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APPENDIX
c
Multistate Life Table
Methodology and Projections
Multistate period life tables were used in this report to develop projections
of numbers of new Ph.D.s that would be needed in the future to sustain certain
growth rates of the labor force. This Appendix describes briefly the method used
to generate these results.
Life table techniques are sometimes the only way to obtain estimates of
certain statistics describing mobility arid career characteristics of a population,
especially those related to rates of occurrence of events, duration of time spent
in an activity, and rates of attrition or exit from a population (due to death,
retirement, job charging, etc.) even when we have not observed the full lifetimes
of the scientists with our data (which is often the case with most data sets).2
Moreover, life table methods provide a useful way of organizing various age-
specific rates (rates of entering the labor force, changing jobs, moving abroad,
retiring, and dying) into a logical framework, which Carl then be used to make
projections of various characteristics of a population, such as its age distribution.
Multistate life tables, an extension of basic life tables, allow greater
complexity to enter the analysis: people can enter as well as exit a population
and can move back and forth across a variety of states within a population. Life
'This originally appeared as Appendix G in the National Research Council report
Meeting the Nation 's Needsfor Biomedical and Behavioral Scientists. Washington, D.C.:
National Academy Press, 1994.
113

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114 / HEALTH SERVICES RESEARCH
tables may be classified as period versus cohort life tables. The Panel on
Estimation Procedures decided that it was more practical, for the purposes of
making projections, to use the former.
Construction of period life tables involves taking age-specific transition
rates Job changing, unemployment, retirement, and death rates) prevailing during
a particular period (e.g., 1989-1991) and applying them to a hypothetical
(synthetic) cohort of people (actually a "synthetic" cohort of new Ph.D.s).
Probabilities are then calculated of entrance or exit from a state (probability of
entering a postdoctoral position, for example) and length of time spent in various
states, as implied by the life table. Statistics of interest were developed for three
time periods (1985-1991, 1979-1985, 1973-1979~. Period life table results are
conventionally referred to as "expected" quantities (i.e., "expected fraction of
people who . . .," "expected length of time . . .") because of the nature of the
methodology: constructing a single hypothetical Ph.D. cohort that experiences the
current transition rates taken from a variety of Ph.D. cohorts.
The data for the life table analysis come from the longitudinal Survey of
Doctorate Recipients (SDR), a sample survey that follows a group of Ph.D.s over
time, interviewing basically the same people every 2 years. A general
description follows (for details, see NRC, 1991 SDR Methodological Report,
forthcoming). In 1973 an initial sample of science or engineering Ph.D.s living
in the United States was drawn, and those sample members have been followed
through time. New Ph.D.s enter the SDR in 1975 and each subsequent SDR year
(1977, 1979, . . . 1991) and are followed over time as well. Individuals are
followed until they reach a certain cutoff point that depends on the survey year
at which they entered "typically 42 years after the Ph.D., although in recent SDR
waves, they are followed until they reach age 70 or until they drop out for other
reasons (nonresponse or deathly.
The form of the data on which the life tables are based consists mostly of
large sets of transition tables constructed from the SDR by National Research
Council (NRC) staff (but death rates are obtained from TIAA-CREF data from
the late 1980s, taken from Bowen and Sosa, 1991~. For every pair of biennial
survey-interview years ("waves") in the SDR (1973-1975 as the first pair,
1989-1991 as the last pair), the number of people moving between various states
within those 2 years were obtained. These states were: postdoctorate, R&D
employments within one's broad Ph.D. field (biomed, behavioral, other), non-
R&D employment within broad Ph.D. field, employment outside of broad Ph.D.
field, out of labor force or unemployed (combined), leaving the country,
retirement, and death. All of the biennial transition proportions were obtained
by 2-year age group by broad Ph.D. field, and by sex. The survey observations
(i.e., people) in one set of biennial transitions are often the same people in
subsequent sets (though older).4

