Violent Physical Bullying Victimization at School: Has ...

Violent Physical Bullying Victimization at School:Has There Been a Recent Increase in Exposureor Intensity? An Age-Period-Cohort Analysisin the United States, 1991 to 2012

Qiang Fu1& Kenneth C. Land2

& Vicki L. Lamb3

Accepted: 18 May 2015# Springer Science+Business Media Dordrecht 2015

Abstract Using data from an annual nationally representative survey of U.S. 8th, 10th,and 12th graders from 1991 to 2012, this paper applies a new two-step method to studytrends in self-reports of victimization during the last year from four forms of violentbullying at school (threatened without injury, threatened with a weapon, injury withouta weapon, injury with a weapon). First, we develop a statistical algorithm for estimat-ing, for each school year, the exposure probability (likelihood or risk of being victim-ized) and intensity rate (rate of victimization among those exposed to the risk of beingvictimized) parameters of zero-inflated Poisson models of truncated and combined self-reported victimization frequency data for the four forms of violent bullying. Estimatesof both the exposure to, and intensity of, the self-reported frequencies for each the fourforms for each of the grades show increases into the middle part of the 2000–2010decade with slight declines in the years 2008–2012. Exceptions are found for intensityrates of threats without injury and threats with a weapon among 12th graders. Second,age-period-cohort analysis was applied to the estimated exposure and intensity param-eters of violent bullying victimization. This analysis reveals: (1) that both the exposureprobabilities and intensity rates decrease from the 8th (typically 13–14 year olds) to the10th (typically 15–16 year olds) to the 12grades (typically 17–18 years old); (2) that theschool years 2006 to 2012 were associated with decreases in time period exposureprobabilities and increases in intensity rates - fewer students victimized per school yearbut those who are victimized are victimized more frequently; and (3) that birth cohorts

Child Ind ResDOI 10.1007/s12187-015-9317-3

This research was supported by research grants from the Foundation for Child Development.

* Qiang [email protected]

1 Department of Sociology, The University of British Columbia, Vancouver, BC V6T 1Z1, Canada2 Department of Sociology, Duke University, Durham, NC 27708, USA3 Department of Human Sciences, North Carolina Central University, Durham, NC 27707, USA

born since the late-1980s had decreases in intensity rates, but their exposure probabil-ities increased until the most recent (1995–1996) cohorts for which the exposureprobabilities have stabilized or declined.

Keywords Victimization . Violent bullying at school . Zero-inflated Poissonmodels .

Age-period-cohort analysis

1 Introduction

Violent physical school bullying has been a focus of research by social scientists fordecades, and such bullying victimization and its negative consequences also havereceived much attention from the media and policy makers in recent years (Berger2007; Srabstein 2008). As compared with their peers, victims of school bullying tend tohave lower self-esteem, fear for their safety at school, and skip classes and avoidcontacts with other students, which can lead to depression and poor school performance(Batsche and Knoff 1994; Bosworth et al. 1999; Olweus 1994; Smith and Myron-Wilson 1998; Smith and Sharp 1994). Furthermore, deaths may result from violentphysical bullying victimization if victims are seriously injured or resort to suicide afteroverwhelming self-devaluation (Olweus 1994).

However, mixed findings from previous research on the trends and patterns ofviolent bullying make it difficult to determine the extent to which awareness of recentincreases in physical bullying victimization in the U.S. is based on recorded data versusincreased media attention. First, a mismatch between prevalence of physical schoolbullying offenders and bullying victims has been widely reported, so the targets ofschool bullying are frequently concentrated among a small proportion of students whoare repeatedly bullied at different levels of frequency (Espelage and Swearer 2003;Olweus 1994; Pellegrini and Bartini 2000; Pellegrini and Long 2002; Pepler et al.2008; Sourander et al. 2000). To reconcile the mismatch, we propose that it is essentialto distinguish: (1) the exposure to bullying victimization from (2) the intensity or rate ofbullying victimization so that the bullying paradigm takes into account both the riskand the intensity of being bullying for those who are at the risk of being bullied.Second, the question of whether trends in bullying victimization has increased ordecreased across recent decades is complicated by the fact that prevalence rates ofviolent physical bullying victimization at school vary by age (or grade level) of thestudents, generally decreasing as students move from the middle school and early highschool years to the later high school years. This raises the possibility that observedprevalence rates of victimization represent an intertwining of age, time period, andcohort effects. And because few researchers have ever conducted an age-period-cohortanalysis of bullying victimization in the U.S., it has been impossible to determine theextent to which trends in the prevalence of violent physical bullying victimization, netof age-dependent effects, are due to childhood experiences of different cohorts, ordifferential temporal contexts.

Based on a nationally representative annual survey of 8th, 10th, and 12th graders inthe United States from 1991 to 2012, we address these questions raised by existingliterature using a new two-step method. In the first step, we develop an algorithm forestimating the exposure and intensity parameters of zero-inflated Poisson (ZIP) models

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of truncated and combined frequencies of four extreme forms of violent physicalbullying victimization for data collected each school year, 1991–2012. In the secondstep of the analysis, age/grade, period, and cohort effects on the annual exposure andintensity parameters obtained from the first step are estimated and analyzed forsubstantive implications. This research thus not only provides a new method forinvestigating the temporal trends of combined and truncated count data but alsocontributes new empirical knowledge about the trends of victimization from school-based physical bullying in the U.S. over the past two decades.

2 Definitions and Characteristics of Violent Physical BullyingVictimization

While much empirical research on bullying victimization in school is guided byOlweus’s (1991: 413) definition that regards a victim of school bullying as Bbeingbullied when he or she is exposed, repeatedly and over time, to negative actions on thepart of one or more other persons^, other scholars have proposed their own definitionsof school bullying. For example, the definition given in a study on British childrenemphasized both the instrumental goal of school bullying and deficiency of socialresources possessed by victims, as Bachieving or maintaining social dominance throughovertly aggressive means which occur because the victims have no sufficient skills orcapacity to integrate with their peer group^(Arora and Thompson 1987). According toSmith and Sharp (1994: 2), bullying victims are targets of Bsystematic abuse of power.^A uniform definition of bullying among youths has recently been put forth by theCenters for Disease Control and Prevention and U.S. Department of Education:

Bullying is any unwanted aggressive behavior(s) by another youth or group ofyouths who are not siblings or current dating partners that involves an observedor perceived power imbalance and is repeated multiple times or is highly likely tobe repeated. Bullying may inflict harm or distress on the targeted youth includingphysical, psychological, social, or educational harm. (Gladden et al. 2014: 7)

Granted that bullying can take a variety of forms, including physical bullying, verbalbullying, cyber bullying and so on, this paper specifically focuses on violent physicalschool bullying. Based on the recent definition of bullying above, physical schoolbullying, like other forms of bullying is associated with a series of harmful behaviorsoccurring repeatedly over time and characterizes an imbalance of power betweenbullies and victims. We thus emphasize three aspects of bullying in the research ofphysical bullying victimization: harm (deliberate, harmful and aggressive behaviorstowards victims), repetition (such behaviors are carried out repeatedly and over time)and power imbalance (the bully is considered to be stronger and more powerful thanthe victim) (Batsche and Knoff 1994; Berger 2007; Olweus 1994).

Although the first two aspects of bullying victimization are defined explicitly, thepower-imbalance aspect should be conceived as fluid and context-specific. First, powerimbalances may change over time when the victims’ means of defending themselves(rocks, sticks and weapons) are readily available (Rigby 2002). Second, it is difficult toidentify the sources of power imbalances or to evaluate whether an overall power

Violent Physical School Bullying in the US, 1991 to 2012

imbalance exists because multiple forms of power (e.g., coercive power and legitimatepower) are associated with interpersonal relations (French and Raven 1959). Third, thepresence of power imbalance in indirect or more subtle forms of bullying has beenquestioned because less powerful individuals may spread rumors against others (Rigby2002). In other words, the power-imbalance aspect is probably more relevant to violentphysical bullying as victims are repeatedly attacked or harassed by students of greaterpower or strength.

