The clustering of bullying and cyberbullying behaviour within Australian schools.
Shaw, Therese ; Cross, Donna
Introduction
Bullying is defined as aggressive behaviour repeated over a period
of time, characterised by a real or perceived imbalance of power
perpetrated with the intent to harm the target (Olweus, 1996). Bullying
between students at school can seriously affect the social, physical and
psychological well-being--as well as the academic achievement--of both
the perpetrators and those who are victimised (Arseneault, Bowes &
Shakoor, 2010; Kaltiala-Heino, Rimpela, Marttunen, Rimpela &
Rantanen, 1999; Kaltiala-Heino, Rimpela, Rantanen & Rimpela, 2000;
Nansel et al., 2001; Wolke, Woods, Bloomfield & Karstadt, 2001). The
Australian Covert Bullying Prevalence Study found that just over one
quarter (27%) of Australian school students aged 8 to 14 years reported
being frequently bullied, and 9% reported frequently bullying others
(every few weeks or more often) (Cross et al., 2009). Approximately 7%
of students in Years 4 to 9 reported being cyberbullied every few weeks
or more often in their last term at school (Cross et al., 2009). While
cyberbullying occurs with less frequency than traditional bullying, its
prevalence is still appreciable and possibly increasing in Australia, as
elsewhere in the world (Smith & Slonje, 2009). The high prevalence
of school bullying and its significant detrimental effects have
prompted, especially in recent years, much research to better understand
this behaviour and to intervene to reduce the harm associated with
bullying (Farrington & Ttofi, 2009).
Research on behavioural phenomena amongst school students, such as
bullying, must take account of the clustering of students within
schools. This is due not only to the study designs used but also to the
contextual influences on the variables of interest. Firstly, cluster
sampling designs, where schools are selected in the first stage of
sampling and individuals in the second, are often used in studies of
young people, as schools facilitate access to the target population and
survey administration (Carlin & Hocking, 1999; Heeringa, West &
Berglund, 2010). Secondly, students' experiences of bullying at
school or within other contexts depend on the behaviour and norms within
the particular group: for example, a number of students in a school may
be victimised by the same perpetrator. Additionally, young people's
behaviour (and particularly problem behaviour) may be influenced by
their peers (Dishion & Owen, 2002; Kiesner, Dishion & Poulin,
2001). Furthermore, in intervention research, wide use is made of
grouprandomised trials, in which whole groups, such as schools, are
randomised to conditions and interdependence of outcomes exist,
particularly if the interventions have a whole-of-school focus.
The consequences of these clustering factors are that students
within a particular school will be more alike with regard to bullying
behaviour than students from different schools. This homogeneity within
schools is measured by the intraclass correlation (ICC). In a two-level
design with students nested within schools, the ICC can be interpreted
as the extent to which students from the same school are more similar
than students from different schools. The ICC is calculated as the ratio
of the variation between schools relative to the total variation (at
school and individual level) in the variable of interest such as
bullying between students. ICC values vary between zero and one. Greater
variation or differences between schools implies greater similarities
within schools and hence a higher ICC value (Twisk, 2006). An ICC of
zero would imply no variation between schools; that is, the variation in
bullying outcomes of students aggregated within schools is equal to the
variation among students across all schools. At the other extreme, an
ICC of one (an unlikely value) would mean that all of the variation
between students is due to school differences; that is, there are no
differences within schools.
ICC values vary according to the outcome measure for which the
value is calculated and the study population (Carlin & Hocking,
1999; Murray, 1998). Additional factors such as the time of the year of
the survey and the gender, ethnicity and year level of the students can
also influence the size of the correlation (Murray et al., 1994;
Resnicow et al., 2010; Siddiqui, Hedeker, Flay & Hu, 1996). ICC
values have been found to be below 0.1 in value for a range of health
outcomes for school-based data, including for measures of tobacco
(Murray et al., 1994; Siddiqui et al., 1996), alcohol and other drug use
(Carlin & Hocking, 1999; Murray, Clark & Wagenaar, 2000;
Scheier, Griffin, Doyle & Botvin, 2002), nutritional intake (Murray,
Phillips, Birnbaum & Lytle, 2001) and physical activity (Murray et
al., 2006). Hutchison (2004) compared ICC values across a range of
variables for primary and secondary students in schools in the Third
International Mathematics and Science Study (TIMSS) survey in the UK and
Wales. Results showed the strongest clustering effects for the various
educational outcomes (mean ICC = 0.18) and ethnic variables (mean ICC =
0.14) compared with demographic variables (mean ICC = 0.02), leisure
activities (mean ICC = 0.04) and home characteristics relevant to
educational attainment (mean ICC = 0.04). Australian data from the 2009
Programme for International Student Assessment (PISA) study similarly
produced ICC values of 0.2 for each of reading, mathematics and science
scores (OECD, 2010).
Limited data are available on ICC values for bullying outcomes and
the extent to which a culture of bullying may thus be stronger in
certain schools compared with others. Bradshaw, Sawyer and
O'Brennan (2009) reported an ICC value of 0.019 for victimisation
amongst elementary school students in a district in Maryland, USA, and
for middle school students for victimisation as 0.006 and bullying
others as 0.009. The evaluation of the KiVa Antibullying Program
conducted in Grades 4-6 in 78 schools in Finland found school-level ICC
values of 0.02 for both victimisation and perpetration (Karna et al.,
2011). Two other cluster-randomised intervention trials presented the
average and ranges, rather than specific values for individual outcomes,
of school-based ICC values. Fonagy and colleagues (2009) reported an
average ICC value of 0.04 for the bullying-related outcomes, which
included measures of aggression and victimisation, in their study of
third to fifth grade students in nine elementary schools in a city of
the mid-west in the USA. Australian data from the Gatehouse Project
(Bond et al., 2004), conducted amongst secondary students in 26 schools
in Victoria, found ICC values between 0.01 and 0.06 for a range of
emotional well-being and drug use outcomes, including bullying
victimisation.
