The Prevalence of Physical Aggression in Canadian Children: A Multi-Group Latent Class Analysis of Data from the First Collection Cycle (1994-1995) of the NLSCY - December 1999

1. Introduction

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Physical aggression is an important concern for public health and human resource development. In this paper we use the term physical aggression to refer to physical acts oriented towards another person which could inflict physical harm. Although there have been numerous studies of children's aggressive behaviour, few studies have focused on the development of aggression. Researchers in this area have tended to aggregate physical aggression with indirect aggression, verbal aggression, opposition, competition, and even hyperactivity (Nagin & Tremblay, in press; Tremblay,1991; Tremblay et al., in press). Recent work on the differences between indirect aggression (e.g. trying to get others to dislike someone you do not like) and physical aggression has shown important age and sex differences in the expression of each of these type of aggressive manifestations (Björkqvist et al., 1992; Crick & Grotpeter, 1995; Lagerspetz, 1988; Tremblay et al., 1996). Physical aggression can be observed as soon as the end of the first year after birth (Tremblay et al., in press), while indirect aggression may appear only once children have acquired some insight into the complexity of social interactions and the means (usually verbal) to manipulate social interactions. Manifestations of physical aggression appear to change with age, and differently for boys and girls (Loeber & Hay, 1997; Loeber & Stouthamer-Loeber, 1998; Tremblay et al.,1996; Tremblay et al., in press).

Children who do not learn to inhibit physically aggressive behaviour during early childhood appear to be at higher risk of becoming the violent juvenile delinquent, the perpetrator of dating violence, the abusing parent and the abusing mate (Cairns & Cairns, 1994; Farrington, in press; Nagin & Tremblay, in press). From this perspective, children who have not learned to inhibit physical aggression are an important handicap for the development of human resources in our modern societies, and a major public health issue. To prevent the development of chronic problems with physical aggression, we need to have reliable means of identifying children at risk, and accurate estimates of the magnitude of the problem in the Canadian population. It is essential to achieve both of these goals if we want to develop effective strategies, policies and intervention programs to promote the healthy development of children in Canada. However, obtaining reliable estimates of the prevalence of physical aggression among children in Canada presents some difficult methodological challenges.

1.1 The Formal Diagnosis of Problem Behaviours and the Problem of an Arbitrary Cutoff Point

Making a formal diagnosis on the basis of a list of behaviour symptoms is one of these difficult challenges. The conventional approach used for the diagnosis of behavioural problems in children consists of using a cutoff point specified in terms of an arbitrary number of behaviour symptoms. This approach is also known as the categorical approach of making a formal diagnosis and is best illustrated by the assessment procedure specified by the American Psychiatric Association's (1994) Diagnostic and Statistical Manual of Mental Disorders (DSM-IV). The clinician's make a yes-or-no judgement as to whether a child has a disorder based on a list of behaviour symptoms, and an arbitrary number is required to make a formal diagnosis. If the number of behaviour symptoms exceeds the minimum specified by the cutoff point then the child is said to have the disorder; otherwise, he or she does not. Another approach is the dimensional approach which consists of specifying an arbitrary cutoff point on a continuous scale that is the (weighted) sum of the observed ratings on the behaviour symptoms (for an illustration of this approach see Achenbach, 1981). Hence, although these two approaches represent different case definition strategies they both use an arbitrary cutoff point to distinguish between disordered and nondisordered individuals.