3 Research on Temporal Patterns of Violent Physical BullyingVictimization

Before evaluating existing literature on temporal patterns of violent physical bullyingvictimization, we discuss the relevance of three related yet distinct temporal varia-tions—age, period, and cohort effects—to the study of bullying victimization. Ageeffects refer to physiological, psychological, and social changes embedded in the agingprocess. It has been shown that the transition from childhood to adolescence ischaracterized by rapid body changes and reestablishment of social relations, whichhave profound influences on physical bullying victimization (Olweus 1994; Pellegriniand Long 2002). In particular, it has been argued that early adolescence, known as aBbrutalizing^ stage of the life course, is critical for the research on physical bullyingvictimization because physical aggression increases in tandem with growth of bodysizes (Cairns and Cairns 1986). Period effects relate to variations in violent physicalbullying victimization taking place at a time period that affect all cohorts simultaneous-ly, and embodies the concurrent influences of socio-economic factors, historical eventsand environmental problems on students from all grades (Glenn 1976; Yang et al.2008). For example, a temporal change in bullying victimization can occur with theintroduction of new technology (such as the Internet and the development and diffusionof social networking sites) or legislation against school bullying, which simultaneouslyaffects behaviors of all students at different ages/grades. Cohort effects pertain tovariations in bullying victimization across groups of students who share a commonyear of birth or other life event such as being enrolled in the same school grade, whichare affected by life-course experiences associated with membership in a particular birthcohort. Although there is a dearth of literature on cohort effects of bullying victimiza-tion, early exposure to overprotective mothers and higher levels of depressive symp-toms may lead to being physically bullied in middle and high schools (Fekkes et al.2006; Pepler et al. 2008; Smith and Myron-Wilson 1998; Sourander et al. 2000). Theepidemic of school bullying can be shifted by differential exposure to risk factors ofbeing bullied across different cohorts.

3.1 Age Trajectories of Bullying Victimization

With regard to age/grade trajectories of physical bullying victimization among adoles-cents, declines in victimization from early adolescent to high-school years have beenwidely documented in both cross-sectional and longitudinal studies worldwide(Espelage and Swearer 2003; Nansel et al. 2001; Olweus 1994; Pellegrini and Bartini2000). For example, a longitudinal study conducted in Toronto, Canada in which

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participants aged 10 to 14 were followed more than 7 years, revealed that self-reportedbullying victimization declined with increasing age regardless of forms of bullyingvictimization (Pepler et al. 2008). Results from a cross-sectional survey of more than130,000 Norwegian students also showed that the percentage of students reported beingexposed to direct physical bullying (relatively open attacks on the victim) decreasedfrom grade 7 to grade 9 (roughly corresponding to ages 14–16) (Olweus 1994).

There are several mechanisms accounting for the decreases of physical bullyingvictimization from early adolescence to young adulthood, especially in the form ofviolent bullying. First, early adolescence is a critical period when physical strength andbody sizes grow rapidly with age. Although younger students tend to be victimized byothers in school due to an imbalance of physical power between themselves and olderstudents, as they age there are fewer older students and thus a lower risk of beingvictimized (Cairns and Cairns 1986; Smith et al. 1999).

Second, direct physical bullying may indicate that younger students lack essentialsocial and verbal skills with which to respond to such bullying. Once they acquire thesecomplex skills in higher grades, direct bullying gives way to more subtle and sophis-ticated styles of aggression (Scheithauer et al. 2006; Smith et al. 1999; Woods andWolke 2004). An age decline in violent forms of bullying is expected accordingly,although the prevalence of indirect bullying may stay constant or even increase.

Third, as students grow up and move from a familiar school environment to a newone with the breakup of existing peer-to-peer social network connections, physicalbullying and violence may be adopted as an aggressive strategy in order to attaindominance and re-establish peer relations in transitions to a new school environment,such as from primary schools to middle schools or from middle schools to high schools(Pellegrini and Long 2002; Staff and Kreager 2008). After dominance relationships areestablished through a series of aggressive or defensive social interactions (such asbullying) and reconciliation, more cooperative strategies are used by older students tomaintain new peer relations in a new school environment (Hawley et al. 2002;Pellegrini and Long 2002). Fourth, physical bullying can have a selection effect onthe composition of higher graders because frequent victims tend to be dropouts andthus fail to go on to higher grades. For example, ten percent of high-school dropoutsidentified fear of being harassed or attacked as the most important reason for leavingschool (Greenbaum et al. 1989).1

Finally, there may also be a deterrence-control process at work in the slowdown ofthe frequencies of severe bullying as students age from the 8th grade (typically 13–14 year old) to the 10th grade (typically 15–16 year old). According to ethnographicinterviews of some juveniles assigned to juvenile court supervision for status offensesdetention, there is recognition among at least some juveniles that, upon reaching their16th birthday (generally corresponding to the 10th grade), persons who engage in thesevere forms of bullying are at risk of being prosecuted in adult courts rather than injuvenile courts, and that the sanctions imposed in the adult courts for such behaviors arelikely to be substantially more severe than these imposed in the juvenile courts (Landet al. 1992: Williams’ field notes). If a Bnothing to lose^ attitude about the future isassociated with adolescent involvement in risky behaviors such as the use of weapons

1 Presumably, this selection effect also applies to bullies. Nevertheless, an age decline in bullying victimizationis expected no matter whether such selection effect applies to both victims and bullies.


(Harris et al. 2002), violent bullying will decrease once its potentially more severe costsare recognized by older adolescents.

3.2 Period and Cohort Patterns in Bullying Victimization

Regarding period and cohort changes in bullying victimization, two issues have beenraised by previous research. First, the prevalence of bullying victimization can bedramatically different from that of bullying offending (Pellegrini and Bartini 2000;Pellegrini and Long 2002; Smith et al. 1999; Sourander et al. 2000). For example, an8-year longitudinal study in Finland showed that the prevalence of bullying victim-ization was consistently higher than that of bullying offending from 1989 to 1997,although the former decreased more rapidly than the latter (Sourander et al. 2000). Alongitudinal study conducted in the U.S. reported that the decreases in self-reportedvictimization in early adolescence were at odds with increases in self-reportedbullying (Pellegrini and Bartini 2000; Pellegrini and Long 2002). To reconcile thesefindings, it was hypothesized that only a specific group of students embedded in alarger pool of students in school is actually at risk of being bullied (Pellegrini andLong 2002; Smith et al. 1999). Likewise, it is possible that some of these targets ofschool bullying may report being bullied more intensely than others (Cairns andCairns 1986; Nansel et al. 2001). Therefore, it is important for researchers todistinguish exposure to school bullying from the intensity and frequency of physicalbullying. Studying violent physical bullying trends among 12th grade students usingthe same Monitoring the Future Study (MTF) dataset, Fu et al. (2013) foundprobabilities for being bullied were higher for students who were male, from single-parent or no-parent families, city dwellers, regarded religion as less important, andshowed worse school performance, while persistently higher intensities were foundfor students who were male, African-American, city dwellers, and from single-parentor no-parent families.

Second, the distinction between period and cohort changes in bullying victimizationhas not been properly addressed in prior research. While news reports of bullyingvictimization and its consequences increased dramatically in the 2000s (Fraire et al.2008; Srabstein 2008), only a few studies examined temporal trends of physicalbullying victimization at school. For example, a study conducted in Norway concludedthat school bullying in the 1980s was more serious and frequent than that 10 to 15 yearsearlier (Olweus 1991). Johnston et al. (1993) found that the percentage of 12th gradersthreatened without a weapon increased by one-third from 1980 to 1992. Based on theNational Crime Victimization Survey, Finkelhor (2013) saw a 74 % decline in theprevalence rates of violent victimization at school for youth aged 12–17 from 1992 to2010. However, few studies conducted in more recent years have examined trends ofviolent physical bullying victimization over a relatively long time span. Moreover, littleresearch has ever attempted to identify and separately estimate the age, period, andcohort effects underlying the trends of physical bullying victimization. If there indeedwas an increase in physical bullying victimization in the U.S. in the 2000s, assuggested by media reports, scholars and policy makers should be cautious ininterpreting such an increase as a period effect (e.g., as a failure of the nation’s recentanti-bullying efforts) if the contributions of age and cohort effects have not beendistilled from the data.

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4 Data

This research is based on a nationally representative annual survey of youth in theUnited States, the Monitoring the Future: A Continuing Study of the Lifestyle andValues of Youth (MTF) study (http://monitoringthefuture.org/). Every year thousands ofstudents from approximately 130 high schools nationwide participate in this survey andrespond to a series of questions on values, behaviors, and characteristics of Americanadolescents. Although an annual survey of 12th graders has been a part of this studysince 1975, annual surveys of 8th and 10th graders were not added to the MTF until1991. In order to study both temporal trends over time as well as to make comparisonsacross the ages (grades), the current study analyzes violent physical bullyingvictimization reports of the 8th, 10th and 12th graders in the MTF study from 1991to 2012. Instructed by MTF research staff, about 250,000 high school students in the8th, 10th and 12th grades participate in this annual survey by completing self-administered and machine-readable questionnaires at school. Because students whowere targets of physical bullying might feel uncomfortable in describing their traumaticexperience to interviewers, the information on bullying victimization is collected by theMTF in a self-report fashion, which is regarded as one of the most reliable methods forcollecting data on school bullying (Ahmad and Smith 1990).

