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

3. Results

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3.1 Reliability Estimates

Within classical true-score theory a test's reliability can be defined as the squared correlation between observed and true scores which is the ratio of true-score variance to observed-score variance (Allen & Yen, 1979). Defined this way a test's reliability is equivalent to the coefficient of determination in structural equation modeling which is equal to 1 minus the ratio of the determinant of the error-score variance to the determinant of the observed-score variance (Bollen, 1989). The coefficient of determination estimates obtained from a one-common factor model are presented in Table 2. These estimates are remarkably high; they vary from .85 at 3 years of age to .99 at 10 years of age for boys and from .78 at 3 years of age to .95 at 10 years of age for girls. These results suggest that the error-score variance is small relative to the observed-score variance in the physical aggression data.

Table 2 Reliability of the Three Behaviour Symptoms Used to Assess Physical Aggression

Age ( years)	Reliability estimate
	Boy	Girl
2	.874	.841
3	.850	.775
4	.881	.842
5	.994	.893
6	.876	.900
7	.865	.882
8	.925	.882
9	.895	.884
10	.999	.947
11	.880	.935
Note: Reliability estimates derived from a one-common factor model using a generally weighted least squares method of estimation (Jöreskog & Sörbom, 1993).

3.2 Choosing an Appropriate Latent Class Model for the Physical Aggression Data

Before we can estimate the prevalence of physical aggression in the Canadian population of children aged 2 to 11 years we have to determine whether we can account for the physical aggression data by postulating the existence of two or more mutually exclusive and exhaustive latent classes of children in the Canadian population. The goodness-of-fit statistics for the various latent class models considered here are presented in Table 3.

First, we considered the model of mutual independence among the behaviour symptoms, the unrestricted one-class model. Within that single latent class, the rating for any one behaviour symptom is assumed to be independent of the rating of the other two behaviour symptoms. The value of the likelihood-ratio chi-square (L²) associated with the unrestricted one-class model for 2-year-old children is 427.07 and 315.69 with 20 degrees of freedom, for boys (p < .00) and girls (p < .00), respectively. Moreover, not surprisingly, this model does not afford an acceptable fit to the physical aggression data for the other age groups as well (see Table 3). This confirms that the ratings are not statistically independent; and therefore, latent class models that assume two or more latent classes are of interest.

Second, we considered the unrestricted two-class model (Bergan, 1983; Dayton & Macready, 1976; Macready & Dayton, 1977; Rindskopf, 1983). This model assumes that the PMKs' ratings on the behaviour symptoms can be explained by a single latent variable made up of two mutually exclusive and exhaustive latent classes of children, a low- and a high-aggressive latent class, respectively. Children in the low-aggressive class would tend not to manifest the behaviour symptoms in question, whereas children in the high-aggressive class would tend to manifest the behaviour symptoms in question. Hence, like the conventional approach to making a formal diagnosis, this model assumes that the childhood disorder, namely, physical aggression, is either present or absent and that no other types of individuals exist in the population. The value of the L² associated with the unrestricted two-class model for 2-year-old boys is 63.50 with 14 degrees of freedom (p < .00). Hence, the unrestricted two-class model does not seem to provide a good fit to the physical aggression data for 2-year-old boys. Similarly, this model yields a L² of 64.80 with 13 degrees of freedom (p < .00) for two-year-old girls. Furthermore, we have obtained very similar results for the other age groups (see Table 3). Hence, the hypothesis that there may be only two exclusive and exhaustive latent classes of children in the Canadian population was rejected for all the age groups except for 8-year-old girls for whom the unrestricted two-class model seems to provide an acceptable fit to the physical aggression data according to both the Pearson and the likelihood-ratio chi-square statistics. These results seem to suggest that there are substantial inter-individual differences within the two categories— potentially useful information—that needs to be considered in order to identify homogeneous groups of children in the population.

Table 3 Pearson and Likelihood-ratio Chi-Square Statistics for Some Latent Class models for Physical Aggression

