1 – Processing complex trajectories

4A range of standard statistical tools can be used to analyse time spent (survival) in a given state: non-parametric estimations are used to measure survival (Kaplan and Meier, 1958; Nelson, 1972; Aalen, 1978), and semi-parametric (Cox, 1972) or parametric [3] models are used to measure the impact of individual characteristics (which are also social properties) on survival in a particular state. These models are meaningful for durations clearly defined by unambiguous start and end dates. However, they are not designed to describe individual trajectories characterized by a complex sequence of changes of state, i.e. when the analysis concerns a succession of repeatable events with numerous possible transitions between states (GRAB, 2006), as is the case for occupational trajectories, for example. How, then, can we explore this type of data?

2 – The problem of repeatable events

5Modern longitudinal methods, which can be applied at the individual level, have already proved useful for studying labour market trajectories. Using historical data from the French employment agency (Agence nationale pour l’emploi, ANPE), we can, for example, study the labour market reintegration of unemployed persons on the basis of individual characteristics (Degenne and Lebeaux, 1999), and in relation to the benefits and support received by each individual (Crépon et al., 2005). Duration models must be used because we are estimating the time between registering as unemployed in the ANPE database and returning to employment. This definition of the duration of unemployment serves to measure the effectiveness of employment policies. However, it disregards the fact that return to employment may be short-lived, and followed rapidly by a further period of unemployment. In other words, duration models cannot directly capture the repeatability of events. [4]

6Likewise, when career interruptions are analysed, their duration is generally taken to be the difference between the start of the nth period of inactivity and the date of subsequent resumption of activity. However, when analysing the complexity of intermittent career trajectories, the analysis of a single event is not sufficient, since individuals may interrupt their career several times over a lifetime. A variety of methods have been suggested for dealing with repeated episodes. First, labour market exits can be classified by exit order. Birth order is traditionally used to study fertility, but unlike the order of births in a family, the order of labour market exits has no particular significance. In terms of the working career, what counts is their duration, or their aggregate duration (Desplanques and Saboulin, 1986; Lelièvre, 1987; Cambois and Lelièvre, 1988). All in all, this method is not entirely satisfactory as the duration of inactivity periods is specified in the same way as in the first definition (time elapsed between start of nth period of inactivity and subsequent resumption of activity). Second, the set of periods can be taken as a whole by considering people who have experienced one or more inactivity periods as a level of aggregation in a multi-level model (Courgeau, 2000). This avenue of research has yet to be explored or rendered operational.

3 – The problem of multiple states and transitions

7The problem of defining duration becomes even more acute when the state in question is not binary, but can be characterized by a variety of situations such as membership of various socio-occupational categories (SOC), full-time or part-time work, unemployment, or economic inactivity.

8Demographers, for example, are interested in measuring the duration of activity and relating it to other events such as birth of children. They ask the classic question of how labour force participation, of women especially, and fertility interact. Event history methods appear better suited for this purpose than cohort analysis, which is based on the assumption that events are independent. However, according to M. Kempeneers and É. Lelièvre (1991), they face three interlinked problems: defining the state whose duration is estimated and the exogenous variables, defining the population exposed to risk and defining the study interval. [5] The choices of definition affect the robustness of the reasoning applied and make the conclusions dependent upon the initial, often implicit, assumptions. The authors therefore recommend a more descriptive analysis taking account of all periods of activity from age 15. This gives rise to a new type of study which not only records labour market exit dates, but also follows life event histories over the entire life course.

1 – The trajectories studied

9The statistical unit of classification is the individual trajectory: clusters are formed on the basis of resemblance between trajectories. Certain choices must be made regarding (i) the study population, (ii) the study interval, and (iii) construction of the analysis variables.

10Only men’s working careers are analysed here (n = 1,341). Classification of the entire population is possible, but has two drawbacks. First, the list of states is different for men and women: military service affects men only. Second, when both sexes are handled together, the necessary simplication involved in classification [6] may sometimes mask the career differences between men and women.