4.1 Measures of Violent Bullying Victimization

It has been shown that measures of bullying victimization tend to be reliable if suchmeasures offer a consistent definition of targeted bullying behaviors, ask respondents toreport frequencies, and have a delimited time interval as the reference period (Bosworthet al. 1999; Olweus 1991). Since 1991, frequencies of four extreme forms of physicalbullying victimization—physically threatening, violent, injurious—during the last12 months have been consistently addressed by the following MTF questions regardingthe school environment.

BThe next questions are about some things which may have happened TO YOUwhile you were at school (inside or outside or in a school-bus). During the LAST12 months, how often …

1. Has an unarmed person threatened you with injury, but not actually injured you?2. Has someone threatened you with a weapon, but not actually injured you?3. Has someone injured you on purpose without using a weapon?4. Has someone injured you with a weapon (like a knife, gun, or club)?^

These four targeted bullying behaviors in the MTF questionnaire are hereinafterreferred as threatened without injury, threatened with a weapon, injury without aweapon and injury with a weapon, respectively. For each of the targeted behaviors,respondents were asked to choose from five frequency categories: 1) not at all, 2) once,3) twice, 4) 3–4 times, and 5) 5+ times. Annual frequency distributions of differentforms of bullying victimization were retrieved from MTF codebooks for each annualsurvey of 8th, 10th and 12th graders.


http://monitoringthefuture.org/

Although the MTF also investigates a series of school violence, misdemeanor, andcriminal acts, only these four questions are relevant to physical bullying victimization,which is the exclusive focus of this research. When students were targeted by theseaggressive and violent behaviors in a repetitive way, these questions corresponded tovictimization from violent bullying at school. In particular, the harmful consequencesassociated with threats deserve attention. When victims are threatened by others, theoption being given to the victims is frequently to either engage in undesired behaviorsor surrender their belongings. Otherwise, threats can escalate to injuries. Even if victimsmanage to escape from threats, repetitive threats per se are associated with a series ofnegative mental-health outcomes, such as self-reported anxiety, depression, loss of self-esteem and suicidal ideation (Rigby and Slee 1999; Salmon et al. 1998).

5 A Two-Step Method for Analyzing the Temporal Characteristicsof Combined and Truncated Count Data

Because the reported bullying victimization frequency counts in the MTF data aregrouped (3–4 times) and truncated (the uppermost response category is 5+ times), theirstatistical analysis is nontrivial. To identify appropriate statistical distributions foranalyzing the combined and truncated frequency count data, recall first that statisticalmodels for dealing with rare events (count data) are built on the Poisson distribution(see, e.g., Fox 2008: 392–394; Long and Freese 2006: 394–396). A property of thePoisson distribution is that its expected value (mean) and variance are equal, but thisstatistical constraint is often violated by over-dispersed (i.e., variance is greater than themean) empirical distributions of rare events. The over-dispersion in empirical distribu-tions often results from excess zeroes, that is, from more observations with zero countsof an outcome event than would be expected if the data were distributed as a Poissonrandom variable. Exploratory data analyses showed that this is the case for the MTFcount data for the four forms of violent physical bullying victimization. In other words,more 8th, 10th, and 12th graders are not at any risk of school-based bullying thanwould be expected if the bullying victimizations were distributed as Poisson variables.

Unfortunately, no existing statistical models or software can be readily applied toconduct age-period-cohort analyses of combined and truncated count data with over-dispersion. To overcome these difficulties in both statistical analyses and data structure,we developed a two-step method for analyzing the temporal (age, period and cohort)characteristics of combined and truncated count data.

Step 1 A Zero-Inflated Poisson Model Estimated by Minimizing Least AbsoluteDeviations

To model the grouped and over-dispersed count MTF data, in the first stepof our empirical analyses we modified the assumption that bullying processesare operative for all high-school students, and instead estimated zero-inflatedPoisson (ZIP) distributions (Fox 2008; Lambert 1992; Long and Freese2006). Consequently, for each specific form of violent bullying victimizationfor each of the three school grades for each year of the MTF data, a pair of

parameters bP;1̂� �, was estimated by minimizing the sum of absolute

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deviations between observed frequencies and frequencies predicted by zero-inflated Poisson distribution:

bP;1̂� �¼ argmin

P;λð Þ

Xx

pobservedx − f x��P;λ� �� ð1Þ

where x=0, 1, 2, 3–4, 5+ and pxobserved represents the observed frequencies.

The probability mass function f(i|P,λ) of the zero-inflated Poisson distributionis defined as follows:

f x��P;λ� �

¼ 1−Pð Þ þ P*Poisson λð Þ ¼ 1−Pð Þ þ P*exp −λð Þ when x ¼ 0

f x��P;λ� �

¼ P * Poisson λð Þ ¼ P *exp −λð Þ*λx

x!when x > 0

8><>: ð2Þ

where P is the proportion of students exposed to school bullying and λ is themean number of occurrences (average number of bullying-victimizationincidents) in a year/grade for any individual exposed to the risk of these formsof violent physical bullying. Conceptually, the ZIP model also specifies a two-stage strategy for analyzing two different aspects of victimization from violentbullying. In the first stage, the exposure to violent bullying is estimated by abinomial distribution with the parameter P indicating the likelihood that astudent is exposed to violent bullying. In the second stage, the intensity (meanrate) of bullying victimization for students exposed to the risk of being bulliedis determined by a Poisson distribution with expectation λ.

In the present study, P is the proportion of the population surveyed who arein the latent class of 8th, 10th, or 12th graders in a given school year at risk ofbeing a victim of one of the four forms of physically threatening, violent,injurious bullying. If the exposure to bullying parameter P is estimated to be1.0, then all students surveyed are at risk of victimization and the zero-inflatedPoisson distribution reduces to a Poisson distribution.2 The exposure to theprocess under study as denoted by parameter P of zero-inflated Poissonmodels is not identical to observed prevalence rates of the outcome eventunder study. In the present case, the observed prevalence rates are theproportions of students who report that they were bullied in an MTF survey.These rates are not identical to P because, for any Poisson distribution withintensity parameter λ, there is a non-zero number of observations for whichthe event count is expected to be zero. In other words, even among a group ofstudents who are at risk of bullying victimization, a non-zero number isexpected to report zero events in any specific time period. In brief, observedprevalence rates experiencing an event in sample surveys are the weightedaverage of: (1) the expected proportion of sample respondents who experiencethe event among those exposed to the event process for a specific Poisson rateparameter λ and (2) the proportion 1-P of the sample members who are in thelatent class of individuals for whom the Poisson process is not operative andthus the outcome event count necessarily is zero. It is the 1-P part of thissummation that enables the ZIP model to account for the excess zeroes or

2 The advantage of ZIP models over Poisson models can be suggested by comparing observed frequencieswith predicted frequencies using different estimation strategies (see Appendix).


number of observations with zero counts in empirical datasets such as theextreme bullying victimization counts in the MTF.

Currently no existing statistical software package can estimate the parametersof ZIP frequency distributions for truncated count data with combined responses.Therefore, a function was written in the R programming language (http://www.r-project.org/) to estimate parameters of ZIP models of frequency distributions fordata with combined and truncated observed frequencies.3 Parameters in this Rfunction are estimated both by the method of least absolute deviations forcombined and truncated data (Cohen 1954), which minimizes the absolutedeviations of observed and expected frequency counts, and by the leastsquares method, which minimizes the sum of squares of deviations ofobserved and expected frequency counts (Fox 2008; Gelman and Hill 2007).Although the least square method is more conventional, the least absolutedeviations method generally is less sensitive to outlying observations (Gelmanand Hill 2007; Zeileis et al. 2008). This was found to be the case in the presentanalyses. As shown in the Appendix for the first and last school yearsinvestigated (1991 and 2012; similar results were found for all school years)4

the predicted frequencies produced by minimization of the absolute deviationsare closer to the observed frequency counts than those generated by the least-square procedure across different years, grades and targeted behaviors.

Step 2 Age-Period-Cohort AnalysesAfter application of the foregoing methods of estimation of the ZIP fre-

quency distributions to the MTF data for each school year, we can array theestimated P and λ parameters in age-by-time-period tables with the rowsdefined by the 8th, 10th, and 12th grades, respectively, and the columnsordered from 1991 to 2012. The diagonals of these tables then correspond tocohorts. In brief, these tables then are similar to the arrays for which theclassical age-period-cohort (APC) multiple classification/accounting statisticalmodel (Mason et al. 1973) was defined—with the outcome variables in thepresent case corresponding to the respective estimated P and λ parameters.5

This then raises the analytic problem of the second step of our empiricalanalysis, that is, estimation of the age, time period, and cohort components oftemporal changes in the exposure (P) and intensity (λ) parameters of the zero-inflated Poisson distributions of bullying victimizations. For this, we appliedtwo statistical methods of estimation of the APC accounting model. The first isa recently developed method of estimation of age, period, and cohort effects ofchanges in age-by-period tables of an outcome variable termed the IntrinsicEstimator (IE). The Intrinsic Estimator method is based on an orthogonal

3 The R source code is available upon request.4 Numerical results for all school years are available from the authors on request.5 In conventional applications of APC multiple classification/accounting models, the cell entries are estimatedpopulation occurrence/exposure rates or proportions, which, in statistical terminology are random variables. Inthe present analyses, the estimated P and λ parameters are functions of the annual grouped and truncatedfrequency counts of the forms of violent victimization in the MTF data. In statistical terminology, thefrequency counts are random variables, and functions of random variables also are random variables. Thus,by arraying the estimated P and λ parameters in grade/age by time period tables, the objects of the APCanalysis in the present study are random variables, as is the case in classical APC accounting modelapplications.

