Social Change and Race-Specific Homicide Trajectories: An Age-Period-Cohort Analysis

Yunmei Lu; Liying Luo; Mateus Rennó Santos

doi:10.1177/00224278221129886

. Author manuscript; available in PMC: 2024 Sep 1.

Published in final edited form as: J Res Crime Delinq. 2022 Oct 11;61(2):224–267. doi: 10.1177/00224278221129886

Social Change and Race-Specific Homicide Trajectories: An Age-Period-Cohort Analysis

Yunmei Lu ¹, Liying Luo ², Mateus Rennó Santos ³

PMCID: PMC10857748 NIHMSID: NIHMS1849302 PMID: 38344105

Abstract

Objectives:

Social change and the aging process are racially bifurcated in the United States, where Black and White populations have long lived in divergent social worlds. This study examines the cohort patterns and life-course trajectories of Black and White homicide involvement over the past four decades.

Data and Methods:

The study uses data from the Supplemental Homicide Reports and Age-Period-Cohort-Interaction (APC-I) models to analyze race-specific trends of (alleged) homicide offending and victimization between 1976 and 2018 in the U.S.

Results:

Results reveal similar patterns in the age, period, and cohort effects on Black and White homicide involvement. However, while the shapes of these trajectories are comparable, the volatility in cohort effects on homicide is much more accentuated for Black cohorts than White cohorts. We also find racial differences for cohorts born after 1990, with a downward cohort pattern among the White group but a flat cohort trend among the Black group.

Conclusions:

Findings suggest that Black cohorts’ homicide involvement is more susceptible than White cohorts’ to the influence of external social changes (e.g., economic downturn, the crack epidemic). In addition, an increasing racial gap between Black and White populations is found among the recent birth cohorts. Possible mechanisms are discussed.

Keywords: Race, Social Change, Homicide, Life Course, Age-period-cohort Analysis

Introduction

In his seminal work, Ryder (1965) argued that cohort is a key concept for understanding social change. Since then, numerous studies have investigated cohort effects on crime trends (e.g. O’Brien, 1989; O’Brien & Stockaftablerd, 2009; Smith, 1986; Steffensmeier et al., 1992; Vogel et al., 2019). However, studies on the subject mostly focus on average differences in crime involvement between birth cohorts and often overlook potential differences in the life-course trajectories of different cohorts (Lu and Luo 2020). Moreover, social change and the aging process can be racially bifurcated, particularly in a context such as the United States where Black and White populations have long been living in “divergent social worlds” (Peterson, Krivo, and Hagan 2010).

The goal of the current study is to advance scholarly literature by examining the potentially distinct cohort patterns in the homicide victimization and offending of Black and White populations in the U.S. A deep history of racial inequalities has effectively bifurcated the life course experiences of individuals across racial delineations, with implications across social domains such as the economy, politics, education, religion, family, and the criminal justice system. This study compares the age, period, and cohort effects on the homicide rates of each racial group, investigating the race-specific nature of macro-level homicide trends in the U.S in light of the country’s history and changing social dynamics from 1976 to 2018. Our age-period-cohort analysis of homicide victimization and offending reveals how racial differences may manifest in different dimensions of time. In particular, using the Age-Period-Cohort Interaction (APC-I) framework developed by Luo and Hodges (2022), we examine two key aspects of cohort effects on the homicide involvement of Black and White cohorts, namely the inter-cohort average difference, which refers to the average differences in crime involvement between cohorts; and the intra-cohort life-course dynamics, which designate the variation in cohort effects over a cohort’s life course. Although both inter- and intra-cohort differences are important cohort-related patterns, estimates of the latter are uncommon and understudied in prior literature.

Our research contributes to the criminological literature in two main ways. First, it connects the literature on macro-level homicide trends with the life course criminological studies on age-crime trajectories. Time operates at the individual- and macro-levels simultaneously, since age is both a point in an individual’s life and a marker of a specific location in history (Elder, Johnson, and Crosnoe 2003). Individuals’ life-course trajectories are embedded in and shaped by their sociohistorical context. The U.S. has gone through great social change over the past several decades, such as the Civil Rights Movement, the Crack Cocaine Epidemic, and periodic economic downturns. Along with these changes, extant research has documented numerous shifts in age-graded norms including delays in ages for entering the labor market, marriage, and parenthood (Shanahan 2000; Twenge and Park 2019). These changing contexts can be key determinants of the pathways of different birth cohorts, with implications to their criminal offending and victimization (Elder and George 2016; Neil and Sampson 2021; Shanahan 2000).

Second, our study estimates and discusses disparities in homicide trajectories between racial groups. Few criminological studies have investigated the race-specific life course patterns, especially at the aggregate cohort level. Slavery in the U.S. was only outlawed in 1865, and it was followed by more than a century of segregationist and discriminatory policies. These policies are an expression of racial differences which are embedded in the social structure of the U.S., impacting the distribution of opportunities and resources across social institutions. Black individuals, in particular, are more likely to have had negative experiences at younger ages related to discrimination across social domains, including neighborhood and school segregation, labor markets delineated by race, mass incarceration, and tough-on-crime policies that affect minorities disproportionately. Given the racialized nature of social context across U.S. history, it is unlikely that Black and White cohorts have had same age-crime trajectories in homicide involvement since the 1970s. However, the actual differences between these trajectories remain underexplored by research.

Using data of Supplemental Homicide Reports (SHR) from 1976 to 2018, we apply age-period-cohort-interaction (APC-I) models to analyze and compare inter-cohort average differences and the intra-cohort life-course trajectories of alleged homicide offending and victimization of Black and White populations. Our findings highlight the differential impacts of social change on the life-course dynamics in crime involvement among racial groups. We explore possible explanations for these differences in light of major social changes that are occurring in the U.S. since the 1970s, and of the racial disparities embedded within these changes over history.

Background

Age-Crime Relationship and the Life-Course Perspective

The age-crime invariance thesis proposed by Hirschi and Gottfredson (1990; 1983) holds that crime involvement increases during adolescence, reaches the peak at the late teens and early twenties, and then declines for the rest of the life course. This age-crime relationship, they argued, is invariant across social conditions, population subgroups, and historical periods. The age-crime invariance thesis is central to many theories on adolescent risky behaviors and the adolescent-adulthood transition. An invariant age-crime schedule implies that a high level of adolescent crime is a natural part of the human development process and raises doubts about the importance of social context as a contributing factor to age effects (Steffensmeier, Zhong, and Lu 2017). Many scholars, however, have questioned the invariance thesis with evidence of divergence in age-crime patterns and with context-specific explanations for the age-crime relationship (O’Brien and Stockard 2009; Steffensmeier, Lu, and Kumar 2019; Steffensmeier, Lu, and Na 2020; Steffensmeier et al. 2017).

At the individual level, longitudinal studies have revealed varying age-crime profiles (Ezell and Cohen 2012; Piquero, Farrington, and Blumstein 2007). While many offenders’ crime involvements peak at late teens, others have late-onset and relatively flat age trajectories. At the aggregate level, evidence suggests that the age-crime patterns vary across offense types (Steffensmeier et al. 1989), time (Greenberg 1985; Matthews and Minton 2018), and countries (Hiraiwa-Hasegawa 2005; Steffensmeier et al. 2019, 2020, 2017). Together, these studies highlight the importance of context in shaping life-course trajectories in crime involvement.

While recognizing the relevance of the biological factors, sociologists have proposed several social explanations for the age effect on crime. Studies using individual-level data found that much of the variance in the age-crime trajectories can be explained by social variables, such as employment, marriage, strain, and peer influence (Benson 2013; Sweeten, Piquero, and Steinberg 2013; Warr 1993). The transition to adulthood often accompanies education and employment attainments, which tend to increase financial accomplishment while reducing status deprivation and strain, thus decreasing the risk of crime involvement (Agnew 1992; Cohen 1955; Merton 1938). In addition, individuals’ bonds to social institutions strengthen with the transition into adulthood. For instance, obtaining full-time jobs, joining the military, forming serious romantic relationships, and having children are all potential turning points to desistance which are tied to age (Laub and Sampson 2003).

These theoretical accounts also imply that age effects on crime are likely to vary across time and population subgroups. For instance, status frustration and stress among youth, including economic or monetary pressures to purchase consumer goods, are likely more prominent in modern industrialized societies than in traditional societies (Agnew 1992; Clinard and Abbott 1973). Also, the opportunities for establishing stronger connections with normative institutions, such as school and employment, are greater in White communities than in Black communities(Peterson et al. 2010). Moreover, social and historical changes may affect different racial groups’ life course experiences in distinct ways, particularly when such groups are so clearly delineated as racial minorities in the U.S. (Pettit and Western 2004; Piquero, MacDonald, and Parker 2002).

Cohort, Social Change, and the Life Course

In the 1980s, Easterlin (1987) proposed the cohort size thesis, positing that large birth cohorts have higher risks of crime involvement as they are more likely to face economic disadvantages and weakened social control. Since then, several studies have found a significant association between cohort size and crime (Menard and Elliott 1990; O’Brien 1989; O’Brien, Stockard, and Isaacson 1999; Savolainen 2000; Smith 1986; Vogel et al. 2019), though others have found only negligible cohort size effects (Steffensmeier, Streifel, and Harer 1987; Steffensmeier et al. 1992). Some studies also have found varying cohort size effects across offense types (O’Brien 1989) and across racial groups (Vogel et al. 2019). However, important conceptual issues of cohort effects are still a subject of debate. We focus on two issues: (1) the difference between cohort effects and cohort-specific mechanisms; and (2) the difference between inter-cohort variation and intra-cohort life-course dynamics.

First, average cohort effects—that is, observed differences in the outcome between cohorts—may occur due to unique cohort characteristics and/or due to cohort-specific early-life experiences which manifest through multiple socio-economic mechanisms. Many prior studies have focused on testing one or two cohort-specific mechanisms, such as relative cohort size or the percent of nonmarital births in a cohort (see, e.g., O’Brien 1989; O’Brien et al. 1999). While Easterlin’s (1987) cohort size thesis is an important theoretical account for APC studies in criminology, it represents only one of many possible mechanisms through which cohorts differ from each other. For instance, the presence of cohort differences between baby boomers and Generation X may be related to the population sizes of these cohorts, but such differences may also be attributed to other cohort-specific experiences such as the military draft to the Vietnam War or the economic recession in the 1980s (Lu and Luo 2020). As a result, the studies that focus on only one or two cohort characteristics may not provide a full assessment of cohort effects on crime involvement.

