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Using Mobility Data to H Develop Occupational Classifications: Exploratory Exercises JOlIN A. HARTIGAN How can occupational mobility data help occupational classification? They may help determine that two occupational titles with slightly different definitions are similar enough to be amalgamated or that some occupation is attracting two distinctly different types of workers and should perhaps be split. They may also supplement the Dictionary of Occupational Titles in suggesting plausible cross-listings for job titles. Occupational mobility data can contribute only a little, however, to the definition of occupations in terms of job tasks: for that, occupational analysis or some alternative methodology is needed. The most significant use of job mobility data is to suggest a suitable hierarchical organization of occupations, given a set of occupational definitions. Mobility data are of value in grouping occupations in a way that reflects the transfer of workers between occupations within a group. Mobility data also are of value in constructing career ladders, that is, hierarchies of occupations up which workers tend to move in the course of successful careers. We have conducted an exploratory analysis of alternative methods of classifying occupations. This analysis assessed the feasibility of developing classifications consisting of groups of occupations between which there are high rates of labor mobility. Our basic data consist of the transfers between the 441 U.S. Census 4~]

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412 WORK, JOBS, AND OCCUPATIONS occupational categories between 1965 and 1970.i Unfortunately, data on various extra-labor-force statuses (e.g., unemployed, in school, in armed forces, etc.) were not available to us. Similarly, no data coded into the 12,099 DOT occupational titles are available, nor are any data available that give complete work histories or short-term transfers between jobs.2 More appropriate data are needed for future work in this area. We use the available census data for our exploratory purposes to illustrate how one might proceed in constructing a classification based on naturally occurring patterns of labor mobility. The first problem we faced in this analysis was the storage and manipulation of the full mobility matrix for the 441 detailed census occupations. A 441 X 441 matrix is formidable (194,481 cells), and the 12,099 X 12,099 matrix for the DOT (more than 146 million cells) is even worse to contemplate. A more manageable way to manipulate such data is to represent them in a list structure, which gives for each 1965 occupation a list of 1970 occupations to which transfers took place and the corresponding counts in each of these occupations. The total storage is reduced without much loss by eliminating very small counts. It is also necessary to carry the transposed list ordered by 1970 occupational categories. STANDARDIZED RATES AND PROBABILITY MODELS In order to adjust tor different numbers of workers in various occupations, Goldhamer (1948) proposed the standardized rate n,.fN tij - ni.n.j where nil number transferring from job i in 1965 to job j in 1970; ni numberin jobiin 1965; n j number in job j in 1970; N total number of workers. Hauser (1978) notes that this measure does not adjust for expected diagonal peculiarities and suggests a measure in which the "margins" nit See Sommers and Eck (1978) for a description of the data used in these analyses. 2Had the work history data routinely collected from Employment Service job applicants been available for analysis, we could have conducted a much more interesting and informative exercise. Unfortunately, although the work history data are initially coded with nine-digit DOT codes, all but the first two digits are dropped when the data are put on tape.

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Using Mobility Data to Develop Occupational Classifications 413 and n.j ignore specified cells such as the diagonal ones, using Goodman's (1969, 1971) quasi-independence techniques. For example, nit, n.j, and N might all plausibly be defined ignoring the diagonals. For a hierarchical structure on the set of jobs, consider the model Pij PieP.i\G (i,j) ' where Pij Pi. P. probability of observing a transfer i to I; probability (roughly) that a worker begins in job i; probability (roughly) that a worker ends in job I; transfer rate corresponding to the smallest group G containing job i and I; there will be a different rate for each group. Following the standard quasi-independence procedure (Haberman, 1974), the maximum likelihood estimates of Pi., P. j, AG are obtained by setting the observed margins and between-group transfers equal to their expected values under the model ni IN = Pi [A P I )\G (i I)] n.i/N = P j ~ Pi. JIG {i, I)] -L G tGti,/)=G j Solutions may be obtained by solving successively for SPi.}, [P.j~, {AG} with the other parameters fixed. The overall fit of the model may be measured by the log likelihood L - I nij log pij . This measure permits comparison of various hierarchies. It also allows construction of new hierarchies by seeking groups G that make L as large as possible. Conceptually, the procedure is straightforward; computation- ally, it would be quite a chore to design iterative parameter estimates for a list data structure and to improve the hierarchy by moving jobs between groups.

