Incidence and Cumulative Incidence as Supplemental Measures of the Scale of Imprisonment

Abstract

Prison growth has primarily been measured as a prevalence over time. We propose cohort-specific supplemental measures: incidence based on the age of first adult admission into prison, and cumulative incidence, based on the proportion of people who will be imprisoned during their lifetime. We present a new estimation method using administrative data. Prior research derived estimates from inmate surveys. The main advantages of this new method are that estimates can be updated every year with little cost and minimal imputation. We present results showing that we have likely reached an inflection point in the growth of cumulative incidence, and the ratio between Blacks and Whites is declining although the disparity is still large—roughly 4.5 to 1.

Keywords

prison incidence cumulative incidence prison prevalence scale of imprisonment mass incarceration

The scale of imprisonment is an important social indicator most commonly expressed as a prevalence—persons incarcerated divided by those at risk of incarceration. In this article, we advocate for supplemental measures—incidence and cumulative incidence of imprisonment. These latter measures are cohort specific. They are based on the age of first adult admission into prison (incidence) and the proportion of people who will be imprisoned during their lifetimes (cumulative incidence). Scholars have previously estimated incidence and cumulative incidence of incarceration using survey data having inherent limitations (Bonczar, 2003a; Bonczar & Beck, 1997; Petit, Western, & Sykes, 2009; Western & Petit, 2010). In this study, we use readily available administrative data to estimate lifetime exposure to incarceration. We acknowledge that prevalence will continue to be the primary indicator of prison scale; however, we show how this new estimation approach provides additional insight into understanding the scale of imprisonment; how it can be updated more frequently than incidence measures based on survey data; and how it provides a more timely representation of trends in incidence and cumulative incidence.

One way criminologists have reported prison prevalence is to use it as a yardstick to measure the scale of imprisonment (Harding, 1992; Mauer, 2017; National Research Council [NRC], 2014; Pfaff, 2017; Zimring, 2010; Zimring & Hawkins, 1991). By scale, researchers want to express the level of incarceration relative to some assumed natural equilibrium, or the level propagated by underlying political, social, or cultural factors. For example, Zimring and Hawkins (1991) express scale by asking “What criteria should govern decisions about how large a prison enterprise should be constructed and maintained?” (p. xi). Without an explicit normative position, criminologists have also attempted to explain growth in prison populations. When prevalence is stratified by race, gender, and other criteria, the level of imprisonment can also be used to study criminal justice disparities and inequities, and to benchmark criminal justice reform.

Despite the importance of this social statistic, criminologists are aware that prison prevalence is sensitive to both historical and contemporary social and policy changes (Beck & Blumstein, 2012; Pfaff, 2017). This is because prevalence represents the stock of prisoners divided by a suitable divisor, typically the number of people in the population. The stock represents an accumulation of people admitted into prison from many different age cohorts. If there are changing trends in admission cohort composition, the rate of entry, or the length of stay for those admitted, these factors are confounded into a single prevalence statistic. Furthermore, in addition to the fact that prevalence does not distinguish between admissions and prison length of stay, it also confounds an understanding of new and repeat incarceration encounters.

These analytical concerns are well known to criminologists who have studied mass incarceration and the drivers of prison growth. Serious discussion of the level of imprisonment incorporate rates of admissions, decomposing them by important strata such as new and repeat commitments, as well as potential changes to length of stay. We argue that additional insight can be achieved by estimating first admission, as well as the lifetime occurrence of a first admission to prison. These social statistics can be used to compare cohorts and therefore can be used to observe trends in imprisonment among cohorts over time. For clarity, we borrow definitions and terminology from the epidemiology literature (Jekel, Katz, Elmore, & Wild, 2007), and label these social statistics as incidence and cumulative incidence.

The new method we developed to estimate incidence and cumulative incidence relies on administrative data. We outline our procedure skimming over details which can be found in a companion methodology paper (Rhodes et al., 2017). In this article, we focus on what these new measures reveal about cohort-specific trends in prison admissions. To provide context, the “Definitions” section introduces core ideas. The “Scholarship on the Scale of Imprisonment” section reviews the use of imprisonment prevalence as a social statistic. This is not an exhaustive review, but it demonstrates the degree to which scholarship to date has depended on prevalence as a social indicator. In this section, we also review some of the limitations of prevalence and admission rates as aggregate social indicia. In the section “Incidence and Cumulative Incidence as Important Supplemental Social Indicators” we explain and justify using prison incidence and cumulative incidence, demonstrating how it provides additional insight that could alter inferences about policy and theory in this field of research.

In “Prior Studies of Incidence and Cumulative Incidence” section, we review how other analysts have developed their measures of incidence and cumulative incidence using Bureau of Justice Statistics (BJS) surveys of prison inmates (SPI). The SPI is a cross-section of inmates confined on a reference date. It is designed to estimate national prison population statistics. These data are combined with Census and administrative data to produce incidence and cumulative incidence estimates. As a shorthand, we refer to these calculations as SPI-estimators. Our new methodology is explained in “Estimation of Incidence and Cumulative Incidence Based Exclusively on Administrative Records” section. Our approach is possible because of recent improvements to the BJS National Corrections Reporting Program (NCRP). As a shorthand, we call these latter statistics NCRP-estimators.

In “Results of the NCRP-Based Estimation of Incidence and Cumulative Incidence” section, we show state-specific and combined state trends in these aggregate measures. We also highlight racial and gender differences. We conclude with a discussion of our findings and their relevance to social indicators of imprisonment.