A second motivation exists for estimation of ICC values for
bullying outcomes. When planning school-based studies, account needs to
be taken of the impact of the non-independence of the subjects in
calculating required sample sizes or under-powered studies will result
(Murray & Short, 1997). The homogeneity within clusters results in
cluster samples having a smaller effective sample size in terms of the
precision with which parameters are estimated-and hence the power to
detect a statistically significant difference-than a simple random
sample of the same number of subjects (Heeringa et al., 2010). The
design effect is a measure of the performance of a complex sampling
design (such as cluster sampling) compared to what would be achieved
with simple random sampling. A means of determining the sample size
required for a cluster sample with sufficient power is to incorporate
the design effect in the calculations. The design effect for cluster
sampling depends on the cluster sizes and the strength of the
correlation within the clusters, hence the need for estimates of ICC
values.
The type of analyses conducted influences the operative ICC value:
that is, the value that will be operating when the results of the study
are determined (Carlin & Hocking, 1999; Murray, 1998) and hence the
value to be accounted for in sample size calculations. In longitudinal
or pre-test-post-test studies, repeated measures analyses or adjustment
for the baseline value of the outcome measure can result in substantive
reductions in operative ICC values compared to cross-sectional designs
(Murray & Blitstein, 2003). Furthermore, the addition of explanatory
variables or covariates in statistical models can reduce the impact of
the ICC and increase the power of a given analysis, if the variables
reduce the variation between schools (Murray, 1998; Snijders &
Bosker, 1999). School level variables correlated with the outcome of
interest are the most likely to reduce this variation.
To our knowledge, to date there are no Australian data published
that describe ICC values for bullying outcomes and there are no
published data from Australia or elsewhere describing ICC values for
cyberbullying outcomes. This article presents ICC values for
self-reported bullying victimisation and perpetration measures,
including cyberbullying, based on a representative sample of Australian
school students. The aim is to explore the extent to which differences
in students' victimisation and perpetration of bullying behaviour
are related to the school they attend: that is, the extent to which
bullying is clustered within specific schools. Differences between
demographic groups are illustrated. A secondary aim is to determine the
impact of clustering effects on sample size requirements to assist
researchers to plan cross-sectional surveys or group-randomised
intervention trials for bullying outcomes. Guidelines for the
calculation of sample sizes are discussed.
Method
Schools and participants
The Australian Covert Bullying Prevalence Study included a
cross-sectional survey of students in years 4 to 9 (typically 9-15 years
of age) conducted in Term 4 in 2007 in 106 schools (55 primary and 51
secondary, 46% response rate) (Cross et al., 2009). A stratified (by
state/territory and location) cluster sampling design was used, with
schools randomly sampled in the first and classes in the second stage of
sampling. Because of differences in school structures between states in
Australia, in four of the eight states and territories (the Australian
Capital Territory, New South Wales, Tasmania and Victoria) Year 7
students were drawn from secondary schools and from primary schools in
the remaining three states (Western Australia, Queensland and South
Australia) and the Northern Territory. The sampling population included
all schools in Australia other than non-mainstream schools, those in
remote areas and schools with fewer than 30 students in 2007 in each of
the sampled year levels. Students with a disability that prevented them
from completing the hard-copy questionnaire were not included in the
sample.
Consent was sought from parents of all students in selected classes
with information and letters mailed directly to parents. Reply-paid
envelopes for the return of consent forms were provided. Active parental
consent was required in government schools in certain states or
territories (36% consent rate), and an active/ passive consent procedure
(where participants actively opt out) was used in all other instances
(96% consent rate).
Students completed hard-copy questionnaires in their classrooms
administered by school staff according to a strict procedural and verbal
protocol. The questionnaire was read aloud to Year 4-6 students.
Alternative learning activities were provided for students without
parental consent and those who declined or were unable to participate.
Of the 8782 students with parental consent, useable surveys were
returned from 7418 students (84%). The sample comprised 52% female
students (n = 3874), 37% (n = 2779) in government schools, 64% (n =
4760) in metropolitan areas and between 14% and 19% in each of Years 4
to 9. Students ranged in age from 8 to 16 years (mean = 12, SD = 1.7
years).
Measures
Bullying victimisation and perpetration were measured using single
items and scales. Consistent with previous research (Solberg &
Olweus, 2003), students were provided with a definition of bullying as
repeated behaviour that happens 'to someone who finds it hard to
stop it from happening', together with examples of different forms
of bullying. All victimisation and perpetration questions referred to
the previous term at school (past 10 weeks) and had specific response
options: 'was not bullied/did not bully', 'once or
twice', 'every few weeks', 'about once a week'
and 'most days'. In each instance, the same items were used
for victimisation and perpetration, with wording adjusted appropriately.
Any type of bullying Items adapted from the Olweus Bully/Victim
Questionnaire (Olweus, 1996) and the Rigby and Slee Peer Relations
Questionnaire (Rigby, 1998) were used to measure any type of
victimisation ('This term, how often were you bullied again and
again by another student or group of students') and perpetration.
Test-retest reliability of these items was moderate (n = 140, [K.sub.w]
= 0.54 and [K.sub.w] = 0.45 respectively). Students were categorised as
having been bullied or as having bullied others if they indicated the
victimisation or perpetration occurred every few weeks or more
frequently in the previous school term (Solberg & Olweus, 2003). Two
12-item scales were included to measure victimisation and perpetration
of different forms of bullying: verbal, exclusion, social (for example,
spreading rumours), physical and threatening bullying behaviour and the
extent of victimisation and/or perpetration. Cronbach's alphas for
these scales were 0.91 and 0.88 respectively. A mean score (0-4) was
calculated for each scale.
