Given Nx,1991, Mx, Dx, and G, then Nx,y can be calculated for any y greater than 1991 (in increments of 2-years) by iterating through the formula:

which expands to:

where I is a k by k identity matrix, S is a k by 1 vector of ones, and the symbol x designates an element-wise matrix multiplication operation. (The operation ((S • D) x I) merely takes the Dx vector and turns it into a matrix with the elements of Dx on the main diagonal and zeros on the off diagonals).

The total number of new entrants is given by the component:

Projections were made separately for each of 4 populations:

  • biomedical Ph.D.'s in biomedical employment fields,

  • non-biomedical Ph.D.'s in biomedical employment fields,

  • behavioral Ph.D.'s in behavioral employment fields, and

  • non-behavioral Ph.D.'s in behavioral employment fields.

To illustrate the use of life table analysis in generating projections of workforce variables, the Panel, as an exploratory exercise, chose to generate estimates of job openings. Given the uncertainty associated with efforts to project demand, the Panel examined three growth rate scenarios based on the average annual growth in the biomedical and behavioral science workforces between 1981 and 1991: zero growth; one-half the 1981-1991 average annual growth; and the average annual growth. 5 Estimates of “net separations” 6 were generated using the life tables. Estimates of needed job openings for the alternative growth scenarios were derived by adding to these separations the number of additional job openings that would need to be created to attain the particular target rate of growth.

In generating the estimates of job openings, the following assumptions were made:

  1. There is never a negative number of new entrants. If there is a surplus in the employed “in field” population at a given year, no new entrants are added to the life table for that year.

  2. The ratio of behavioral Ph.D.'s to non-behavioral Ph.D.'s employed in behavioral fields remains constant. That is, both Ph.D. populations increase or decrease at the same rate. The same assumption is made for the models of biomedical and non-biomedical Ph.D.'s in biomedical employment.

  3. The age/destination state proportionate distribution, Dx, is taken from the age/destination distribution observed among new entrants between 1985 and 1991.

  4. The age/origin state distribution for the current population, Nx,1991, is calculated by taking the age-specific origin state distribution among the current Ph.D. population between 1985 and 1991 and applying it to the age distribution of the 1991 current population.

  5. The age-specific 2-year transition proportions, Mx, used to survive the current age-distribution is taken from the observed transition proportions between 1985 and 1991.

The committee's Panel on Estimation Procedures will extend this work and will prepare a separate report for release in 1994 on the role of multistate life table methods in the estimation of national need.


1. Methodological detail is available on request from NRC/OSEP Studies and Surveys Unit (Memorandum by Peter Tiemeyer, September 30, 1993). A general discussion of multistate life tables can be found in Keyfitz (1985).

2. In our particular project, however, we began with transition rates as the basic input data, and derived other life table statistics from those rates.

3. We define R&D employment to be basic or applied research, management of R&D, or development and design of systems and products; it is based on the individual's self-report of primary or secondary work activity.

4. With respect to the treatment of missing data: in general, to enter into the calculation of a biennial transition table, an individual case was required to have valid survey data on age and Ph.D. field and valid data for both of the survey years (for that transition table) on employment field (biomedical, behavioral, etc.) and employment status (postdoctorate, employed, retired, etc.). We developed decision rules for the treatment of all of these variables to handle various conditions (available on request). For example, work activity (i.e., R&D vs. non-R&D) could be missing if the person's employment field was other than biomedical or behavioral (because one of the “states” of the model is “employed outside of Ph.D. field” and those who are out of labor force, retired, or out of the country could be missing employment field and “work activity ”.

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