11As the individuals leave the observation at the survey date, the data are right-censored. [7] With duration models it is possible to control for the effect of censoring, but not the descriptive statistics. There is no technical reason to prevent qualitative harmonic analysis (Barbary and Pinzon Sarmiento, 1998) or optimal matching methods (Macindoe and Abbott, 2004) from being applied to trajectories of varying lengths. However, by doing so, we move away from the description of a single population over time. The description of individual trajectories implies that all individuals are observed over an identical period, delimited by the same boundaries.

12Our study, for example, concerns occupational mobility between age 14 – the legal school-leaving age for the cohorts studied – and age 50, [8] the age of the youngest respondents. [9] The analysis could have been extended beyond age 50 by working on a survey sub-population. Its usefulness would have been limited, however, as the number of respondents drops rapidly with age: only 67.4% of the population are still in the sample at age 55, 42.5% at age 60, 21.3% at age 65 and 4.6% at age 70.

13We describe the various states that constitute the trajectories using eight socio-occupational categories defined by INSEE. A more detailed breakdown of states would not be especially useful, since changing from a clerical worker to a sales worker does not have the same significance as moving up to a higher-level occupation. By construction, the sample does not include any retirees, as the description ends at age 50. We should thus have reasoned on the basis of seven SOCs (see note 1), i.e. the six active categories and the category of “other economically inactive persons”. However, we thought it would be useful to divide this latter category between students and persons who are inactive for family or health reasons. [10] A further category was needed to describe the trajectories of men who had experienced the Algerian war. We therefore added the “military conscript” category based on the 1954 list of SOCs but which has since been removed.

2 – Creating a typology by qualitative harmonic analysis

14Harmonic analysis is a branch of mathematics widely applied in physics and biology. Its use in the social sciences is more recent, and dates back to the 1970s (Deville, 1974, 1977). The aim at that time was to use the notion of duration to explain social phenomena via data on individual trajectories. As pointed out by Deville (1977, p. 18): “Faced with such abundant data, the statistician feels somewhat perplexed. Increasingly complex tables become uninterpretable without the help of ‘automatic’ analysis methods. So he tries to define an analysis method to help him get the most from his data. Here, ‘the most’ has a specific, quantifiable, meaning, linked to the method he applies”. This technique was then adapted for the exploratory statistical analysis of complex trajectories (Deville and Saporta, 1980; Deville, 1982), and became known as qualitative harmonic analysis (QHA).

15QHA involves defining an observation period, dividing it into a finite number of intervals then measuring for each individual the proportion of time spent in each of the states in each interval. A correspondence analysis on the matrix [11] thus formed provides a means to summarize the information by selecting the factors with the highest inertia (Deville, 1982). This eliminates statistical “noise” without eliminating the individual. Progress in classification techniques has made it possible, among other things, to use ascending hierarchical classification to determine trajectory typologies (Barbary, 1996; Degenne et al., 1996; Barbary and Pinzon Sarmiento, 1998). The typology thus constructed takes account not only of the sequence of states, but also the time when events occur and the duration in the various states. It also keeps all individuals in the analysis.

16QHA has rarely been used until now, due to a lack of suitable data. J.-C. Deville (1982), for example, created an ad hoc sample on the basis of rejected questionnaires from the INSEE family surveys of 1962 to 1975. [12] The sample was by no means representative and its sole purpose was to provide a field of application for a method which until then had none. Over a twenty-year period, the method was used only once on French data [13] (Degenne et al., 1996). This research focuses on a part of the occupational trajectory with the aim of studying labour market integration. In fact, it was not until the emergence of renewed interest in event history data collection that practical applications were developed, initially on Latin American data (Dureau et al., 1994; Barbary, 1997; Barbary and Pinzon Sarmiento, 1998). The Biographies et entourage survey enables us to track migration trajectories using this method (Bonvalet et al., 2008), but also entire working careers. [14]