Second, the handful of studies that have estimated cohort effects (see, e.g., Greenberg and Larkin 1985; Kim, Bushway, and Tsao 2016; Maxim 1985; O’Brien 2019; O’Brien and Stockard 2009) were framed under a traditional APC paradigm. The traditional APC paradigm typically involves two critical yet rarely tested underlying assumptions: (1) cohort effects can arise independently from the effects of social change (indexed by period) and individuals’ aging process; and (2) cohort effects are stable across the life course (Lu and Luo 2020; Luo and Hodges 2022; Neil and Sampson 2021). These assumptions depart from Ryder’s (1965: 843–861) seminal conceptualization that cohort effect should be age-time-specific. In other words, cohort effects can be dependent on specific ages and periods. In response, Luo and Hodges (2022; see also Lu and Luo 2020) proposed the APC-I paradigm which empirically defines a cohort effect as the degree to which age and period effects are moderated by one another. Within this new paradigm, there are two types of cohort effects that can be estimated and tested in an APC analysis. The first type, the inter-cohort average deviation, has been the intended focus of traditional APC models and estimates the overall or average differences between cohorts. In addition, the APC-I estimates a second type of cohort-related variation, called intra-cohort life-course dynamics, which quantifies deviations in cohort effects over each cohort’s life course. That is, the APC-I model relaxes the assumption of stable cohort effects and allow researchers to empirically examine the intra-cohort life-course dynamics. Under this framework, cohort effects can be either “permanent” (i.e., a cohort’s higher or lower crime involvement remains higher or lower than other cohorts throughout its life course) or “contingent” (i.e., a cohort’s risk of crime involvement is only tied to a specific age during a specific period).

The APC-I approach is particularly valuable and suitable for studying how Black and White cohorts’ life-course trajectories have been differentially shaped by social and historical context. Each birth cohort grows up in a unique historical location that may lead to unique trajectories which are shaped by the social norms and institutions of a period (Elder and George 2016; Shanahan 2000). Moreover, the between-cohort differences and the intra-cohort life-course dynamics are likely interrelated. For instance, if a cohort’s age at onset of crime is delayed, this cohort’s overall crime involvement could be relatively low even if this cohort commits more crimes later in their life course. While numerous studies have examined the age-crime trajectories of individuals, few has attended to the potentially different crime trajectories among birth cohorts.

Empirically, critical events in the past fifty years have affected life-course homicide trajectories of cohorts in the U.S. First, the racial turmoil in inner cities in the 1970s resulted in deep social transformations and contributed to an increase in violent crime (Wilson 1987). Second, the level of violent crime, particularly youth homicide, increased dramatically in the late 1980s and early 1990s during the crack cocaine epidemic. These historical events may be treated exclusively as period effects by the traditional APC framework, but they may have differential impacts on different age groups and result in cohort effects. Cohorts who were teenagers and young adults during this time, particularly those most directly exposed to the inner-city violence of the period, likely had their homicide offending and victimization trajectories impacted by this experience relative to other cohorts.

Social change in the past decades has also contributed to shifts in age-graded expectations and lifestyles. First, the transition to full-time employment and marriage occurred much later in life in the 2000s than in the 1960s, partially because of the expansion of higher education, the shift from an industrial to a service-based economy, and the growth in cohabitation (Bumpass and Sweet 1989; Piquero et al. 2002; Shanahan 2000). A recent study by Twenge and Park (2017) found that American adolescents are currently less likely than earlier generations to engage in adult activities such as dating, drinking, going out without parents, and driving. With the changing age-graded expectations and youth lifestyles, cohorts have different socialization experiences and are exposed to different levels of social control, which are potential mechanisms contributing to the decline in youth violence in recent decades (Baumer, Cundiff, and Luo 2020).

Race, Social Change, and the Life Course

The social change and institutional context that affect different birth cohorts’ life-course trajectories are also embedded in the issues of race and social inequality in the U.S. (Ousey 1999; Piquero et al. 2002). The history of racial segregation and institutional segmentation has ensured that Black and White populations continue to live in two separate and unequal societies: they reside in segregated neighborhoods, attend separate schools, and seek employment in a labor market delineated by race (Vogel et al. 2019). Numerous studies have revealed disadvantages faced by Black individuals at different stages of their life such as the transition to adulthood or parenting (Barr et al. 2018). Compared to other racial groups, Black individuals are more likely to disconnect with the social institutions of school, work, and the military (DeLuca, Clampet-Lundquist, and Edin 2016; Settersten 2005), to experience more hostile romantic relationships (Hardie and Lucas 2010; Kurdek 2008), and to have contact with the juvenile justice system (Pettit and Western 2004). Due to residential segregation, Black youth are more likely to grow up in disadvantaged neighborhoods with poverty, scarcity of community resources, single-headed households, high crime rates, and low collective efficacy (Sampson, Raudenbush, and Earls 1997; Sampson and Wilson 1995).

One consequence of these differences is that social and historical transformations are likely to have differential impacts on White and Black cohorts’ life-course trajectories of homicide involvement. First, Black individuals residing in metropolitan areas are more susceptible to the impact of inner-city transformations since the 1970s and the crack cocaine epidemic in the late 1980s. The decline in employment opportunities in such locations and the increase of mass incarceration since the 1970s worsened Black youths’ economic prospects and contributed to higher risks of homicide offending compared to White youth, who were more likely to reside in suburban areas (Bischoff and Reardon 2014). Similarly, the crack cocaine epidemic of the late 1980s affected Black youth disproportionately (Blumstein, Rivara, and Rosenfeld 2000; Blumstein and Rosenfeld 1998). Lacking proper supervision, legitimate opportunities, or socioeconomic resources, Black youth residing in disadvantaged neighborhoods were more likely to be recruited to the illicit drug industry and to be exposed to gang violence related to the drug trade at the time.

Second, the War on Drugs, the Tough on Crime campaign, and the mass incarceration approach in the U.S. criminal justice system in the past three decades have disproportionately affected Black cohorts. The U.S. penal population has increased sixfold between the 1970s and 2000s and the incarceration rate for Black individuals is about eight times higher than for White individuals (Pettit and Western, 2004). Although rates declined slightly after 2009, incarceration is still a common experience for low-income Black men (Turney and Goodsell 2018; Western and Wildeman 2009). Furthermore, research has shown that incarceration is associated with low income, limited job opportunities, and family instability (Pettit and Western 2004; Western, Kling, and Weiman 2001; Western and Wildeman 2009). Young Black men who have had contact with the criminal justice system and have been imprisoned at early ages are likely to face negative outcomes throughout their life course, which could influence their risks of crime involvement.

Third, another important social change is the crime drop since the 1990s. Homicide rates dropped by 50% between 1991 and 2014, and this decline has been consistently recorded across different data sources, geographical regions, and racial groups (Baumer, Vélez, and Rosenfeld 2018; Zimring 2006). Researchers have proposed several explanations for this decline, including the growth of police forces, the stabilization of drug markets, an aging population, and changes in security technology (see a review in Baumer et al., 2018). A recent study has revealed that the decline in youth offending since the 1990s can be explained by a decrease in unstructured socializing, a decrease in alcohol consumption, and lower preferences for risky activities among youth (Baumer et al. 2020). Another study documented substantial quality-of-life improvement in the neighborhoods of American cities after the great decline in violence (Sharkey 2018). However, it is unclear whether these social changes had similar implications for Black and White cohorts.

Expectations

Drawing from extant literature, we develop the following expectations regarding the cohort effects on Black and White homicide involvement. First, we expect age-homicide trajectories to vary between Black and White cohorts. The racial turmoil in the 1970s and the crack epidemic in the early 1990s exacerbated the risks of homicide involvement (as offenders and victims) among cohorts who were teenagers and young adults during those periods. Such interactive effects of critical ages and social environments operate to differentiate cohorts’ homicide involvement. The early exposure to a criminogenic environment may not only lead to an early age of onset but also to higher rates of offending throughout the life course, resulting in inter-cohort differences. Alternatively, a cohort’s high level of crime involvement may be contingent on or limited to a specific age (e.g., early adulthood) or period (e.g., the 1980s), but otherwise similar to other cohort’s trajectories at older ages. For instance, cohorts born after 1980 may have flatter age trajectories than older cohorts, which may be linked to the delayed transition to adult status and the declining adult activities among U.S. adolescents (Twenge and Park 2019). Such nuances in cohort effects on homicide involvement are lost in traditional APC models due to their focus on the average cohort difference. In contrast, the APC-I framework used in the current study is specifically designed to reveal these nuances in cohort effects.

Second, we expect varying degrees of cohort difference in homicide trajectories between Black and White populations. With the history of racial segregation, labor segmentation, and criminal justice policies that disproportionately affect racial minorities, we expect Black cohorts to be more susceptible to social change. That is particularly the case for negative events, as disadvantaged groups may lack a protective buffer against their harmful impacts, be it in terms of higher job security to withstand an economic downturn, a financial cushion from savings or family wealth, and/or the capital to depart from an increasingly violent neighborhood during the crack cocaine epidemic (Currie 1997; Rogers and Pridemore 2013). Hence, historical events may have stronger effects on Black cohorts’ life-course homicide involvement. For instance, Black cohorts born between the 1970s and 1980s (teenagers and young adults during the crack epidemic) are expected to have a greater homicide involvement than White cohorts born around the same time.

Data and Methods

Homicide data for this study are collected from the Supplementary Homicide Report (SHR), and the race-age-period-specific population estimates are from the Current Population Survey (CPS). The compiled dataset extends from 1976 to 2018. The SHR is an extension of the Uniform Crime Report (UCR) and provides detailed information on homicide reported by law enforcement agencies in the U.S..¹ Though the UCR publishes historical data on other types of crimes, data about homicides is particularly appropriate for longitudinal analysis as homicides have a clear objective definition – the willful killing of a person by another – which is relatively stable over time, and which produces verifiable evidence (United Nations Office on Drugs and Crime 2019). In addition, disaggregated counts by race are available for homicides, but not for other crimes. Finally, though homicide is less common than other crimes in the UCR, it is the most serious, it is irreversible, and rates are very high in the U.S. compared to other western democracies, particularly amongst minorities (Messner and Rosenfeld 2012).

