Relationship between Victimization at School and Achievement

The Relationship between Victimization at School and Achievement: The Cusp Catastrophe
Model for Reading Performance
Author(s): Georgios D. Sideridis, Faye Antoniou, Dimitrios Stamovlasis and Paul L. Morgan
Source: Behavioral Disorders, Vol. 38, No. 4 (August 2013), pp. 228-242
Published by: Sage Publications, Inc.
Stable URL: https://www.jstor.org/stable/23890650
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Behavioral Disorders
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The Relationship between Victimization at School and
Achievement: The Cusp Catastrophe Model for
Reading Performance
Georgios D. Sideridis
Boston Children’s Hospital, Harvard Medical School
Faye Antoniou
University of Athens
Dimitrios Stamovlasis
Aristotle University of Thessaloniki
Paul L. Morgan
The Pennsylvania State University
ABSTRACT: We evaluated the relationship between victimization and academic achievement
from a nonlinear perspective using a cusp catastrophe model. Participants were 62 students with
identified learning disabilities (LD) using statewide criteria in Greece. Students participated in a 2
year cohort-sequential design. Reading assessments involved measures of word accuracy,
pseudoword accuracy, and reading comprehension using a standardized reading battery.
Victimization was assessed using the Olweus (1993) questionnaire. Analysis involved estimation
of the cusp model with the dependent variable (word accuracy or reading comprehension) being
predicted by pseudoword or word decoding (the asymmetry factor) and victimization (the
bifurcation or splitting factor). We hypothesized that the relationship between word and
pseudoword decoding or reading comprehension and word decoding would be disrupted by the
presence of high levels in victimization. That is, increased victimization following a critical point
would be associated with unpredictable, sudden, and nonlinear changes in students’ achievement
in reading. Results were consistent with this hypothesis. Specifically, achievement in word
decoding and reading comprehension entered a state of uncertainty (chaos) when victimization
reached a certain threshold value. Consequently, victimization may not only affect emotional
functioning but also seriously disrupt both reading achievement and self-regulatory processes
related to reading.
â–  Learning to read is fundamental across
many aspects of a student’s life (Torgesen &
Hudson, 2006). Thus, identifying factors influ
encing reading acquisition represents a long
standing priority of educational research. Most
efforts have followed the paradigm of identi
fying positive predictors of reading acquisition.
Notwithstanding, of equal importance is the
investigation of negative predictors, particular
ly for students with learning disabilities (LD)
for whom lack of motivation and repeated
failures have been associated with serious
emotional risks (e.g., Maag & Reid, 2006).
Students with LD are at risk for atypical
228/August 2013
behavioral functioning generally (Stanovich,
1986), as well as concomitant emotional/
behavioral disorders specifically (Rock, Fess
ier, & Church, 1997). For example, Sideridis
(2007) linked specific motivational tëndencies
and school failures to a cognitive vulnerable
scheme for students with LD that may subse
quently be linked to depression (Fluntington &
Bender, 1993; Sideridis, 2009). Other research
studies have reported that students with LD are
at greater risk of engaging in both internalized
and externalized behaviors (Faraone et al.,
1993), mood disorders (Cantwell & Baker,
1991), anxiety disorders (Prior, Sanson, Smart,
Behavioral Disorders, 38 (4), 228-242
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& Oberklaid, 1999), or conduct problems
(John, 1989; Nabuzoka & Smith, 1993).
Recently, the National Joint Committee on
Learning Disabilities (NJCLD) stressed the
importance of emotional factors as comorbid
characteristics of LD, suggesting that “LD may
coexist with other disorders. For example,
individuals with LD also may manifest social
emotional, behavioral, or attention difficulties,
which may be either concomitant or second
ary to LD” (p. 2). On a different white paper on
the assessment of LD, they stated,
Learning disabilities vary in their manifesta
tions depending on task demands and may
include difficulties in language (i.e., listening,
written and oral expression, spelling, reading),
mathematics, handwriting, memory, percep
tion, cognition, fine motor expression, social
skills, and executive functions (e.g., attention,
organization, reasoning), (p. 8)
Thus, the accumulated research evidence
suggests substantial comorbidity between
learning and behavioral disabilities. For exam
ple, Greenbaum et al. (1996) reported that
increasing percentages of student with emo
tional/behavioral disorders (EBD) were reading
below level in their study’s 7-year time span,
from 54% to 85%. Similarly, Nelson, Benner,
Lane, and Smith (2004) reported that 83% of
their study’s sample of children with EBD
scored below the norm group on a standard
ized measure of reading skill. Thus, students
with EBD may be especially at risk of
experiencing chaos catastrophic functions in
their reading acquisition, particularly given
their greater susceptibility to victimization
(Swearer, Wang, Maag, Siebecker, & Frerichs,
2012).
A negative factor experienced by a large
number of school children, including those
with LD and EBD, is victimization and bullying
(Rose & Espelange, 2012). For example, the
National Center for Education Statistics (NCES;
2007) reported that one third of all students
have been victimized at school. Some children
experience victimization and bullying on a
daily basis. Sixth graders were the most likely
candidates to sustain an injury from bullying,
with the most common type experienced being
emotional bullying. The report went on to
present long-term consequences for victims,
which involved low self-esteem, difficulty of
trusting others, lack of assertiveness, aggres
sion, difficulty controlling anger, and isolation.