Definitions

Imprisonment prevalence is the number of people incarcerated divided by the at-risk population. Prevalence is an easy calculation. The numerator is a statistic captured by almost every federal, state, or local jurisdiction. The denominator, the number of people in the population or the number of people at risk to be imprisoned, is also readily available from decennial censuses interpolations, or sources of crime statistics.

In contrast, there is no simple method for computing the incidence and cumulative incidence of imprisonment. One of the difficulties is that estimation requires distinguishing first-time admissions from repeat admissions. Previously, criminal justice researchers have had to cobble together survey, administrative, and census data to construct these incidence statistics. While researchers have done a credible job of estimating these social indicia, the survey-based approach has limitations that make estimation uncertain. This is primarily because national prison surveys are designed to answer questions about prevalence, not incidence. Analysts must make analytical accommodations and must use other sources of data to fill in the incidence gaps not covered by the survey.

Rather than survey responses, we use administrative data to compute the numerator of prison incidence as the number of cohort members admitted to prison for the first time at a specified age. Prison cumulative incidence is the sum of incidence over the lifetime of the cohort. To make this concrete, consider a 1982 birth cohort. If 100 members of this cohort enter prison for the first time in 2000, 200 more enter for the first time in 2001 and 300 more entered prison for the first time in 2002; then as of 2002, the cumulative incidence numerator for the 1982 age cohort would be 600. The denominator is the number of people in the age cohort. Although the definition is straightforward, estimation is challenged by conceptual and computational difficulties which we discuss when we explain the NCRP-estimator method. To emphasize the differences in these social indicia, consider calculations made in 2002. Incidence for the 1982 cohort provides a metric of all those admitted to prison for the first time at age 20. The cumulative incidence for the 1982 cohort in 2002 provides a measure of all those admitted to prison at least once by age 20. Prevalence on some referent date in 2002 is a count of everyone in prison regardless of age, cohort, or number of prior stays in prison.

In summary, incidence, cumulative incidence, and prevalence have different meanings and each provides a useful but unique insight into incarceration levels and patterns. This article is focused on incidence and cumulative incidence. As surveys provide uncertain estimates of incidence and cumulative incidence, this article advocates that researchers use administrative data to overcome survey-based limitations. Before we show the utility of incidence and cumulative, we demonstrate the central role prevalence has played in the scholarship on the scale of imprisonment.

Scholarship on the Scale of Imprisonment

Criminologists use prison prevalence as a key aggregate social statistic when trying to ascertain some presumed normative level of incarceration, and when analyzing the causes of the level and distribution of incarceration. Sometimes researchers stratify their data by race, offense (primarily violent, property, drug), and other subgroups to shed light on the possible drivers of the level of imprisonment. Our brief review shows the strengths and limits of prison prevalence as a social statistic, thereby providing background for how incidence and cumulative incidence enhance the study of the scale of imprisonment.

Defining and Interpreting Prevalence

Pfaff (2017) has shown that specifying the denominator is crucial to interpretation of the prevalence statistic. He demonstrates that changing the denominator from the total population to the crime rate changes the perspective on when the era of mass incarceration began, skipping over the 1970s and, instead, starting in the 1980s. Similarly, Mauer (2017) notes that when comparing jurisdictions on prison prevalence, the analyst and policy maker ought to be aware of differences among jurisdictions in such factors as the rate of violent crime; the extent to which jurisdictions use alternative institutions (such as mental hospitals), or other forms of social control (such as probation); and differences among jurisdictions in their use of highly restrictive community sanctions (such as home confinement). These factors can account for marked differences in jurisdiction-specific prison prevalence. Rather than make a naïve comparison, Mauer (2017) emphasizes using this broader theoretical and empirical context.

Norms of Imprisonment Levels

Some authors have postulated or implied existence of a normative level of incarceration (Blumstein & Cohen, 1973; Harding, 1992). For example, Harding (1992) implicitly assumed a normative level of Australian State imprisonment based on the national average across all Australian States. He was concerned that one of the States, Western Australia, had a history of excessive incarceration. He placed responsibility on the lower courts where judges were not utilizing community-based sanctions. He also argued that judges who imposed sentences were irresponsible because they ignored or minimized the costs of imprisonment. This is a similar point made by Pfaff (2017), Lynch (2011), and Zimring and Hawkins (1991). Pfaff argues that state prison systems bear the costs of charging decisions made by counties—a type of moral hazard—because the county prosecutors are protected against the risk and cost of incarceration by the state. Harding and Zimring & Hawkins (1991) colloquially called this the “custodial free lunch.”

Harding stipulated that the State of Western Australia, which exceeded the national average, should reduce its prison population to that average. Of course, after reforming the outlier State, the average itself would decline thereby establishing another norm. More importantly, this presupposes that the national average itself is the proper norm. It is well known that nations and states within nations have widely disparate rates of incarceration (see, for example, Jacobson, Heard, & Fair, 2017; Mauer, 2017) with the United States ranking near the top among nations, and Louisiana at the top of U.S. State jurisdictions (Carson, 2016). Which nation and which state has the correct norm? More importantly, how do scholars reconcile state or national differences without exploring the causes of the scale of imprisonment?

We do not try to resolve whether there is, or should be, a normative level of imprisonment. Some criminologists might propose a normative level of incidence and cumulative incidence. Our sole purpose here is to show how this literature has used prevalence as the normative benchmark.