Different from the aggregated data reported in the UCR, the SHR reports homicide cases at the incident level, including the known demographic characteristics of alleged offenders and victims. Homicide offending is “alleged” because the one key variable missing from SHR is whether the offenders involved in the homicide incident were arrested. Therefore, we use the terms alleged offenders or alleged offending rates, rather than arrestees or arrest rates (Kaplan 2021). In addition, we estimate effects separately for trends in both alleged offending and victimization, which serve as a reference of comparison and as a robustness check for estimates.

The SHR data enables us to estimate age-race-specific rates of alleged homicide offending and victimization for each year across the study period. Because some incidents involve multiple offenders, we transformed the dataset from incident-level to offender-level. After dropping cases that do not involve White or Black individuals (i.e., Asian, Pacific Islanders, etc.), and cases with missing information on the age and race variables, a final analytic sample contained 262,934 Black alleged offenders, 247,925 White alleged offenders, 299,110 Black victims, and 272,159 White victims between the ages 15 to 49.²

Because our focus is on the aggregate level cohort effects, the unit of analysis is the aggregate level age-period-specific homicide offending and victimization counts rather than individual-level homicide risks. Therefore, for each racial group, the data of alleged homicide offenders and victims are aggregated into 7 five-year age categories (i.e., age 15–19, 20–24…45–49) and 9 period categories (i.e., 1976–79, 1980–84… 2015–18).³ Data has been aggregated using 5-year ranges to ensure a sufficient count of incidents in each age-by-period intersection, while avoiding broader year ranges (e.g., 15 years) that could omit important variation. We measure homicide involvement using two dependent variables: the race-age-period-specific alleged homicide offending counts and the race-age-period-specific victim counts. To account for the changes in the population at risk within each racial group, age group, and year, we merged the SHR data with population estimates from the CPS. The homicide offending or victimization counts are entered as a dependent variable in a Poisson model and the race-specific-population data are included as an offset term. This approach is essentially the same as using homicide rates per fixed population size.

To demonstrate the data structure, we display Black and White age-period-cohort-specific alleged homicide offending rates in Table 1. Each cell in this table corresponds to an age-period-cohort-specific alleged homicide offending rate for a race group. The total sample size (i.e., the number of cells) for each race group is 63 (i.e., 63 age-period-cohort-specific rates). Across all periods and age groups, the homicide rate of Black individuals is much higher than the homicide rate of White individuals. The upper-left-to-the-lower-right diagonals of the table are the homicide offending rates of each specific cohort over different ages. For instance, the alleged homicide offending rate per 100,000 population of Whites born around 1960 was 6.82 at ages 15–19, 9.91 at ages 20–24, 7.43 at ages 25–29, and 2.08 at ages 45–49. The time range of the data is suitable for our research questions because it covers periods when U.S. homicide rates were extremely high (late 1970s and late 1980s) as well as a period of a dramatic homicide decline after 1990. More importantly, based on the nine period groups and seven age groups, we have 15 five-year cohort groups from 1930 to 2000, capturing the cohorts of baby boomers (1946–64), generation X (1965–1980), and millennials (1981–1996). This scope allows us to compare cohorts who grew up in both high and low violence eras, and to examine more recent cohorts who came of age in an era with considerable changes in policies and public perceptions about racial inequality. The data structure for homicide victimization rates is the same as the alleged offending rates shown in Table 1.

Table 1.

Race-Specific Alleged Homicide Offending Rates by Age, Period, and Cohort.

	Race-specific rates	Period
Age	(Cohort)	1975	1980	1985	1990	1995	2000	2005	2010	2015
15–19	Black	41.77	39.68	48.39	100.48	56.10	36.20	43.28	32.67	35.61
	White	6.82	7.16	7.03	11.15	8.24	5.33	5.26	3.77	4.04
	(Cohort)	(C1960)	(C1965)	(C1970)	(C1975)	(C1980)	(C1985)	(C1990)	(C1995)	(C2000)
20–24	Black	76.26	64.09	62.15	94.50	66.62	56.73	57.13	48.24	47.04
	White	9.69	9.91	9.13	10.37	8.60	8.01	7.56	5.94	5.73
	(Cohort)	(C1955)	(C1960)	(C1965)	(C1970)	(C1975)	(C1980)	(C1985)	(C1990)	(C1995)
25–29	Black	77.80	62.15	49.39	50.72	33.71	35.21	38.23	32.15	34.46
	White	8.64	8.68	7.43	7.40	5.73	5.66	5.90	5.34	5.56
	(Cohort)	(C1950)	(C1955)	(C1960)	(C1965)	(C1970)	(C1975)	(C1980)	(C1985)	(C1990)
30–34	Black	63.12	50.96	37.85	32.77	21.83	18.84	22.59	21.60	24.26
	White	7.37	7.26	5.78	5.49	4.21	4.17	4.53	4.76	5.08
	(Cohort)	(C1945)	(C1950)	(C1955)	(C1960)	(C1965)	(C1970)	(C1975)	(C1980)	(C1985)
35–39	Black	49.83	41.18	31.13	23.88	14.71	13.44	13.63	13.78	17.06
	White	6.12	6.05	4.75	4.14	3.18	3.02	3.60	3.70	4.15
	(Cohort)	(C1940)	(C1945)	(C1950)	(C1955)	(C1960)	(C1965)	(C1970)	(C1975)	(C1980)
40–44	Black	39.25	33.32	25.09	18.42	10.77	9.70	10.19	9.25	11.65
	White	4.78	4.86	4.06	3.26	2.57	2.43	2.61	2.77	3.11
	(Cohort)	(C1935)	(C1940)	(C1945)	(C1950)	(C1955)	(C1960)	(C1965)	(C1970)	(C1975)
45–49	Black	31.45	24.95	18.05	14.26	8.06	7.12	7.31	6.64	7.94
	White	3.61	3.90	3.07	2.71	1.93	1.82	2.08	2.14	2.22
	(Cohort)	(C1930)	(C1935)	(C1940)	(C1945)	(C1950)	(C1955)	(C1960)	(C1965)	(C1970)

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Note: Rows are defined by age groups and columns period groups. Cohorts follow the upper-left-to-lower-right diagonals and are indicated in parentheses.

Analytical Strategy

An intuitive approach to estimating age, period, and cohort (APC) effects is to include the three time-related measures as independent variables in a regression model. Unfortunately, such models are unidentifiable because of the exact linear dependence of the three variables—any two among the three completely determines the value of the third (e.g., cohort = period - age). Many estimation methods have been proposed to address the identification problem, including forcing the effects of two adjacent age groups to be equal or constraining the linear effects to be a function of their nonlinear terms (see Fosse & Winship, 2019 for a review). However, such constraints imply assumptions that are difficult to verify in empirical research (Lu and Luo 2020; Luo et al. 2016; Luo and Hodges 2020, 2022; Rodgers 1982).

We use the age-period-cohort-interaction (APC-I) model (Luo and Hodges 2022) to examine the cohort-specific life-course trajectories of alleged homicide offending and victimization among aggregate groups of Black and White persons. As described in the literature review section, the APC-I approach is both a new conceptual framework for understanding cohort effects and a methodological approach to modeling cohort effects. Whereas prior methods aim at recovering independent and additive effects of age, period, and cohort, the APC-I method instead acknowledges the interdependence among these factors and is designed to estimate cohort effects in relation to age and period. Specifically, Luo and Hodges (2022) argued that because the influence of social events differs between age groups, cohort effects should accordingly be modeled as age-by-period interactions.

In its simplest form, the APC-I can be specified as a generalized linear model:

{g (E (Y}_{i j})) = μ + α_{i} + β_{j} + {α β}_{i j (k)}

(1)

where ${g (E (Y}_{i j}))$ denotes the link function of the expected offending/victimization rates $Y$ for the $i$ th age group in the $j$ th period of time; $μ$ denotes the global mean of the dependent variable; $α_{i}$ denotes the main age effects associated with ith age category; $β_{j}$ denotes the main period effects associated with the $j$ th period; ${α β}_{i j (k)}$ denotes the interaction of the $i$ th age group and $j$ th period group, which corresponds to the effect of the $k$ th cohort. We also include the age-period-specific population as an offset term in the model to account for the variation in the population at risk. Note that model (1) is identified just like any ANOVA model with two main effects and an interaction term between the two. Also note that the effect of one cohort may include multiple age-by-period interaction terms ${α β}_{i j (k)}$ . As our outcome variables are represented by counts of events (offenses or victimizations), we use a Poisson regression to generate the estimates.⁴

The APC-I method permits a decomposition of cohort-related variation into two parts: inter-cohort average deviations and intra-cohort life-course dynamics. The possibility of estimating intra-cohort life-course dynamics is a unique strength of the APC-I model in that it can be used to test whether cohort effects on the outcome are contingent on specific ages or persisting over the life course. In Appendix I, we provide detailed discussions on the conceptual and methodological advantages of the APC-I model compared to traditional APC models.

We implement the APC-I analysis in three steps (Lu and Luo 2020; Luo and Hodges 2022). First, we estimate the age and period main effects on alleged homicide offending and victimization rates of each racial group, comparing their similarities and differences. Second, we estimate the inter-cohort average deviations in each race-specific model. These inter-cohort average deviations indicate whether a specific cohort is generally more or less likely to involve in homicide than other cohorts. Third, we examine the intra-cohort life-course trajectories for each cohort in each racial group by estimating the linear trend in the interaction terms contained in that cohort and plotting the race-cohort-specific age trajectories.

Results

Age and Period Main Effects

Table 2a reports the estimated age and period main effects on the race-specific (A) alleged homicide offending and (B) victimization in the APC-I models. The intercept in each model represents the overall mean in each sample, and the slope represents the deviation of each age and period from that overall mean. For instance, the intercept of −8.165 in model 1 represents the average homicide offending rate among the Black population, which is 57 per 100, 000 (e^−8.165 * 100,000). The intercept −9.940 in model 2 represents the average alleged homicide offending rate among the White population, which is 5 per 100,000 (e^−9.94 * 100,000). The estimated age coefficient for the Black age group 15–19 is .468, meaning that being in the age group of 15–19 increases the risk of homicide offending by 60% (e ^.468 - 1) relative to the mean. ⁵ The 15–19 age group is also associated with an increase in homicide offending among the White population by 29% (e ^.250 - 1), a smaller effect than for the Black population. The race-specific victimization models can be interpreted in the same way.