Cyberbullying Cyberbullying perpetration and victimisation
behaviour were measured using two 8-item scales tackling bullying
behaviour perpetrated via mobile phone, email or the internet (for
example, 'sent nasty messages on the internet', 'mean or
nasty comments or pictures posted to websites', 'ignored or
left out of things over the internet'). Cronbach's alphas of
0.86 and 0.88 were found for the cyber victimisation and perpetration
scales respectively. Mean scores (0-4) were calculated for the
cyberbullying victimisation and perpetration scales. The scale scores
were also dichotomised (0 and > 0) to obtain binary measures of any
exposure to and involvement in cyberbullying behaviour. These variables
hence measure any involvement in cyberbullying behaviour (even a single
instance such as being sent a nasty text message or hurtful comment on a
social networking site) and should not be interpreted as defining
students who are or are not cyberbullied, or did or did not cyberbully
others.
Both binary and continuous measures of bullying behaviour are
illustrated in this article as the sample sizes required for each can
differ markedly.
Demographic variables considered in this article include student
gender as well as those that represent ways of grouping schools commonly
included in research studies and, hence, where separate ICC values may
be of interest. These are school sector (government versus
non-government), location (metropolitan versus non-metropolitan), school
level (primary versus secondary) and school size. Schools were
dichotomised into two equally sized groups according to the numbers of
schools in the sample; smaller primary schools had up to 410 primary
students and smaller secondary schools up to 666 secondary students.
These groupings were chosen to ensure sufficient numbers of schools for
the calculation of ICC values in each.
Calculation of ICC values and standard errors
The ICC values in this article were calculated using the
'analysis of variance' approach. This estimator of the ICC in
a two-level design is the ratio of the variation due to differences
between schools ([[sigma].sup.2.sub.g]) to the total variation in the
outcome measure, where the total variation is the sum of the variation
between individuals in the same school ([sub.e][sup.2]) and the school
level variation:
ICC = [sub.g][sup.2]/([sub.g][sup.2] + [sub.e][sup.2])
(Snijders & Bosker, 1999). Binary outcomes are commonly
analysed by means of logistic regression, in which case the level 1
error term is assumed to follow a logistic distribution with a constant
variance of [sub.e][sup.2] = [sup.2]/3 = 3.29 (Rabe-Hesketh &
Skrondal, 2008; Snijders & Bosker, 1999; Twisk, 2006).
Maximum likelihood estimates of the ICC values and their standard
errors were calculated in Stata10 using the xtreg (with the mle option)
and xtlogit procedures, fitting linear and logistic regression models
respectively and including random intercepts to account for the
school-level clustering (StataCorp, 2007). The ICC standard error for
continuous outcomes is calculated using the delta method in xtreg
(StataCorp, 2011). For binary outcomes, in xtlogit, estimates of the
standard errors are derived from the second derivative of the likelihood
function (Rodriguez & Elo, 2003).
Only schools with five or more students were included (Rabe-Hesketh
& Skrondal, 2008) and in most instances the ICC values are estimated
based on at least 40 schools, to ensure adequate estimation of the
variance components, ICC values and standard errors (Donner & Klar,
2004; Murray, Varnell & Blitstein, 2004).
A simple means of ascertaining the required sample size for a
cluster sample is to calculate the required sample size based on simple
random sampling and then inflate this figure by the design effect,
obtaining an adjusted sample size with the power required for the study.
The design effect (also known as the variance inflation factor or VIF)
is the increase in between-school variance due to the homogeneity of
students within the same school. For a simple cluster sample it is:
design effect = 1 + (m -1) x ICC
where m is the average number of students sampled per school (Kish,
1965). This approach to calculating a sample size will be illustrated in
this article.
Results
ICC values
The ICC values and standard errors for the binary and continuous
bullying measures for the entire sample and broken down by demographic
variables are presented in Table 1.
For the total sample, the ICC values range from 0.015 to 0.031 for
the victimisation and between 0.037 and 0.071 for the perpetration
measures. The standard errors varied between 0.003 and 0.033. The two
largest ICC values also had the largest standard errors, indicating the
most uncertainty with regard to these estimates. Similarities in ICC
values were found for the binary and continuous scale measures for any
type of victimisation (0.025 and 0.023 respectively), the binary any
type of perpetration and cyber perpetration measures (0.071 and 0.067
respectively) and the continuous scale measuring any type of
perpetration and cyber perpetration measures (0.039 and 0.037
respectively) (Table 1).
The estimates are based on between 43 and 106 schools, with cluster
sizes ranging from 35 to 85 students per school (Table 2). The actual
cluster sizes varied appreciably between schools: for example, for the
total sample, in the smallest school 11 students and in the largest
between 181 and 186 students responded (depending on the outcome
measure). Note that, while the binary measures for any type of bullying
perpetration and victimisation (the third to sixth columns of Table 1)
represent behaviour that occurs every few weeks or more often, those for
cyberbullying perpetration and victimisation (the last four columns of
Table 1) represent any exposure to or involvement in cyberbullying
behaviour.
The ICC values give some interesting insights into the clustering
of bullying behaviour within schools. Higher ICC values indicate greater
disparities between schools with regard to bullying behaviour and thus
higher concentrations of bullying behaviour within particular schools.
Lower ICC values indicate commonalities between schools. As the total
ICC values are close to zero (range from 0.015 to 0.071), variation
between schools is low and the occurrence of bullying behaviour seems,
therefore, not particular to only certain schools. In almost all
instances, the ICC values for the perpetration measures were higher than
those for the victimisation measures. Differences between schools
therefore accounted for a greater percentage of the variation in
perpetration than their contribution to the victimisation measures. The
exception was for secondary schools, where the ICC values for the
matching perpetration and victimisation measures were similar.
Group differences in values
The ICC values were higher for girls on each of the bullying
measures; girls' perpetration of bullying and bullying
victimisation behaviour thus differed to a greater extent between
schools than was the case for boys.
When comparing ICC values for primary and secondary schools, the
values were higher in primary schools for perpetration of bullying but
lower for bullying victimisation. This means that differences between
primary schools were more pronounced and that students within the same
primary school were more similar in their perpetration behaviour than
those within secondary schools, where students across schools were more
similar. But less diversity existed between primary schools on bullying
victimisation than between secondary schools.