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http://www.r-project.org/

http://www.r-project.org/

decomposition of the parameter space of the classical age-period-cohort multipleclassification/accounting statistical models (Fu et al. 2011; Yang et al. 2004,2008). It estimates an estimable function6 of the underidentified coefficient vectorof the APC accountingmodel that corresponds to theMoore-Penrose generalizedinverse solution to the normal equations of the model. To deal with perfectcollinearity among indicators of age, period and cohort (e.g., Age=Period—Cohort), the IE assumes a qualitative null-vector constraint, which is exclusivelydetermined by dimensionality of the data table (Fu 2008; Fu et al. 2011; O’Brien2011). After Fu (2000) showed that the IE can be viewed as a special case of theclassical ridge estimator for the conventional linear regression model that is usedwhen the regressors are highly collinear, statistical properties of the IE wereinvestigated and empirical studies based on this method have accumulated (Fuand Hall 2006; Schwadel 2011; Yang et al. 2004, 2008). A recent papercorroborates that the IE produces estimated age, period, and cohort effects thatare replicable by another method based on a maximum entrophy estimationcriterion (Browning et al. 2012). Because recent methodological discussions (Fuet al. 2011; O’Brien 2011) of the Intrinsic Estimator and classical equality-constrained estimators have emphasized the virtues of cross-method validationof empirical estimates, a second method of estimation of the APC accountingmodels for the P and λ coefficients, an equality-constrained least-squares methodwith time period coefficients for two consecutive time periods for which each ofthe estimated P or λ coefficients showed little change, was applied. Estimatesfrom this second approach thus serve as an assessment of the robustness of the IEestimates of the age, time period, and birth cohort effect coefficients. Theestimated age, time period, and birth cohort effect coefficients from this methodof estimation were virtually identical to those from the IE; accordingly, in theAPC analyses reported here, we report the IE estimates.7

6 Results

Our first research question is: Has there been a recent increase in physically threatening,violent, injurious forms of bullying in middle and high school contexts in the U.S. asobserved in the MTF victimization data? To address this question, we first examinedthe observed relative frequency counts (percentages) of responses to each of the fourbullying victimization questions in the MTF for each of the annual 8th, 10th, and 12thgrade samples, 1991–2012. Even though the MTF samples are quite large, thesepercentages have a good deal of stochastic variability from year-to-year that complicatethe determination of trends over time. In order to smooth the year-to-year fluctuationsand make more evident the trends over time, three-point moving averages (see, e.g.,Kendall and Stuart 1976, pp. 380–418) were applied to the observed relative frequencydistributions. Table 1 reports the results for 8th graders, while Tables 2 and 3 give thecorresponding estimates for the 10th and 12th graders.

6 Estimable functions are linear functions of the effect coefficients of an underidentified linear model that areinvariant to whatever solution to the normal equations assumed for estimation of the coefficients.7 The numerical estimates for both sets of APC analyses are available from the authors on request.