There are many problems associated with using an arbitrary cutoff point to make a formal diagnosis. First, because the cutoff point for number of behaviour symptoms in the conventional approach is dictated by prevailing custom rather than being systematically derived from research (for some recent attempts, see Lahey, Applegate, Barkley et al., 1994; Lahey, Applegate, McBurnett et al., 1994), it may well be that two diagnostic categories created by this procedure do not constitute homogeneous groups of children in the population. This is also a problem with the dimensional approach. As a result, there may be substantial inter-individual differences within the two categories—potentially useful information—that these two approaches simply chooses to ignore. Moreover, the potential lack of internal validity of the two diagnostic categories created by these two approaches may cause many problems when it comes to compare children from different diagnostic categories on external validating factors. For instance, it may be difficult interpret the results from studies aimed at validating the diagnostic categories using psychosocial factors, demographic factors, biological factors, family genetic factors, family environmental factors natural history, and response to therapeutic intervention if the internal validity of the diagnostic categories is unknown (Cantwell, 1996). Moreover, the conventional as well as the dimensional approach assume that the childhood disorder is either present or absent, although there may be more than two mutually exclusive and exhaustive categories of individuals in the population. For instance, children form a third diagnostic category may be subsyndromal for the disorder but nonetheless experience a considerable amount of functional impairment. Having these individuals mixed within the other two diagnostic categories would have the effect of reducing, if not annihilating, the correlation that may otherwise exist between the disorder and any external validating factor (Robins, 1985).

Second, the use of an arbitrary cutoff point in the conventional approach relies on the implicit assumption that all behaviour symptoms are equally good indicators of the behavioural or emotional problem in question. However, children who display the same number of behaviour symptoms do not necessarily constitute an homogeneous group of individuals.

Finally, the problem of using an arbitrary cutoff point is exacerbated by the common practice of using the same cutoff point, irrespective of the age or the sex of the child. As a result, any difference in the prevalence of the behaviour symptoms between, let say, boys and girls, is interpreted within the conventional approach as a true difference in the prevalence of the behavioural or emotional problem in question. However, part if not all of the observed differences in the prevalence of the behaviour symptoms between boys and girls may be due to the fact that the behaviour symptoms are not functioning in the same way for the two groups (i.e., the propensity to manifest the behaviour symptoms for a given diagnostic category may well vary between the two groups). In contrast, any difference in the prevalence of the behaviour symptoms between boys and girls is interpreted as differential symptom functioning within the dimensional approach. However, part if not all of the observed differences in the prevalence of the behaviour symptoms between the two groups may be due to a true difference between the two groups in the prevalence of the behaviour or emotional problem in question. Hence, these two approaches for making a diagnosis hampers the achievement of one of the main objectives of developmental epidemiology, which consists of comparing estimates of the prevalence of behavioural or emotional problems in childhood across groups of children who may differ in age or sex. Therefore the conventional as well as the dimensional approach lack an objective procedure for combining behaviour symptoms into a formal diagnosis that is appropriate both for boys and girls, irrespective of their age.

This study has two principal aims, one methodological and the other substantive. The first aim is to demonstrate the use of a statistical technique, latent class analysis, for diagnosing behaviour problems. Latent class analysis provide an empirical means of discerning how many diagnostic categories or classes are evident in a sample, based on the patterns of responses to several behaviour symptoms. In the case of physical aggression, for example, it determines whether children naturally cluster into two classes (e.g., aggressive and non-aggressive), or into three (e.g., aggressive, mildly aggressive and non-aggressive) or more classes. The technique also determines the probability than an individual belongs to each class, given his or her pattern of responses to the set of behaviour symptoms. Thus, it avoids the problem of setting arbitrary cutoff points for diagnosis. Also, because the technique can be applied separately for boys and girls, and for each age cohort, it allows for the possibility that the weight accorded to particular behaviour symptoms can vary, depending on the age and sex of the child. In demonstrating latent class analysis, we also note that the technique provides an objective means for assessing the relative importance of each behaviour symptom, enabling one to identify a small set of behaviour symptoms that could be used to make reliable diagnoses. The substantive aim of this study is to obtain estimates of the prevalence of physical aggression in the Canadian population, based on data from the National Longitudinal Study of Children and Youth. The analysis yields estimate of prevalence for boys and girls aged 2 through 11.

1.2 Latent Class Analysis as an Alternative Approach to Making a Formal Diagnosis

The issue of making a formal diagnosis on the basis of a list of behaviour symptoms can be addressed objectively within the framework of latent class analysis. Latent class analysis stems from Paul A. Lazarsfeld's early work on latent structure analysis investigating the dependence of a set of manifest categorical variables on a small number of unobservable or latent variables (Lazarsfeld, 1950a, 1950b, 1954; Lazarsfeld & Henry, 1968). The work pioneered by Lazarsfeld has found a variety of applications in the educational, social and behavioural sciences (for reviews see Clogg, 1995; Langeheine, 1988). An area of special interest has been medical diagnosis, where latent class analysis has been used to obtain objective cutoff scores for psychiatric classification—case identification (Hudziak et al., 1998; Rindskopf & Rindskopf, 1986; Uebersax & Grove, 1990; Young, 1983).