For most of the measures, the value of the ICC for the total sample
lay within the range of the values for school level, school size and
school sector, indicating variation between schools within groupings was
greater or similar to the variation across the entire sample of schools.
The exceptions were the ICC values for the two cyber perpetration
measures (binary and continuous), where the ICC values for the total
sample were higher than those for each of the primary and secondary
school samples. For example, the value for the binary cyber perpetration
measure of 0.067 is higher than the primary school value of 0.040 and
secondary school value of 0.032, indicating greater diversity between
the entire range of schools sampled than between schools within the
primary and secondary groupings. This may be due to the relatively low
proportion of students, particularly in primary schools, who report
perpetrating cyberbullying behaviour, leading to the seemingly more
pronounced clustering of this behaviour within schools.
Apart from the two binary perpetration outcomes, the ICC values for
the different sized schools did not differ by more than 0.01. In smaller
primary schools, school-level variation accounted for an estimated 9.2%
of the total variation in perpetration of any type of bullying
behaviour. While the numbers of primary and secondary schools within
each school size grouping were not sufficient to estimate separate ICC
values adequately, subsequent analyses showed that the large ICC for
perpetration of any type of bullying in smaller schools may be due to a
high level of variability on this binary measure, particularly among the
28 smaller primary schools. School differences made up 8.3% of the total
variation in involvement in cyberbullying in larger schools. This result
could not be attributed to primary or secondary schools in particular
(both levels had ICC values well below 0.083) and seems to be a
consequence of differences between larger primary and secondary schools
in perpetration of cyberbullying behaviour.
The ICC values were similar for the government and non-government
sectors. Apart from the binary measure for perpetration of any type of
bullying, the non-metropolitan schools had higher or similar values to
the metropolitan schools on all the measures, signifying more pronounced
clustering of bullying and cyberbullying behaviours within certain
non-metropolitan schools.
The impact of other factors on ICC values
The power of an analysis can be improved by lowering the value of
the operative ICC value through judicious statistical modelling, such as
including variables that explain school-level variation in analyses
(Murray & Blitstein, 2003). The reductions in ICC values resulting
from the addition of covariates to statistical models are illustrated in
Table 3 for logistic or linear regression models including different
demographic variables. In particular, the impact of the addition of
gender and the Australian state or territory is shown. The inclusion of
gender does not have a large effect on the ICC values (comparing Models
1 and 2, and Models 3 and 4) whereas the inclusion of Australian state
or territory does (comparing Models 1 and 3, and Models 2 and 4). This
finding is because gender is measured at the student level and it is not
able to explain or reduce much of the variation between schools, unlike
variables measured at the school level. In fact, increases in ICC values
may result from adjustment for student level variables when there is an
imbalance in the variable among schools, such that they appear more
similar than they are (Murray & Blitstein, 2003).
Importantly, some of the variation between the states and
territories is likely to be due to differences in parental consent
processes, with government sectors in certain states or territories
requiring active parental consent (rather than allowing active/ passive
consent procedures) before students could participate in the surveys.
These requirements resulted in markedly lower participation rates among
active consent-only schools and hence likely greater homogeneity of
responding students. A further explanation may be differences in the
location of Year 7 students, mostly in primary schools in certain states
and territories, and in secondary schools within others.
Cluster sizes and design effects
While the ICC values seem negligible and they indicate small
clustering effects, their impact in terms of the design effect and
therefore on the power of a study is not able to be ignored, especially
when large numbers of students are sampled per school (Table 4). A
selection of ICC values for primary and secondary schools from Table 1
have been used for illustrative purposes, together with increments of 25
students (roughly one class) per school.
As expected, the design effects increase as the ICC values and the
cluster sizes increase. The greater the homogeneity of students within
schools and the more students sampled per school, the less independent
information to be gained from each individual and the sample. Even for a
small ICC of 0.006 and an average of 200 student respondents per school,
the required sample size to achieve the same power for a cluster sample
is more than double (2.2 times) that of a simple random sample.
The importance of an ICC value with regard to power and sample size
determination is related to the number of students who will be sampled
per school. A small ICC of 0.006 is not an issue if 25 students per
school are sampled, as only a small increase in sample size is needed to
attain the required power for the study. But it is evident from the
first column of the table the degree to which a larger sample is
required for that same ICC value, as the number of students per school
increases. Sample sizes need to be inflated by a factor of at least 1.5
for the higher ICC values, regardless of whether the numbers of students
per school are 25 or 250. Thus, if design effects are ignored when
designing studies aimed at measuring and testing bullying outcomes,
underpowered samples will result.
Calculation of required sample size
Studies of bullying behaviour in schools are conducted for multiple
reasons and the purpose of the study is a determining factor in deciding
on the form of the bullying measure to be used. Commonly, the prevalence
of such behaviour is estimated or compared: for example, in studies of
anti-bullying interventions. Alternatively, a researcher may wish to
explore the relationship between bullying behaviour and other
individual--level factors such as students' mental health or
academic outcomes. If prevalence is the focus, single questions that can
be categorised to identify students who have been bullied or have
bullied others would be appropriate. When investigating associations
between individual characteristics and bullying, a multi-item scale-from
which a continuous composite score can be calculated as a measure of
involvement in bullying behaviour--would give greater sensitivity and
variability than a binary outcome.
