Data.

We use the Survey of Doctorate Recipients (SDR), a longitudinal survey of the science workforce population with doctoral degrees in science, engineering, and health earned in the U.S (http://www.nsf.gov/statistics/srvydoctorates/). The SDR is sponsored by the National Science Foundation (NSF), and is usually administered every two years. The sampling frame is the Survey of Earned Doctorates (SED), an annual census of individuals receiving a research doctorate. Once in the sample, individuals are surveyed repeatedly until age 76. The sample is refreshed with new doctorate recipients at each wave. The SDR includes rich information of doctoral recipients such as date of birth, educational history, employment status, field of degree, geographic place of employment, occupation, labor force status, race/ethnicity, salary, sex, and many others. The SDR’s sample size is one of its strengths. For example, the overall sample size for the 2010 survey was more than 30,000, with the overall sampling rate around 5.2%. Response rates are considered to be good; for example, 79.8% of those surveyed completed it in 2010. Several other studies have used this data set (e.g., [5]) to study the population characteristics of the science workforce.

Our study sample includes U.S. science and engineering doctorates employed as tenure- track and tenured faculty members in academia. Scientific fields of the faculty include the life sciences (biology, medical science, etc.), physical sciences (chemistry, physics, astronomy, and geology), engineering, computer science and mathematics, and social science (economics, psychology, etc.). In study 1, we use eight survey waves conducted from 1995 through 2010. We primarily used 1995–2010 survey data of tenured and tenure-track faculty to calibrate our simulation model. The data contains 65,208 observations on approximately 19,518 faculty aged 76 or less, with an average of 3.3 observations per sample member. SDR survey weights, provided by the NSF, are used in our analysis to adjust for attrition bias, resulting in more representative data of the population. To calibrate our simulation model, we use the SDR survey to extract detailed information such as the faculty members’ age distribution, hiring rate, and exit rate at each age.

Model.

We extend a previously published simulation model [5] by formulating hiring PhD workforce in faculty positions as affected by limited university capacities. This helps demonstrate where the variation of our results from previous studies emerges. We will later conduct an independent empirical examination to depict fidelity of our model predications. Fig 1 shows a conceptual diagram of the simulation model. The large center rectangle represents professors employed in tenure-track or tenured positions in academia. Inflows are hiring at different ages and outflows are exit rates at different ages that include retirement, tenure denial, or attrition for other reasons such as taking a job outside academia. The number of new PhD graduates, postdocs, or similar populations that could be interested in academia, as well as academic position openings, affect the inflow. The latter is formulated as the sum of openings to fill positions of people who exit academia (in the Fig 1, Openings to fill the exit rate) and additional new positions that open due to an increase in university capacity (in the Fig 1, Openings for adjustment).

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Fig 1. A simple representation of a stock-flow model of faculty members.

Note: the value for faculty positions is not necessarily constant; numbers depict age; N: population; i: hiring inflow; e: exit rate; o: aging; h: hazard rate of attrition.

https://doi.org/10.1371/journal.pone.0208411.g001

We simulate the model from 1995 to 2010. For each iteration, the model generates value for each variable and transition rates between them endogenously. The only measure we change exogenously over time in the simulation is the exit rate and total faculty positions. The total number of faculty positions in the base runs is set based on 1995–2010 data. We simulate this model using the following inputs: age distribution of faculty, exit rate, and hiring rate, all in 1995.

To model aging, the central rectangle in the conceptual model (Fig 1) is formulated following the same logic described in reference [5]. If we represent the total number of people in academia with N_total, this population includes faculty of different ages. We use 46 stocks that represent faculty members aged 32–75 (44 stocks), one stock for 31 years old and younger, and one stock for 76 years old and older. If we define N_a(t) as the number of individuals employed as faculty members at age a at time t, the total faculty population, N_total, is: (1)

People age through the pipeline, moving from the left side of the figure to the right. For each age category, N_a(t), there is an inflow that represents new hires, and an outflow that represents the retire/exit rate of that specific age.

As stated, in this model, hiring is formulated endogenously, which makes it different from Blau & Weinberg’s (2017) model [5]. Specifically, the total number of faculty position openings each year is comprised of (i) openings for adjustment, which are new positions due to an increase in university capacity, and (ii) openings to fill the exit rate, which are the positions of ones who have left.

We define h_a as the annual hazard rate of attrition for faculty with age a. Note that this hazard rate includes retirement, transition to a non-science job, and transition to a science but non-faculty job. In each time period, i_a is the number of faculty hired at age a, and o_a is the rate at which people are aging from a-1 years old to a years old. Age-specific transition can then be characterized as (2)

i_a is estimated by the total hiring i_total multiplied by p(hiring, a), the proportion of new hires at a years old in 1995. Data show that the age distribution of new hires does not change much during our analysis. Thus we set: (3)

For endogenous formulation of hiring, we have (4)

The first term, , is the difference between capacity and the total number of faculty members, which represents the faculty capacity gap. The second term, , is the total number of faculty members leaving academia at all ages at time t. If , it represents a steady state for the number of faculty members, and hiring will only replace the exit rate (). For , we use the past data and assume a +1.3% annual growth rate (the average growth rate in 1995–2010). Our main counter-factual run in which the only difference with the base run is the constant retirement rate is referred as scenario s1 in the paper. We conduct a sensitivity analysis for growth rates of 0.65% to 1.95% in , results represented as scenarios s2 and s3 in the paper. We also simulate for conditions when increase in hiring presumably results in hiring older faculties. Scenarios s4 and s5 in the paper represent average hiring age of one and two years older than the base run. These conditions are implemented by shifting the hiring distribution to the right, one or two years older, respectively.