These consequences are quite serious for
students experiencing victimization, as they
Behavioral Disorders, 38 (4), 228-242
also struggle with the daily demands of school.
Yet, and despite its likelihood of being ex
perienced in school, no research studies to
date have examined how victimization may
impair the self-regulation or achievement of
students with LD, although the characteristics
of this population suggest a vulnerability to
victimization (Mishna, 2003). One hypothesis
relates to the emotional profile of the bully
victim1 student, whose behavior is governed
by high levels of reactive aggression (e.g.,
victims reacting to bully behaviors) or proac
tive aggression (engage in bullying behaviors;
Rose & Espelage, 2012). That is, when other
students target their resources to productive
tasks and engage in activities that promote
learning, victims would allocate resources for
dealing with an imminent threat or recovering
from a negative event associated with a
bullying episode. Recent evidence, however,
suggests that children with LD may be more
likely to bully other children at school (Twy
man et al., 2010). Children displaying atten
tion difficulties are also more likely to be
bullied and victimized (Twyman et al., 2010;
Unnever & Cornell, 2003; Weiner & Mak,
2009).
Victimization in Disabilities
Victimization, or purposeful physical or
psychological injury of another person (Kum
pulainen, Räsänen, & Henntonen, 1999), may
interfere with effective se If-regulation at school.
Baumeister, Storch, and Geffken (2008) found
that victimization was positively correlated
with reports of general and social anxiety,
loneliness, withdrawal, anxiety, depressive
symptoms, social problems, and disruptive be
havior. The presence of those behaviors is
exacerbated by the fact that in only 18% of
such events are teachers are present, who might
help mitigate the negative impact of victimiza
tion on student well-being (Atlas et al., 1998)).
Furthermore, students with LD who have been
victimized are more likely to themselves bully
other students (Estell et al., 2009). Moreover,
many students with LD experience sibling
bullying as well (Little, 2002) so victimization
does not necessarily end at school dismissal.
The main hypothesis put forth by Baumeister
and colleagues is that repetitive incidences of
bullying seriously interfere with the develop
ment of healthy self-concepts and interpersonal
skills, thereby leading to greater feelings of
loneliness and isolation. Increasing isolation
August 2013 / 229
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can then exacerbate victimization and bullying
as a result of diminishing peer support and
protective social networks.
The Present Study
The present longitudinal study assessed
the effects of victimization on the performance
of students with LD in word accuracy and
reading comprehension. Specifically, we eval
uated the hypothesis that adverse social
experiences for students with LD with their
peers would interfere with two well-estab
lished reading acquisition mechanisms. These
two mechanisms are the reciprocal interrela
tions between (a) pseudoword decoding and
word decoding and (b) word decoding and
reading comprehension. Pseudoword decod
ing involves a letter-based model of visual
word recognition. In the absence of word
meaning and sight recognition, pseudowords
are recognized by use of a letter-sound
representation system (Grainger, Bouttevin,
True, Bastien, & Ziegler, 2003) and sounding
out the word. If these processes fail, less
effective guessing occurs. In word decoding,
the process involves feedback from the word to
the letter representation system (McClelland &
Rumelhart, 1981) and, in the presence of
unknown words, a return to the letter repre
sentation system. Research based on cascaded
activation indicates that neural activity for
higher-order representations (word level) is
faster compared with lower-level representa
tions (i.e., pseudoword identification; McClel
land, 1979). Alternative models for explaining
the mechanisms that underlie word and
pseudoword decoding have been proposed
but are beyond the scope of the present study
(Hooper & Paap, 1997). Increasing skill in
letter-sound correspondence and word decod
ing is important in becoming a proficient
reader (National Reading Panel, 2008). In the
present study, we hypothesized that the
established and automatic relationship be
tween word and pseudoword decoding would
be negatively affected by high levels in
victimization. This should occur because
victimization results in strong emotional reac
tivity that we hypothesize is linked to a
systemic emotional shutdown. If instances of
victimization are frequent and repetitive, then
the victim’s cognitive and emotional resources
and coping mechanism are likely allocated
toward dealing with the threat and not with
engaging with positive academic behaviors.
230/August 2013
The proposed interference would be a function
of predispositions, vulnerabilities, individual
differences, and interactions of those with the
different social contexts and levels. This is why
behavior is predicted to become erratic and
reach a state of uncertainty, which is why the
cusp model is mostly appropriate to evaluate
such changes (i.e., by predicting the full range
of variable responses during the catastrophic
phase).
The relation between word decoding and
reading comprehension may also be moderated
by levels in victimization (Torgesen & Hudson,
2006). On one hand, word decoding is
considered a relatively automatic process in
which students use cognitive resources to
comprehend written text (Compton et al.,
2005). Subsequent reading involves an appli
cation of decoding rules to new material and is
one of the most important discriminators
between effective and ineffective readers (Lyon
& Moats, 1997; Stanovich, 1991; Torgesen,
2000). On the other hand, reading comprehen
sion is a process that is affected by greater
ongoing cognitive requirements and skills. For
example, the ability to construct and recognize
coherent representations within text is a pre
requisite to reading comprehension (Solari &
Gerber, 2008).