Apart from the measurement scale of the bullying outcome and the
design effect (as determined by the ICC and cluster size), the required
sample size for a cluster sample of schools depends on the size of the
effect to be determined and, for categorical outcomes, the prevalence of
the outcome. Table 5 summarises the calculation of the required numbers
of students and schools for cluster samples for the four measures of any
bullying, the corresponding ICC values for each measure and different
prevalence rates and effect sizes. The calculations are conducted
separately for primary and secondary schools, assuming an average of 100
responding students per school (after accounting for consent and
non-response rates), power of 80% (conventionally the minimum acceptable
value) and based on simple two-sided tests of proportions or means in
two independent samples. Prevalence rates of 10% to 30% were chosen in
line with the rates found in the Australian Covert Bullying Prevalence
Study (9% and 27% for any perpetration and victimisation respectively)
and small (0.25) and moderate (0.5) effect sizes for continuous outcomes
(Cohen, 1988). In most cases the required numbers of schools are rounded
up. To achieve power greater than 80%, larger sample sizes than
presented here would be required. As an illustration of the use of the
design effect estimate to determine the required sample size for a
cluster sample, consider a study with the major outcome of comparing the
prevalence of bullying victimisation in two groups (for example, in an
intervention trial) in primary schools (ICC = 0.019 from Table 1).
Assuming that the prevalence of bullying victimisation is 20% in the
group with the lower rate and wishing to have 80% power to detect a
difference of 5% between the groups (that is, 20% in one and 25% in the
other), the required number of students per group for a simple random
sample is 1140 students. With 100 students per school, the anticipated
design effect is 2.9, resulting in a total required sample of 3306
rather than 1140 students per group. Given the assumption of 100
respondents per school, this equates to about 33 schools per group and
66 schools in total. Note that, while the number of schools does not
figure directly in the calculation of the design effect, it is
implicitly determined by the numbers of students to be sampled per
school and it is therefore advantageous to sample fewer students per
school and more schools, rather than more students in fewer schools.
Within Table 5 the values of the input parameters to the
calculations are adjusted as appropriate for the various measures, but
also to illustrate their impact on the sample size calculation. Firstly,
the lower prevalence of the two groups was varied between 10%, 20% and
30% to show how the required sample size increases as this rate
increases. Thus, for a binary outcome, 730, 1140 and 1420 students are
required in a simple random sample to detect a difference of 5% for
prevalence rates of 10%, 20% and 30% respectively. Secondly, the impact
of effect size is illustrated in terms of both differences in
percentages and means. For 20% prevalence, to detect a smaller
difference of 5% requires a larger simple random sample size of 1140
compared to that of 320 to detect a difference of 10% between two
groups. Similarly, a sample of 255 is required to determine a small
effect size of 0.25 for a continuous outcome measure as statistically
significant compared with 64 if only a moderate effect size of 0.5 was
considered important. Thirdly, the much lower sample size requirements
for testing outcomes measured on continuous compared with categorical
scales are illustrated.
It is important to note the differences in required cluster sample
sizes for the victimisation and perpetration outcomes measured on the
same scale, due to the differences in ICC values for the perpetration
outcomes. For example, a sample of 2117 was required for the binary
bullying victimisation measure compared with 6716 for the perpetration
measure. As sample size calculations need to account for all the key
outcome measures of a study, the largest ICC value is most pertinent.
As a practical example of the effects of ignoring school-level
clustering on the conclusions drawn from a study, consider the case of
an intervention trial of an anti-bullying program in secondary schools.
Based on the Australian Covert Bullying Prevalence Study data, one could
assume that the prevalence of bullying victimisation is about 30%
(conservatively rounded up from 27%). Assume that a researcher, ignoring
clustering effects, determines the sample size required for the trial as
380 students in order to have 80% power to detect a 10% decrease in
bullying behaviour as statistically significant--that is, if the program
results in a reduction from 30% to 20% of students victimized--it should
be seen as effective. But with a sample of 380 students, the program
would actually need to achieve a reduction of at least 22%--that is,
from 30% to 8% of students victimised before a statistical test would
show the program as having a significant impact.
Discussion
Good estimates of ICC values offer insights into bullying behaviour
in schools and are vital for planning group-randomised trials or studies
using cluster sampling (Donner & Klar, 2004; Murray et al., 2006;
Scheier et al., 2002; Siddiqui et al., 1996). To our knowledge, limited
information has been published regarding ICC values for bullying
behaviour and nothing to date for cyberbullying. This article presents
ICC values for bullying and cyberbullying outcomes based on a large
representative Australian sample. Each calculation is based on more than
40 schools to ensure stability of the estimates (Donner & Klar,
2004).
Few studies reporting ICC values for bullying-related outcomes for
mainstream students were identified in the literature. Values reported
by Karna and colleagues (2011) are not directly comparable with those
found in this study as they also accounted for classroom-level
clustering, which is of greater importance in Finland than in Australia
due to the stability of classroom structures in the Finnish system. The
outcome measures used by Bradshaw and colleagues (2009) were similar to
the single-item binary any bullying measures used in this study, with
similar dichotomisations following the work of Solberg & Olweus
(2003). The value obtained here for victimisation for primary school
students of 0.019 was the same as that found by Bradshaw and colleagues
(2009) for elementary students (Grades 4 and 5, n = 76 schools). Their
reported values for middle school students (Grades 6-8), based on only
19 schools, of 0.006 for victimisation and 0.009 for perpetration were
substantially lower than those in this study for similar aged students
(0.032 and 0.031 respectively). In contrast, the values for secondary
students of 0.03 found here are within the range of 0.01-0.06 reported
for a range of outcomes including bullying victimisation in the
Gatehouse Project conducted amongst secondary students in Australia
(Bond et al., 2004). No published ICC values for cyberbullying-related
outcomes were found.
The low ICC values obtained in this study (below 0.1) show that
little variation exists between schools; that is, bullying behaviour is
not more concentrated in certain schools but is prevalent to a similar
extent across all schools. This is possibly a surprising finding, given
bullying often occurs within the school context and may be perpetrated
by a relatively small number of students. Indeed, the values are in line
with those of other health outcomes such as nutritional intake and
physical activity (Carlin & Hocking, 1999; Murray et al., 1994;
Murray et al., 2000; Murray et al., 2001; Murray et al., 2006; Scheier
et al., 2002; Siddiqui et al., 1996) that one would possibly expect to
be less influenced by the school environment than bullying. In contrast,
ICC values for academic outcomes do display much stronger clustering
effects (Hutchison, 2004; OECD, 2010). This lack of evidence that some
schools have a stronger 'bullying culture' than others,
together with the fact that about a quarter of Australian students are
bullied every few weeks or more often, highlights the need for bullying
reduction and management programs in all schools.