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414 CLUSTERING ANALYSES WORK, JOBS, AND OCCUPATIONS Alternative procedures are available. A hierarchical clustering has been carried out by Dauffenbach (1973~. He discusses the 1970 Census classification and principles for constructing a new classification. For job i a vector is constructed equal to the proportion that transfer from job j to job i for all j. Distance between jobs is the Euclidean distance between these vectors. (Some other distances and data vectors are also considered.) Thus two jobs are similar if there are similar patterns of movement into them. Complete linkage clustering (cf. Hartigan, 1975) was then used to construct a binary tree of clusters on the set of all jobs. The results are not very different from the census classification. The measure of distance and the data vector of proportional transfers used by Dauffenbach are not wholly adequate. In particular, there will be large transfers from jobs with many workers, and such jobs will tend to make large contributions; there will be many entries near zero in every vector, and it seems wrong to ignore this property of the vectors; the essential information in the data is carried by the transfers from each job to just a few other jobs. The problem with the measure of distance is that after we have computed Euclidean distance between two vectors of length 441, we do not know what we have. Complete linkage is statistically inconsistent. Nevertheless, Dauffenbach's clusters are suggestive. An alterative method of constructing clusters uses a quasi-independence model (see Appendix G). This would require advanced programming that has not been done. A simpler method is to use the standardized transfer rates - nit [if ni.n.; where ni total number transferring from job i; n; total number transferring into job j; N total number of transfers. Any two jobs i and j are similar if tij and tji are both high; the measure of distance between i end j is dij = 1/mint, tji). The single linkage technique constructs clusters by linking together jobs for which the transfer rate exceeds some threshold; a cluster is made up of jobs linked together. Varying the threshold produces a hierarchy of clusters.

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Using Mobility Data to Develop Occupational Classifications 415 We have applied this technique: the clusters obtained are shown in Table H-1. Like Dau~enbach's clusters they draw together different levels of skills, such as librarian and library attendant or health record technician and medical secretary. They also show some absurd associations, such as dentist and flight engineer, which are due in part to single linkage chaining together a number of slightly related jobs and in part to the unreliability of transfer rates that (because diagonal terms are removed) may be rather high for jobs with high retention rates, from which people transfer to just a few other jobs. CAREER LADDERS We would like a classification scheme not only to group together occupations between which transfers are likely but also to order occupations so that transfers tend to take place from lower-ranked jobs to higher-ranked jobs. In order to accommodate both aims and to explain the transfer data succinctly, it would seem desirable to put jobs close together in the structure whenever there are many transfers in either direction. The small groups should therefore consist of families of jobs within which a career ladder exists; there may only be a weak ladder relationship between the larger groups. (In the census scheme there are strong ladder relations between the large groups.) A probabilistic model constructs an ordering and a hierarchical classification of all jobs. The probability of a transfer i to j is Pit Pi.P.iiii, where Pi is the probability (roughly) that a person is in job i in 1965, p j is the probability (roughly) that a person is in job j in 1970, and kit is constant over all i < j such that G is the smallest group containing iti. To estimate the parameters given the order and hierarchy, it is sufficient to know the marginal numbers of workers, the number of transfers from lower-status to higher-status occupations within each group, and the number of transfers within each group.

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418 WORK, JOBS, AND OCCUPATIONS Sophisticated programming is required to construct a hierarchical clustering and an ordering according to this model. A quick but less adequate way to construct an ordering is as follows. Let si be the level of the ith job. Compute {si } so that most transfers from i are to jobs j, where (sj-si) is small. The easiest criterion to minimize is the sum of (Si-SO over all transfers, subject to the condition that the sum of Si2 over all workers be fixed. This criterion leads to the iterative 3 equals the average Sj over all transfers to and from i, equal to JO (nijSj + Isis/ :~,~nji + n,,) for obtaining improved estimates Si given the old estimates Si. The starting point for the estimates would be the original numbering for the jobs, which will give a crude rank order by level in the standard classifications. The procedure should be repeated several times. Another simple procedure is to reorder the jobs so that as many transfers as possible take place to increase the ordering; this is simpler conceptually but more complicated in computation than the procedure described above. FEASIBILITY Our analyses were carried out to explore the feasibility of using mobility data to construct an occupational classification. Our tentative conclusions are the following: 1. Mobility data can be useful for constructing a hierarchical clas- sification and ordering of occupations, but the basic occupational titles on which the mobility data are collected must be defined by other procedures. 2. There are formidable statistical and computational problems in- volved in constructing a classification in this way. In particular, in developing classifications for job-worker matching, it is crucial to pay careful attention to activities before entry and after exit from the work force. In addition, computations should be carried out using list structures; a standard matrix representation is not feasible. 3. Some plausible statistical models for transfers are available and could be used as a guide in evaluating and generating classifications and career- ladder orderings. 4. Crude reclassifications and orderings suggest that the 1970 Census classification had many pairs of similar jobs in quite different groups, owing to its emphasis on socioeconomic status. 5. It would be feasible to construct occupational groupings so that most transfers take place within relatively small groups and so that most transfers take place upon a career ladder.