We evaluated the effects of social and
emotional dysfunction, as demonstrated by
high levels in victimization, on the reading
achievement of students with LD. We hypoth
esized that the effects of emotional dysregula
tion are systemic and not only affect emotional
well-being but also interfere with other pro
cesses such as a student’s ability to use both
more automatic decoding skills and cognitive
ly demanding reading comprehension skills.
Catastrophe Theory: A Description
Catastrophe theory (CT) is a branch of the
nonlinear dynamical systems (NDS) theory and
originates with Thorn’s (1975) pioneering work
attempting to explain morphogenesis (Brown,
1995). Catastrophe theory describes discontin
uous changes in a system, where qualitatively
different states of behavior are observed under
gradual increases in a number of control
variables (Tesser, 1980; Thelen & Smith,
1994; van Geert, 2003). The most well-known
and simple model of catastrophe theory is the
cusp model (Guastello, 2001, 2002) positing
that behavioral changes between two stable
states of a systems are predicted by two
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•
o
<6V°°
Figure 1. Visual display of the cusp model for
word accuracy (dependent vari
able) as a function of pseudoword
decoding (asymmetry factor) and
victimization (bifurcation). The sys
tem undergoes a state of disorgani
zation as levels of victimization
increased beyond a specified point.
Controlling variables, the asymmetry, and bifurca
tion factor. The former is linearly related to the
outcome, whereas the latter is related nonlinearly
(Van Gelder & Port, 1995). Figure i graphically
displays the response surface of cusp model,
which mathematically is described by the function
5f/8y= — y3 +by + ct (1)
The cusp model predicts discontinuous
transitions between two equilibrium states or
two modes of behavior as a function of
variable values in the two control parameters.
The dependent variable y bifurcates into two
divergent behavioral forms as a function of the
bifurcation variable b at the divergence point
(i.e., the bifurcation area) and the asymmetry
variable a. The prediction is that for low values
in the bifurcation variable, the system is rather
stable, and a linear relationship best describes
the link between the asymmetry variable and
the dependent variable (i.e., behavior changes
gradually and smoothly). However, when the
bifurcation variable takes on high values, the
behavior is predicted to become bimodal2
depending on the levels of the asymmetry
factor. That is, beyond a critical point, the
dependent variable (behavior) becomes overly
stressed, unstable, and unpredictable. It then
Behavioral Disorders, 38 (4), 228-242
begins oscillating between the two divergent
modes (low and high).
Catastrophe theory has had many influential
applications in psychological and behavioral
science, especially because it provided better
interpretation of empirical data and explana
tions of human-behavior phenomena, which are
predominately nonlinear in nature (Zhao et al.,
2008). Works with CT applications includes
theoretical endeavors, empirical research, or
both contributing to the development of the new
scientific paradigm of the NDS theory. Some
characteristic works include the following: the
connection of CT to Piagetian stagewise devel
opment (Molenaar & Oppenheimer, 1985; Van
der Maas & Molenar, 1992), to motivation and
academic performance (Guastello, 1987), atti
tude change (van der Maas, Molenaar, & van
der PI igt, 2003), modeling cognitive overload
phenomena (Stamovlasis, 2006, 2011), and
approach-avoidance motivation dynamics (Sta
movlasis & Sideridis, in press), to mention a few.
Methodologically, there are three different
approaches and statistical modeling proposed by
Oliva, Desarbo, Day, and Jedidi (1987), Guas
tello (1982; 1992; 2011) and Grassman, van der
Maas, and Wagenmaker (2009), respectively. In
the present study, we followed the latter
method, which we considered more appropri
ate. However, discussion on the differences,
objections, or controversies concerning the
modeling and statistical issues can be found
elsewhere (e.g., Alexander, Herbert, DeShon, &
Hanges, 1992; Cobb et al., 1983; Guastello,
1992; Grassman et al., 2009).
We expected that catastrophe models
would be very appropriate for modeling
victimization’s relation with reading achieve
ment, as we expected it would result in both
sudden and abrupt changes. Stewart and
Peregoy (1983) stated that whenever we
expect sudden, compared with smooth, and
abrupt changes in the dependent variable in
response to linear and smooth changes in the
independent variables, catastrophe models
should be the methodology of choice. The
term catastrophe is actually implemented to
define such sudden, nonlinear changes. These
changes could be large unexpected shifts in
the values of a dependent variable, which
destroy the linear relation with a predictor, due
to the operation of a third variable. In this
research, the well-established relationships of
(a) word decoding with pseudoword decoding
and (b) word decoding with reading compre
hension are reexamined under the effect of
August 2013 / 231
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victimization. The main thesis of the present
article is that the relation between pseudoword
decoding and word decoding and between
word decoding and reading comprehension
could be described by a cusp catastrophe
model, when simultaneously accounting for
the contribution of victimization. The choice
of victimization as a bifurcation is theory
driven. Prior work indicates that victimization
should be considered an especially strong
stressor on children’s behavioral regulation
and so might be expected to function as a
chaos-inducing factor interfering with cogni
tive processes related to reading acquisition.