3 – Constructing a typology by optimal matching

20The optimal matching method is based on a set of dynamic algorithms used mainly in molecular biology to analyse similarities between DNA sequences. It was introduced into the social sciences by Andrew Abbott in the 1980s (Abbott and Forrest, 1986; Abbott and Hrycak, 1990). It serves to measure the degree of similarity or dissimilarity between sequence pairs. This is done by evaluating the “cost” of transforming one sequence into the other. This transformation involves different operations: insertion (an element is inserted into the sequence), deletion (an element is removed) and replacement (an element is replaced by another). In practice, optimal matching is based on just two elementary operations. In the first operation, known as indel (a contraction of insertion and deletion), the same cost is assigned to both insertion and deletion, so that changing from sequence 1 to sequence 2 by inserting an element is equivalent to changing from sequence 2 to sequence 1 by deleting one. [19] The second operation, replacement, is a combination of an insertion and a deletion, [20] though the cost of a replacement does not have to be double that of an indel: the replacement cost is chosen by giving priority to reducing the distance either between sequences that are identical but which occur at different times, or to sequences that occur simultaneously but have one or more differences.

21The cost of a series of operations is equivalent to the sum of costs of the corresponding elementary operations. The distance between two sequences is thus defined as the minimum cost of transforming one sequence into another. Specific dynamic algorithms ensure that the minimum cost is rapidly obtained (Sankoff and Kruskal, 1983). By matching all sequence pairs, a distance matrix is created which can then be used to group the most similar sequences, using classification methods for example, and produce a typology. Choosing the costs of elementary operations is a key stage in optimal matching analysis. It is this freedom to determine costs which gives the method its flexibility and adaptability (Lesnard and Saint-Pol, 2004).

Constructing the cost matrix

22We work on sequences of occupations held by respondents over annual periods. Replacement costs can be varied according to the elements replaced.

23Given the limited theoretical grounding on the question, certain researchers prefer to use fixed replacement costs (Dijkstra and Taris, 1995). However, many studies adopt differentiated replacement costs based on hypotheses specific to the study topic: the more similar the elements, the lower the replacement cost. For example, in some studies of working careers, the replacement costs are based on the relative hierarchical positions of different occupations (Stovel et al., 1996; Halpin and Chan, 1998; Blair-Loy, 1999; Scherer, 2001; Solis and Billari, 2002). An alternative solution is to derive replacement costs from probabilities of transition between elements: the lower the transition probability, the higher the replacement cost (Rohwer and Pötter, 2005). In the absence of any preset hierarchy between socio-occupational categories, we chose the second solution.

24We then focus on the relation between replacement cost and indel cost. According to the literature, several options are available to us. First, as a replacement is equivalent to a combined insertion and deletion, the indel cost can be set at half the replacement cost. We can then set the indel cost at a value slightly above half the maximum replacement cost, thereby avoiding the use of indels. This approach is justified when priority is given to the position of the elements in the sequence (i.e. the timing of events), rather than their respective order. If priority is given to the sequencing of elements, it is preferable to set the indel cost at 1/10 of the maximum replacement cost (Macindoe and Abbott, 2004). In a trajectory characterized by intra-generational mobility, the order of the various states is crucial. We therefore chose the second option (Appendix 1).

25The coding of trajectories with each method is illustrated in Box 1.

Box 1. Representing trajectories with OM and QHA

Let us take 7 years of a career as an example (from age 14 to age 20): at ages 14 and 15, the respondent is a student; at ages 16-18 he is a manual worker, and at ages 19-20 he holds a higher-level occupation. To simplify the presentation, we will limit the analysis to these three states (S, M and H). This trajectory is represented differently by the two methods.
• With QHA, if the period is divided into two intervals, one of 4 years (ages 14-17) and one of 3 years (ages 18-20), we get the following matrix:

Ages 14-17 Ages 18-20 S M H S M H 0.5 0.5 0 0 0.33 0.67 • With OM, we get the sequence SSMMMHH in which each letter represents the respondent’s state in a particular year. The sequence could be changed to SSSMMHH by inserting an M and deleting an S or by replacing an M with an S. We select the series of operations with the lowest cost.