Table 2a.

Estimated Age and Period Main Effects on Alleged Homicide Offending and Victimization by Race.

	(A) Alleged Offending								(B) Victimization
	Black (M1)				White (M2)				Black (M3)				White (M4)
	Coef.	IRR	S.E.		Coef.	IRR	S.E.		Coef.	IRR	S.E.		Coef.	IRR	S.E.
Intercept	−8.165	0.000	0.006	^***	−9.940	0.000	0.005	^***	−7.912	0.000	0.005	^***	−9.807	0.000	0.004	^***
Age Main Effects
15–19	0.468	1.596	0.010	^***	0.250	1.285	0.010	^***	−0.122	0.885	0.010	^***	−0.143	0.866	0.011	^***
20–24	0.783	2.188	0.009	^***	0.526	1.693	0.009	^***	0.501	1.651	0.008	^***	0.353	1.423	0.009	^***
25–29	0.434	1.544	0.010	^***	0.312	1.366	0.010	^***	0.423	1.527	0.009	^***	0.244	1.276	0.010	^***
30–34	0.051	1.053	0.012	^***	0.093	1.098	0.011	^***	0.183	1.201	0.010	^***	0.091	1.095	0.010	^***
35–39	−0.282	0.754	0.014	^***	−0.145	0.865	0.012	^***	−0.072	0.930	0.011	^***	−0.046	0.955	0.011	^***
40–44	−0.575	0.563	0.017	^***	−0.387	0.679	0.014	^***	−0.329	0.720	0.013	^***	−0.173	0.841	0.012	^***
45–49	−0.879	0.415	0.020	^***	−0.650	0.522	0.016	^***	−0.584	0.557	0.015	^***	−0.325	0.722	0.013	^***
Period Main Effects
y1975	0.593	1.810	0.014	^***	0.284	1.329	0.013	^***	0.403	1.496	0.012	^***	0.328	1.388	0.012	^***
y1980	0.415	1.514	0.014	^***	0.305	1.356	0.013	^***	0.263	1.301	0.013	^***	0.344	1.410	0.011	^***
y1985	0.237	1.267	0.015	^***	0.141	1.152	0.013	^***	0.145	1.157	0.013	^***	0.170	1.186	0.012	^***
y1990	0.269	1.308	0.015	^***	0.149	1.160	0.013	^***	0.347	1.416	0.012	^***	0.275	1.317	0.011	^***
y1995	−0.209	0.812	0.017	^***	−0.116	0.890	0.014	^***	−0.093	0.911	0.013	^***	−0.043	0.958	0.012	^***
y2000	−0.354	0.702	0.017	^***	−0.215	0.806	0.015	^***	−0.255	0.775	0.014	^***	−0.196	0.822	0.013	^***
y2005	−0.277	0.758	0.017	^***	−0.153	0.858	0.014	^***	−0.229	0.795	0.013	^***	−0.198	0.821	0.013	^***
y2010	−0.399	0.671	0.017	^***	−0.226	0.798	0.015	^***	−0.359	0.698	0.014	^***	−0.384	0.681	0.014	^***
y2015	−0.275	0.760	0.016	^***	−0.169	0.845	0.014	^***	−0.223	0.800	0.013	^***	−0.296	0.744	0.014	^***
Cohort (Age-by-period Interaction)
	(See Table 2b)				(See Table 2b)				(See Table 2c)				(See Table 2c)

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Notes:

p<.05

^**

p<.01

^***

p<.001.

IRR=Incident Rate Ratio; S.E.=Standard Error. N=63. The unit of analysis in each APC-I model is the age-period specific homicide offending/victimization rate. Because we use population data and we do not intend to make statistical inferences outside of the range of the study period of our data, statistical significance is included only as a reference. We focus on the effect sizes because they represent population parameters rather than sample statistics. All APC-I models are estimated with the sum-to-zero coding, under which the main effect is interpreted as the deviation from the grand mean and the interaction term as the deviation from the main effects (see details in Appendix I).

Figures 1 and 2 graphically present the age, period main effects, and the inter-cohort deviations across all of the four APC-I models in Table 2. We first compare the Black and White models of homicide offending in Figure 1 (based on Model 1 and 2 in Table 2a). The horizontal solid line represents the overall mean (i.e., the intercept). A positive deviation from the horizontal line represents a higher-than-average offending rate, while a negative deviation represents the opposite. Both Black and White teenagers and young adults (i.e., ages 15–34) display higher risks of homicide offending than other age groups, and homicide offending declines linearly with age after the peak. These findings are consistent with the age-crime literature that reveals high violent crime concentration among teenagers and young adults (Hirschi and Gottfredson 1983; Steffensmeier et al. 1989). Although the overall shape of the age effects is consistent across models, the heightened risks are notably greater for the Black group than for the White group. Additionally, while older individuals from both racial groups are less likely to be involved in homicide offending, age differences are more pronounced among the Black group than among the White group.

Figure 1b illustrates the estimated period effects on Black and White alleged homicide offending. Similar to the age main effects, the direction of period main effects is also consistent between racial groups. Both groups have higher-than-average rates from 1975 to 1990, but both have a steep decline to a lower-than-average level in 1995. However, the magnitudes of period coefficients vary between racial groups. The estimate for the period 1975–1979 is at 0.593 for the Black sample and at 0.284 for White sample (Table 2a), suggesting the homicide rate in this period is about 81% higher than the grand mean for Black offenders, but only 33% higher than the grand mean for White offenders. Conversely, period effects since 1995 are more sharply negative for the Black group than for the White group, reaching a minimum of −32.9% for Black offenders in 2010, against −20.2% for White offenders.

Figure 2 presents the age and period main effects on Black and White homicide victimization estimated from Models 3 and 4 of Table 2. The overall patterns of the age and period effects are very similar to those of the homicide offending models. Homicide victimization risks also increase among teenagers and young adults, reach the peak at the early twenties, and then decline gradually. These age differences, however, are slightly more pronounced among Black victims than White victims. In addition, the age patterns of victimization for both racial groups are slightly flatter than the age patterns of offending. This is consistent with prior literature that found the age-crime distribution of offender and victims are similar but not identical (Jennings et al. 2010). In terms of the period main effects on victimization, the overall patterns are also similar to offending, with positive period main effects before the 1990s and negative period effects after the 1990s. However, different from the racial gap we observed in the period effects on offending, the magnitudes of period main effects on victimization are almost identical between race.

Overall, age and period effects show consistent patterns between racial groups. However, the volatility in those effects is more pronounced for the Black population than for the White population, especially for alleged homicide offending, suggesting that Black individuals may be more susceptive to age and period-specific influences in terms of their homicide involvement. In addition, while APC effects are generally more accentuated for the Black population than for the White population, the effects for the two racial groups tend to be more similar when using victimization data than when using offending data. That is particularly the case for period effects on homicide victimization trends, which are indistinguishable between the two racial groups. All other characteristics of the APC effects, including direction and magnitude, were remarkably similar when using offending and victimization data. However, the fact that effects are more similar across racial groups in victimization than offending is consequential. A possible explanation, which will be expanded upon in the discussion session, is that racial disparities in the criminal justice system may contribute to some of these differences, as they could impact alleged homicide offending more strongly than victimization.

Inter-Cohort Deviations

Tables 2b and 2c present the age-by-period interaction effects in each race-specific model on the two homicide outcomes. The set of interaction terms on each diagonal corresponds to the mean deviations of a given cohort in each period across its life span. For example, the interaction terms from the 1^st column 1^st row in Table 2b for Blacks (ages 15–19 in 1975–79) to the 7^th column 7^th row (ages 45–49 in 2005–09) correspond to the deviations of the Black cohort born around 1960 (i.e., coefficients −.677, −.386, …, −.202). If the cohort’s age-specific homicide rate is not affected by cohort differences, the age-by-period interaction term is small and approximates zero.

Table 2b.

Coefficient Estimates of the Age-by-period Interaction Terms in the APC-I Model on Alleged Homicide Offending by Race.

Black (M1)	Period

	1975–79		1980–84		1985–89		1990–94		1995–99		2000–04		2005–09		2010–14		2015–19
Age Group	Coef.	Sig.	Coef.	Sig.	Coef.	Sig.	Coef.	Sig.	Coef.	Sig.	Coef.	Sig.	Coef.	Sig.	Coef.	Sig.	Coef.	Sig.
15–19	−0.677	^***	−0.550	^***	−0.174	^***	0.525	^***	0.419	^***	0.127	^***	0.229	^***	0.069	^*	0.031
20–24	−0.391	^***	−0.386	^***	−0.239	^***	0.149	^***	0.276	^***	0.261	^***	0.191	^***	0.144	^***	−0.006
25–29	−0.022		−0.068	^**	−0.120	^***	−0.125	^***	−0.056		0.133	^***	0.138	^***	0.087	^**	0.032
30–34	0.152	^***	0.116	^***	−0.003		−0.179	^***	−0.108	^**	−0.109	^**	−0.005		0.072		0.064
35–39	0.249	^***	0.237	^***	0.135	^***	−0.162	^***	−0.169	^***	−0.114	^*	−0.177	^***	−0.044		0.045
40–44	0.303	^***	0.318	^***	0.212	^***	−0.129	^**	−0.188	^***	−0.147	^**	−0.175	^***	−0.150	^**	−0.044
45–49	0.386	^***	0.333	^***	0.188	^***	−0.080		−0.174	^**	−0.152	^*	−0.202	^***	−0.177	^**	0.122	^*

White (M2)	Period

	1975–79		1980–84		1985–89		1990–94		1995–99		2000–04		2005–09		2010–14		2015–19
Age Group	Coef.	Sig.	Coef.	Sig.	Coef.	Sig.	Coef.	Sig.	Coef.	Sig.	Coef.	Sig.	Coef.	Sig.	Coef.	Sig.	Coef.	Sig.
15–19	−0.189	^***	−0.160	^***	−0.014		0.439	^***	0.402	^***	0.065	^*	−0.011		−0.270	^***	−0.260	^***
20–24	−0.113	^***	−0.111	^***	−0.030		0.090	^***	0.168	^***	0.197	^***	0.076	^**	−0.092	^**	−0.185	^***
25–29	−0.013		−0.029		−0.021		−0.033		−0.024		0.063	^*	0.043		0.015		−0.001
30–34	0.047		0.011		−0.054		−0.112	^***	−0.112	^***	−0.024		−0.003		0.120	^***	0.127	^***
35–39	0.098	^**	0.067	^*	−0.012		−0.158	^***	−0.157	^***	−0.109	^**	0.005		0.104	^**	0.162	^***
40–44	0.094	^*	0.090	^*	0.074	^*	−0.153	^***	−0.125	^**	−0.083	^*	−0.073		0.060		0.116	^**
45–49	0.075		0.133	^**	0.057		−0.074		−0.151	^**	−0.110	^*	−0.036		0.063		−0.042

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Notes: All APC-I models are estimated with the sum-to-zero coding, under which the main effect is interpreted as the deviation from the grand mean and the interaction term as the deviation from the main effects.

p<.05

^**

p<.01

^***

p<.001.