This is particularly true in the case of students
experiencing both frequent but also multiple
forms of victimization (polyvictimization in the
terms of Soler, Paretilla, Kirchner, & Forns,
2012). Those effects can be of two forms,
developmental and localized (Finkelhor,
1995). The former are associated with long
lasting effects due to disturbances in develop
ment during a sensitive period that result in the
creation of a vulnerability for bullying or
victimization (e.g., insecure attachment). Lo
calized effects are context specific and involve
instances of increased apprehension and vig
ilance when a stressor is present (e.g., a bully;
Canton-Cortes & Canton, 2010; Crosby, (Deh
ler, & Capaccioli, 2010). Both developmental
and localized effects create an immense stress
over victimized students’ efforts to cope with
the demanding school environment. Thus, as
Figure 1 shows, the cusp is a function of the
asymmetry factor (pseudoword decoding),
whose function is regulated by the bifurcation
factor (victimization). That is, as levels in the
bifurcation variable (victimization) increase
beyond a critical level for a given high level in
the asymmetry factor (pseudoword decoding),
behavior is expected to become discontinu
ous (Rogosch, Dackis, & Cichetti, 2011). We
hypothesized that these mechanisms function
during the sometimes cognitively or behavior
ally demanding classroom environments along
with other processes (e.g., anxiety and low
motivation) but that victimization and bullying
would place an especially strong and stressful
strain on children’s cognitive functioning that
would be linked to systemic failure and
performance decrements in reading.
Importance of the Present Study
It is important to employ catastrophe
theory to test for self-regulation failure based
232 /August 2013
on emotional factors relating to LD. This is
the case for at least four reasons. First, the
NJCLD now considers emotions as a factor
that potentially influences the school func
tioning of students with LD. Second, emo
tions have long been suspected as affecting
self-regulation evidenced by experimental
mood studies. Third, the use of nonlinear
methods may elucidate the potentially non
linear relationship between emotionality and
achievement, which may inform when, to
whom, and under what conditions behavioral
interventions may be necessary to maximize
children’s academic achievement. Fourth,
this exploration may help us refine our
theories of self-regulation relating to school
achievement by incorporating types and
valence levels of emotional variables. As
argued earlier, this work should also have
great applicability to students with EBD
because reading and behavioral difficulties
frequently co-occur (e.g., Hinshaw, 1992;
Reid, Gonzalez, Nordness, Trout, & Epstein,
2004; Trout, Nordless, Pierce, & Epstein,
2003) suggesting the presence of a pathogen
ic relationship (Hinshaw, 1992; Lane, 2004)
that has been described as transactional
(Cook et al., 2012). According to Hinshaw
(1992), one path to reading failure involves
children who go to school with well-estab
lished behavior problems that interfere with
their ability to focus on reading instruction
and ultimately learn to read.
Hypotheses Tested
We evaluated the hypothesis that victim
ization negatively affects reading achieve
ment. Specifically, we hypothesized that two
widely established interrelations between (a)
word and pseudoword decoding and (b) word
decoding and reading comprehension would
be disrupted by victimization. That is, levels of
victimization beyond a critical point would be
associated with negative effects on reading
achievement that are explained by a disorga
nization in self-regulation and lower levels of
reading achievement, as indicated by bimodai
or multimodal distributions.
Method
Participants
Data were collected from 62 elementary
and secondary students with LD from Grades
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TABLE t
Descriptive Statistics of the Characteristics of the Participating Students
Grade Total Boys Girls Age (Years) Bilingual Students
Attended n n n Mean (SD) n
5 29 20 9 10.41 (0.37) 5
6 11 6 5 11.54 (.44) 2
7 10 8 2 12.48 (.75) 2
8 10 7 3 14.08 (1.07) 0
9 2 2 0 13.85 (0.88) 0
Total 62 43 19 11.67 (1.53) 9
Grade Total Boys Girls Age (Years) Bilingual Students
Attended n n n Mean (SO) n
5 29 20 9 10.41 (0.37) 5
6 11 6 5 11.54 (.44) 2
7 10 8 2 12.48 (.75) 2
8 10 7 3 14.08 (1.07) 0
9 2 2 0 13.85 (0.88) 0
Total 62 43 19 11.67 (1.53) 9
5 through 9 grade 29, grade 1 11
f^7th grade 10, Nßth grade 10, /Vgth grade 2).
Students were selected from 16 public schools in
the area of southern Greece. There were 43 boys
and 19 girls, and their mean age was 11.67 years
(SD = 1.53; minimum = 9.96, maximum =
16.89). The sample, although relatively small, is
adequate for a medium effect size with a power
of .80 (Maxwell, 2000). Fifty-three children were
Greek monolingual learners, whereas 9 were
bilinguals, with Albanian being their mother
tongue. Thirty-six of the participants had been
classified as students with LD from state
diagnostic agencies, whereas the other 26 had
to meet the criteria based on a discrepancy
model (Fletcher, Morris, & Lyon, 2003) for
classifying students with LD. The criteria used
were (a) adequate intelligence (>85), (b) dis
crepancy between ability (as reflected in IQ
scores) and achievement in the subscales of
decoding and spelling of the Software for
Screening Learning Skills and Difficulties
(LAMDA; Protopapas & Skaloumbakas, 2008),
and (c) absence of physical disabilities. Further
more, all students were identified as having LD
using a normative rating scale for the screening
of LD (Learning Disabilities Screening for
Teachers questionnaire, Padeliadu & Sideridis,
2008). All elementary school students attended
the resource rooms of their schools. Table 1
overviews the characteristics of the participants.