1 – Similar resulting cluster distributions

26Working on all the male occupational trajectories, we chose two partitions [21] (6 and 10 clusters of trajectories) which offered a good compromise between the imperative of synthesis and the need to present the heterogeneity of individual trajectories. The 6-cluster solutions obtained with QHA and OM are relatively similar. Five of the clusters have very comparable profiles, though their sizes are slightly different (Table 1). The category names indicate their main characteristic, making their description more readily understandable. [22]

Table 1

Six-cluster career distribution of men from the Paris region, 1930-1950 cohorts

No. Cluster (main characteristic) QHA (%) OM (%) 1 Intermediate occupations 29 27 2 Manual workers 26 26 3 Higher-level occupations 26 26 4 Clerical and sales workers 13 9 5 To self-employed 5 5 6 From intermediate to higher-level occupations – 6 7 From farming to manual occupations 1 – Total 100 100 Population: 1,341 working careers of men in the 1930-1950 cohorts living in the Paris region at the time of the survey. Source: Biographies et entourage survey (INED, 2001).

Six-cluster career distribution of men from the Paris region, 1930-1950 cohorts

Cluster 7. From farming to manual occupations (1% with QHA)

33This last cluster, specific to QHA, comprises individuals who were farm workers in their youth. At the start of their trajectory, most of them work as family helpers, then very quickly, between ages 20-30, they move on to become manual workers in most cases.

34The six-cluster QHA and OM solutions seem to be based on the state occupied for the longest period. For this reason, mobility is only very marginally identified (clusters 5, 6 and 7). A ten-cluster distribution provides a more balanced picture (Table 2).

Table 2

Ten-cluster career distribution of men from the Paris region, 1930-1950 cohorts

No. Cluster (main characteristic) QHA (%) OM (%) 1 Higher-level occupations 25 26 2 Intermediate occupations 22 16 3 Manual workers 19 19 4 Clerical and sales workers 12 6 5 From manual to intermediate occupations 10 6 6 To self-employed 5 5 7 From farming to manual occupations, before age 25 2 – 8 From farming to manual occupations, after age 25 1 – 9 Economically inactive 1 – 10 Self-employed 2 – 11 From intermediate to higher level occupations – 6 12 From manual worker to intermediate occupation or self employed – 7 13 From clerical to intermediate occupations – 5 14 From manual to clerical occupations – 2 Total 100 100 Population: 1,341 working careers of men in the 1930-1950 cohorts living in the Paris region at the time of the survey. Source: Biographies et entourage survey (INED, 2001).

Ten-cluster career distribution of men from the Paris region, 1930-1950 cohorts

35The ten-cluster distributions are fairly similar. In particular, although the sizes sometimes differ, the first six clusters (which represent 93% of respondents with QHA and 78% with OM) have very comparable profiles. Clusters 1-4 include the stable trajectories of higher-level, intermediate, manual and clerical occupations. Clusters 5 and 6, on the other hand, correspond to occupational mobility: transition from manual to intermediate occupation, and to the status of self-employed.

36The other QHA clusters are small and reflect marginal trajectories. Clusters 7 and 8, for example, both concern individuals who started in farming occupations and then become manual workers. The two clusters are distinguished by age at transition. Cluster 9 is of particular interest as it is the only one to place emphasis on economic inactivity: it comprises men who have experienced one or more periods of inactivity whose total duration is significantly longer than is the case for men in the other clusters.

37Last, the clusters identified by OM only are larger and concern mobile trajectories only: from intermediate to higher-level occupations; from manual to intermediate or self-employed (transition after age 35); from clerical to intermediate; from manual to clerical. OM thus seems to be more sensitive to transitions and is capable of capturing intra-generational mobility processes in relatively homogeneous clusters.

38The emergence in the ten-cluster distributions of mobility trajectories that are invisible in the six-cluster distributions illustrates the importance of exploring a trajectory cluster at different partition levels, to see, for example, whether the population grouped in a particular trajectory type could be usefully subdivided into several separate types. A distribution with a large number of clusters reveals a broad range of trajectories, but may be difficult to characterize. The aim of a cluster approach is not only to explore complex data, but also to synthesize the broad variety of individual behaviours. The choice of the number of clusters thus represents a compromise between concision and exhaustivity, depending on the research hypotheses and questions under study. These results represent basic analyses of occupational trajectories that can be refined by matching different individual characteristics (birth cohort, place of birth, father’s occupation, etc.) with the respondent’s career type using descriptive statistics. Regression models including the trajectory cluster as a dependent or independent variable can also be used, although the diachronic nature of the relations of causality calls for a careful approach.