Theoretical Explanations for the Scale of Imprisonment

There is a great deal of scholarship on the level imprisonment and its historical, cultural, social, and political roots (Garland, 2001; Gottschalk, 2006; Savelsberg, 1994; Simon, 2007; Tonry, 2004; Wacquant, 2009; Whitman, 2003). Some theorists examine incarceration in relation to other forms of punishment contrasting prison levels among different nations. The prime motivation for most of these theorists is to explain the era of U.S. mass incarceration. Zimring and Hawkins’s (1991) The Scale of Imprisonment is also a significant contribution exploring theories that address the political, social, and historical factors that affect the level of imprisonment. They review theories centered on punishment and economic development (Rusche & Kircheimer, 1939); the constancy of punishment prior to the era of mass incarceration (Blumstein & Cohen, 1973); historical forces that placed the scale of imprisonment in relation to the institution of incarceration (Rothman, 1971); a theory of the carceral network that views imprisonment as an extension of societal power in which prison is only one of many controlling institutions such as schools, factories, barracks, and hospitals (Foucault, 1977); and a theory that views the level of imprisonment as a response to social disorder (Ignatieff, 1978). These theoretical approaches all rely upon prevalence measures of prison, probation, and other social and criminal justice control mechanisms.

In a more recent exposition, Zimring (2010) explores whether there is evidence for a unitary political process affecting the aggregate rise in the U.S. incarceration levels, or 51 processes—state and federal—driving the growth. He concludes that there is one underlying process using growth rates in the prevalence of prison to undergird his argument. Disagreeing with Zimring, other scholars (Lynch, 2011; Pfaff, 2017) argue there are 3,144 separate processes driving prison growth and decline. The locus of control is located within the 3,144 county prosecutor’s offices. Again, it is not our purpose to decide among competing theoretical expositions, but to show how dependent the exposition has been on prevalence rates. Alternatively, scholars could derive more contemporary explanations by examining incidence, but the literature has not gone in that direction.

The Scale of Imprisonment and Mass Incarceration

Even if criminologists cannot make a strong case either theoretically or empirically for a normative scale of imprisonment, explaining mass incarceration is practical. There is an abundance of literature on this question. One of the most comprehensive and critical expositions of this problem was the report produced by the NRC (2014), “The Growth of Incarceration in the United States: Exploring Causes and Consequences.” The report observes that the policy response to rising crime rates post–World War II increased both the probability of imprisonment and the length of stay. Policy changes included harsher sentencing structures including mandatory minimum sentences and career criminal statutes, but half the growth was based on changes in the likelihood of imprisonment given an arrest. According to the NRC, in an increasingly punitive political climate, incarceration became the key solution to crime control. The NRC report also highlights internation and intranation difference in prison prevalence rates (see Figures 2-1, 2-2, 2-3, 2-4, and 2-5 in the NRC report).

There has been a wealth of research on criminal justice processes that may be related to levels of imprisonment (Blumstein & Beck, 1999; Blumstein & Beck, 2005; Liedka, Piehl, & Useem, 2006; Lynch, 2011; Pfaff, 2017; Raphael & Stoll, 2013; Western, 2006; Zimring, 2010). The NRC report highlights the work of Beck and Blumstein (2012) who compared the prison prevalence rates in relation to changes in the imprisonment per unit arrest and changes in the length of stay over the decades 1980-1990, 1990-2000, and 2000-2010. These authors concluded that incarceration rates and time-served were the local drivers of mass incarceration. Pfaff (2017) has challenged these conclusions. He claims almost all the prison growth is attributable to prison admission rates. It is not our intention to arbitrate this dispute. Rather, we note that incidence statistics, by providing more contemporaneous explanations, could provide new insight.

As many scholars and the NRC report point out, admission rates are a key component to understand prison prevalence. But admission rates themselves involve complexities in distinguishing whether the admission is a return while under supervision (Gaes, Luallen, Rhodes, & Edgerton, 2016), and the rate at which offenders are recycling through prison (Rhodes et al., 2014). These factors must be understood to explain not only the underlying processes of mass incarceration but also the range of people affected by this phenomenon. The higher the number of returns, the lower the number of unique people who are affected by the carceral experience (Pfaff, 2017; Rhodes et al., 2014). Distinguishing single from repeat returns is not important for determining prison prevalence, but it is evidence of the extent to which imprisonment affects a narrow or wide range of people. As incidence and cumulative incidence identify unique individuals, it has an important role to play in our understanding of the scale of imprisonment.

Racial Disparities in Prevalence

In addition to scholarship explaining the scale of imprisonment, a rich literature highlights differences in imprisonment rates by demographic strata, a great deal of which is devoted to explaining racial differences (Alexander, 2010; Blalock, 1967; Cole, 1999; Dollar, 2014; Horowitz, 1985; Liska, 1992; Stolzenberg, D’Allesio, & Eitle, 2004; Tonry, 1995; Western, 2006). The level of racial disparity has been documented in the NRC’s (2014) report using prison prevalence and rates of admission. During the period 1925 to 1980, Black prison admission rates varied between 75 and 125 per 100,000, while White prison admission rates varied between about 25 and 45 admissions per 100,000, roughly a three to one ratio of Black to White incarceration rates (Beck & Blumstein, 2012; Langan, 1991). In 1980, prison admission and imprisonment rates rose dramatically for both Black and White offenders. By 2000, the ratio of Black to White prison prevalence rates was 6.3 to 1, and as recently as 2010, the ratio was 4.6 to 1. In the next section, we show how incidence and cumulative incidence, disaggregated by racial and other important social strata, can augment our understanding of the scale of imprisonment.