Procedures
Students’ participation in the study was
assured after informed consent was provided by
both them and their guardians. Their participa
tion was voluntary, and researchers emphasized
the anonymity of their responses. Assessments
took place in students’ classrooms during
regular school hours and in the special com
puter room of each school. All students were
Behavioral Disorders, 38 (4), 228-242
tested individually in reading comprehension
by trained psychology students. The reading
comprehension test lasted for 30 minutes. The
LAMDA software for screening for LD was
administered in a whole-group administration
on personal PCs and with the use of head
phones. The duration of administration was
approximately 40 minutes. All students were
aware that they could withdraw their participa
tion at any time during the tests. Students were
told that the purpose of these assessments was
to obtain a picture of the current level of
students’ performance to evaluate whether new
teaching procedures would be effective.
Research Design
A 2-year longitudinal design with pre-post
measures per year was implemented. Thus,
there were four measurement points between
the winter semester of 2009 and the spring
semester of 2011. Of interest was the predic
tion of end-of-first-year relations between
pseudoword and word decoding as a function
of levels of victimization. We were also
interested in evaluating the relationship of
decoding to higher levels of reading ability
(reading comprehension) as students advanced
a grade level and their decoding abilities
became more automatized.
Measures
Word Decoding
Students’ ability to correctly decode words
(Griffiths & Snowling, 2002) was assessed
using a list of 57 words. These words were
presented in order of ascending difficulty and
had one to eight syllables. Students were asked
to read aloud by paying attention to accurate
word reading and stressing. Any decoding
mistakes (missing or added letters/syllables,
August 2013 /233
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word replacement, and/or wrong stressing)
were scored with 0, whereas phonologically
correctly read words were scored with 1. The
subtests discontinue rule was implemented
when students committed five consecutive
decoding errors, and all following items were
scored with 0. Students were then presented
the following subtest of pseudoword decoding.
The internal consistency (Cronbach’s alpha) of
the word decoding scale was .85.
Pseudoword Decoding
Students’ phonological ability to decode text
without meaning (Share & Stanovich, 1995) was
assessed using a list of 40 orthographically
regular, pronounceable nonwords (i.e., stagara
instead of brvdsta). These words without mean
ing were presented in ascending difficulty, and
the number of syllables ranged from one to six.
Failed responses involved errors of letter or
syllable additions, omissions, and substitutions.
This subtest discontinue rule involved five con
secutive decoding errors. All subsequent items
were scored as failed. The scale’s internal con
sistency (Cronbach’s alpha) estimate was .83.
Victimization at School
Victimization was assessed using the
questionnaire by Olweus (1993, 1997, 2005).
The scale involves the measurement of nine
types of victimization behaviors (i.e., physical,
verbal, racial, sexual, electronic/cyber, indirect
bullying, social exclusion, stealing/damaging of
belongings, threat/coercion, and other types) on
a frequentist scale from 1 to 5 ranging from “It
hasn’t happened to me in the past few months,”
to “It happens to me several times a week.” The
alpha of the subscale was .74. The total score
was used as the moderating variable.
Data Analysis
The statistical methods appearing in the
literature for testing the cusp catastrophe employ
either least squares regression or maximum
likelihood (Cobb, 1981; Cobb & Zacks, 1985;
Guastello, 2001, 2002; Oliva et al., 1987). In
the present work, we implemented the method
developed by Grassman et al. (2009), which is a
modified maximum likelihood (Cobb, 1981;
Cobb & Zacks, 1985) and is performed in R. The
method employs more robust statistics and so
was considered the most suitable approach for
the preset data and for testing our hypotheses.
234/August 2013
The estimated nonlinear model is com
pared with the linear and logistic counterparts.
For a cusp model to be superior to a linear or a
logistic model, several conditions need to be
met: (a) their difference using a chi-square test
should first be significant in favor of the cusp
model, (b) the indices of Akaike information
criterion (AIC) and Bayesian information crite
rion (BIC) should be lower for the cusp model,
(c) all coefficients should be significant, and (d)
the R2 of the cusp model should be superior.3
All models were fitted by use of the cusp
library in R. The level of significance was set to
5% for a one-tailed test to increase power to
the acceptable level of .80.
Moreover, to evaluate the stability of the
samples’ point estimates, bootstrapping was
employed (Efron, 1982,1985) using the formula
of Hesterberg, Monaghan, Moore, Clipson, and
Epstein (2003):
mboot = ^J2m”
with mboot representing the mean of the
bootstrap distribution and m” the mean of each
individual sample for a k number of replica
tions. Results from bootstrapped estimates are
shown in Table 2. The findings provided
confidence that the cusp model results are
stable and representative of the population of
students with LD.
Results
Descriptive statistics
Descriptive statistics concerning selected
characteristics of the participating students,
such as sex, age, grades, and lingual/bilingual
group, are shown in Table 7.
Year 1: Cusp Model 1—Effects of
Pseudoword Decoding and Victimization
on Word Decoding
The model tested posited that word
decoding is linearly related to pseudoword
decoding (asymmetry factor), with the levels
of victimization being the bifurcation variable
(or splitting factor; see Figure 7). All models
were run using the cusp package in R (see
Appendix A).