Tab

le1

Frequencydistributio

nsof

four

form

sof

victim

izationfrom

violentbully

ingreported

by8thgraders,MTF,

1991

to2012

12

Threatenedwith

outinjury

Threatenedwith

aweapon

Injury

with

outaweapon

Injury

with

aweapon

N=108,618

N=108,860

N=108,781

N=109,028

Never

Once

Twice

3–4tim

es5+

times

Never

Once

Twice

3–4tim

es5+

times

Never

Once

Twice

3–4tim

es5+

times

Never

Once

Twice

3–4tim

es5+

times

1991

69.7

%14.6

%5.7%

3.7%

6.3%

81.3

%11.0

%3.5%

2.0%

2.3%

75.8

%12.9

%5.0%

3.0%

3.3%

90.8

%5.3%

1.9%

1.0%

1.0%

1992

70.0

%14.4

%5.7%

3.7%

6.2%

81.3

%10.9

%3.5%

2.0%

2.3%

76.2

%12.7

%4.9%

2.9%

3.3%

90.9

%5.3%

1.9%

1.0%

0.9%

1993

70.3

%14.0

%5.7%

3.8%

6.1%

81.6

%10.8

%3.6%

1.9%

2.2%

77.0

%12.3

%4.7%

2.8%

3.2%

91.2

%5.1%

1.8%

1.0%

0.9%

1994

70.3

%13.9

%5.8%

3.9%

6.1%

81.6

%10.7

%3.5%

2.0%

2.2%

77.7

%11.9

%4.5%

2.7%

3.2%

91.3

%5.0%

1.8%

1.0%

0.9%

1995

70.1

%14.0

%5.8%

4.0%

6.2%

81.9

%10.6

%3.3%

2.0%

2.3%

78.2

%11.6

%4.2%

2.6%

3.3%

91.5

%4.8%

1.8%

0.9%

1.0%

1996

69.9

%14.1

%5.9%

3.9%

6.3%

82.1

%10.5

%3.1%

1.9%

2.4%

78.7

%11.4

%4.1%

2.4%

3.4%

91.5

%4.7%

1.8%

0.9%

1.1%

1997

70.1

%14.1

%5.8%

3.8%

6.2%

82.4

%10.4

%3.0%

1.8%

2.4%

79.1

%11.1

%4.1%

2.3%

3.4%

91.7

%4.6%

1.8%

0.8%

1.1%

1998

70.7

%13.8

%5.6%

3.6%

6.2%

82.3

%10.4

%3.1%

1.6%

2.5%

79.4

%10.8

%4.1%

2.3%

3.4%

91.7

%4.6%

1.8%

0.8%

1.1%

1999

71.6

%13.3

%5.3%

3.5%

6.3%

82.1

%10.4

%3.1%

1.7%

2.6%

79.5

%10.5

%4.2%

2.3%

3.5%

91.7

%4.6%

1.7%

0.8%

1.0%

2000

72.6

%12.7

%5.0%

3.4%

6.3%

82.2

%10.2

%3.2%

1.7%

2.7%

79.7

%10.1

%4.2%

2.4%

3.5%

92.0

%4.5%

1.6%

0.9%

1.0%

2001

73.5

%12.2

%4.9%

3.3%

6.2%

82.6

%9.9%

3.0%

1.8%

2.7%

79.9

%9.9%

4.2%

2.5%

3.5%

92.4

%4.2%

1.4%

0.8%

1.1%

2002

74.0

%12.1

%4.8%

3.1%

6.0%

83.2

%9.6%

2.9%

1.7%

2.6%

80.1

%9.8%

4.1%

2.6%

3.4%

92.9

%4.0%

1.3%

0.8%

1.1%

2003

74.1

%12.3

%4.6%

3.1%

6.0%

83.3

%9.6%

2.9%

1.7%

2.6%

79.4

%10.2

%4.1%

2.8%

3.6%

93.1

%3.9%

1.2%

0.8%

1.0%

2004

73.5

%12.8

%4.6%

3.2%

6.0%

83.2

%9.6%

2.9%

1.7%

2.6%

77.7

%11.1

%4.3%

3.0%

3.9%

93.1

%4.0%

1.2%

0.7%

1.0%

2005

73.0

%13.0

%4.7%

3.2%

6.1%

82.8

%9.9%

3.0%

1.7%

2.6%

75.2

%12.5

%4.7%

3.4%

4.3%

93.1

%4.1%

1.2%

0.7%

0.9%

2006

72.4

%13.1

%5.0%

3.2%

6.3%

82.3

%10.1

%3.1%

1.8%

2.7%

73.1

%13.6

%5.1%

3.6%

4.6%

92.9

%4.3%

1.2%

0.7%

1.0%

2007

72.4

%12.8

%5.0%

3.4%

6.3%

81.8

%10.3

%3.3%

1.9%

2.8%

71.9

%14.1

%5.5%

3.7%

4.8%

92.8

%4.4%

1.2%

0.7%

0.9%

2008

72.5

%12.6

%4.9%

3.6%

6.3%

81.8

%10.2

%3.3%

2.0%

2.8%

71.6

%14.3

%5.7%

3.5%

4.9%

92.8

%4.4%

1.2%

0.8%

0.9%

2009

73.0

%12.3

%4.7%

3.7%

6.3%

82.0

%10.0

%3.3%

1.9%

2.8%

72.0

%14.1

%5.6%

3.4%

4.9%

93.1

%4.1%

1.2%

0.8%

0.8%

2010

73.6

%11.9

%4.6%

3.7%

6.2%

82.7

%9.7%

3.1%

1.8%

2.7%

72.8

%13.8

%5.4%

3.2%

4.8%

93.7

%3.8%

1.1%

0.7%

0.8%

2011

74.2

%11.6

%4.5%

3.6%

6.1%

83.2

%9.4%

3.0%

1.7%

2.7%

73.8

%13.4

%5.2%

3.1%

4.5%

94.2

%3.5%

1.0%

0.5%

0.7%

2012

74.6

%11.5

%4.5%

3.5%

6.0%

83.6

%9.3%

2.9%

1.6%

2.6%

74.3

%13.2

%5.1%

3.0%

4.4%

94.5

%3.4%

0.9%

0.4%

0.7%

1Because

athree-pointm

oving-averagedata-smoothingprocesswas

appliedto

thedata,the

percentagesshow

nin

thetableareslightly

differentfrom

theselistedin

MTFcodebooks

2Asfrequencydistributio

nswerereported

inthistable,thesum

ofpercentagesover

thesameform

ofviolentbullyingin

asingleyear

is100%

Q. Fu et al.

Tab

le2


nsof

four

form

sof

victim

izationfrom

violentbully

ingreported

by10th

graders,MTF,1991

to2012

12

Threatenedwithoutinjury

Threatenedwith

aweapon

Injury

withoutaweapon

Injury

with

aweapon

N=107,767

N=107,939

N=107,890

N=108,025

Never

Once

Twice

3–4tim

es5+

times

Never

Once

Twice

3–4tim

es5+

times

Never

Once

Twice

3–4tim

es5+

times

Never

Once

Twice

3–4tim

es5+

times

1991

71.2

%13.8

%5.9%

3.7%

5.4%

83.8

%9.4%

3.1%

1.7%

1.9%

82.2

%10.1

%3.8%

2.0%

2.0%

92.9

%4.6%

1.4%

0.7%

0.5%

1992

71.5

%13.7

%5.8%

3.7%

5.3%

84.1

%9.2%

3.1%

1.7%

1.8%

82.7

%9.8%

3.7%

1.9%

1.9%

93.1

%4.4%

1.4%

0.7%

0.5%

1993

71.9

%13.4

%5.7%

3.8%

5.3%

84.5

%8.8%

3.2%

1.7%

1.8%

83.7

%9.1%

3.6%

1.8%

1.8%

93.5

%4.0%

1.3%

0.6%

0.6%

1994

72.0

%13.4

%5.6%

3.8%

5.2%

85.0

%8.5%

3.1%

1.7%

1.8%

84.5

%8.7%

3.3%

1.7%

1.6%

93.9

%3.7%

1.2%

0.6%

0.6%

1995

72.0

%13.4

%5.7%

3.8%

5.1%

85.4

%8.4%

2.9%

1.6%

1.8%

85.1

%8.4%

3.1%

1.7%

1.6%

94.1

%3.5%

1.2%

0.6%

0.7%

1996

72.3

%13.4

%5.7%

3.7%

5.0%

85.7

%8.3%

2.8%

1.5%

1.8%

85.6

%8.2%

3.0%

1.7%

1.5%

94.3

%3.3%

1.2%

0.6%

0.7%

1997

72.9

%13.2

%5.6%

3.4%

4.9%

85.8

%8.2%

2.7%

1.5%

1.8%

85.9

%8.0%

2.9%

1.7%

1.5%

94.4

%3.2%

1.2%

0.6%

0.7%

1998

73.4

%13.1

%5.4%

3.3%

4.8%

85.8

%8.0%

2.8%

1.6%

1.8%

86.0

%7.9%

3.0%

1.6%

1.5%

94.3

%3.2%

1.2%

0.6%

0.7%

1999

73.9

%12.9

%5.2%

3.3%

4.9%

85.6

%8.0%

2.8%

1.7%

1.9%

85.7

%7.9%

3.1%

1.6%

1.6%

94.1

%3.4%

1.2%

0.6%

0.6%

2000

74.3

%12.4

%5.1%

3.3%

4.9%

85.5

%8.1%

2.9%

1.8%

1.8%

85.7

%7.7%

3.2%

1.6%

1.8%

94.0

%3.5%

1.2%

0.7%

0.6%

2001

75.0

%11.8

%5.0%

3.2%

5.0%

85.4

%8.1%

2.9%

1.7%

1.9%

85.7

%7.5%

3.2%

1.8%

1.9%

93.9

%3.5%

1.2%

0.7%

0.6%

2002

75.7

%11.2

%4.8%

3.2%

5.2%

85.4

%8.0%

3.0%

1.6%

2.0%

85.9

%7.1%

3.1%

1.8%

2.0%

94.0

%3.4%

1.2%

0.7%

0.6%

2003

76.0

%10.9

%4.6%

3.1%

5.4%

85.3

%8.0%

3.0%

1.5%

2.1%

85.2

%7.4%

3.2%

1.9%

2.2%

94.1

%3.4%

1.2%

0.7%

0.6%

2004

75.8

%11.0

%4.5%

3.2%

5.5%

85.2

%8.1%

2.9%

1.5%

2.2%

83.8

%8.3%

3.5%

1.9%

2.5%

94.4

%3.3%

1.1%

0.6%

0.6%

2005

75.1

%11.4

%4.5%

3.4%

5.6%

84.8

%8.6%

2.7%

1.6%

2.3%

81.4

%9.8%

3.9%

2.1%

2.8%

94.5

%3.3%

1.0%

0.6%

0.6%

2006

74.6

%11.7

%4.6%

3.5%

5.6%

84.4

%8.9%

2.7%

1.6%

2.3%

80.0

%10.8

%4.1%

2.2%

2.9%

94.7

%3.2%

1.0%

0.5%

0.6%

2007

74.3

%11.8

%4.8%

3.5%

5.5%

83.9

%9.3%

2.8%

1.7%

2.4%

79.0

%11.4

%4.2%

2.5%

2.9%

94.7

%3.2%

0.9%

0.5%

0.6%

2008

74.4

%11.7

%4.9%

3.5%

5.5%

83.7

%9.3%

2.9%

1.7%

2.4%

78.9

%11.4

%4.2%

2.6%

2.9%

94.8

%3.2%

0.9%

0.5%

0.6%

2009

74.4

%11.7

%5.0%

3.5%

5.4%

83.8

%9.2%

3.0%

1.7%

2.3%

78.8

%11.3

%4.3%

2.7%

2.9%

94.8

%3.1%

0.9%

0.5%

0.6%

2010

74.9

%11.6

%4.8%

3.4%

5.2%

84.5

%8.7%

3.0%

1.7%

2.2%

79.4

%11.0

%4.2%

2.6%

2.8%

95.1

%2.9%

0.9%

0.5%

0.6%

2011

75.7

%11.3

%4.8%

3.4%

4.8%

85.2

%8.3%

2.9%

1.7%

2.0%

80.3

%10.5

%4.0%

2.5%

2.6%

95.5

%2.6%

0.9%

0.4%

0.6%

2012

76.3

%11.1

%4.7%

3.4%

4.6%

85.7

%7.9%

2.8%

1.7%

1.9%

81.0

%10.2

%3.9%

2.4%

2.5%

95.7

%2.5%

0.9%

0.4%

0.5%

1Because

athree-pointm


appliedto

thedata,the

percentagesshow

nin

thetableareslightly

differentfrom

theselistedin

MTFcodebooks


nswerereported

inthistable,thesum

ofpercentagesover

thesameform

ofviolentbullyingin

asingleyear

is100%


Tab

le3


nsof

four

form

sof

victim

izationfrom

violentbully

ingreported

by12th

graders,MTF,1991

to2012

12

Threatenedwith

outinjury

Threatenedwith

aweapon

Injury

with

outaweapon

Injury