Incidence and Cumulative Incidence as Important Supplemental Social Indicators

In contrast to prevalence statistics, or rates of admissions, incidence and cumulative incidence statistics are cohort specific. To provide intuition, see Table 1. This table shows cohorts from 1982 to 1987 along the rows, and the columns represents a period over which we can observe prison admissions—the observation window. Members of the 1982 cohort turn 18 in the year 2000, 19 in the year 2001, and so on. Successive cohorts turn age 18 and older in later years and are represented in successive rows of Table 1. The cells of the table represent the age of a person when admitted to prison for the first time in a specific year of the observation window. Some small proportion of each of these cohorts, ranging from birth years 1982 to 1987, will enter prison for the first time during one of the years of our observation window. For each cell, the reader should mentally replace the age value with a proportion representing the percentage of the age cohort entering prison at that specific age.

Table 1.

Age Cohorts 1982 to 1987 and the Age of Admission to Prison During the Observation Window 2000 to 2015.

Age cohort	2000	2001	2002	2003	2004	2005	2006	2007	2008	2009	2010	2011	2012	2013	2014	2015
1982	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33
1983		18	19	20	21	22	23	24	25	26	27	28	29	30	31	32
1984			18	19	20	21	22	23	24	25	26	27	28	29	30	31
1985				18	19	20	21	22	23	24	25	26	27	28	29	30
1986					18	19	20	21	22	23	24	25	26	27	28	29
1987						18	19	20	21	22	23	24	25	26	27	28

Analysts might propose hypotheses on whether the observed changes are due to a cohort effect, namely, characteristics of people in the same age group, or characteristics of the period, specifically a change to the social/policy environment that affects everyone simultaneously regardless of age, and therefore regardless of cohort. Changes in sentencing policy are examples of a potential period effect and are emphasized in the NRC (2014) report and other scholarship (Lynch, 2011; Western, 2006). Exposure to lead during early childhood and its attendant effects on crime may be an example of a cohort effect, as lead abatement would reduce this risk for later generations (Aizer & Currie, 2017; Nevin, 2007; Reyes, 2007). However, because of linear dependence (age = period – cohort), methodologists (see especially Fienberg, 2013; Luo, 2013a, 2013b) argue that cohort and period effects are conflated, although there is some disagreement on this point (Held & Riebler, 2013; Obrien, 2013; Yang & Land, 2013). Regardless, cohort patterns in and of themselves are inherently interesting in this criminal justice context, so we avoid getting distracted by distinguishing cohort from period effects. Our focus in this article is to present an easily computed measure of incidence and cumulative incidence useful for thinking about these hypotheticals.

Prior studies have been primarily designed to study trends in cumulative incidence over time. Western and Pettit (2010) compare birth cohorts born just after World War II and as recently as 1975-1979. Their interest was to describe the growth in racial and socioeconomic disparity from the earlier to the later cohorts. Bonczar (2003a) examines birth cohorts both farther back and closer in time using cohort birth years spanning 1901-1983. His paper emphasizes the increasing level of cumulative incidence from 1974 to 2001. Incidence and cumulative incidence provide this unique perspective on social control by estimating the proportion of the cohort sanctioned with prison at a particular age, or over an entire lifetime.

By focusing on incidence, rather than prevalence or admission rates, we address a different set of questions. What is the age at which first-incarceration is most likely to occur? Is the first admission as an adult more likely to occur during late teens and early 20s? What is the shape of this distribution? How fast is the decline in the rate of change of first admission as cohorts age? Are there racial differences? We answer these incidence questions when we present findings. We also examine cumulative incidence, addressing the following question: “Does the cumulative incidence of prison change over cohorts?” If we are observing trends based on cohorts, is there a theoretical explanation?

We show cohort differences graphically, and we introduce permutation tests of the null hypothesis of no systematic cross-cohort changes of incidence over time. The graphics and permutation analyses are extended to stratification of incidence and cumulative incidence by race and gender.

Prior Studies of Incidence and Cumulative Incidence

Despite the importance of estimating incidence and cumulative incidence of incarceration, until recently there has been no single source to calculate these statistics with administrative data. For that reason, criminologists have had to construct these statistics primarily from survey data. Pettit, Western, and Sykes (2009) estimate cumulative incidence up to age 34. As we show later in this article, by that age, the incidence rate is approaching zero. Western and Pettit (2010) summarize this analysis showing the influence of gender, race, and education. They constructed their analysis to examine cumulative incidence for specific cohorts: the cohort born 1945-1949 for which cumulative incidence was measured in 1979; the cohort born 1955-1959 for which cumulative incidence was measured in 1989; the cohort born 1965-1969 for which cumulative incidence was measured in 1999; and the cohort born 1975-1979 for which cumulative incidence was measured in 2009. The report by Pettit et al. (2009, Table 38) shows that the cumulative risks of imprisonment by age 30 to 34 for the 1945-1949 cohort were 1.35% for non-Hispanic Whites, 2.83% for Hispanics, and 10.35% for non-Hispanic Blacks. For the 1975-1979 cohort, the cumulative incidence to ages 30 to 34 were 5.35% for non-Hispanic Whites, 12.16% for Hispanics, and 26.84% for non-Hispanic Blacks. This shows a substantial increase in cumulative incidence over time. Unfortunately, they did not publish aggregate cumulative incidence estimates, so we cannot compare their overall estimates with the ones reported in this article. Nonetheless, their work shows how disaggregating cumulative incidence by important demographic and social strata can provide additional insight.