Table 2c.

Coefficient Estimates of the Age-by-period Interaction Terms in the APC-I Model on Homicide Victimization by Race.

Black (M3)	Period

	1975–79		1980–84		1985–89		1990–94		1995–99		2000–04		2005–09		2010–14		2015–19
Age Group	Coef.	Sig.	Coef.	Sig.	Coef.	Sig.	Coef.	Sig.	Coef.	Sig.	Coef.	Sig.	Coef.	Sig.	Coef.	Sig.	Coef.	Sig.
15–19	−0.601	^***	−0.514	^***	−0.095	^**	0.352	^***	0.328	^***	0.110	^***	0.202	^***	0.116	^***	0.102	^***
20–24	−0.323	^***	−0.295	^***	−0.139	^***	0.127	^***	0.188	^***	0.198	^***	0.139	^***	0.101	^***	0.004
25–29	−0.022		−0.046		−0.033		−0.091	^***	−0.045		0.125	^***	0.051	^*	0.059	^*	0.002
30–34	0.083	^**	0.110	^***	−0.006		−0.114	^***	−0.129	^***	−0.046		0.032		0.045		0.023
35–39	0.190	^***	0.174	^***	0.085	^**	−0.094	^***	−0.133	^***	−0.137	^***	−0.123	^***	−0.023		0.062	^*
40–44	0.311	^***	0.248	^***	0.112	^**	−0.048		−0.110	^**	−0.122	^**	−0.164	^***	−0.175	^***	−0.052
45–49	0.362	^***	0.321	^***	0.077		−0.131	^**	−0.099	^*	−0.128	^**	−0.138	^**	−0.122	^**	0.142	^**

White (M4)	Period

	1975–79		1980–84		1985–89		1990–94		1995–99		2000–04		2005–09		2010–14		2015–19
Age Group	Coef.	Sig.	Coef.	Sig.	Coef.	Sig.	Coef.	Sig.	Coef.	Sig.	Coef.	Sig.	Coef.	Sig.	Coef.	Sig.	Coef.	Sig.
15–19	−0.187	^***	−0.206	^***	−0.160	^***	0.255	^***	0.289	^***	0.119	^***	0.116	^***	−0.101	^**	−0.126	^***
20–24	−0.136	^***	−0.100	^***	−0.043		0.072	^**	0.103	^***	0.154	^***	0.063	^*	−0.033		−0.080	^**
25–29	−0.030		−0.008		0.055	^*	−0.017		−0.042		0.045		0.014		0.005		−0.020
30–34	−0.020		0.012		0.028		−0.013		−0.041		−0.068	^*	−0.012		0.057		0.057
35–39	0.101	^***	0.059	^*	0.030		−0.063	^*	−0.069	^*	−0.075	^*	−0.095	^**	0.019		0.094	^**
40–44	0.139	^***	0.137	^***	0.046		−0.140	^***	−0.096	^**	−0.085	^*	−0.061		−0.003		0.063
45–49	0.135	^***	0.106	^**	0.044		−0.093	^**	−0.144	^***	−0.089	^*	−0.026		0.055		−0.013

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p<.05

^**

p<.01

^***

p<.001.

In Table 3 we summarize the information presented in Table 2b and 2c into aggregated measures of cohort estimate: inter-cohort average deviations. The inter-cohort deviations correspond to the average of the age-by-period interactions for each cohort in Tables 2b and 2c (i.e., the diagonals). A positive inter-cohort deviation indicates that the cohort has higher risks of homicide offending or victimization than the expectation based on age and period main effects, while a negative inter-cohort estimate suggests the opposite. We also use z-tests to examine whether these inter-cohort deviations are statistically different from zero.⁶

Table 3.

Estimated Inter-cohort Average Deviations^a of Alleged Homicide Offending and Victimization by Race.

	(A) Alleged Offending								(B) Victimization

	Black				White				Black				White

Cohort	Coef.	IRR	S.E.	Sig.	Coef.	IRR	S.E.	Sig.	Coef.	IRR	S.E.	Sig.	Coef.	IRR	S.E.	Sig.
1930	0.386	1.471	0.046	^***	0.075	1.078	0.044		0.362	1.436	0.038	^***	0.135	1.145	0.034	^***
1935	0.318	1.374	0.032	^***	0.114	1.120	0.030	^***	0.316	1.372	0.027	^***	0.122	1.130	0.024	^***
1940	0.251	1.286	0.026	^***	0.081	1.085	0.023	^***	0.172	1.187	0.022	^***	0.094	1.099	0.019	^***
1945	0.130	1.139	0.022	^***	0.029	1.029	0.019		0.059	1.061	0.018	^***	−0.002	0.998	0.016
1950	−0.015	0.986	0.019		−0.064	0.938	0.016	^***	0.005	1.005	0.015		−0.055	0.947	0.014	^***
1955	−0.160	0.852	0.017	^***	−0.098	0.907	0.015	^***	−0.118	0.889	0.013	^***	−0.061	0.941	0.012	^***
1960	−0.269	0.764	0.016	^***	−0.101	0.904	0.013	^***	−0.205	0.814	0.013	^***	−0.061	0.941	0.011	^***
1965	−0.212	0.809	0.016	^***	−0.065	0.937	0.014	^***	−0.185	0.831	0.013	^***	−0.056	0.946	0.012	^***
1970	−0.056	0.945	0.015	^***	0.007	1.007	0.013		−0.031	0.970	0.012	^*	−0.044	0.957	0.012	^***
1975	0.140	1.151	0.015	^***	0.148	1.159	0.014	^***	0.104	1.109	0.013	^***	0.079	1.082	0.014	^***
1980	0.187	1.206	0.015	^***	0.184	1.203	0.014	^***	0.137	1.146	0.013	^***	0.122	1.130	0.014	^***
1985	0.117	1.124	0.015	^***	0.071	1.073	0.015	^***	0.083	1.086	0.014	^***	0.061	1.063	0.016	^***
1990	0.135	1.144	0.015	^***	−0.035	0.966	0.018	^*	0.101	1.107	0.015	^***	0.021	1.021	0.018
1995	0.032	1.032	0.019		−0.228	0.796	0.023	^***	0.060	1.062	0.019	^**	−0.091	0.913	0.025	^***
2000	0.031	1.031	0.029		−0.260	0.771	0.034	^***	0.102	1.108	0.030	^***	−0.126	0.881	0.038	^***

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Notes:

p<.05

^**

p<.01

^***

p<.001. IRR=Incident Rate Ratio; S.E.=Standard Error.

An Inter-cohort average deviation represents a cohort’s average deviation from the predicted rate determined by age and period main effects. A positive inter-cohort deviation represents higher-than-expected risks, whereas a negative inter-cohort deviation represents lower-than-expected risks. A small inter-cohort deviation that is not significantly different from zero suggests that on average, a cohort does not deviate from the expected rate determined by age and period main effects.

The inter-cohort deviation patterns are almost identical for alleged homicide offending (Table 3 (A) and Figure 1c) as for homicide victimization (Table 3 (B) and Figure 2c). In both cases, the inter-cohort deviation patterns are consistent across Black and White groups, except for the younger cohorts born after 1990. For example, the inter-cohort estimates of homicide offending for Cohort 1960 are −.269 in the model about the Black group and −.101 in the model about the White group, suggesting that (though at notable different levels) both the Black and White Cohort born around 1960 have lower alleged homicide offending risks (−24% and −10%). These patterns are graphically presented in Figures 1c and 2c, where cohorts with higher-than-expected risks are presented as positive deviations from the horizontal line, and cohorts with lower-than-expected risks are presented under the horizontal line. For both offending and victimization, most of the baby boom cohorts born between 1945 and 1965 demonstrate lower-than-expected risks, while cohorts born between 1975–1985 have higher-than-expected risks.

Figure 4. — Predicted Cohort-Specific Homicide Victimization Trajectories by Race (Deviations from the Mean)

Notes: The horizontal line in each graph represents zero deviation from the global means of Black or White homicide. Each point in the figure represents the total deviations from the race-specific mean rates under the influence of age, period, and age-by-period interaction effects estimated by the APC-I models reported in Table 2a and 2c.

Once again, we find that the general patterns of effects are consistent across the two racial groups in homicide outcomes, though the effect sizes of the cohort deviations are, again, notably more accentuated for Black cohorts. The heightened homicide risks of cohorts born before the 1940s are greater among the Black group than among the Whites group. In addition, though baby boomers (Cohort 1945–1965) from both racial groups are less likely to commit homicides, these effects are more negative among boomers who are Black.

We also observe one key difference in cohort effects between the two racial groups. Though the risks of homicide offending and victimization have declined substantially among White cohorts born after 1985, the pattern for the young Black cohorts has remained relatively flat, despite a smaller decline. The gap is greater in offending than in victimization. Black cohorts born after 1975 displayed higher-than-expected risks of homicide involvement, while young White cohorts have declined to a lower-than-expected level. The discrepancy between White and Black cohorts born after 1985 is the most notable difference between racial groups, as it shows a substantial decrease in homicide risks for White cohorts (at −23% for offending and −12% for the victimization of Cohort 2000) that has no match for Black cohorts, who experience a relatively high risk of homicide involvement. We elaborate on this divergence in the discussion.