Latent class analysis provides an empirical means to identify a set of mutually exclusive and exhaustive latent classes of individuals that account for the distribution of responses to a set of manifest discrete variables. The basic assumption of this model is that within any single latent class the manifest variables are independent of one another (i.e., the assumption of local independence). Thus, the association among the manifest variables results from the differences between the two or more latent classes. For example, the behaviour symptoms may be indicators of a latent variable, such as, physical aggression. Further, physical aggression may be comprised of two or more latent classes. On the one hand, there may be a latent class whose members do not tend to manifest the behaviour symptoms in question (i.e., low-aggressive). On the other hand, there may be a second latent class whose members tend to manifest the behaviour symptoms in question (i.e., high-aggressive). Each individual is assumed to be in one, and only one, of the latent classes. In essence, the population of interest is assumed to be made up of two (or more) qualitatively different types of children that may differ markedly in their propensity to manifest the behaviour symptoms in question.

The objective of latent class analysis is to reproduce the observed frequencies associated with the response patterns to the set of manifest variables using two kinds of parameters: (a) the probability that a randomly selected individual belongs to a given latent class, and (b) the conditional probability of a rating in a rating category for a particular behaviour symptom given the individual membership in a latent class. In the example given above, the outcome of a latent class analysis would comprise estimates of the proportion of children in the population who belong to the low- and high-aggressive latent classes. The outcome of the latent class analysis in this example would also comprise the estimates of the probability of manifesting each behaviour symptoms given membership in either the low- or the high-aggressive latent class.

The extent to which these parameters can be used to reproduce the response patterns to the set of manifest variables can be assessed empirically using either the Pearson chi-square statistic, X², or the likelihood-ratio chi-square statistic, L². Both the Pearson and the likelihood-ratio chi-square statistic have a large sample ² distribution under certain conditions (see Clogg, 1979). When the expected cell frequencies obtained under the latent class model are close to the observed cell frequencies, then the X² and/or L² value will be small and the model being examined can be said to provide an adequate fit to the data. A large value of X² and/or L², on the other hand, corresponds to a value in the right-hand tail of the ² distribution and is indicative of a poor fit of the model to the data. A standard by which to judge whether the X² and/or L² is large or small is given by the degrees of freedom. The degrees of freedom associated with the specified latent class model can be determined by subtracting the number of independent parameters to be estimated from the number of nonredundant observed cell frequencies. In the same vein, two hierarchically related latent class models (two latent class models are hierarchically related if one includes all the parameters of the other plus some additional ones) can be compared using the likelihood-ratio chi-square statistic since it can be partitioned exactly (Fienberg, 1980). That is, it is possible to subtract the L² of a latent class model based on the larger number of parameters (e.g., a three-class model) from the latent class model based on the smaller number of parameters (e.g., a two-class model). The degrees of freedom associated with the resultant L² are given by subtracting the degrees of freedom for the less restricted model from those for the more restricted model. A large difference in L², compared to the change in the degrees of freedom, suggests that the added parameters of the less restricted model have real significance. On the other hand, a drop in L² from the more restricted model to the less restricted model close to the change in number of degrees of freedom indicates that there is no significant improvement in fit when the less restricted latent class model is chosen to represent the data. Thus, unlike the conventional approach latent class analysis offers a systematic approach to test for the existence of two or more qualitatively different types of individuals in the population of interest. That is, it provides the means for deciding on an appropriate number of latent classes on the basis of the available data.