The methods and data used by Pettit et al. (2009) are similar to the survey-based approach of Bonczar (2003a), and as we noted at the beginning, we refer to these as the SPI-estimators, as the primary source of data for first imprisonment is based on the survey of prison inmates conducted by BJS about every 6 or 7 years since 1974.¹

Analysts at the BJS have published cumulative incidence estimates (Bonczar, 2003a; Bonczar & Beck, 1997a). We focus on the more recent report as both reports appear to use the same methodology. Readers interested in the details of the BJS SPI-estimator methodology should consult Bonczar (2003b).² Bonczar (2003a) estimates that as of 2001, nearly 17% of Black male Americans then alive had been incarcerated at some time during their lives. Moreover, between 1974 and 2001, the estimated percentage had nearly doubled. Bonczar (2003a) projects that “if rates of incarceration remain unchanged, 6.6% of all persons born in the United States in 2001 will go to state or federal prison during their lifetimes, up from 5.2% in 1991, and 1.9% in 1974” (p. 1).

There are limitations to the SPI-estimator. We highlight concerns here, but the methodology paper provides a more comprehensive critique (Rhodes et al., 2017). One concern is that the SPI-estimator assumes stability across cohorts. Given observed cohort changes over time, the stability assumption seems unreasonable and inconsistent with study objectives.³ Another concern is that the survey has nonsampling and sampling errors. Although sampling errors could be estimated,⁴ they are not reported, and nonsampling errors are likely high.⁵ We mention these problems because our proposed approach requires no stability assumptions for short-term projections and modest stability assumptions for long-term projections. Because we use administrative data of high quality, sampling error is not an issue although long-term projections are model-based and subject to error. This new approach is described next.

Estimation of Incidence and Cumulative Incidence Based Exclusively on Administrative Records

The BJS NCRP is the only current, readily available source of administrative records that can be used to calculate prison incidence and cumulative incidence. The NCRP reports the beginning and end of all prison terms for offenders who served any time during an observation window that spans multiple years.⁶ The work reported here is based on data collected through December 2015. Offenders in prison as of December 2015 have missing release dates as their release had yet to occur at the time the data were last updated. Offender records are linked with a unique identifier, so first-time admissions are easily distinguished from repeat admissions.

The NCRP data allow us to examine more recent cohorts than those studied by Bonczar or by Petit, Western, and Sykes. Our data span age cohorts born between 1982 and 1987. Because the NCRP is updated yearly, each year analysts can extend this span by one year—to the 1988 cohort, the 1989 cohort, and so on.

We provide intuition on how the NCRP-term records are used to estimate prison incidence and cumulative incidence, and a complete explanation is provided in Rhodes et al. (2017). We limit our calculations to people 18 years or older. This allows us to calculate first-time admission without having to account for juvenile records and people who may have been admitted to an adult prison under the age of 18. While this constraint limits our calculations to first-time admission beginning with age 18, it removes a degree of uncertainty regarding juvenile admissions. For most of the data, we have an observation window starting in 2000 and ending in 2015. However, this window is extended every year as an additional year’s worth of data are added to the administrative database. There is no national collection of jail records. Anyone who is admitted to jail and does not receive a prison term is not included in these incidence calculations.

The denominator for these social statistics is used to express incidence as a proportion of the at-risk pool for a specific cohort. Analysts computing SPI-estimators have used U.S. birth cohorts gathered from U.S. Census data. The algorithm we have developed uses age cohorts also coming from U.S. Census sources. Everyone born during a specific year and currently living in a specific jurisdiction regardless of their citizenship is a member of the age cohort. We view this choice as a reasonable one, as the United States has a large pool of noncitizens who can commit crime and are at risk to be imprisoned. Age cohorts may grow over time until death rates ultimately dominate net immigration rates, so the at-risk pool for the denominator applies to an ever-changing population. But this is also true of a U.S. born birth cohort because of changes in rates of mortality, and one of the challenges when using birth cohorts is to estimate prisoner death rates, which are not identified in Census mortality tables.⁷

Our measures of the numerator for incidence and cumulative incidence rely on counting first admissions during our observation window 2000 to 2015 and imputing first admission beyond the observation window. Table 1 helps to understand the algorithm. Start with the 1982 age cohort, whose members turn 18 in 2000 at the start of our observation window. We count the number of 1982 age cohort members who enter prison for the first time at the age of 18, at the age of 19, and so on. This allows us to compute rather than estimate incidence and cumulative incidence for this age cohort through the age of 33.⁸ We limit our calculation beginning with the 1982 age cohort because we cannot observe first time admission for people older than 18 in 2000. For example, we cannot observe admissions during 1999 for the 1981 age cohort.

For age cohorts after 1982, the counting exercise is no more difficult, but we cannot track incidence through the age of 33. For the 1983 age cohort, we track incidence through the age of 32. For the 1984 age cohort, we track incidence through the age of 31, and so on. Because we cannot track cohorts beyond the observation window, we employ a method of imputation. Our method of imputation relies on older age cohorts to project or impute incidence for younger cohorts beyond the age of 33, attempting thereby to answer the following question: What is the likely lifetime cumulative incidence for age cohorts born in 1982 and later? To provide intuition, consider the 2016 incidence for the age cohort born in 1982. We assume that the incidence (measured as a population proportion rather than a raw number) for the 1982 age cohort would be about the same in 2016 as was the incidence for the 1981 age cohort in 2015. To restate this another way, the person in the 1982 age cohort who has never been incarcerated by age 33 (the last year of our observation window) will have the same rate of first time admission at age 34 as the person born in 1981. Now extend this logic going forward for the 1982 age cohort. Someone who has not been incarcerated by age 35 will be imprisoned at the same rate as the first imprisonment at age 35 for the 1980 age cohort. This is the logic of imputation that we use for the 1982 through 1987 cohorts. Subtleties and calculation issues are explained in Rhodes et al. (2017).