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APPENDIX C / 1 15
Another data ingredient for the life tables is the distribution of states (same
as above) of"new entrants to the SDR" for SDR waves 1975-1991, again by
age, sex, and Ph.D. field. These are used as estimates of numbers of new Ph.D.s
in each survey year.
These transition data sets constructed by NRC staff were transformed into
proportions to be used as input into a Multistate Life Table program (Tiemeyer
and Ulmer, 1991~. Initial work involved explorations of data quality, sample
sizes, and the stability of rates over time. To have large enough sample sizes for
(what we would hope to be) reliable estimates of sex differences in career
patterns as well as estimates of how the career patterns have changed over time,
it was necessary to aggregate the data into three broad time periods (as opposed
to looking at a larger number of time periods): 1985-1991, 1979-1985,
1973-1979.
Projection Models
Life table construction begins by calculating a matrix containing the
proportion of individuals exiting an origin state for each possible destination state
between ages x and x+2 (in our case). This matrix is called Mx.
Our projection models hold the population of those employed "in field" to
some constant growth rate. The following algorithm is used:
1. Survive the current specified Ph.D. population forward 2 years.
2. Calculate the number of individuals employed "in field."
Calculate the differences between the target "in field" population and the
number "in field" in the survived current population. This yields the number of
new entrants needed to increase the "in field" population to its target size.
4. Divide the result of (3) by the proportion of new entrants who enter an
"in field" employment state on receiving their Ph.D.
5. Use the result of (4) as the number of new entrants who would have had
to enter the population between year y and y+2 to attain the target "in field"
population. Add these individuals into the life table, distributed approximately
by age and destination state.
3.
Let Nx y represent the number of individuals in the specified Ph.D.
population in each employment state at age X for a given year Y. Nxy is a k
by k matrix, where k equals the number of states in the model. The columns
indicate origin states (in the base year) and the rows destination states. So
Nx ~995~4,1] would equal the number of people who were in the 4th state (out of
field employment) in 1995 who were in the 1st state (in field post-doe) in 1991.

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1 16 / HEALTH SERVICES RESEARCH
For the base year, the off-diagonal elements Of Nx am are all O and the on-
diagonal elements are equal the number of individuals age X in 1991 in the
specified Ph.D. population in each employment state.
Let N-xy represent the number of individuals in each state (by origin state
in 1991) in year y, BEFORE new entrants between year y and y-2 are added into
the life table. Then N-xy is given by:
NX-Y=NXY-2-~I+ 2X~ PI- 2~
Let Figs, represent the total number of individuals in the specified Ph.D.
population employed "in field" in year 1991. Then F,g9~ is given by:
71 3 8
Fl991 = ~ ~ ~ NX,lg91[d'0]
x=25 o=1 d=1
where x represents age, o represents origin of state, d represents destination state,
Nx ~g'~[o,d] represents the dth row and the oth column of Nx ~99~, and where states
1 through 3 represent the employed "in field" states.
Let F-y represent the total number of individuals in the specified Ph.D.
population employed "in field" in year Y who were in the specified Ph.D.
population (although not necessarily employed "in field") in year Y-2. Then F-y
. .
lS glVeI1 Dy:
71 3 8
Fy-=~Nxty[d~o]
x=25 o=1 d=1
Let G represent the assumed 2-year growth rate for Fv.
employed "in field" population size for any given year is:
Target "In Field" Population Size (y) =
Flggl .(l +G3y 1991
Then the target
Let Dx represent the proportionate distribution by age and state of new
entrants to the specified Ph.D. population over the two year period between Y
and Y+2. Dx is a 1 by k vector with each column representing the proportion

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APPENDIX C / 1 17
of all new entrants who are age X who enter that state on receiving their Ph.D.
Summing Dx across all ages and states should equal one.
Finally, let R represent the proportion of all new entrants who enter an "in
field" state on receiving their Ph.D. Then R is given by:
7~ 3
R= ~' ~ DXRQ]
x=2s `=t
Given Nx age,, 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:
NX';iggi=
N- ((First (1+G) ~ Fy) ((S Dxjxl)
which expands to:
Nx't>1991=Nx,y-2 (I 2 ~ ( 2 ~
7i 3 8 1 ( M )-1 ( M )\J
[d,o]
R
((S Dxjx~
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).

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1 18 / HEALTH SERVICES RESEARCH
The total number of new entrants is given by the component:
(F1g91 (1 +G)Y )- ~ ~ ~ lN.,.Y 2 (I+ 2 ) (I- 2 )\J[d~o]
R
·((S D )xt
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 rates scenarios based on the
average annual growth in the biomedical arid 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:
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.

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APPENDS C / 1 19
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.
NOTES
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 Tom 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, etch. 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."
5. The 1981-1991 average annual growth rates were: 4.25 percent per year
for the biomedical sciences workforce and 3.5 percent per year for the behavioral
sciences workforce.
6. Net separations are defined in this analysis as losses arising from death,
retirement or outmobility to another state minus gains from inmobility of
experienced scientists from other states of employment. Alternative definitions
will be explored in subsequent work by the Panel on Estimation Procedures.

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REFERENCES
Borden, W.G~ Id J.A. Sow 1989. Fro~ ~, ^~/~ /~ ~e ~ ~d
a. Princeton, a: Prince10n University Press.
KeyAtz, K. 1985. -/~d ~/ ~ 2nd Ed. Hen Yolk:
Springer-Verlag.
Notional Research Council. Forthcoming. 7997 ~R a// R~r
Washington, D.C.: Notional Academy Press.
Tiemeyer, P., ad G. Ulmer. 1991. ~F ~ ^~, ~e CO~~f/
-~ [~ /. Center for Demography Wowing Pier 91~4,
. . ^ -~ ,. .
OnlVerSl~ OI WlSCOnSlD.