The results are shown in Table 3. Ail
intercepts and slopes (except the intercept of
Victimization) are significant. The slope of
Victimization (b = .544) was significant at the
Behavioral Disorders, 38 (4), 228-242
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TABLE 2
Descriptive Statistics and Bootstrapped Point Estimates Based on 1,000 Replications of Sam Sizes of n = 100
Variable Mean M-Bias SDboo. 95th CIs
Word decoding Year 1 43.246 0.012 9.822 41.292-45.224
Pseudoword decoding Year 1 25.508 0.032 7.974 23.945-27.136
Word decoding Year 2 48.492 0.028 6.829 47.154—49.887
Reading comprehension Year 2 32.853 0.009 5.761 31.711-34.017
Victimization Year 1 1.760 0.001 0.766 1.607-1.911
Victimization Year 2 1.598 0.001 0.550 1.490-1.708
Variable Mean M-Bias SDboot 95th CIs
Word decoding Year 1 43.246 0.012 9.822 41.292-45.224
Pseudoword decoding Year 1 25.508 0.032 7.974 23.945-27.136
Word decoding Year 2 48.492 0.028 6.829 47.154^19.887
Reading comprehension Year 2 32.853 0.009 5.761 31.711-34.017
Victimization Year 1 1.760 0.001 0.766 1.607-1.911
Victimization Year 2 1.598 0.001 0.550 1.490-1.708
Note. The purpose of this analysis was to ascertain t population of students with learning disabilities.
5% level for a one-tailed test. The amount of
variance explained by both models was similar
(R2 = 38% for the linear vs. 36% for the cusp
model3). However, because of the limitations
in accurately measuring Cobb’s pseudo-R
parameter in R, we relied on model superiority
by use of the AIC and BIC estimates. For the
cusp model, the estimates were AIC = 136.64
and BIC = 149.41, whereas the respective
estimates for the linear model were 436.29
and 444.80 points, respectively. Moreover, the
cusp model was compared with the linear
model by use of a chi-square difference test.
Their difference exceeded levels of signifi
cance for a change of 2 degrees of freedom in
the model, x2(2) = 303.60, p < .001. Figure 2
shows the fit of the cusp model with the actual
data at Time 1, where the move from stable to
unstable states of organization is apparent as
the observations fall into the bifurcation area
(folded upper surface). Figure 3 displays the
lower surface with the desirable amount of
observations within the bifurcation area, and
Figure 4 displays bimodality in responding,
which is another criterion related to the
presence of a cusp. The above support our
hypothesis that victimization functions as the
bifurcation variable exerted a significant effect
over the relationship between pseudoword
and word decoding.
Year 2: Cusp Model 2—Effects of
Word Coding and Victimization on
Reading Comprehension
The second model tested posited that
reading comprehension is linearly related to
word decoding (asymmetry factor) with the
levels of victimization being the bifurcation
variable (or splitting factor). When fitting the
cusp model to examine the relationship
between word decoding and reading compre
hension with Year 2 data, results once again
fully supported the cusp model. These results
are shown in Table 4. All intercepts and slopes
(except the intercept of Victimization) are
significant. The slope of Victimization (b —
— .505) was significant at the 5% level.
Furthermore, the difference between the two
models, based on the AIC and BIC criteria, was
TABLE 3
Parameter Estimates of the Cusp Model for the Presence of a Performance Goal Structure at Time 1
Variable B SE Z Value P
a(lntercept) -2.247 0.741 -3.032 .0024″
a(Pseudoword Decoding) 0.151 0.042 3.558 .0004″*
b(lntercept) -0.210 0.852 -0.247 .805
b(Victimization) 0.544 0.290 1.872 .061f
w(lntercept) -3.006 0.506 -5.937 .0001″*
w(Word Decoding) 0.0913 0.010 9.074 .0001***
Variable B SE Z Value P
a(lntercept) -2.247 0.741 -3.032 .0024″
a(Pseudoword Decoding) 0.151 0.042 3.558 .0004″*
b(lntercept) -0.210 0.852 -0.247 .805
b(Victimization) 0.544 0.290 1.872 .061f
w(lntercept) -3.006 0.506 -5.937 .0001″*
w(Word Decoding) 0.0913 0.010 9.074 .0001***
***p < .001; “p < .01; *p < .05; fp < .05 (one-tailed).
Behavioral Disorders, 38 (4), 228-242 August 2013 / 235
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% * \ ^
%,%• \ \
/ O r\C. vAO r V \ \ 1 ^
V ԉۢ
,e
d*
\ ,
–
%\\1 ^
*r
Figure 2. Fit of the cusp model with the
actual data at Time 1. Word accu
racy as a function of pseudoword
decoding and victimization. The
move from stable to unstable states
of organization is apparent as the
observations fall into the bifurca
tion area (folded upper surface).
significant in favor of the cusp model (AlCLjnear
= 384.716; AICCU5P = 159.624; BICLinear =
393.224; BICr.,T = 172.386), *2(2) = 229.100, p
< .001. Figure 5 shows the fit of the cusp model
with the actual data at Time 2, where the move
from stable to unstable states of organization is
apparent as the observations fall into the bifurca
tion area. Multimodality in behavior is shown in
Figure 6. The above support our hypothesis that
victimization functions again as the bifurcation
variable exerted a significant effect over the
relationship between pseudoword and word
decoding and in reading comprehension.