While this methodology only provides incidence measures for age cohorts born in 1982 and later, these more recent cohorts are especially useful because the prison experiences of these people are relatively recent, ongoing, and subject to public policy intervention. The projections for these cohorts are most credible because incidence for early ages is observable, estimated incidence for middle ages is imputed from closely matched older cohorts, and estimated incidence for older ages—while based on distantly matched older cohorts—is relatively low and has little ultimate impact on the total calculation.⁹

Results of the NCRP-Based Estimation of Incidence and Cumulative Incidence

In this section, we show prison incidence and cumulative incidence estimates for the 1982 through 1987 age cohorts. We focus on one state, then show national estimates based on 43 states, and finally, we show strata-specific estimates based on race and gender.

Georgia as an Illustration

Calculations for Georgia illustrate the approach. Georgia has provided NCRP-term records dating back to 1971. Those older records are useful for diagnostics which are covered in the methodology paper (Rhodes et al., 2017). For this article, we restrict the Georgia data to the 2000-2015 window.

Figure 1 has four panels. We discuss each panel, from left to right, for the two top panels, then left to right, for the bottom two panels. The first panel is the tabulated cumulative incidence for the 1982 age cohort through age 33. The curve starts on the graph’s x axis at age 18 because we only consider prison terms as of age 18. The curve shows that, for the 1982 age cohort, (a) cumulative incidence increases at an increasing rate through the early 20s and thereafter increases at a decreasing rate,¹⁰ (b) more than five of every 100 Georgians in the 1982 age cohort have experienced prison by the age of 33, and (c) cumulative incidence is increasing past age 33 at a lower rate.

Figure 1.

Illustrations from Georgia: Incidence and cumulative incidence across recent age cohorts.

The second panel shows the cumulative incidence for the age cohorts 1982 through 1987. For improved resolution, we do not show a legend, but the cohorts are easily distinguished. The longest curve pertains to—and replicates—the curve for the 1982 age cohort that is displayed in the first panel. The shortest curve pertains to the 1987 age cohort. Cumulative incidence has remained stable across these six contiguous age cohorts. The finding is not surprising, but it is important. If there were no stability, there would be little justification for imputation. However, there appears to have been a slight decrease in cumulative incidence. We provide statistical tests later in this paperto infer whether there is a change in incidence over time.

The lower left panel shows the incidence for the 1982 age cohort. Incidence peaks at age 21 and declines thereafter at least until the age of 33. The lower right panel shows incidence instead of cumulative incidence over the six contiguous age cohorts. The impression is that incidence is similar for the six age cohorts, but there is undeniable year-by-year fluctuation.¹¹

Using imputation, Figure 2 projects the curves shown in Figure 1 forward to an age approaching 70. As before, the panels are explained from left to right for the two top panels, then left to right for the bottom two panels.

Figure 2.

Illustrations from Georgia: Incidence and cumulative incidence across recent age cohorts with projections.

The first panel shows projections for the 1982 age cohort. Through age 33, Panel 1 in Figure 2 is the same as Panel 1 in Figure 1. For age 34, cumulative incidence is based on the average incidence at age 34 of earlier age cohorts. Similar projections apply to age 35, 36, and so on. Because of this imputation, we call this the 1982 synthetic cohort. The second panel shows the projections for the 1982 through 1987 age cohorts. To distinguish the curves, we have made the curves progressively thinner for younger cohorts (the 1982 age cohort has the thickest curve; the 1987 age cohort has the thinnest curve). In fact, the curves are practically indistinguishable visually.

The third panel shows incidence for the 1982 age cohort. This curve is identical to its counterpart in Panel 1 of Figure 1 until the age of 33; after that point, incidence is based on the average incidence for older age cohorts. Notice that there is a slight bump in incidence at age 34. Because the imputations look backward toward older cohorts, and because incidence seems to be trending downward according to Panel 2 of Figure 1, the bump implies that at age 34, estimated incidence is slightly biased upward. This bump will appear in other figures that aggregate across states. The final panel shows the observed and projected incidence for the 1982 through 1987 age cohorts. The curves differ slightly until age 33. Thereafter, they are necessarily the same because they are based on the same imputations. For example, we cannot observe age 34 incidence for any of the cohorts born in 1982 and later. For them, we impute age 34 incidence based on cohorts prior to 1982.