with

aweapon

N=42,143

N=42,236

N=42,183

N=42,269

Never

Once

Twice

3–4tim

es5+

times

Never

Once

Twice

3–4tim

es5+

times

Never

Once

Twice

3–4tim

es5+

times

Never

Once

Twice

3–4tim

es5+

times

1991

75.1

%12.5

%4.8%

3.5%

4.1%

85.3

%8.7%

3.2%

1.6%

1.2%

86.2

%8.3%

2.8%

1.5%

1.3%

94.3

%3.6%

1.2%

0.4%

0.5%

1992

75.5

%12.5

%4.6%

3.4%

3.9%

85.1

%8.7%

3.3%

1.6%

1.2%

87.0

%7.7%

2.8%

1.4%

1.2%

94.6

%3.3%

1.1%

0.4%

0.5%

1993

76.1

%11.9

%4.7%

3.3%

4.0%

85.1

%8.7%

3.3%

1.6%

1.3%

87.8

%7.1%

2.7%

1.3%

1.1%

95.0

%3.0%

1.0%

0.5%

0.5%

1994

76.6

%11.7

%4.5%

3.2%

3.9%

85.5

%8.6%

3.0%

1.5%

1.3%

88.3

%6.7%

2.6%

1.3%

1.1%

95.2

%2.8%

1.0%

0.6%

0.4%

1995

77.2

%11.3

%4.4%

3.1%

4.0%

86.3

%8.3%

2.6%

1.3%

1.4%

88.3

%6.8%

2.4%

1.3%

1.2%

95.1

%2.8%

1.1%

0.5%

0.4%

1996

77.9

%11.2

%4.1%

3.0%

3.8%

87.3

%7.9%

2.4%

1.2%

1.2%

88.3

%7.0%

2.3%

1.1%

1.3%

95.1

%2.9%

1.1%

0.5%

0.4%

1997

78.4

%11.0

%4.1%

2.8%

3.7%

88.1

%7.4%

2.2%

1.2%

1.1%

88.5

%7.1%

2.2%

1.1%

1.3%

95.1

%2.9%

1.1%

0.4%

0.4%

1998

78.6

%11.0

%4.0%

2.7%

3.6%

88.4

%7.2%

2.2%

1.2%

1.0%

88.8

%6.8%

2.1%

1.1%

1.2%

95.3

%2.9%

0.9%

0.5%

0.4%

1999

78.4

%11.2

%4.1%

2.7%

3.6%

88.3

%7.3%

2.1%

1.3%

1.0%

88.8

%6.6%

2.1%

1.3%

1.2%

95.4

%2.8%

0.8%

0.5%

0.5%

2000

78.6

%11.3

%3.9%

2.6%

3.5%

88.3

%7.2%

2.1%

1.3%

1.1%

88.8

%6.4%

2.3%

1.3%

1.2%

95.6

%2.7%

0.6%

0.6%

0.5%

2001

79.2

%11.1

%3.8%

2.4%

3.5%

88.4

%7.1%

2.2%

1.2%

1.2%

88.6

%6.5%

2.4%

1.2%

1.3%

95.6

%2.6%

0.6%

0.6%

0.6%

2002

80.1

%10.5

%3.8%

2.2%

3.5%

88.8

%6.7%

2.2%

1.1%

1.2%

88.7

%6.3%

2.5%

1.1%

1.3%

95.8

%2.5%

0.6%

0.4%

0.6%

2003

80.5

%9.9%

3.9%

2.3%

3.5%

88.7

%6.7%

2.2%

1.1%

1.2%

88.8

%6.3%

2.6%

1.0%

1.2%

95.9

%2.5%

0.8%

0.4%

0.5%

2004

80.7

%9.4%

4.0%

2.4%

3.6%

88.7

%6.7%

2.3%

1.0%

1.3%

88.9

%6.3%

2.5%

1.1%

1.1%

96.0

%2.5%

0.8%

0.3%

0.4%

2005

80.9

%9.2%

4.1%

2.3%

3.5%

88.4

%6.7%

2.4%

1.1%

1.3%

89.1

%6.3%

2.3%

1.3%

1.1%

96.0

%2.5%

0.8%

0.3%

0.3%

2006

81.0

%9.2%

4.2%

2.2%

3.5%

88.2

%6.7%

2.6%

1.2%

1.3%

89.0

%6.4%

2.3%

1.3%

1.2%

95.8

%2.6%

0.9%

0.3%

0.4%

2007

81.0

%9.2%

4.1%

2.2%

3.5%

87.9

%6.9%

2.6%

1.3%

1.3%

88.5

%6.6%

2.4%

1.2%

1.4%

95.7

%2.6%

1.0%

0.3%

0.4%

2008

81.0

%9.1%

4.1%

2.4%

3.6%

87.6

%7.2%

2.5%

1.3%

1.3%

87.8

%6.8%

2.7%

1.2%

1.6%

95.5

%2.5%

1.0%

0.4%

0.5%

2009

81.1

%8.9%

3.9%

2.4%

3.6%

87.5

%7.3%

2.4%

1.3%

1.4%

87.4

%6.8%

2.9%

1.2%

1.6%

95.4

%2.6%

1.0%

0.4%

0.6%

2010

81.2

%8.8%

3.9%

2.5%

3.6%

87.7

%7.1%

2.3%

1.3%

1.5%

87.4

%6.8%

3.0%

1.1%

1.6%

95.3

%2.6%

1.0%

0.4%

0.6%

2011

82.5

%7.5%

3.9%

2.6%

3.5%

88.1

%6.6%

2.4%

1.3%

1.5%

87.7

%6.8%

2.8%

1.1%

1.6%

95.4

%2.6%

0.9%

0.5%

0.6%

2012

83.3

%7.4%

4.0%

2.3%

3.1%

88.3

%6.4%

2.4%

1.3%

1.5%

87.9

%6.8%

2.7%

1.1%

1.6%

95.5

%2.6%

0.9%

0.5%

0.5%

1Because

athree-pointm


appliedto

thedata,the

percentagesshow

nin

thetableareslightly

differentfrom

theselistedin

MTFcodebooks


nswerereported

inthistable,thesum

ofpercentagesover

thesameform

ofviolentbullyingin

asingleyear

is100%

Q. Fu et al.

It can be seen from Tables 1, 2 and 3 that, by and large, fewer studentsacross different grades reported being victims of extreme physical bullyingvictimization in 2012 as compared to these in 1991, regardless of the form ofbullying. Such decreases in the prevalence of bullying victimization at schoolare consistent with declining percentage of violent victimizations at schoolreported elsewhere (Molcho et al. 2009; Robers et al. 2012). For example, overthe period of the study the percentage of 8th graders who reported never beingthreatened without injury increased from 69.7 % to 74.6 % (Table 1). Inaddition, the percentage of 12th graders who reported never being threatenedwithout injury also increased by 8.2 % (Table 3).

On the other hand, while lower percentages of both 8th and 10th graders reportednever being injured without a weapon in 2012 than in 1991, higher percentages of 8thand 10th graders reported being injured without a weapon frequently (3+) in 2012 thanin 1991 (Tables 1 and 2). For example, the percentages of 8th and 10th gradersreporting being injured without a weapon three or more times increased from 6.3 %and 4.0 % in 1991, respectively, to 7.4 % and 4.9 % in 2012. Understanding thesemixed findings regarding the occurrence and frequencies of physical bullying victim-ization requires a more refined analysis of the observed relative frequency distributionswith the analytic methods described above.

Accordingly, we next applied the ZIP models described above to the observedrelative frequency distributions in order to differentiate and estimate time trends inboth exposure to, and intensity of, bullying victimization. Figures 1, 2, 3, 4, 5 and 6exhibit graphs of the annual estimates of the estimated exposure (P) and intensityparameters (λ) of the ZIP models for each of the 8th, 10th and 12th grade samples forthe years 1991–2012.

Fig. 1 Exposure to Violent Bullying Victimization: 8th Graders, 1991–2012


These graphs contain some key findings that address our research questions. First,across this entire period of the study, it can be seen that more serious forms of physicalbullying victimization involving a weapon generally had lower levels of both exposureand intensity. Second, although the estimated exposure parameters for the four differentforms of bullying victimization largely showed long-term declines over the period ofstudy, the declines in exposure leveled off in the early 2000s, in the 2001–2003 school

Fig. 2 Intensity of Violent Bullying Victimization: 8th Graders, 1991–2012


Q. Fu et al.

years, and then trended up to higher levels in the 2007–2009 years before tapering off(see Figs. 1, 3 and 5). Third, similar to what we found above in the tables of relativefrequency distributions, there were large increases in exposure to being injured withouta weapon for both 8th and 10th graders from 2003 to 2008. Fourth, the graphs of trendsin the estimated intensities of bullying victimization (see Figs. 2, 4 and 6) showenduring, albeit fluctuating, increases for almost all forms of bullying victimization




across the three grades. Thus those who were experiencing violent physical bullyingvictimization were victimized with greater frequency.

Turning next to our second research question, that is, the extent to which there aredistinct temporal patterns of age, periods, and cohort effects in the MTF bullyingvictimization data, note that the graphs of the estimated ZIP parameters in Figs. 1, 2,3, 4, 5 and 6 are suggestive: 1) the levels of both exposure to, and intensity of, physicalbullying victimization decrease across the grades from the 8th to the 10th to the 12th,which is indicative of age/grade effects; 2) the simultaneous increases in the intensity ofmost forms of violent physical bullying victimization across the different grades aroundthe year 2006 suggests the existence of period effects; and 3) in the early 2000s, theincreases (or leveling off) in the exposure to being threatened without injury for lowergraders preceded the corresponding changes for the 10th and 12th graders, whichindicates cohort effects.