Model Interpretation, Discussion,
and Implications
The cusp model suggests that both linear
and nonlinear changes in the dependent
variable (e.g., reading comprehension) might
be expected, which are described by the
control variables. When victimization values
are low, reading comprehension retains its linear
relation with word decoding. That is, students
with higher word-decoding abilities perform
better in reading comprehension, whereas stu
dents with lower word-decoding ability perform
worse in reading comprehension. However, the
middle-range word-decoding abilities might
236 / August 2013
*r –
Figure 3. Visual display of low surface of
the cusp model. The shaded area
is the bifurcation set. In this area,
for fixed values of the indepen
dent variable, the dependent var
iable becomes bimodal or multi
modal. Based on van der Maas et
al. (2003), for a catastrophe to be
evident, 10% of the observations
must fall in the bifurcation area.
behave in unpredictable way, when victimiza
tion exceeds a certain value, the bifurcation
point, beyond which the system enters the
bifurcation set, the area of unpredictability
where discontinuous changes occur. In this area,
subjects with the same parameter values of
word-decoding abilities and victimization might
succeed or fail in reading comprehension. The
above is called the hysteresis effect, which is due
to the sensitivity of the parameters and the
underlying dynamical process.
Thus, one interpretation of these findings is
that during a cognitive process such as
reading, a student with middle-range value of
word-decoding abilities might enter a chaotic
regime when a shift from success to failure
might occur as a result of the negative effect of
the bifurcation variable, which here is victim
ization beyond a particular level. This may
occur because victimization results in negative
emotional feelings, which then disorganize
self-regulation processes, interfere with the
activated mental resources, and subsequently
inhibit reading comprehension. The effect
should be more pronounced for students with
low ability levels, such as those with LD or
EBD.
Behavioral Disorders, 38 (4), 228-242
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Conditional density
00
3
CD O
o O 1—I—r~-r
-10 1 2 3
N = 41 Bandwidth = 0.2141
Conditional density
N = 18 Bandwidth = 0.4581
Conditional density
N = 3 Bandwidth = 0.429
Conditional density
00
3ZT
« o
o o I r r
-1 o 1
N » 41 Bandwidth = 0.2141
Conditional density
-2 0 12 3
N = 18 Bandwidth = 0.4581
Conditional density
N = 3 Bandwidth = 0.429
Figure 4. The lower panel displays the be
havior of the observations in the
bifurcation set. In accordance with
the expectations of the cusp model,
the behavior is bimodal. The upper
and middle panels display the
observations at other points of the
low-response surface. Actual data
at Year 1.
Discussion
We evaluated the hypothesis that victimi
zation negatively influences academic perfor
mance. Investigating whether and to what
extent this occurs is important in part because
so many students, including those with disabil
ities, currently experience victimization and
bullying in schools (NCES, 2007; Rose &
Espelange, 2012). We selected two empirically
validated relationships (that of word and pseu
doword decoding and between word decoding
and reading comprehension) and predicted that
those established relations would be significant
ly negatively moderated by victimization. That
is, levels of victimization beyond a critical point
would disrupt these cognitive processes and
result in student achievement levels that take all
possible forms (i.e., academic achievement
moves from a predictable to an unpredictable
stage). These hypotheses were confirmed in the
present study.
The most important finding of the present
study was that the presence of victimization
negatively affected the reading achievement of
students with LD. In fact, those changes were
not predicted by a linear model (suggesting the
presence of a negative linear trend) but by the
cusp model in which increases in the levels of
victimization beyond a point were associated
with abrupt and sudden, nonlinear changes in
reading performance. Victimization was linked
to a state of disorganization in the dependent
variable for which the established relation
between word and pseudoword decoding or
word decoding and reading comprehension no
longer held (Cobb [1981] described it as the
“anti-prediction” state). The cusp model was
significantly more predictive of the proposed
relationships compared with the linear model
TABLE 4
Parameter Estimates of the Cusp Model at Year 2 for the Prediction of Reading Compreh from Word Decoding
Variable B SE Z Value P
a(lntercept) -2.580 1.146 -2.250 .0244*
a(Word Decoding) 0.092 0.024 3.883 0001***
b(lntercept) 1.117 0.762 1.466 .143
b(Victimization) -0.505 0.208 -2.426 .015*
w( Intercept) -2.793 0.433 -6.453 .0001***
w(Reading Comprehension) 0.117 0.011 10.792 0001***
Variable B SE Z Value
a(lntercept) -2.580 1.146 -2.250 .0244*
a(Word Decoding) 0.092 0.024 3.883 .0001***
b(lntercept) 1.117 0.762 1.466 .143
b(Victimization) -0.505 0.208 -2.426 .015*
w( Intercept) -2.793 0.433 -6.453 .0001***
w(Reading Comprehension) 0.117 0.011 10.792 0001***
***p < .001; **p < .01; *p < .05; fp < .05 (one-tailed).
Behavioral Disorders, 38 (4), 228-242 August 2013 / 237
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-o ,<A ** (S*\ ^
Figure 5. Fit of the cusp model with the
actual data at Year 2. Reading
comprehension as a function of
word decoding and victimization.
