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The clustering of bullying and cyberbullying behaviour within Australian schools.

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
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