For a more systematic evaluation of the extent to which age, period, and cohorteffects are detectable in the MTF bullying victimization data, we applied the IntrinsicEstimator method. For this, we arrayed the estimated exposure (P) and intensityparameters (λ) from the ZIP models for each of the four bullying victimizationquestions and each of the school years 1991 to 2012 in age/grade-by-period tables.As an illustration, Table 4 shows the estimated exposure (Panel A) and intensityparameters (Panel B) for the threatened without injury victimization question.8 Therows of Table 4 correspond to the age/grade categories, the columns correspond to thetime periods, and cohorts then correspond to the diagonals of the tables. As regards theage/grade categories, the modal ages for 8th, 10th and 12th graders surveyed during thespring of each year are 14, 16, and 18, respectively, as indicated in the labels of the

8 Corresponding tables for the other three MTF bullying questions are available from the authors on request.


Q. Fu et al.

Tab

le4

Estim

ated

exposure

andintensity

parametersof

thezero-inflatedpoissonmodelsforthreatened

with

outinjury

victim

izationfrequencydistributio

ns

PanelA.E

stim

ated

exposure

parameter,P

(%)

Year

Year

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

2011

2012

8/14

41.4

40.9

40.0

40.1

40.1

40.5

40.3

39.6

38.2

36.7

35.3

34.9

35.1

36.2

36.8

37.4

36.9

36.5

35.7

34.8

33.9

33.4

Grade/Age

10/16

39.2

38.9

38.2

38.2

38.2

37.9

37.3

36.8

36.2

35.1

33.7

32.4

31.7

32.0

33.0

33.8

34.2

33.9

34.0

33.5

32.5

31.8

12/18

34.9

34.7

33.3

32.7

31.7

31.2

30.6

30.5

31.0

31.1

30.5

28.9

27.6

26.6

26.1

26.1

26.1

25.9

25.2

24.3

23.2

22.7

PanelB.E

stim

ated

intensity

parameter,λ

Year

Year

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

2011

2012

8/14

1.31

1.32

1.35

1.35

1.37

1.36

1.35

1.34

1.36

1.37

1.39

1.37

1.34

1.32

1.32

1.34

1.37

1.39

1.41

1.42

1.43

1.42

Grade/Age

10/16

1.32

1.32

1.33

1.32

1.32

1.31

1.29

1.28

1.27

1.31

1.35

1.38

1.41

1.41

1.40

1.39

1.39

1.40

1.39

1.38

1.37

1.37

12/18

1.25

1.22

1.26

1.25

1.27

1.23

1.22

1.21

1.19

1.16

1.14

1.16

1.22

1.29

1.31

1.30

1.30

1.32

1.36

1.41

1.45

1.47


rows of Table 4. These modal ages then can be used to calculate the modal birth yearsfor cohort analysis, with the earliest birth cohort being 1973 and the most recent being1996. Since demographers term birth cohorts born in the years 1965 to 1982 as theBaby Bust or Generation X cohorts and in the years 1983 to 2001 as the Echo Boom,New Boomer (children of the Baby Boomers born in the 18 years from 1946 to 1964),or Millennials (Carlson 2009), it follows that the earliest 10 birth cohorts in the MTFdata analyzed here, born 1973 to 1982, are late Gen Xers and the last 12 birth cohorts,born 1983 to 1994, are Millennials.

Estimates of the age, period, and cohort effect coefficients obtained from applicationof the Intrinsic Estimator to tables of the form of Table 4 for each of the four forms ofviolent physical bullying victimization are shown in Fig. 7, from which several findingsemerge. First, as expected from the foregoing examination of the patterns in Figs. 1, 2,3, 4, 5, and 6, both exposure to and intensity of bullying victimization virtually showedmonotonic decreases with grade (age) from the 8th to the 10th to the 12th (Figs. 8 and9). The decreases for 10th to 12th graders tend to be sharper than these for 8th to 10thgraders. Second, more dramatic decreases across the grades were associated with themost severe forms of physical bullying (injury with or without a weapon), which werelikely to due to more severe legal punishment of the bullying offenders (Land et al.1992). Third, as shown by Figs. 9, 10, 11 and 12, both period and cohort effects alsochanged more dramatically for the more serious forms of violent physical bullyingvictimization (injury with or without a weapon), which illustrates the importance ofdistinguishing different forms of physical bullying victimization. Fourth, the estimatedperiod effects show salient decreases in exposure to physical bullying victimization,especially for more severe forms of bullying behaviors, after the year 2005 when aseries of anti-bullying campaigns, such as National Bullying Prevention Month, wereinitiated (see Fig. 9). Fifth, such decreases in exposure after 2005 coincided with

Fig. 7 Estimated Grade of School (Age) Effects on Estimated ZIP Exposure Parameters (P) for Four Types ofViolent Bullying Victimization, MTF, 1991–2012

Q. Fu et al.

dramatic increases in the intensity of violent physical bullying victimization (seeFig. 10), which means that, as the percentage of students exposed to school bullyingdeclined in the years after 2005, the intensity of physical bullying among those still

Fig. 8 Estimated Grade of School (Age) Effects on Estimated ZIP Intensity Parameters (λ) for Four Types ofViolent Bullying Victimization, MTF, 1991–2012

Fig. 9 Estimated Time Period (Year) Effects on Estimated ZIP Exposure Parameters (P) for Four Types ofViolent Bullying Victimization, MTF, 1991–2012


exposed to bullying increased. Sixth, Figs. 9 and 10 show that this pattern of inversechanges in period effects of bullying exposure and intensity occurred not only after2005 but over the entire period of the study.

Fig. 10 Estimated Time Period (Year) Effects on Estimated ZIP Intensity Parameters (λ) for Four Types ofViolent Bullying Victimization, MTF, 1991–2012

Fig. 11 Estimated Birth Cohort Effects on Estimated ZIP Exposure Parameters (P) for Four Types of ViolentBullying Victimization, MTF, 1991–2012

Q. Fu et al.

7 Discussion and Conclusions

Using data from the MTF annual survey of 8th, 10th, and 12th graders for the years1991 to 2012, the first question addressed in this paper is: Has there been a recentincrease in school-based extreme forms of bullying victimization—physically threat-ening, violent, injurious—in the United States in recent years? Our examination oftemporal trends in the relative frequency distributions of responses to MTF victimiza-tion questions concerning four forms of extreme violent bullying indicated that, relativeto the early 1990s, fewer students reported being victims. On the other hand, higherpercentages of 8th and 10th graders reported being injured without a weapon frequently(3+) in 2012 than in 1991.

To more systematically study the temporal trends in the relative frequencydistributions, we developed a least absolute deviations method using the Rprogram for estimating the exposure and intensity parameters of zero-inflatedPoisson models of truncated and combined frequency distributions. Applicationof the algorithm to the MTF data showed that both estimated exposure to, andintensity of, violent physical bullying victimization increased in the mid-2000s.Standing out among these increases are very large increases in exposure tobeing injured without a weapon for both 8th and 10th graders from 2003 to2008, a period of rising public concern about a Bbullying epidemic.^ At thesame time, the ZIP analyses showed a long-term decline in the exposure toinjury with a weapon from the early 1990s to 2012. Thus, the answer to ourresearch question is yes: There is evidence of a short-term increase in extremeviolent forms of bullying victimization in the mid- to late-2000s, but with thequalifier that on the longer term—the two decades for which data are

Fig. 12 Estimated Birth Cohort Effects on Estimated ZIP Intensity Parameters (λ) for Four Types of ViolentBullying Victimization, MTF, 1991–2012


available—there has been an overall decline in exposure. By contrast, the longterm trend for the intensity of physical bullying among those bullied is eitheran increase or no trend, depending on the form of the violent bullying and thegrade of school.

As the first step of analyses may conceal a great deal of heterogeneity in temporaleffects embedded in bullying victimization, even seemingly constant levels of victim-ization from violent bullying (e.g., the exposure to being injured with a weapon) can bedriven by differential temporal effects operating in different directions along the age,period, and cohort time dimension. Accordingly, we then addressed the ques-tion: is it possible to disentangle the intertwined age, period, and cohort effectsin the observed relative prevalence rate distributions of physical bullyingvictimization? For this, we applied the recently developed Intrinsic Estimatormethod for age-period-cohort analysis to age/grade-by-period tables of theestimated exposure and intensity parameters of the ZIP distributions. Theseanalyses clearly showed that the age/grade effects on bullying victimizationmonotonically decline from the 8th to the 10th to the 12th grade. They alsoshow that the recent increases in violent physical bullying victimization were amixture of period effects on intensity of physical bullying victimization andcohort effects on exposure to violent bullying victimization. Moreover, theperiod effects on exposure to, and intensity of, violent physical bullyingvictimization changed in opposite ways and this pattern was more evident inmore recent years. In other words, fewer students are exposed to bullyingvictimization as in the late 2000s; however those victims were bullied morefrequently.