Based on the first panel of Figure 2, we project that seven to eight of every 100 Georgians born in 1982 will spend some time in prison during their lifetimes. The risk of first-time incarceration is greatest during an offender’s early 20s and decreases progressively thereafter. That is, about seven to eight of 100 Georgians in the 1982 age cohort will spend some time in prison provided immigrants into Georgia and emigrants leaving Georgia have the same risk of prison as do native born Georgians. This may be false, so the seven to eight of 100 estimate is approximate. Also, the approximation is based on observed prison admissions until the age of 33, but past the age of 33, the approximation presumes that the incidence for older cohorts is the same as the incidence for the 1982 age cohort. Because the approximations are based on the nearest neighbor older cohorts, the approximations are likely reflective of reality, but of course there is no way to place a confidence interval on that presumption. Finally, we are unable to precisely identify first-time admissions for offenders who were admitted and released from other states, moved to Georgia, and were then re-admitted to prison in Georgia. Our inability to identify first-time admissions when an offender is an immigrant from another state means that the seven to eight per 100 estimate is probably biased upward. Rhodes et al. (2017) argue that the bias is small because empirical evidence is that cross-state recidivism is small.

National Estimates of Incidence and Cumulative Incidence for the 1982 to 1987 Age Cohorts

Figure 3 has the same structure as Figures 1 and 2, but here, the synthetic cohort curves are based on data from all the states that have contributed admission data to the NCRP since at least 2012. For details on how we accommodated states that did not have data throughout the 2000 to 2015 observation window, see Rhodes et al. (2017). Eighteen of the 50 states have data through most of the observation window, and this subset is composed of many of the larger correctional systems such as California and New York State. About seven of 100 of individuals who are members of the 1982-1987 age cohorts will serve some prison time over their lifetimes. The highest rate of incidence is still about 21 years of age. Visually, the cumulative incidence seems very similar for each of the age cohorts 1982 through 1987. A permutation test confirms this. Because data are aggregated, cross-state immigration and emigration play a smaller role than they do in Georgia, although population movement into and out of the United States remains potentially important.

Figure 3.

National estimates of incidence and cumulative incidence—1982 to 1987 age cohorts.

Racial and Gender Breakdowns in Incidence and Cumulative Incidence

We can stratify the NCRP-estimates by meaningful population groups. Figure 4 shows results for White males and Figure 5 shows results for Black males. These estimates are based on the large subset of the states that have data through most of the 2000 to 2015 observation window.

Figure 4.

Illustrations from an 18-state composite: Incidence and cumulative incidence across recent cohorts (White males).

Figure 5.

Illustrations from an 18-state composite: Incidence and cumulative incidence across recent cohorts with imputations (Black males).

Using the 1982 age cohort as a point of comparison, Figure 5 shows that, across these 18 states, more than four of every 100 White males will be incarcerated by the age of 33. In comparison, nearly 20 of every 100 Black males will be incarcerated by the age of 33. For both groups, the highest risk for first-time incarceration is somewhat near the age of 20, but that risk is lower for White males than for Black males. Also, compared with the peak risk at about age 20 or 21, the risk is considerably higher between the ages of 18 and 19 for Black males than for White males. These estimates confirm similar estimates by Bonczar (2003a) and Western and Petit (2010).

However, new information comes from inspecting Panel 2 of Figures 4 and 5. Comparing the 1982-1987 cohorts, there may be a trend toward lower cumulative incidence for White men, but if so, the pattern is muddled by some reversals of the lines, some apparent ties, and in general no large differences. In contrast, for Black men, the patterns seem distinct. Older cohorts always have higher cumulative incidence (as of age 28) than do all younger cohorts. Based on a permutation test, the probability that this pattern would emerge by chance is 1/720. It appears that the burden of prison has been declining especially for Black males, although cumulative incidence for Black males remains much higher than the cumulative incidence for White males.

We turn next to differences by gender. Many fewer women than men are held in state prisons as reflected by their prevalence rates. Figure 6 shows prison incidence and cumulative incidence for women. Contrasts between Figure 6 and earlier figures are apparent. At least through the age of 33, incidence and cumulative incidence are much lower for women than for men. The period of highest risk is later for women than for men, and we see the period of heightened risk as being dispersed, suggesting that first-time admissions are more delayed for women. In addition, we see no evidence that incidence has been increasing or decreasing across the 1982-1987 age cohorts for women.

Figure 6.

Illustrations from an 18-state composite: Incidence and cumulative incidence across recent cohorts (women).

Discussion

Aggregate indicators of imprisonment are important social statistics. Prison prevalence has been the dominant social indicator used to document the mass incarceration movement and to compare jurisdictions on their scale of imprisonment. Prevalence has also been used to document disparities in racial groups and other meaningful strata. To disentangle the root causes of prison growth or decline, analysts disaggregate the components of the prison stock into rates of admission and length of stay by subpopulations. Despite the voluminous literature on mass incarceration, there is still a question about the most important drivers. In this article, we make a case for supplementing our understanding of the scale of imprisonment with measures of incidence and cumulative incidence. These latter statistics can be used to represent changes in age cohorts over time. Incidence and cumulative incidence focus our attention on possible social, political, and cultural changes affecting these cohorts. While we have not directly tested any theory, we have attempted to provide a framework for theory testing. Incidence and cumulative incidence are cohort specific. Therefore, they can be used to evaluate potential cohort changes affecting mass incarceration and the scale of imprisonment.

Our findings show that in the aggregate, the highest rate of incidence occurs at age 21, and by age 32, the rates are a fraction of what they are at age 18. The reason this is important is because we can count without inference actual incidence up to age 28 to 33 for the 1982-1987 cohorts, and when we impute incidence beyond those years, imputation is a small fraction of the cumulative incidence. Nationally, about seven in 100 people in the 1982 to 1987 age cohorts will be imprisoned in their lifetime. For women, the cumulative incidence is much lower. By age 33, the cumulative incidence for women is about 1%; while for Black men, it is almost 20%; and for White men, about 4%. These estimates are derived from recent age cohorts, cover the period of peak incarceration rates, and represent data on age cohorts more contemporaneous than prior publications by Western and Petit (2010), Petit et al. (2009), and Bonczar (2003a).