Once we have obtained parameter estimates for a particular latent class model prediction can be made concerning latent class membership for each individual based on his or her observed pattern of responses to the set of manifest variables. One assignment rule would be to assign each individual to the latent class t (t = 1, 2, ...T) which maximizes the probability of observing his or her response pattern to the set of manifest variables. In the example given above, an individual with a given response vector would be classified as low-aggressive if his or her posterior conditional probability of membership in this latent class is higher than his or her posterior conditional probability of membership in the other latent class; otherwise, he or she would be classified as high-aggressive. Hence, unlike the conventional approach, the latent class classification scheme is not based on an arbitrary cutoff point but rather on an optimal classification rule that insures that the expected (i.e., theoretical) proportion of misclassified individuals is minimized.

Only rarely will a symptom be perfectly sensitive (i.e., the symptom is always present among individuals who are disordered), and therefore, a symptom often presents a false-negative rate above 0. Similarly, rarely will a symptom be perfectly specific (i.e., the symptom is never present among individuals who are nondisordered), and therefore, a symptom often presents a false-positive rate above 0. Generally speaking, a symptom gives valuable but not perfect diagnostic information as to the true disorder state of the individual. Consider the situation where a cutoff point has been chosen which is intended to distinguish between two disorder states; namely, disorder present or absent. The symptom may sometimes indicate the disorder when, in fact, the individual does not have the disorder and/or it may sometimes fail to identify the disorder when it is present. As a result, it is impossible to define a cutoff point that distinguishes perfectly all those with a disorder from all those without it, and therefore, any classification rule will yield some misclassifications. Epidemiologists distinguish between the predictive value positive and the predictive value negative of a test (Galen & Gambino, 1975; Weinstein et al, 1993). In the case of two disorder states the predictive value positive of the test is the posterior conditional probability of having the disorder given a positive test result. Conversely, the predictive value negative of the test is the posterior conditional probability of not having the disorder given a negative test result. Epidemiologists also define the diagnostic information the test conveys as the ratio of the conditional probability of having the disorder given a positive test result to the conditional probability of not having the disorder given a positive test. To the extent that a test result conveys diagnostic information it can be used effectively to revise our prior probability that an individual does or does not have the disorder—the prior probability is equal to the prevalence of the disease in the population. Within the latent class framework the predictive value positive and the predictive value negative of the symptoms that are used simultaneously to predict latent class membership can be defined for each latent class. The predictive value positive can be defined as the posterior conditional probability of membership in a particular latent class for the individuals who are predicted to belong to this latent class. In contrast, the predictive value negative can be defined as the posterior conditional probability of nonmembership in a particular latent class for the individuals who are not predicted to belong to this latent class. In addition, the diagnostic information provided by the symptoms can be defined for any given latent class as the ratio of the posterior conditional probability of membership in this latent class to the posterior conditional probability of membership in the other latent classes for the individuals who are predicted to belong to this latent class. Hence, unlike the conventional approach, the latent class approach provides a framework to optimize the diagnostic choices—the trade-off between false positives and false negatives—given the inherent uncertainty attached to the process of making a formal diagnosis (i.e., a child's diagnosis can only be arrived at in a probabilistic manner). Further, this optimal trade-off between false-positive rates and false-negative rates can be arrived at while taking into account the consequences of misclassifications (i.e., minimize the negative consequences of misclassifications and/or maximize the positive consequences of correct classification).

Another advantage of latent class analysis is that it can include information about categorical concomitant variables (i.e., variables like sex or age that can be used to group individuals) (Clogg & Goodman, 1984, 1985, 1986; Dayton & Macready, 1988). Simultaneous latent class analysis across several groups can be used to test whether the proportion of individuals in each latent class vary from one level of the covariate to another. This can be accomplished by imposing inter-group homogeneity constraints on the estimated latent class proportions. In the example given above, simultaneous latent class analysis could be used to test whether the prevalence of physical aggression is the same for boys and girls. In the case where the increase in L² is small compared to the increase in the degrees of freedom this would suggest that two randomly selected individuals from the two groups have the same probability to belong to the low- and high-aggressive latent classes. Hence, unlike the conventional approach, the multi-group latent class analysis provides the means to compare the prevalence estimates across groups of individuals without having to assume that the propensity to manifest the behaviour symptoms are the same across groups (i.e., it allows for differential conditional behaviour symptom rating probabilities for individuals within a given latent class who are at specified level of the covariate).