We estimate the annual hazard rates of attrition by looking at the difference between net changes in each stock variable in two consecutive surveys adjusted for hiring rates and divided by 2. We use the data from 1995–1999 and 2008–2010 to estimate the hazard rates of attrition in our simulation model; for the period between, we linearly interpolate.

Finally, if we focus on the outflow of each stock every year, h_a(t) proportion of faculty leave academia, and (1-h_a(t)) proportion of them age while still employed. Thus: (5)

We simulate the model from 1995 to 2010, using the Vensim DSS software package, which is generally a good package for differential equation-based models (including system dynamics models). The integration type is Euler. Our time unit is one year which provides a more systematic comparison of the model with previous models (For numerical estimations, we set dt = 0.125 of time unit and ensured that the results are robust for smaller values of dt). We also kept the model details on a level at which we can see age distribution (not just average age) and then compare our results with previous models. The model and its inputs are provided as supplementary material (S4 File). For readers interested in potential extensions of the model for future work, and building more complex system dynamics models of a faculty workforce, we report causal loop diagrams of several dynamic hypotheses that can influence the process of aging and hiring in academia (simple vs. medium complexity [this paper] vs. high complexity), depicted in Figures A-C in S3 File respectively. In this paper, our intention was not to incorporate all of these mechanisms, however, the provided map in Figure C in S3 File has been very useful in exploring potential hypotheses in our initial stages of modeling.

Data.

We use the same data source as used in the simulation model. In study 2 we included 1983–2010 survey data of tenured and tenure-track faculty to study the long-term age trend, which contains a total of 109,499 observations.

Identification strategy.

Although most institutions had enforced mandatory retirement by December 31, 1993, some schools had eliminated mandatory retirement before that date, and most of these decisions were driven by a state law that prohibited mandatory retirement [12, 23]. As such, there are two subgroups of institutions: those that did not eliminate the mandatory retirement policy until the federal law took effect in 1994 (late uncap), and those that eliminated mandatory retirement policy somewhat earlier (around 1990s) (early uncap). By comparing early uncapped schools with late uncapped schools prior to 1994 we can estimate the effect of elimination of mandatory retirement. The early uncapped group includes all institutions in Wisconsin, Maine, Utah, Montana, and Nevada, and all public institutions in Alabama, Arizona, Connecticut, Florida, Idaho, Louisiana, New Hampshire, New York, Texas, Virginia, and Wyoming. If the intervention of uncapping has any effect on the average age, we expect to see an increase in the average age of early uncapped schools prior to 1994.

To empirically test how the elimination of mandatory retirement has contributed to the increase in the age of tenure-track faculty, we use an individual-level SDR data set and faculty age as our main outcome. We employ two identification strategies. First, we conduct a difference-in-differences analysis. We compare the average faculty age of early uncapped schools with that of late uncapped schools in a difference-in-differences analysis. We use SDR data from 1983–1993 in which early uncapped schools (treatment group) have eliminated the mandatory retirement around 1990, and late uncapped schools (comparison group) have not eliminated the policy in the time horizon. For most of the early uncapped schools the mandatory retirement policies were eliminated around 1988 to 1991 [23]. As we do not have observations within this period in our sample, we define our treatment period as after 1990. Here the deviation from the trend in the comparison group will reflect other hard-to-observe factors (e.g., the economy, or other higher education reforms) that may have influenced faculty age in the absence of the elimination of mandatory retirement. We also controlled for the year trend, faculty characteristics such as rank, field, salary, gender, race, marital status and citizenship, and so forth, as well as characteristics of the institution where faculty are employed such as the type of institution, whether it is privately controlled, and Carnegie classification.

Second we look at effects of years-since-intervention, i.e., treatment as dosage. In the previous analysis we could only use SDR data up to 1993 because late uncapped schools had eliminated mandatory retirement from 1994 onwards. Here we consider treatment as dosage—that is, we define the treatment variable as the number of years a school has eliminated the mandatory retirement policy. This method allows us to include more years of data in our analysis (1983–2010), and we can statistically compare the faculty age of a school that has eliminated mandatory retirement for a certain number of years. In addition, because schools in the two groups (late and early uncapped) eliminated mandatory retirement at different time points, we also gain the statistical power to estimate the effect of elimination of mandatory retirement within the same year. In this analysis we include year fixed effects and other control variables as in the previous model. Note that in both analyses we have also controlled for institutional fixed effects or use institution or state level clustered standard error as alternative specifications, which our results are robust to.

Finally, to estimate the effect of elimination of mandatory retirement on the number of new hires, we construct an institution-level data set and use the number of new hires for an institution in a year as the main outcome, employing a difference-in-differences analysis. Since hiring data were available from 1993, we consider the removal of mandatory retirement due to the federal law as the exogenous intervention. This, in contrast to the previous analysis, makes the late uncapped schools the treatment group, and the early uncapped schools the comparison group. The assumption is that the hiring of early uncapped schools is arguably much less affected by the elimination of mandatory retirement at the federal level from 1994, as such schools had already eliminated this policy considerably earlier than 1994. Thus, the additional difference in the number of new hires between the schools in the two groups after the elimination of mandatory retirement on the federal level can be attributed to the effect of the federal policy. As some lags might exist in the impact of the elimination of mandatory retirement on new hires (i.e., it takes time for faculty positions to saturate and thus decrease the number of new hires), we test how our estimates are sensitive to different specifications of the post-policy period. Other control variables are the same as before, with the exception of using aggregate measures of faculty characteristics (e.g., number of assistant professors, number of females, and so forth.) at the institutional level. Note that here the institutional-level data is aggregated using individual-level data with weights. One potential limitation of this approach is that the SDR sample might not be fully representative on the institutional or regional level. However, the difference-in-differences estimates would still provide reasonable estimates for the magnitude and direction of the effect under the assumption that representativeness of the sample on the institutional level does not vary across years.