Our data also show that the rate of cumulative incidence for Black males is declining while the rate is stable for White males. However, these rates are still widely disparate. The cumulative incidence for the most recent cohort—the 1987 cohort—shows that by age 28, the ratio for Blacks to Whites is still about 4.5 to 1. We cannot make precise racial comparisons with prior estimates by Bonczar (2003a) and Petit, Western, and Sykes (2009). Although we all make separate estimates for men and women, the prior studies used Hispanic or Latino as a racial/ethnic category. Nevertheless, Petit et al. (Table 37) show that the cumulative incidence through ages 30 to 34 for all racial and ethnic subgroups was much larger in the most recent cohort than in the earliest birth cohort. Bonczar (2003a) shows a similar pattern of growth in cumulative incidence for all racial/ethnic groups. As our cohorts are more recent, we may be at an inflection point or beyond in the growth curve representing the rate of change of cumulative incidence for Black residents. It is worth emphasizing that this finding illustrates a major advantage of using the NCRP administrative data. As NCRP is updated every year with new calendar year records, the most recent incidence and cumulative incidence trends can be evaluated with each successive addition.

We have drawn comparisons between this new NCRP-estimator method and the SPI-estimator developed by other researchers. The SPI-estimator agrees broadly with our estimate of cumulative incidence of prison over a lifetime. As reported earlier, Bonczar projected a cumulative incidence of 6.6% of all persons born in the United States in 2001. This assumes a steady state process from 1983 to 2001 which appears to be unlikely. Bonczar’s estimate is similar to the estimate provided in Figure 6; however, there are differences in the baseline populations. Bonczar’s targeted baseline population is people born in the United States during 2001; however, he must use data for birth cohorts born in earlier periods and assume the rates are stable. Our targeted population is people born between 1982 and 1987 and living in the United States. Because net immigration only accounts for about 5% of the population, and because 1987 is not far removed from 2001, the differences in the targeted populations should not be large. Bonczar’s projections are based on incarceration patterns using five inmate surveys: 1974, 1979, 1986, 1991, and 2001. Our projections are based on incarceration patterns found in administrative data between 2000 and 2015. Bonczar’s estimates include incarceration in federal facilities, but we have argued (Rhodes et al., 2017) that a large plurality of the federally incarcerated population is composed of noncitizens who are here legally or illegally (and hence would not be part of Bonczar’s estimates), or for repeat offenders (and hence would not be first-time offenders). Nevertheless, our estimates currently do not account for federal offenders. Our estimates also exclude seven states for which offender records could not be linked,¹² but these compose a small fraction of the imprisoned population. Bonczar’s estimates conceptually include all states.

Both estimators have their flaws. However, despite the differences in methodology, the survey-based estimates and the NCRP-based estimates are similar, and we emphasize the two methods are complementary in the sense they give us two bites at the apple of estimating incidence and cumulative incidence. This is analogous to getting two different estimates of crime using the National Crime Victimization Survey and the administrative data on crime reported to the FBI’s Uniform Crime Reporting program. These programs are designed to give somewhat different insights into crime levels and crime trends. While the SPI-estimator cannot be estimated between administrations of the inmate survey, the NCRP-based estimates can be updated every year providing a useful platform for annual updates of trends in incidence. This allows scholars to more accurately track trends in incidence and cumulative incidence to identify the historical and cohort-specific causes of mass incarceration. Furthermore, as illustrated with Georgia, the NCRP-based estimator can provide state-specific estimates, and with one of the primary debates in the field centered on whether mass incarceration is the product of national shifts or more regional and local changes, this geographic specificity is an essential tool. Finally, the NCRP-estimator is not subject to the same sampling and nonsampling errors, making it an important alternative to SPI-estimators. While we acknowledge limitations of the NCRP-estimator, many of these limitations will decrease with ongoing improvements to the NCRP as state data are added, and as federal data are incorporated into the estimate.

Footnotes

Authors’ Note

Points of view in this document are those of the authors and do not represent the official position of the U.S. Department of Justice.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by Grant No. 2015-R2-CX-K135 awarded to Abt Associates by the Bureau of Justice Statistics, Office of Justice Programs, U.S. Department of Justice.

Notes

Author Biographies

William Rhodes is a public policy consultant, and an economist, specializing in program evaluation and quantitative analysis. He is co-principal investigator for the Bureau of Justice Statistics’s (BJS) National Corrections Reporting Program (NCRP).

Gerald G. Gaes is a consultant and visiting faculty member in the College of Criminology and Criminal Justice, Florida State University. He is co-principal investigator for BJS’s NCRP and focuses on corrections and punishment research.

Jeremy Luallen is a part-time faculty at Tufts University. He is an applied econometrician with experience in program evaluation and quantitative analysis and currently working as a data scientist at Athena Health.

Ryan Kling is a senior associate at Abt Associates. He is a programmer and an applied econometrician with research interests in post-incarceration employment and using large data sets to answer policy questions.

Tom Rich is a senior associate at Abt Associates. He is project director for BJS’s NCRP. He has previously directed projects involving school safety, law enforcement, and emergency preparedness and disaster response.

Christopher Cutler is a senior analyst at Abt Associates. He is a programmer with research interests in incarcerated populations, justice systems, and the handling of immigration violations.

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