The ICC values were higher for the perpetration than the
corresponding victimisation outcome measures, indicating students across
all schools were more homogeneous in respect to their reports of
bullying victimisation than perpetration of bullying. This was
particularly evident for primary rather than secondary schools. These
trends occurred for both the any type of perpetration and cyber
perpetration measures, implying that clustering of cyberbullying
behaviour is similar to that of bullying behaviour in general. This
greater variability between schools with regard to perpetration than
victimisation may be reflective of lower rates of self-reported bullying
perpetration, highlighting differences between schools. Additionally,
these differences may be related to school-level social norms or
normative expectations related to the reporting of victimisation and
perpetration of bullying behaviour. It is possible that students in some
schools are less likely to report bullying perpetration or victimisation
than is the case in other schools, perhaps due to school climate or
unhelpful staff responses, adding to the variability between schools.
Self-serving attribution bias may be more evident for perpetration than
victimisation, suggesting students report bullying targeting them more
highly than their own perpetration of bullying (Osterman et al., 1994).
Differences in ICC values were noted for various demographic
groups. For example, higher ICC values for girls on all the measures
indicated contextual effects were stronger for girls, with a greater
concentration of bullying behaviour in certain schools for girls while
occurring more commonly across all schools for boys. Whereas few studies
report ICC values for gender separately, or for the other demographic
groups considered in this study, Siddiqui and colleagues (1996) also
reported larger clustering effects for female than male students on
current smoking status. One conclusion from this finding is that
bullying between girls is more context dependent than is the case for
boys. Thus, the need for bullying prevention and management
interventions is uniform across all schools for boys, but the need may
be greater within certain schools than others for girls.
Similarly, a trend towards higher ICC values for non-metropolitan
than metropolitan schools signifies a greater clustering of bullying
behaviour in certain non-metropolitan schools. This may be attributable
to the diversity of schools and environments in non-metropolitan areas
in Australia, which include schools in rural areas as well as large
regional centres. School size and sector did not greatly influence ICC
values.
The reductions in ICC values that can be achieved through the
addition particularly of school-level variables to regression models,
noted by Murray and colleagues (2001; Resnicow et al., 2010), are also
demonstrated here. Apart from demographic variables as considered in
this study, the inclusion in models of other school-level factors
correlated with the outcome of interest--such as teacher:student ratios,
anti-bullying policy implementation and the quality of school
leadership--may also result in reductions in ICC values. Additionally,
if a longitudinal study and repeated measures analyses are planned,
lower ICC values are operative than those from a cross-sectional study
(Murray & Blitstein, 2003). Therefore, the values presented here are
likely an upper limit of those that would apply should such models be
applied in planned studies.
Nevertheless, while ICC values are typically less than 0.1 and
appear negligibly small, their influence in reducing precision and thus
the power of a study are substantial if the number of students sampled
per school is large. The impact of an ICC as low as 0.006 in terms of
the design effect and resultant increase in sample size required for a
cluster sample to achieve the same level of power as a simple random
sample has been illustrated. For bullying-related outcomes, design
effects are not negligible when samples of about 25 or more students are
sampled per school. Sample sizes need to be inflated by a factor of at
least 1.5 and sometimes substantially more, or imprecise estimates and
underpowered studies will result. A lack of power will lead to erroneous
conclusions: for example, a failure to identify factors associated with
bullying outcomes or to demonstrate effective interventions.
Further, the practice of assuming clustering effects may be ignored
on the basis of the non-significance of a hypothesis test that the ICC
is zero is not recommended as such tests have limited power (Donner
& Klar, 2004) and are thus unlikely to detect ICC values that do
differ significantly from zero.
A simplified approach to the calculation of the required sample
size for a cluster sample is presented. While the calculations are valid
for testing individual-level effects, often the number of schools
sampled is also of relevance. From a sampling perspective, it may be
difficult to obtain a representative sample of a target population of
students from a limited number of schools. In studies testing contextual
school-level variables (for example, school size or policy), it is
recommended that a minimum of 40 schools be sampled as a sufficient
sample size to assess school-level outcomes (Donner & Klar, 2004;
Murray et al., 2004). In intervention trials where schools are assigned
to study conditions, this equates to 20 or more schools per study
condition.
This study is subject to a number of limitations that restrict the
applicability of the presented ICC values. Firstly, no account was taken
of possible classroom clustering effects. In Australia, classroom
clustering would only apply to primary schools where--unlike in
secondary schools where students move between classes throughout the
day--in a single school year students largely stay with the same
classroom of students and teacher throughout the school day. The extent
to which classroom-level effects will be present depends on the extent
that bullying behaviour tends to be perpetrated between students in the
same classroom rather than more broadly. Unfortunately information on
class membership was not available for the sample analysed for this
paper, but analyses of data from another project held by the Child
Health Promotion Research Centre revealed classroom-based ICC values two
to five times higher in magnitude than school-based values in a sample
of 20 primary schools. While the ICC values presented in this article
are thus appropriate for use when designing studies in secondary
schools, they are underestimates of the relevant values for primary
schools where class-level clustering is of importance and may need to be
accounted for. Secondly, the ICC values represent a combination of
school and cohort effects. As this is a cross-sectional study of a
specific cohort of students, these effects could not be separated
(Smolkowski, Biglan, Dent & Seeley, 2006). Thirdly, due to the
relatively small numbers of students per year level and likely cohort
effects, it was not possible to reliably estimate ICC values per year
level. Consequently the applicability of the values presented for
primary and secondary school students is limited by the extent to which
clustering effects are similar in year levels within primary and
secondary schools. Fourthly, the data used in this study were collected
in the last term of the school year. The time of the year students were
surveyed was found to influence the ICC values related to physical
activity outcomes (Murray et al., 2006). Clustering effects may also
differ by school term for bullying outcomes, especially in certain year
levels such as the first year of secondary school (Pellegrini &
Bartini, 2000; Rigby, 1998; Smith, Madsen & Moody, 1999).
