it. In our example, this involves calculating the correlations between dA and dBC, dB and dAC, and dC and dAB. If dA and dBC are correlated, that means that information contained in Dimension A is also found in the combination of the other dimensions and, thus, that it is statistically related to them.
II. Typological Methods
9The diverse strategies for developing typologies of multidimensional sequences found in the literature can be said to belong to one of five groups.1 The strategies in the first group are based on creating a new state variable that combines the possible states on each of the dimensions (Aassve et al., 2007; Chaloupkova, 2010; Dijkstra & Taris, 1995; Elzinga, 2003; Elzinga & Liefbroer, 2007; Lesnard, 2008). For example, in the case of conjugal and parental histories, the possible combined states could include “no partner, no children”; “no partner, one or more children”; “in a union, no children”; and “in a union, one or more children.” This can quickly lead to a very large alphabet (number of states). In the case of four dimensions, each with three states, the combined variable could potentially take 3 x 3 x 3 x 3 = 81 values. This can be impractical with optimal matching (OM) if we want to set substitution costs that are specifically adapted to each pair of states; however, it is easy to avoid this difficulty by setting a constant substitution cost or by using transition rates between states (Lesnard, 2008).
10The second strategy is a more refined approach that makes an extensive alphabet unnecessary. It is the most widespread and is often called MSA or MCSA. It can only be used with one dissimilarity measure: OM. It is based on combining the substitution costs for the different dimensions. For example, the substitution cost for “no partner, no children” / ”in a union, one or more children” would be defined as some combination of the substitution costs for “no partner” / “in a union” and “no children” / “one or more children.” One possible principle for this combination is the sum (or the mean) of the costs defined for each dimension individually (Blair-Loy, 1999; Gauthier et al., 2010; Pollock, 2007; Salmela-Aro et al., 2011; Stovel et al., 1996): for example, the sum (or mean) of the cost of substituting “in a union” for “no partner” and the cost of substituting “one or more children” for “no children.” It is also possible to devise a more complex linear combination of the different dimensions (Abbott & Hrycak 1990), for example, by weighting them (Gauthier et al., 2010, p. 34). If there is a single substitution cost that applies to all dimensions, then substituting one combined state for another is equivalent to counting the number of dimensions that differ (Robette, 2010). For example, the cost of substituting “no partner, no children” with “in a union, one or more children” would be 2, while the cost of substituting “no partner, no children” with “in a union, no children” would be 1. This second strategy can be considered a special case of the first, with substitution costs that are set in a simple and efficient way.
11The third strategy is to calculate a distance matrix for each dimension separately and then summarize them as a single distance matrix through linear combination (Blanchard, 2005; Han & Moen, 1999).
12The fourth strategy involves first producing typologies of sequences for each dimension and then combining them (Blanchard, 2005), for example, using cross tabulation. Types of marital trajectories and of parental trajectories could then be analyzed jointly.
13The fifth and final strategy, known as globally interdependent multiple-sequence analysis, was developed recently (Robette et al., 2015). Here, as in the third strategy, we begin by calculating a distance matrix for each dimension. Each of the dimensions is then summarized using multidimensional scaling (MDS). The dimensions are then jointly analyzed using a technique for the analysis of multiple tables (canonical partial least squares2), which takes into account the correlations between them, producing a single distance matrix.
14These five strategies can be systematically compared and classified according to three criteria: (1) multidimensionality, (2) parsimony, and (3) interdependence (Table 13).3
Table 13. Taxonomy of Strategies for Multidimensional Sequence Analysis

15Multidimensionality refers to the fact that each dimension’s contribution to the overall results may or may not be explicit (unequivocal) and flexible (with user-definable parameters). The first strategy, like the others, integrates multiple dimensions. But by masking them within a single composite state variable, it is the only one of the five that does not, in fact, emphasize multidimensionality. For example, it does not allow us to define parameters specific to each dimension or to give more emphasis to one dimension in particular, weighting it or evaluating its specific impact on the results.
16Parsimony refers to how likely an approach is to yield a limited and manageable number of types of trajectories.4 Combining typologies produced separately for each dimension (as in strategy 4) can lead to an awkwardly large number of types.
17Interdependence refers to the fact that the relations between the dimensions can be masked (strategy 3); taken into account locally—that is, the emphasis is on the relations between dimensions at each moment in the sequences (strategies 1 and 2); or taken into account globally—where the emphasis is on the interrelation between the dimensions by way of whole sequences as units of analysis in their own right (strategies 4 and 5).