Intra-Cohort Life-Course Trajectories

To examine whether cohort effects in homicide offending persist, diminish, or accumulate over the cohort’s life span, we first test the linear change in the interaction terms contained in each cohort in Table 4. The coefficients presented in the table are the estimated linear orthogonal polynomial contrast of the age-by-period interaction terms contained in each cohort. Because the first and the last cohorts only have one age-by-period interaction estimate available, they do not have an intra-cohort slope. While some cohorts (e.g., Cohort 1970) have relatively stable cohort effects across the life course, that is, non-significant intra-cohort life-course dynamics test, many cohorts (e.g., Cohort 1975) display significant life-course changes in cohort effects at different ages. The intra-cohort slopes should be interpreted along with the inter-cohort deviations shown in Table 3. A significant positive slope suggests the cohort effects tend to increase across the life course, while a significant negative slope suggests the cohort effects tend to diminish across the life course. For instance, Black Cohort 1990 has significantly higher alleged offending risks (as shown in Table 3), but their higher-than-expected risk appears to diminish with age (Table 4, b=−0.139, p<.001).

Table 4.

Intra-cohort Life-Course Dynamics Tests^a of Alleged Homicide Offending and Victimization by Race

	Alleged Offending						Victimization

	Black			White			Black			White

Cohort	Slope	S.E.	Sig.	Slope	S.E.	Sig.	Slope	S.E.	Sig.	Slope	S.E.	Sig.
1930	NA	NA		NA	NA		NA	NA		NA	NA
1935	0.021	0.045		0.027	0.041		0.007	0.037		–0.023	0.033
1940	–0.043	0.045		–0.029	0.040		–0.080	0.038	^*	–0.040	0.033
1945	–0.161	0.044	^***	–0.079	0.037	^*	–0.158	0.035	^***	–0.052	0.030
1950	–0.174	0.046	^***	–0.139	0.037	^***	–0.099	0.034	^**	–0.120	0.030	^***
1955	0.081	0.045		–0.045	0.036		0.083	0.034	^*	–0.015	0.030
1960	0.350	0.043	^***	0.072	0.035	^*	0.309	0.035	^***	0.074	0.031	^*
1965	0.238	0.044	^***	0.096	0.035	^**	0.204	0.036	^***	0.130	0.032	^***
1970	0.032	0.043		–0.022	0.036		0.005	0.035		0.045	0.034
1975	–0.471	0.037	^***	–0.223	0.032	^***	–0.328	0.031	^***	–0.152	0.032	^***
1980	–0.297	0.033	^***	–0.176	0.030	^***	–0.217	0.028	^***	–0.154	0.031	^***
1985	–0.066	0.031	^*	0.028	0.031		–0.076	0.029	^**	–0.055	0.032
1990	–0.139	0.028	^***	0.007	0.030		–0.141	0.026	^***	–0.096	0.031	^**
1995	–0.053	0.028		0.060	0.032		–0.079	0.027	^**	0.014	0.034
2000	NA	NA		NA	NA		NA	NA		NA	NA

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Notes:

p<.05

^**

p<.01

^***

p<.001. S.E.=Standard Error.

The intra-cohort life-course dynamic test examines the linear change in the interaction terms at different ages within a cohort. Because the first and the last cohort only has one age-by-period interaction estimate available, they do not have an intra-cohort slope. If the intra-cohort slope is statistically significant, the cohort effects are not the same (or “permanent”) across a cohort’s life course. If the intra-cohort slope is not statistically different from zero, the cohort effect is stable across a cohort’s life course. The intra-cohort slope should be interpreted along with the inter-cohort deviation. For example, if a cohort has positive inter-cohort deviation but a negative intra-cohort slope, the cohort’s higher-than-expected risk of arrest declines with age.

To better compare the cohort-specific life-course trajectories of Black and White involvement in homicide, we summarize the estimated age, period, and age-by-period interaction effects for each birth cohort in Figures 3 and 4. Because of the imbalanced data structure (e.g., Cohort 2000 only has one age-specific rate available at the end of the study period in 2015), we present six cohorts’ race-specific homicide patterns. These cohorts have data available for at least ages 15 to 34, thus capturing the age groups with the highest risks of homicide offending and victimization.⁷ The mean difference between the Black and White homicide offending/victimization rate is subtracted from these curves, so the Black and White cohort trajectories can be compared using the same scale despite the absolute differences across racial groups. The age distributions of the predicted offending and victimization rates for each cohort and racial group without adjustment for the absolute differences in race-specific rates are in Appendix IV.

Figure 3 displays how the age trajectories of homicide offending of the six race-specific cohorts are shaped by social change and cohort-specific experience. Risks are higher among young age groups in all cohorts and racial groups and similarly decline with age after the peak. However, estimated trajectories also show important differences across cohorts. First, the peak age of homicide is at 22 for most cohorts except for Cohort 1970 and Cohort 1975—when teenagers (age 15–19) had almost the same level of homicide risk as young adults (age 20–24). The relatively high teenager homicide offending risks can be attributed to the age-by-period interaction—teenagers being exposed to greater homicide risks during the crack epidemic in the late 1980s and early 1990s. Second, some birth cohorts display a rapid decline in homicide risks after reaching the peak age, while for others the decline is more gradual. For instance, the age offending curve of Cohort 1970 dropped quickly to the mean after reaching the peak at the early twenties, while the curve of Cohort 1975 declined more slowly and stayed positive until the early thirties. Third, although different racial groups born around the same time demonstrate consistent age-homicide trajectories, the effect fluctuations are, again, greater for Black cohorts than for White cohorts. For instance, both Black and White cohorts born around the 1960s are subject to greater risks of homicide offending at young ages, but these heightened risks are much greater for the Black cohort than for the White cohort. Both Black and White cohorts’ homicide offending declines with age after the peak, but the declining slope is steeper for most Black cohorts than White cohorts. In sum, consistent with expectation, there is variation in age-homicide trajectories across different cohorts and between racial groups.

Figure 4 presents the age-homicide victimization trajectories of six Black cohorts and six White cohorts. Consistent with the offending patterns, these trajectories confirm that homicide victimization trajectories vary across cohorts and racial groups. Different racial groups born around the same time share similar victimization trajectories, but the effect fluctuations are greater for Black cohorts than White cohorts. However, the Black-White differences in victimization seem smaller than the racial differences in offending. In addition, overall, all of the six cohorts have flatter age trajectories in homicide victimization compared to homicide offending. For instance, the peak age for homicide victimization of Cohort 1965 is in the late twenties while the peak age for homicide offending of Cohort 1965 is in their early twenties. For Cohort 1970, the risks of homicide offending are very similar during adolescence and early adulthood, but the risks of homicide victimization are much higher in their early twenties. Again, these findings show many commonalities in the age and race profiles of victims and offenders (i.e. the victim-offender overlap), but also show that homicide offending and victimization have important differences in their trends and effects (Jennings et al. 2010).

Supplemental Analyses

A major concern of using official statistics collected by law enforcement agencies such as the SHR is that the data may be affected by law enforcement practice changes rather than reflecting actual changes in behaviors. In response, we conduct supplemental analyses with homicide mortality data collected in the National Vital Statistics System by the Centers for Disease Control and Prevention (CDC). As shown in Appendix V and VI, these results demonstrate almost identical patterns to the results of the homicide victimization data from SHR in our main analysis. To be clear, the CDC does not contain data on homicide offenders.

In addition, we also fit an additional model using the traditional APC-Mixed Effects method proposed by O’Brien and colleagues (2008) as a robustness check in Appendix VII. Findings using this alternative model support similar substantive conclusions in regards to general APC effects. However, the mixed effects model does not allow the estimation of intra-cohort life-course trajectories as it was developed within the traditional APC framework with an assumption of constant cohort effects. Detailed discussions on the differences between the APC-I model and the traditional models are included in Appendix I.

Discussion

The current study applied the APC-I model to analyze the age trends in alleged homicide offending and victimization of Black and White birth cohorts. By analyzing the homicide data across 45 years, we find both consistency and differences in the age, period, and cohort effects on alleged homicide offending and victimization between the two racial groups. Our results reveal that the overall shape of the age, period, and cohort distributions are mostly similar between racial groups. This is an important finding which emphasizes the degree to which social forces affecting homicide involvement traverse racial delineations, both when aggravating and when mitigating risks. Within a cohort, the two racial groups showed a comparable shape of age trajectories, with homicide concentrated among teenagers and young adults. Similarly, a sharp decline in period effects during the 1990s affected both racial groups, as did the subsequent plateau since the mid-2000s. Finally, both racial groups’ cohort effects increased and decreased in relative tandem.

However, while the overall patterns of APC effects are comparable across Black and White populations, our results reveal important differences. The directions of the effects are similar, but the effect sizes vary substantially. Generally, homicide involvement by Black persons, particularly alleged homicide offending, was much more volatile. Holding constant the absolute level differences across the two racial groups, Black offenders display greater variation across age groups, periods, and cohorts. Although our current study cannot directly identify or estimate the mechanisms underlying these differences, we offer elaborations around two possible contributors: (1) the structural disadvantages resulting from racial segregation and institutional segmentation; and (2) the racial disparities and biases in the criminal justice system that disproportionately affect the Black population.

First, although the macro-level events (e.g. an economic downturn, the crack cocaine epidemic) may affect all population subgroups simultaneously, young Black males are particularly vulnerable to the criminogenic effects of some social changes (Peterson et al. 2010; Sampson and Wilson 1995; Vogel et al. 2019). Marginalized segments of the population may suffer from a saturation of other social disadvantages (e.g. poverty and family disruption). This can prevent them from achieving desired goals or from protecting themselves from additional strain, and reduces the buffer from the harmful influences of negative social change (Currie 1997; Peterson et al. 2010; Sampson and Wilson 1995). For example, individuals who lose a job during an economic downturn can support themselves should they have accumulated assets, savings, and/or a supporting family. In contrast, for individuals living paycheck-to-paycheck, the loss of employment carries the overwhelming risk of immediate hardship and strain. Similarly, a well-off individual facing an increase in the crack trade or violence in a given neighborhood can simply choose to move, reducing their risk of homicide offending and victimization, where many others do not have such choices. Prior literature also has revealed that the deindustrialization of central cities in the 1970s (e.g., the transformation from the manufacturing industry to the service industry) has brought especially adverse effects on Black communities since, at the time, racial minorities were disproportionately employed in the traditional manufacturing industries (Peterson et al. 2010; Wilson 1987). Consequently, many young Black males were trapped in disadvantaged neighborhoods with limited access to job opportunities and/or quality education, where some were exposed to the drug trade and gang violence (Hagan 1997).