Some further considerations in the interpretation of the results
presented here are pertinent. The data analysed were self-reported, and
clustering effects are likely to be higher for peer-report of bullying
involvement. Karna and colleagues (2011) reported an ICC value of 0.13
for peer nomination of victimisation. Greater homogeneity of peer
nominations may be partly due to a phenomenon known as reputation bias,
where students' perceptions of some of their peers as
'victims' or 'bullies' persist despite behavioural
changes that may occur (Hymel, Wagner & Butler, 1990). If
teacher-report is used, teacher-level clustering is an added strong
source of variation to be accounted for.
Some authors have used linear procedures to calculate ICC values
for binary outcomes. We compared the values obtained using xtlogit and
xtreg and found the values using the linear procedure were lower. Taking
a conservative approach, we have presented the ICC values obtained from
the logistic regression procedure, as this is in accordance with how the
data are likely to be analysed and therefore arguably the more relevant
ICC value.
Clearly, smaller sample sizes are required with continuous than
with categorical outcomes, and studies can be powered to detect small
effect sizes with relatively few schools. But, as mentioned, the number
of schools sampled is also a critical consideration. In addition, the
choice of outcome measure to be used should be based on theoretical
considerations and the study's research questions.
Murray and colleagues (2004) have described the need for
researchers to use ICC estimates 'in their power analyses that
closely reflect the endpoints, target population, and primary analyses
planned for the trial'. The values reported in this article may
have assisted in this regard. A simplified method of determining the
required sample size for a school-based cluster sample targeted at the
measurement and testing of bullying outcomes is also described. The
approach is applicable to any outcome measure and setting for which
appropriate ICC values are available.
Conclusions
Results from this study suggest that bullying behaviour is
relatively uniform across schools in Australia, with no marked
differences in the bullying culture between schools. Indeed, school
context is more strongly associated with academic outcomes than
bullying.This highlights the importance of providing anti-bullying
interventions in all schools, for both boys and girls and regardless of
school level, size, geographic location or sector.
A number of factors affect the required sample size for a cluster
study design to achieve a certain precision and power. Greater
homogeneity of students within schools, as measured by higher ICC
values, leads to larger design effects and thus the amount by which the
sample needs to be inflated to achieve the same precision and power as a
simple random sample. Similarly, the larger the number of students that
will be sampled per school, the larger the sample size for a cluster
sample will need to be. Bullying outcomes measured on a continuous scale
often require substantially smaller sample sizes than binary outcomes.
Larger sample sizes are required if greater precision in estimation is
desired or if it is important for the study to detect a smaller effect.
Although the ICC values for bullying outcomes are small, they are
not able to be ignored and need to be accounted for when designing
studies, particularly when large numbers of students are sampled per
school. Sample sizes need to be inflated by a factor of at least 1.5 and
sometimes substantially more, or estimates of bullying outcomes will
lack precision and underpowered studies will result.
When designing studies to test school contextual variables or for
intervention trials, the number of schools sampled is of vital
importance to the validity of the findings. Samples of 40 schools are
recommended--20 per study condition in group-randomised trials--to test
for school-level variables or intervention effects adequately. Studies
based on small numbers of schools are likely to be underpowered and
subject to a number of biases. While in general it is advantageous to
sample more schools with fewer students in each, rather than fewer
schools with more students, in intervention trials the sample size
requirements also need to be assessed in light of the resources
available to the research team to support intervention implementation in
study schools.
Acknowledgements
The Australian Covert Bullying Prevalence Study was funded by the
Australian federal Department of Education, Employment and Workplace
Relations.
We would like to thank the students, parents and staff at
participating schools and Melanie Epstein and other staff at the Child
Health Promotion Research Centre (CHPRC) at Edith Cowan University for
their contributions to the Australian Covert Bullying Prevalence Study.
We would also like to thank the editor and reviewers, as well as
Professor Stephen Zubrick, for helpful comments that have resulted in
improvements to the paper.
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Therese Shaw
Donna Cross
Edith Cowan University
Authors
Therese Shaw is a biostatistician at the Child Health Promotion
Research Centre, School of Exercise and Health Sciences, Edith Cowan
University. Email: t.shaw@ecu.edu.au
Donna Cross is Professor of Child and Adolescent Health at the
Child Health Promotion Research Centre, School of Exercise and Health
Sciences, Edith Cowan University.
Table 1 ICC values (and standard errors)--total sample and by
demographic group
Any type of bullying:
victimisation or perpetration
Vict. Perp. Vict. Perp.
Variable Yes/No Yes/No scale scale
Total 0.025 0.071 0.023 0.039
(0.007) (0.017) (0.005) (0.007)
Gender Female 0.035 0.128 0.048 0.056
(0.011) (0.033) (0.011) (0.012)
Male 0.022 0.044 0.008 0.035
(0.010) (0.019) (0.006) (0.009)
School Primary 0.019 0.083 0.018 0.041
level (0.007) (0.024) (0.006) (0.010)
Secondary 0.032 0.031 0.028 0.025
(0.013) (0.017) (0.010) (0.009)
School Smaller 0.021 0.092 0.023 0.033
size (0.008) (0.027) (0.007) (0.009)
Larger 0.025 0.033 0.020 0.040
(0.010) (0.018) (0.008) (0.011)
School Government 0.024 0.063 0.020 0.040
sector (0.010) (0.024) (0.008) (0.011)
Non- 0.022 0.077 0.023 0.036
government (0.008) (0.023) (0.007) (0.009)
Area Metropolitan 0.018 0.076 0.018 0.032
(0.007) (0.023) (0.006) (0.008)
Non- 0.029 0.057 0.025 0.049
metropolitan (0.012) (0.023) (0.009) (0.014)
Cyberbullying:
victimisation or perpetration
Exp. Involv. Exp. Involv.