As expected, the monotonic decreases in bullying victimization with age can beattributed to several mechanisms (such as physical and psychological transition and re-establishment of peer relations) proposed by existing research. In particular, bullyingperpetuated by older students in school and the reestablishment of social relations in thetransition from middle school to high school can account for the sharp decreases from10th to 12th graders.

The tradeoff between violent physical bullying victimization exposure andintensity in more recent school years clearly suggests the resilience of thephenomenon as manifested across different forms of violent bullying. Theseperiod trends can be situated in the context of events and developments focusedon violent offending and school violence that received widespread media,research, or policy attention in late-1990s and 2000s. For instance, in 1998Matthew Sheppard was tortured and killed bringing attention not only to hateissues related to sexual orientation but also to bullying (Brooke 1998). In 2004,a joint United States Secret Service and Department of Education report linkedviolent school attacks to bullying victimization, noting that in over 70% of seriousacts of school violence the perpetrators had been victims of long-term bullying (Vossekuilet al. 2004). This was followed by the designation of October as NationalBullying Prevention Month in 2006 (PACER’S National Bullying PreventionCenter 2013) and a general diffusion of zero-tolerance policies on the misbe-havior of youth in school with particular attention to violence and violentvictimization in school districts across the county in the 2000s. This increasing

Q. Fu et al.

attention focused on bullying from the late-1990s through the 2000s and theassociated anti-bullying campaigns and intervention programs are consistentwith the reductions in exposure to violent physical bullying victimization foundin our period effect estimates. But, at the same time, physical bullying becameconcentrated on a smaller fraction of middle and high school students with theaverage intensity of bullying rising correspondingly.

Furthermore, different cohorts show differential exposure to, and intensity of,violent physical bullying victimization. In our age-period-cohort analysis, thecohort effects of exposure to bullying victimization generally decline to a lowpoint for the 1986 and 1987 cohorts. The cohort effects then begin to increasewith the 1988 cohort. The 1988 cohort was at about the 8th grade level at thetime of the September 11 terrorist attack and subsequent responses and con-flicts; thus, at an age when they were most susceptible to aggressive behaviorsand violence (in middle school). This cohort and those in the following 5 yearsthrough 1993 also grew up with the proliferation of the Internet, social net-working sites, and mobile phones. The present research cannot identify a causallink between this spread of the electronic media and the violent physicalbullying victimization wave of the 2002–2008 years, but the temporal concor-dance is evident. Most recently, as the attention of the public, parents, andschool administrators has been brought to focus on physical bullying in schoolenvironments, recent birth cohort effects show substantial declines in intensityand stabilization of exposure rates. Whether these downturns will continue fornew student cohorts entering middle schools, as was the case for the birthcohorts from 1973 to 1987, remains to be seen.

Limitations of the present research should be acknowledged. First, theaggregated MTF dataset does not allow us to identify what personal andhousehold characteristics are more vulnerable to violent physical school bully-ing. Further research on the performance of students from different demograph-ic, socio-economic and behavioral groups is required. Second, the MTF onlycollects data on extreme forms of physical bullying victimization, which fails toincorporate other forms of bullying victimization, such as cyber bullying.Although victims who are repeatedly and overtly bullied by their peers alsotend to be bullied in other ways (Rigby 2002; Scheithauer et al. 2006), resultsreported in this research should by no means be interpreted as representative ofoverall bullying victimization in the US as the prevalence of all forms ofbullying victimizations should be higher than that of victimization from phys-ical bullying at school. Third, more recent research has suggested that asizeable group of victims of school bullying should be classified as bully/victims, because they sometimes resort to bullying their peers as well (Woodsand Wolke 2004). Unfortunately the MTF survey instruments do not ask aboutthe individual’s bullying behaviors.

Violent physical bullying victimizations remain a problem for school childrenand a concern for students, parents, teachers and other school officials, and thepublic. The effects of bullying victimization can be emotionally and physicallydamaging to the victims. The listed limitations to the current study point toimportant avenues for future research on bullies, victims, and bully/victims.


Appendix

Table 5 Observed frequencies and predicted frequencies using different estimation strategies

Threatened without injury Injury with a weapon

Never Once Twice 3–4times

5+times

Never Once Twice 3–4times

5+times

Grade 8 1991 Observed 69.7 % 14.6 % 5.7 % 3.7 % 6.3 % 90.8 % 5.3 % 1.9 % 1.0 % 1.0 %

ZIP: Sum ofabsolute deviations1

69.8 % 14.6 % 9.6 % 5.6 % 0.5 % 90.8 % 5.3 % 2.7 % 1.1 % 0.1 %

ZIP: Sum of squaresof deviations2

71.1 % 15.3 % 8.9 % 4.4 % 0.3 % 91.0 % 5.5 % 2.5 % 0.9 % 0.0 %

Poisson: Sum ofabsolute deviations3

74.1 % 22.2 % 3.3 % 0.4 % 0.0 % 92.3 % 7.4 % 0.3 % 0.0 % 0.0 %

Grade 8 2012 Observed 74.6 % 11.5 % 4.5 % 3.5 % 6.0 % 94.5 % 3.4 % 0.9 % 0.4 % 0.7 %


74.6 % 11.5 % 8.2 % 5.2 % 0.5 % 94.5 % 3.4 % 1.5 % 0.5 % 0.1 %


75.8 % 11.9 % 7.6 % 4.3 % 0.3 % 94.7 % 3.6 % 1.3 % 0.4 % 0.0 %


74.8 % 21.7 % 3.1 % 0.3 % 0.0 % 95.1 % 4.8 % 0.1 % 0.0 % 0.0 %

Grade 10 1991 Observed 71.2 % 13.8 % 5.9 % 3.7 % 5.4 % 92.9 % 4.6 % 1.4 % 0.7 % 0.5 %


71.3 % 13.8 % 9.1 % 5.3 % 0.4 % 92.9 % 4.6 % 1.9 % 0.6 % 0.0 %


72.4 % 14.4 % 8.6 % 4.4 % 0.3 % 93.0 % 4.7 % 1.8 % 0.5 % 0.0 %


75.6 % 21.2 % 3.0 % 0.3 % 0.0 % 94.2 % 5.7 % 0.2 % 0.0 % 0.0 %

Grade 10 2012 Observed 76.3 % 11.1 % 4.7 % 3.4 % 4.6 % 95.7 % 2.5 % 0.9 % 0.4 % 0.5 %

ZIP: Sum of absolutedeviations1

76.4 % 11.1 % 7.5 % 4.6 % 0.4 % 95.7 % 2.5 % 1.3 % 0.5 % 0.0 %


77.2 % 11.4 % 7.1 % 3.9 % 0.3 % 95.8 % 2.6 % 1.2 % 0.4 % 0.0 %


76.3 % 20.6 % 2.8 % 0.3 % 0.0 % 96.1 % 3.8 % 0.1 % 0.0 % 0.0 %

Grade 12 1991 Observed 75.1 % 12.5 % 4.8 % 3.5 % 4.1 % 94.3 % 3.6 % 1.2 % 0.4 % 0.5 %


75.1 % 12.5 % 7.8 % 4.3 % 0.3 % 94.3 % 3.6 % 1.5 % 0.5 % 0.0 %


76.0 % 12.9 % 7.3 % 3.5 % 0.2 % 94.4 % 3.7 % 1.4 % 0.4 % 0.0 %


79.5 % 18.3 % 2.1 % 0.2 % 0.0 % 95.1 % 4.8 % 0.1 % 0.0 % 0.0 %

Grade 12 2012 Observed 82.5 % 7.7 % 4.0 % 2.5 % 3.4 % 95.5 % 2.6 % 0.9 % 0.5 % 0.5 %

ZIP: Sum of absolutedeviations1

82.6 % 7.7 % 5.6 % 3.7 % 0.4 % 95.5 % 2.6 % 1.3 % 0.5 % 0.0 %


83.2 % 7.9 % 5.4 % 3.2 % 0.3 % 95.6 % 2.7 % 1.2 % 0.5 % 0.0 %


82.7 % 15.7 % 1.5 % 0.1 % 0.0 % 96.1 % 3.8 % 0.1 % 0.0 % 0.0 %

1 Frequencies in this row are estimated by minimizing the sum of absolute deviations between observedfrequencies and frequencies predicted by zero-inflated Poisson distributions2 Frequencies in this row are estimated by minimizing the sum of squares of deviations between observedfrequencies and frequencies predicted by zero-inflated Poisson distributions3 Frequencies in this row are estimated by minimizing the sum of absolute deviations between observedfrequencies and frequencies predicted by Poisson distribution

Q. Fu et al.

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