Another possible explanation for the greater magnitude of age, period, and cohort effects observed among the Black group are racial disparities in criminal justice processing, or more broadly the racial biases in society. Black persons residing in low-income neighborhoods may be subject to greater police surveillance (Kochel, Wilson, and Mastrofski 2011; Warren et al. 2006). Consequently, part of the difference between Black and White alleged offending patterns may be a consequence of differences in enforcement. Sentencing research also reveals that Black defendants are more likely to be sent to prison, receive long sentences, be disconnected from family and community, have long criminal histories, and suffer many other collateral consequences related to a conviction (Doerner and Demuth 2010; Steffensmeier, Ulmer, and Kramer 1998; Tonry and Melewski 2008). These consequences accumulate with other social disadvantages associated with race, affecting individuals, their families, and even future generations (Wakefield and Uggen 2010). Moreover, persons who are Black are more likely to be mistakenly identified as crime suspects than White persons (Oliver and Fonash 2009; Smalarz et al. 2016). As the SHR data does not differentiate between alleged homicide offenders who were arrested from those who were not arrested, the demographic information of suspects who are not arrested relies on the account of victims and eyewitnesses who may make mistakes. Such issues related to racial disparities or discrimination can partially affect our results that showed greater Black-White differences in alleged offending than in victimization patterns, as the demographic characteristics of homicide victims are usually verifiable (from the body), and are much less subjective than the characteristics of suspected offenders.

In short, compared to their White counterparts, young Black males are more likely to grow up in disadvantaged neighborhoods, live in households with parental incarceration, have greater risks of surveillance by law enforcement, and be mistakenly identified as suspects. While all youth are subject to greater risks of crime involvement, young Black males might also be subject to added risks of contact with the criminal justice system because of their race, with collateral consequences to their entire life course (Travis, Western, and Redburn 2014; Warren et al. 2006).

We also find that although all cohorts’ homicide risks decline with age after reaching the peak, the decline after age 30 is more accentuated for Black than White cohorts. This is particularly notable for alleged homicide offending. Based on prior research we propose three selection processes that may serve as explanations: (1) mass incarceration that incapacitates the Black population disproportionately; (2) higher mortality risks for Black individuals than White individuals; and (3) differential life-course transitions and opportunity between Black and White populations. First, incarceration removes a large number of young homicide offenders from the population at risk. Many cohort members incarcerated at young ages may not return to their communities for the rest of their life or until older ages. That is particularly true for serious violent crimes like homicide, which carry longer sentences. Research revealed that cohorts who were teenagers and young adults in the 1980s and 1990s were substantially more likely to be incarcerated than other cohorts (Shen et al. 2020). This selection effect is greater for Black individuals than White individuals because of the racial disparities in the criminal justice system and the differential involvement in crimes (Steffensmeier et al. 1998; Warren et al. 2006). Thus, harsher sentences for Black persons, regardless of the reason, can increase incapacitation, and consequently lead to a steeper decline in the homicide trajectory at older ages.

A second selection effect that may explain the greater decline in Black homicide involvement with age is mortality. Crime and delinquent behaviors are associated with excess mortality (Laub and Sampson 2003; Repo-Tiihonen, Virkkunen, and Tiihonen 2001). For instance, Karmen’s (2000) study of the New York homicide decline in the 1990s suggests that untimely death caused by murder by fellow criminals, drug overdoses, and AIDS permanently removed a large number of potential offenders from the city’s population. Other studies also revealed that the increase in homicide deaths in the 1990s played an important role in widening the Black-White life expectancy gap (Geronimus, Bound, and Colen 2011; Harper et al. 2007). The greater decline in homicide offending at older ages observed among Black cohorts in our finding may be related to the heightened risks of mortality revealed in prior studies. However, this hypothesis is speculative, and can only be empirically tested when more life-course data for young cohorts (who grew up after the 1990s) becomes available. In fact, recent research suggests that homicide declines since the early 1990s have reduced the Black-White life expectancy gap by 17% (Sharkey and Friedson 2019). If the mortality selection argument is true, the Black-White gap at older ages (i.e., after age 30) in the trajectories should decrease for young cohorts born after the 1990s, as they were exposed to lower homicide risks in recent years.

Another selection effect worth mentioning is the Black-White differences in life-course transitions and labor market opportunities. Prior literature suggests that Black men tend to marry and have stable employment later than White men (Manning, Brown, and Payne 2014; Payne 2012). As a result, young Black men are exposed to greater criminogenic risks for a longer part of their young ages (e.g., drugs and gangs) but soon transition to social institutions that protect them from crime involvement at older ages. This timing differential may lead to greater effects of getting older on Black cohorts’ homicide involvement than for their White counterparts.

A third important finding from this study is the variation in age-homicide trajectories across birth cohorts of homicide offenders, a finding which contradicts the age-crime invariance thesis. In our results, alleged homicide offending peaks at around ages 20–24 for most cohorts, but Cohorts 1970 and 1975 demonstrate higher homicide offending rates during their teenage years (15–19) relative to other cohorts. The Black cohorts born in 1975–79 have higher homicide offending at ages 15–19 than at ages 20–24, thus shifting the peak age of this cohort’s homicide trajectory to a younger age group. These changes are possibly linked to the crack epidemic in the late 1980s and early 1990s, which had a profound impact on these cohorts’ teenage years. During the epidemic, members of these cohorts were drawn to a circle of weaponization and violence related to the crack trade. As older dealers were incapacitated by the tough-on-crime policies at the time, much of the trafficking was operated by young traders, who were close to their peak age of criminal propensity (Blumstein and Rosenfeld 1998). Thus, these youth might have been exposed to much greater risks of gang violence when they reached their late teens than any preceding or succeeding cohort (Blumstein and Rosenfeld 1998; Blumstein and Wallman 2006). In addition, as shown in Figure 1, this change in teenage offending is particularly prevalent for the homicide trajectories of Black persons, who were more likely to reside in disadvantaged communities.

Finally, another Black-White difference that hints at a particularly concerning trend is the differential decline in cohort effects since the 1990 Cohort in both alleged offending and victimization (Figure 1c and 2c). This finding highlights the importance of differentiating cohort effects from period effects. If we focus on the racial differences in the period pattern only, it may lead us to a conclusion that the Black population benefited more from the recent social changes and demonstrate greater homicide decline than the White population in alleged offending (see Figure 1b). However, after decomposing the inter-cohort deviations, we find that cohort effects for White individuals born after the 1990s have declined to the lowest levels seen in our entire series, but effects for Black individuals have declined much less and remained above the mean until the end of our series. Consequently, there is a growing difference between the cohort effects of Black and White populations. That difference is unprecedented in the entire series, and is consistent for both offending and victimization. Despite improvements in American urban centers and the consistent plummeting crime rates observed across cities and racial groups until the Covid-19 pandemic (Sharkey 2018), young Black cohorts do not seem to have benefited as much as young White cohorts from these positive social changes. Much of the inequities we observed can be the result of disparities in crime and justice of the past (Pettit 2012). Although mass incarceration has declined in recent years, it may still disproportionately affect minority communities, including children of incarcerated parents.

This study has several limitations. First, official crime statistics may have problems such as underreporting issues or influenced by law enforcement policies which may not reflect patterns of offending. The underreporting issue is less of a concern of the current study as our focus is the relative distribution of homicide across age, period, and cohort rather than the absolute level of homicide. As shown in prior research, although the SHR has consistently lower reported homicide rates than the homicide mortality rates from the CDC, the overall time trends are identical across the two data sources (Regoeczi et al. 2014). For the problem of law enforcement changes, we believe that homicide in particular, because of its severity and its relatively unambiguous definition, is less sensitive to disparities in enforcement. Nonetheless, the demographic characteristics of alleged offenders are subject to personal accounts, can be racially biased, as discussed previously. In this regard, the fact that victimization and alleged offending share similar patterns and effects serves as a robustness check. In addition, we have conducted supplemental analyses using homicide mortality data from the CDC, with consistent findings compared to our SHR victims data. For the alleged offender data, we argued that racial biases or discrimination in the criminal justice system or among the victims and witnesses who provided race information for some alleged offenders may serve as one possible explanation for the effects we have identified. Future studies may use alternative sources, such as self-report survey data, and possibly attempt to parse out the influence of racial biases and discrimination. In addition, about 30% of alleged offenders in the SHR are missing data about their race and age. Therefore, we have no information about the age and race distribution among these missing cases or about the exact precision of existing age and race information reported, which are issues that can affect our estimates.

Second, the use of aggregated data at the national level can omit sub-national trends and patterns that are associated with the mechanisms underlying our findings. This can be particularly relevant in a country as large and diverse as the U.S., where race relationships and crime may have local determinants. While our findings correspond to a national average, future research can investigate the extent of the heterogeneity in these effects across lower levels of aggregation.

Third, because of the inconsistency in the historical reporting practices in the SHR, we are unable to disaggregate the sample into more race and ethnicity groups. Considering, for instance, the rapid increase of the Hispanic population in the U.S., future research with appropriate measures of race and ethnicity should differentiate the cohort trajectories of Hispanic homicide offending and victimization.

Finally, while the APC-I model enables the estimation of age, period, and cohort effects, our analysis does not speak about the specific mechanisms underlying the effects that we identified. We provide detailed discussions about the cohort effects and racial differences, but our explanations for these differences are speculative and are based on an informed reading of extant research, rather than being a result of the current empirical analysis. Future research can supplement ours in investigating specific drivers using methods such as fixed effects regression and time-series models.

Despite these caveats, the current study has important implications on the life course and age-period-cohort literature in criminology. It further underscores the importance of social and historical context in shaping the life-course trajectories of different population subgroups—an important principle of the life course paradigm (Elder 1998). With the new APC-I model, our study provides a novel extension of the age-period-cohort analysis in criminology, highlighting the importance of both between-cohort differences and within-cohort life course dynamics. This research reaffirms Ryder’s (1965) seminal work decades ago: cohort is an important concept for understanding social change and a vehicle to connect individual-level life course processes and macro-level social context. More importantly, our results suggest that the influence of external social forces may unfold differently depending on race. The long history of racial segregation and discrimination in American society has left imprints on the distributions of opportunities and resources in a wide range of institutions. Racially differentiated punishments and rewards in education, the labor market, and criminal justice may be mutually reinforcing and contributing to greater disadvantages in the life course trajectories of certain Black cohorts.