The move from stable to unstable
states of organization is apparent as
the observations fall into the bifur
cation area (folded upper surface).
across all tests and models. All requirements of
the analytical methodology were met with
the cusp model, suggesting that victimiza
tion interfered with attentional regulation and
cognitive processes necessary for proficient
reading. It should be stressed that although the
present findings demonstrate that the selected
variables displayed asymmetrical and bifurca
tion functions within the cusp catastrophe
model, other variables of cognitive or affec
tive origin are not excluded from having an
analogous role.
Limitations
The present study is limited by a number
of factors. First, the sample size is relatively
small. Although we bootstrapped the param
eters of interest, there are concerns relating to
the use of resampling and the possibility that
sample idiosyncrasies are reproduced (Hes
terberg et al., 2003). Second, we looked at
the aggregate victimization term, although it
consisted of several different types of victim
ization. An analysis by type with a larger
sample could provide more detailed informa
tion. Third, the present findings are correla
tional in nature, and no causal statements
should be drawn regarding the above rela
tionships. Fourth, our study’s findings were
238 / August 2013
Conditional density
CO
c
Ol) O
a> -TVT o – -yH
ü ö i I I r
-10 12
N = 50 Bandwidth = 0.18
Conditional density
N = 10 Bandwidth = 0.1738
Conditional density
00
d
• 1J I
> – i
i i—i—r
-10 12
N = 50 Bandwidth = 0.18
Conditional density
o –
=» ~7\
1 Ki Is
-10 12
N = 10 Bandwidth = 0.1738
Figure 6. Multimodality is present in reading
comprehension. Actual data at Year 2.
based on only a sample of students with LD
who may have comorbid behavioral prob
lems. Last, potentially other variables related
to social and emotional adjustment may also
function under the lens of the cusp model.
Future studies, however, should examine
whether the present findings extend to stu
dents with EBD.
Conclusions, Contributions,
and Recommendations
The present study provided a nonlinear
perspective to our understanding of a complex
set of interrelations that explain academic perfor
mance using the nonlinear effects of emotional
processes. These findings may help explain the
observed co-occurrence between reading and
behavioral difficulties generally (Hinshaw, 1992),
as well for students with LD and EBD specifically
(e.g., Greenbaum et al., 1996). We consider it
highly plausible that the complex set of relations
surrounding personal and situation factors in
effective self-regulation can be explained more
fully by nonlinear methods and NDS theory.
Our findings, with a sample of students
with LD, are highly relevant for students with
learning and behavior problems such as
students with EBD. These students are likely
to display comorbid learning difficulties and
disabilities, particularly in reading (e.g.,
Behavioral Disorders, 38 (4), 228-242
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Greenbaum et al., 1996; Rock et al., 1997;
Trout et al., 2003), as well as greater risk for
victimization (Swearer et al., 2012). Theoret
ically, our findings highlight a possible mech
anism by which the reading difficulties of
students with EBD might be exacerbated—as
these students are victimized, their reading
difficulties may worsen. This then might itself
increase their risk of social isolation over time,
reducing their peer supports and so furthering
their victimization and its attending impact on
their academic and behavioral functioning
(Morgan, Farkas, & Maczuga, 2012). Con
versely, our findings may also help explain the
positive effects of interventions targeting socio
emotional learning on academic achievement
(Durlak, Weissberg, Dymnicki, Taylor, &
Schellinger, 2011), as these interventions may
in part lower the likelihood of victimization
but at the same time provide a platform to
promote academic achievement (by prevent
ing the emotional shutdown due to victimiza
tion). From a practical standpoint, interven
tions that reduce the likelihood of victi
mization by students with disabilities may be
one potential way to help increase their
academic and behavioral functioning and
thereby their educational opportunities over
time (Durlak et al., 2011 ; Durlak & Weissberg,
2011; Morgan et al., 2012). Clearly, and
despite their correlational nature, our results
support a contention that victimization consti
tutes a particularly pernicious impediment to
the reading achievement of students with
disabilities.
NOTES
1. Bully or victim, as it is highly likely that both
roles are assumed at one point or another, suggesting
the fluidity and complexity of the behaviors associ
ated with bullying and the continuously changing
dynamics of the social environment (Espelage &
Swearer, 2011). Several theorists attempted to
extend Bronfenbrenner’s (1986) socioecological
model to explain behavior as a function of nested
contextual systems (e.g., Rose, Allison, & Simpson,
2012), but they are beyond the scope of the present
study.
2. Bimodal in the sense that changes in the splitting
or bifurcation variable from one point on are
associated with two divergent predicted responses
compared with one as in the unimodal case.
3. This last criterion is not relied upon heavily as
pseudo R2 values can become negative in the case of
nonnormal distributions. It was evaluated in con
junction with the values of AIC and BIC with the
decision making in favor of the cusp model being
based heavily on the latter factors.
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AUTHORS’ NOTE
Address correspondence to Georgios D.
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MANUSCRIPT
Final Acceptance: 11/10/13
Appendix A
In R, the syntax used to run the model
using the cusp package in R (Crassman et al.,
2009) was the following:
fit <- cusp(y ~ Word Identification, alpha
~ PseudoWord Decoding, beta ~ Victimiza
tion, data = database) with the dependent
variable being word identification, the asym
metry factor pseudo word decoding, and the
bifurcation factor victimization.
242 / August 2013 Behavioral Disorders, 38 (4), 228-242
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