18Global interdependence removes the constraint of contemporaneity. Here, it is the overall form of a dimension that is linked to the others. In strategies 4 and 5, dissimilarity is measured separately for each dimension in an initial step, before the relationships between the regularities within each dimension are examined. As the distance matrixes are calculated independently for each dimension, different dissimilarity measures can be used in each case: for example, a timing-based metric (e.g., the Hamming distance) for one dimension and an order-based metric (e.g., the longest common subsequence) for another. Different time windows and time units can also be used.
19Strategy 3 also allows the use of different dissimilarity measures, time windows, and time units; however, simply summing the distance matrixes does not correctly handle the relations between the dimensions. For example, with two-dimensional sequences, if two individuals have a dissimilarity of 1 on one dimension and are identical on the second, they will have the same global distance as two individuals who are identical on the first dimension and have a dissimilarity of 1 for the second, although what differs between the two individuals is substantively absolutely distinct.
20It is important to keep in mind that applying these criteria—multidimensionality, parsimony, and local and global interdependence—may or may not be desirable, depending on our research questions and our data. No single strategy is inherently the best. The choice of a strategy should be based on sociological and empirical criteria.
21A relatively small number of sequence analysis studies to date have looked at one of the key elements of the life course paradigm (Giele & Elder, 1998): the fact that individual life courses are embedded in social relations—in other words, that different lives are linked. Most studies in this line have investigated the transmission of trajectories from parents to children (Falcon, 2012 ; Fasang & Raab, 2014; Liefbroer & Elzinga, 2012; Robette et al., 2015), although some have examined the trajectories of spouses/partners (Lelièvre & Robette, 2010; Lesnard, 2008; Pailhé et al., 2013).
22To assess the extent to which parents’ trajectories are passed on to their children, Liefbroer and Elzinga (2012) set aside typologies and analyzed the dissimilarities between the trajectories of parents and their children. It could be argued, however, that in some cases perfect similarity between parent and child sequences is not suitable for use as evidence of transmission. The median age at parenthood, for example, may be 20 for the parental cohort and around 25 for their children’s cohort. The sequences of parents and children may, thus, differ formally in cases where their life histories are perfectly equivalent, given the structural changes in the historical context in which their lives unfolded. Furthermore, the reasons for a given degree of dissimilarity can differ. For example, an age difference of two years at the time of marriage may be treated as equivalent regardless of whether one is earlier or later than the other, although in reality the meaning of the discrepancy in the two directions may differ.
23Some researchers have adopted the second of the strategies presented above, combining substitution costs (Fasang & Raab, 2014; Pailhé et al., 2013). This strategy has the advantage of allowing contrasting types of intergenerational transmission to be identified—that is, groups of dyads in which the sequences of the parents and children are distinct but frequently associated (Fasang & Raab, 2014). This strategy is particularly appropriate if we want to synchronize the sequences within each dimension within a dyad (e.g., comparing the timing and pace of transitions between parents and children) or to characterize the situation of a dyad at a given moment (local interdependence). This, however, is not always the case. For example, the nature of the various dimensions may sometimes differ considerably. Take the study of the relationship between parents’ overall career and the start of their children’s careers. In this example, the parents' sequences could span 45 years—between ages 14 and 60 (with years as time units)—with the alphabet (i.e., the set of possible states) based on a socio-occupational nomenclature. Their children’s initial career trajectories, on the other hand, might span only 3 years after they left school (with months as time units), and the alphabet consist of employment statuses (e.g., being in education, unemployed, in part-time employment, or in full-time employment). In purely technical terms, the difference in the length of parents’ and children’s sequences could be managed using missing values. But in this hypothetical example with such widely differing time windows and units, doing things in this way would, at the very least, be inelegant—even impractical—and would significantly obscure the results. More importantly still, from a substantive perspective, “pasting” one sequence onto another and aligning them locally would simply not be a fruitful approach. What we want to study instead is the overall form of the different dimensions, their regularities, and the relationship between those regularities (global interdependence).5 In this case, strategy 5 is recommended (Robette et al., 2015). Moreover, it allows us to choose a dissimilarity measure that is focused on order or duration for parents and a measure based on timing for the sequences of the children, for example.
Notes de bas de page
1 Note that some authors have compared the results of using multiple strategies on the same dataset (Blanchard, 2010).
2Sometimes called symmetric partial least squares.
3 These criteria are inspired by Gauthier et al. (2010).
4 More precisely, one typology can be considered more parsimonious than another if it preserves the same amount of information with fewer clusters or if it preserves more information with the same number of clusters. In practice, parsimony is a balance between the amount of information and the number of clusters, and the decisions involved are not always simple ones.
5 Another example might be the dyad of sequences formed by workdays (or weeks) as collected in the Emploi du temps (Time use) surveys (Lesnard & Kan, 2011), and past occupational trajectory.
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