Variable Yes/No Yes/No scale scale
0.031 0.067 0.015 0.037
(0.007) (0.013) (0.004) (0.007)
Gender 0.038 0.081 0.022 0.053
(0.011) (0.018) (0.007) (0.012)
0.011 0.064 0.012 0.027
(0.010) (0.017) (0.007) (0.008)
School 0.029 0.040 0.006 0.030
level (0.010) (0.013) (0.003) (0.008)
0.032 0.032 0.019 0.017
(0.012) (0.012) (0.007) (0.007)
School 0.031 0.048 0.012 0.032
size (0.011) (0.014) (0.005) (0.009)
0.028 0.083 0.019 0.042
(0.010) (0.021) (0.007) (0.013)
School 0.029 0.062 0.011 0.034
sector (0.011) (0.019) (0.005) (0.010)
0.031 0.066 0.019 0.036
(0.010) (0.016) (0.007) (0.010)
Area 0.023 0.069 0.013 0.037
(0.008) (0.017) (0.005) (0.010)
0.041 0.065 0.019 0.037
(0.014) (0.020) (0.008) (0.011)
Note: Binary measures for any type of bullying perpetration
and victimisation represent behaviour that occurs every few
weeks or more often, binary measures for exposure to (Exp.) and
involvement (Involv.) in cyberbullying include single instances of
such behaviour (once or twice a term or more often). ICC given
first, with standard errors in brackets.
Table 2 Numbers of schools and students
Number Total Mean
of number of cluster
Grouping schools students size
Total 106 7238-7312 69
Gender Female 101 3795-3836 38
Male 99 3416-3454 35
School level Primary 55 4569-4606 83
Secondary 51 2669-2711 53
School size Smaller 54 3843-3882 72
Larger 52 3384-3430 66
School Government 52 2708-2749 53
sector Non- 54 4530-4565 84
government
Area Metropolitan 63 4640-4701 74
Non- 43 2592-2615 61
metropolitan
Minimum Maximum
cluster cluster
Grouping size size
11 181-186
Gender 5 108-109
5 98-101
School level 20-21 181-186
11 125-129
School size 11 181-186
12 128-131
School 11 148-152
sector 12 181-186
Area 11 181-186
15 148-152
Note: Mean cluster size rounded to the nearest whole unit. Ranges given
as numbers of students varied by outcome measure.
Table 3 ICC values adjusted for demographic variables
Any type of bullying:
victimisation or perpetration
Vict. Perp. Vict. Perp.
Variable Yes/No Yes/No Scale Scale
Unadjusted 0.025 0.071 0.023 0.039
Model 1 0.019 0.057 0.018 0.030
Model 2 0.019 0.055 0.020 0.029
Model 3 0.010 0.040 0.013 0.020
Model 4 0.010 0.037 0.015 0.020
Cyberbullying:
victimisation or perpetration
Exp. Involv. Exp. Involv.
Variable Yes/No Yes/No Scale Scale
Unadjusted 0.031 0.067 0.015 0.037
Model 1 0.028 0.037 0.011 0.022
Model 2 0.026 0.036 0.010 0.022
Model 3 0.013 0.026 0.003 0.018
Model 4 0.009 0.026 0.002 0.018
Model 1: Adjusted for area, sector, year level, school size
Model 2: Adjusted for gender, area, sector, year level, school size
Model 3: Adjusted for area, sector, year level, school size, Australian
state or territory
Model 4: Adjusted for gender, area, sector, year level, school size,
Australian state or territory
Table 4 Design effect sizes for different cluster sizes
ICC values
m 0.006 0.019 0.032 0.041 0.083
25 1.1 1.5 1.8 2.0 3.0
50 1.3 1.9 2.6 3.0 5.1
100 1.6 2.9 4.2 5.1 9.2
150 1.9 3.8 5.8 7.1 13.4
200 2.2 4.8 7.4 9.2 17.5
250 2.5 5.7 9.0 11.2 21.7
m: cluster size or number of students per school
Table 5 Required sample sizes for cluster samples
Parameters
Design (% prevalence
effect and
School ICC (m = difference)/
Measure level value 100) Effect size
Any vict. Primary 0.019 2.9 10, 5
Yes/no Secondary 0.032 4.2 10, 5
Primary 0.019 2.9 20, 5
Secondary 0.032 4.2 20, 5
Primary 0.019 2.9 30, 5
Secondary 0.032 4.2 30, 5
Primary 0.019 2.9 20, 10
Secondary 0.032 4.2 20, 10
Primary 0.019 2.9 30, 10
Secondary 0.032 4.2 30, 10
Any perp. Primary 0.083 9.2 10, 5
Yes/no Secondary 0.031 4.1 10, 5
Any vict. Primary 0.018 2.8 Effect size 0.25
scale Secondary 0.028 3.8 Effect size 0.25
Primary 0.018 2.8 Effect size 0.5
Secondary 0.028 3.8 Effect size 0.5
Any perp. Primary 0.041 5.1 Effect size 0.25
scale Secondary 0.025 3.5 Effect size 0.25
Primary 0.041 5.1 Effect size 0.5
Secondary 0.025 3.5 Effect size 0.5
Students Schools
[n.sub. [n.sub. Per
Measure SRS] cluster] group Total
Any vict. 730 2117 22 44
Yes/no 730 3066 31 62
1140 3306 33 66
1140 4788 48 96
1420 4118 42 84
1420 5964 60 120
320 928 10 20
320 1344 14 28
380 1102 12 24
380 1596 16 32
Any perp. 730 6716 68 136
Yes/no 730 2993 30 60
Any vict. 255 714 8 16
scale 255 969 10 20
64 180 2 4
64 244 3 6
Any perp. 255 1301 13 26
scale 255 893 9 18
64 327 4 8
64 224 3 6
m: cluster size; [n.sub.SRS]: sample size required for simple random
sample;[n.sub.cluster]: sample size required for cluster sample;
prevalence percentage shown in bold