APPENDIX

Appendix I. The APC-I Model

How the APC-I Model differs from traditional APC methods

Many methods have been developed under the traditional APC framework including the Intrinsic Estimator (IE) and the APC-Mixed model. Such methods attempt to estimate the independent and additive effects of age, period, and cohort on crime trends. In doing so, they implicitly assume that cohort effects can occur when the influence of social changes on the outcome is uniform across age groups. That is, regardless of the identification strategies, they all attempt to estimate the independent and additive effects of age, period, and cohort on crime trends. However, as Luo and Hodges (2022) and Lu and Luo (2021) argued, this traditional framework contradicts the demographic literature that clearly defines cohort effects using an age-period specification (Hobcraft, Menken, and Preston 1982; Ryder 1965). The APC-I model conceptually differs from the traditional APC framework by explicitly modeling cohort effects as a structure of the interaction terms between age and period main effects; that is, considering cohort effects as differential period effects depending on age. By so doing, the APC-I model acknowledges and accommodates for the inherent interdependence between age, period, and cohort.

As a sensitivity analysis, we compared the results of our results using the APC-I model against another APC method. Specifically, we fit the APC-mixed effects model (O’Brien, Hudson, and Stockard 2008)—a traditional APC method that has been used in criminological research. Results from the two modeling strategies yield similar findings in terms of the age main effects, period main effects, and the inter-cohort differences compared to the main results in Figure 1 in the manuscript. Detailed results are available in Appendix VII. Because the other popular methods including the Intrinsic Estimator (IE) have been subject to criticism and generates estimates that are highly dependent on otherwise innocuous choices such as coding schemes (Bell and Jones 2014; Fosse and Winship 2019; Luo et al. 2016; Luo and Hodges 2020), we do not use these models in the sensitivity analysis.

Despite the similarity in numeric estimates, two cautionary notes are necessary to help readers understand the difference between the APC-I model and the APC-mixed effects model. First, the estimation results based on the APC-mixed effects model may be difficult to interpret because there is no variation in the third variable (e.g., cohort) while holding the other two (e.g., age and period) constant. In contrast, the main effects and interaction effects estimated using the APC-I model are meaningful because the model explicitly acknowledges the interdependence of the three predictors. Second, most extant methods including the APC-mixed effects model assume cohort effects to be additive and constant across ages, so they focus on inter-cohort differences (as shown in Figure 1c and 2c) and cannot provide estimates of potential life-course changes (as shown in Figure 3 and 4). In contrast, the APC-I model allows researchers to estimate and test potential life-course dynamics in crime involvement by evaluating variation in cohort effects across age groups.

Most importantly, the APC-I approach implemented in the current study is qualitatively different from extant APC models. We consider the APC-I method as a new conceptual framework for examining cohort effects in that it recognizes the interdependence of the three APC factors and thus treats cohort as interactive rather than additive effects in relation to age and period effects. While in the current study the APC-I model has yielded comparable inter-cohort difference estimates to a traditional method, we emphasize that it differs conceptually from these approaches (so do their interpretations) and that inter- and intra-cohort patterns in crime involvement have not been simultaneously examined in most prior APC studies.

The Sum-to-Zero coding scheme

All APC-I models are estimated using the sum-to-zero coding in R (R-3.6.2). Different from the dummy coding scheme that sets one age or period category as the reference, the sum-to-zero coding (i.e., the effect coding, ANOVA coding) for categorical variables is, loosely speaking, equivalent to the mean centering for continuous variables, where a coefficient represents the deviation from the overall mean and an interaction term represents the deviation from the main effects. This approach also allows us to adjust for the absolute differences in the homicide rates between Black and White populations when comparing age, period, and cohort effects across racial groups. Specifically, two population subgroups may have different average levels of crime involvement but still share similar trends across time or similar distribution across age groups. Hence, factors that contribute to the between-group differences are not necessarily the same as the factors that contribute to differences in trends. Since the focus of the current paper is on the distribution across cohorts and age groups, the sum-to-zero coding allows us to adjust for differences in the level before comparing trends. Predicted cohort trajectories without the adjustment for the race-specific means are included in Appendix IV.

Appendix II. Percent cases missing race and age information by period in the data of alleged homicide offenders and victims

	Alleged Offenders			Victims

Period	Total # Cases	% Unknown for Race	% Unknown for Age	Total # Cases	% Unknown for Race	% Unknown for Age
1975	82,576	21.80%	24.30%	76,717	0.25%	1.45%
1980	111,260	26.65%	28.60%	101,289	1.09%	1.99%
1985	102,957	25.93%	28.90%	94,623	0.75%	2.21%
1990	128,557	28.94%	33.50%	113,736	0.87%	1.81%
1995	94,198	28.27%	33.50%	83,239	1.08%	2.09%
2000	82,758	28.25%	33.90%	73,181	1.33%	2.21%
2005	88,344	27.23%	31.60%	76,181	1.38%	1.92%
2010	78,730	26.56%	30.20%	67,228	1.20%	1.21%
2015	70,690	29.40%	31.80%	61,323	1.55%	1.17%
Total	840,070	27.06%	30.80%	747,517	1.03%	1.82%

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Notes: There are 840,070 alleged offenders in the dataset, 258,550 cases (31%) have no age information, and 227,307 cases (27%) have no race information. Most of these missing cases (222,504) are overlapping and have no information for both race and age. There are 747,517 victims, 13,627 (1.8%) have no age information and 7,667 (1%) have no race information, 2,153 cases have no information for both.

Appendix III — Predicted Cohort-Specific Age Trajectories of Alleged Homicide Offending and Victimization by Race, All Cohorts

Notes: The horizontal line in each of these graphs represents zero deviation from the global means of Black or White homicide offending. Dots and triangles shown in the figure represent the total deviations from the mean race-specific rates under the influence of age, period, and age-by-period interaction effects estimated by the APC-I model.

Appendix IV. — Predicted Cohort-Specific Age-Homicide Trajectories by Race, without Adjusting for Race-Specific Means

Appendix V. — Estimated Age, Period Effects and Inter-cohort Deviations on Homicide Mortality by Race (Data from National Vital Statistics System)

Notes: The horizontal solid lines in 1a and 1b represent zero deviation from the global mean. The horizontal solid line in Figures 1c represents zero deviation from the expectation determined by age and period main effects.

Appendix VI. — Predicted Cohort-specific Age Trajectories of Homicide Mortality by Race (Data from National Vital Statistics System)

Notes: The horizontal line in each of these graphs represents zero deviation from the global means of Black or White homicide. Dots and triangles shown in the figure represent the total deviations from the mean race-specific rates under the influence of age, period, and age-by-period interaction effects estimated by the APC-I model.

Appendix VII. — Estimated Age, Period and Cohort Effects on Alleged Homicide Offending by Race, APC-Mixed Models

Notes: The horizontal solid lines in 1a and 1b represent zero deviation from the global mean. The horizontal solid line in Figures 1c represents zero deviation from the expectation determined by age and period main effects.

Footnotes

Homicide counts include murder and nonnegligent manslaughter and exclude suicides, traffic fatalities, and fetal deaths. Manslaughter by negligence cases were dropped from the dataset.

About 30% of the cases in the SHR data have no demographic information for the offender. We analyzed missing cases across the study periods in Appendix II. The proportions of missing cases for race and age are generally consistent across periods except for 1975–1979, which has slightly fewer missing cases. However, the discrepancies are relatively small. We discussed this limitation in the discussion section.

The SHR data includes offenders for all ages from age 10 or younger to age 80 or older. However, we decide to truncate the sample to ages 15 to 49 because very few cases fall outside this age range. We conducted supplemental analyses using data from ages 15 to 54 and ages 15 to 59, and results are consistent.

⁴

When applying the APC-I model to a normally-distributed outcome in aggregated data, the age-by-period interaction terms are confounded with the error terms as there is no replication per cell. However, because our outcome is a count variable and can be analyzed with Poisson regression models, we can still test the interactions when the model is saturated. Even if the outcome variable is continuous, one can still examine the presence of cohort effects with the global deviance test or using Tukey’s test of additivity (Tukey, 1949; See Luo & Hodges, 2020:37). All Tukey’s test of additivity conducted within each race-specific sample are statistically significant, suggesting the presence of interaction effects.

⁵

Because we use population, and not sample data, and because we do not attempt to make any statistical inferences outside the scope of our data, the statistical significances included in the tables are for reference only. Readers should focus on effect sizes when comparing differences across age, period and cohort or between racial groups (Alexander 2015; Berk, Western, and Weiss 1995; Hartley and Sielken 1975).

⁶

Due to the imbalanced data structure, one has to be cautious when interpreting inter-cohort estimates of cohorts that have only one or two observations available. For instance, Cohort 1995 only has data available at age 15–19 and 20–24 within the study period, so the inter-cohort deviations for Cohort 1995 represents the cohort’s deviations from the mean homicide risks when they were young. This truncated data problem is inherent in all aggregate-level APC data and cannot be addressed by statistical models. However, this issue does not affect between-race comparison.

⁷

Race-specific homicide trajectories of all cohorts are presented in Appendix III.

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PERMALINK

Social Change and Race-Specific Homicide Trajectories: An Age-Period-Cohort Analysis

Yunmei Lu

Liying Luo

Mateus Rennó Santos

Abstract

Objectives:

Data and Methods:

Results:

Conclusions:

Introduction

Background

Age-Crime Relationship and the Life-Course Perspective

Cohort, Social Change, and the Life Course

Race, Social Change, and the Life Course

Expectations

Data and Methods

Table 1.

Analytical Strategy

Results

Age and Period Main Effects

Table 2a.

Figure 1.

Figure 2.

Inter-Cohort Deviations

Table 2b.

Table 2c.

Table 3.

Figure 4.

Intra-Cohort Life-Course Trajectories

Table 4.

Figure 3.

Supplemental Analyses

Discussion

APPENDIX

Appendix I. The APC-I Model

How the APC-I Model differs from traditional APC methods

The Sum-to-Zero coding scheme

Appendix II. Percent cases missing race and age information by period in the data of alleged homicide offenders and victims

Appendix III.

Appendix IV.

Appendix V.

Appendix VI.

Appendix VII.

Footnotes

References

References:

ACTIONS

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