2 – Measuring differences

39The degree of similarity between the cluster distributions obtained using qualitative harmonic analysis and optimal matching can be illustrated using a correspondence matrix which cross-tabulates the population distributions obtained with two different partitions. It shows whether the individuals of a cluster from the first partition are concentrated in a single cluster of the second partition or scattered between several different ones. Here, the six-cluster OM distribution spans the rows and the QHA distribution spans the columns (Table 3). The correspondence matrix of the ten-cluster distributions is given in Appendix 3.

Table 3

Correspondence matrix between OM and QHA six-cluster distributions

QHA 1 2 3 4 5 6 Total OM 1 291 16 4 45 5 1 362 2 7 320 0 5 13 7 352 3 17 5 311 18 0 0 351 4 2 11 1 98 0 5 117 5 13 0 9 2 49 0 73 6 59 1 23 2 1 0 86 Total 389 353 348 170 68 13 1,341 Population: 1,341 working careers of men in the 1930-1950 cohorts living in the Paris region at the time of the survey. Source: Biographies et entourage survey (INED, 2001).

Correspondence matrix between OM and QHA six-cluster distributions

40The diagonal of the table illustrates the strong similarity between the clustering obtained with the two methods, with the exception of cluster 6, which is small. [23] A correspondence rate summarizing the degree of similarity between the two partitions can be calculated by summing the mode of each line and of each column and dividing it by twice the total sample size. We thus calculate the mean of the correspondence of the OM clustering to the QHA clustering and of the QHA clustering to the OM clustering:

41where n_ij represents the number of trajectories belonging simultaneously to the la i^th cluster of the first partition and the j^th cluster of the second partition, and where N is the total population.

42The correspondence between the distributions obtained with the two methods is 82% for a six-cluster partition and 75% for a ten-cluster partition. The two methods thus give strongly converging results, a good indication of their robustness.

43There are several nuances, however, which enable us, on the basis of our observations, to point up the comparative advantages of each method. Comparison of the six- and ten-cluster distributions reveals several differences, which can be explained in two ways.

44First, the OM clusters are more sensitive to transitions, notably at the start of the trajectory when changes of occupational state are more frequent. They specifically distinguish between stable trajectories and mobile ones, and in this respect are more homogeneous than the QHA clusters. This is because OM analyses the sequence year by year, while QHA measures the proportion of each period spent in each state and does not include the sequence of transitions from one state to another in the analysis.

45In addition, QHA produces several smaller clusters grouping very specific types of trajectory, i.e. differing substantially from the others by the nature and sequence of states over the trajectory as a whole. We observe, for example, a cluster comprising individuals who have experienced short and scattered periods of inactivity, along with trajectories including one or more periods in a farming occupation. By contrast, the OM partition tends to incorporate these marginal trajectories into larger clusters.

3 – Comparative advantage of each method

46QHA provides a means to select for clustering only the main factors resulting from factor analysis. It is thus possible to eliminate some of the “noise” contained in trajectory data, i.e. to tease out the most structuring characteristics of the information, making it easier to interpret. But analysts cannot know what part of the statistical information is ignored as a consequence. They can only identify ex-post the factors disregarded by the procedure. In addition, by dividing the observation period into intervals of varying length, attention can be focused on the stages where the events of interest to the analyst are most frequent. This may be useful for occupational trajectories in which most events occur before age 30. In other words, QHA provides a means to concentrate on the periods when specific events occurred and on the duration of states. In particular, the occupations held for the longest durations have a stronger influence on cluster distribution than in OM, thus placing emphasis on socio-occupational stability, which still remains high today (Goux, 1991).

47Conversely, the OM analysis conserves the full detail of the sequence, rather than simplifying it (Lesnard and Saint-Pol, 2004). It is thus the analyst’s task to fix the costs on the basis of theoretical hypotheses, thereby favouring one pattern of trajectory clustering over another. Depending on the choice of indel cost, for example, interest can be focused on the type of transition or rather on the timing of an event. Compared to QHA, optimal matching thus places more emphasis on the sequence of events and the types of transition occurring within it. Consequently, it gives more weight to transitions from one SOC to another, and is able to detect intragenerational occupational mobility.