Model	Pearson chi-square (X2)		Likelihood-ratio chi-square (L2)		Degrees of freedom		pa		% of variance		% correctly allocated
	M	F	M	F	M	F	M	F	M	F	M	F
	2-year-old
Independence	1,399.27	1,807.84	427.07	315.69	20	20	.00	.00
Unrestricted two-class	116.64	134.49	63.50	64.80	14b	13	.00	.00	85.13	79.48	91.70	93.81
Unrestricted three-class	7.70	10.29	8.38	10.93	7c	8c	.30	.21	98.04	96.54	88.78	91.37
	3-year-old
Independence	1,180.66	1,456.27	520.06	338.23	20	20	.00	.00
Unrestricted two-class	80.22	153.77	68.07	40.90	14b	14b	.00	.00	86.91	87.91	96.20	90.76
Unrestricted three-class	9.40	9.09	13.01	8.33	8c	6	.11	.22	97.50	97.54	88.83	90.14
	4-year-old
Independence	1,534.00	6,038.16	582.17	384.92	20	20	.00	.00
Unrestricted two-class	96.03	309.70	51.97	49.90	14b	14b	.00	.00	91.07	87.04	95.89	94.08
Unrestricted three-class	9.02	8.11	10.14	10.68	9c	10c	.34	.38	98.26	97.23	94.70	93.69
	5-year-old
Independence	1,080.67	7,188.99	459.38	273.46	20	20	.00	.00
Unrestricted two-class	75.41	412.19	50.47	9.62	14b	15b	.00	.84	89.01	96.48	95.26	94.53
Unrestricted three-class	11.58	1.32	13.7	1.54	9c	12c	.13	1.0	97.02	99.44	95.31	94.26
	6-year-old
Independence	2,373.66	13,543.70	463.58	368.40	20	20	.00	.00
Unrestricted two-class	105.45	374.98	68.34	41.75	15b	15b	.00	.00	85.26	88.67	95.68	96.46
Unrestricted three-class	41.34	16.09	35.88	16.75	9c	11c	.00	.12	92.26	95.45	94.67	96.37
	7-year-old
Independence	910.50	5,527.58	369.84	256.73	20	20	.00	.00
Unrestricted two-class	37.79	97.43	32.79	14.86	14b	15b	.00	.46	91.13	94.21	95.16	96.64
Unrestricted three-class	18.64	5.13	15.89	7.66	9c	14c	.07	.91	95.70	97.02	94.06	96.45
	8-year-old
Independence	1,275.93	816.65	537.84	286.74	20	20	.00	.00
Unrestricted two-class	39.40	20.60	34.45	22.79	15b	14b	.00	.11	93.59	92.05	96.69	97.29
Unrestricted three-class	5.87	10.74	8.59	12.88	9c	10c	.48	.23	98.40	95.51	94.00	97.11
	9-year-old
Independence	3,897.88	939.78	442.04	212.21	20	20	.00	.00
Unrestricted two-class	148.79	132.35	46.02	16.97	15b	15b	.00	.32	89.59	92.00	95.60	97.17
Unrestricted three-class	6.14	4.66	8.23	4.50	9c	13c	.51	.99	98.14	97.88	94.88	97.28
	10-year-old
Independence	1,948.50	13,799.73	382.39	382.8	20	20	.00	.00
Unrestricted two-class	64.05	172.52	37.93	45.69	15b	15b	.00	.00	90.08	88.06	96.86	98.17
Unrestricted three-class	7.68	4.96	10.00	5.16	10c	12c	.44	.95	97.39	98.65	96.76	97.99
	11-year-old
Independence	4,556.44	7,275.56	344.19	200.74	20	20	.00	.00
Unrestricted two-class	166.86	180.31	52.48	15.35	14b	14b	.00	.35	84.75	92.75	95.67	97.24
Unrestricted three-class	4.57	7.23	5.14	4.47	11c	12d	.92	.97	98.50	97.77	94.32	97.04
Note: a Significance level associated with the likelihood-ratio chi-square statistic. b Strictly speaking there are 13 degrees of freedom for this model, but a terminal solution was obtained in which one or more conditional behaviour symptom rating probabilities converged to either 0 or 1. Therefore, a new solution was estimated in which these parameters were set a priori equal to that value. This procedure is described and utilized in Clogg (1979) and Goodman (1974a, 1974b). This allows us to assume that L2 has nonetheless a large sample 2 distribution (i.e., L2 is asymptotically distributed as chi-square). c Strictly speaking there are 6 degrees of freedom for this model, but a terminal solution was obtained in which one or more conditional behaviour symptom rating probabilities converged to either 0 or 1. Therefore, a new solution was estimated in which these parameters were set a priori to that value (see justification above). d For one reason or another the unrestricted three-class model was empirically under identified for 11-year-old girls; and therefore, restrictions were imposed on the conditional behaviour symptom rating probabilities such that π j(1)|1 = π j(3)|3, π j(2)|1 = π j(2)|3 and π j(3)|1 = π j(1)|3.

These results led us to consider a latent class model that includes three exclusive and exhaustive latent classes of children, namely, the unrestricted three-class model described above. This model yields a L² of 8.38 with 7 degrees of freedom for 2-year-old boys (p < .30). Hence, the unrestricted three-class model seems to fit the physical aggression data for 2-year-old boys remarkably well. Similarly, this model yields a L² of 10.93 with 8 degrees of freedom (p < .21) for 2-year-old girls. Furthermore, the unrestricted three-class model seems to provide an excellent fit to the physical aggression data for the other age groups except for 6-year-old boys where the model is rejected (see Table 3).

Another way to evaluate the fit of the unrestricted three-class model to the physical aggression data is to compare it to the model of mutual independence since these two models are hierarchically related. We found that by postulating the existence of three mutually exclusive and exhaustive latent classes of children we could account for a great deal of the variance in the PMKs' ratings on the physical aggression behaviour symptoms; that is, 98 per cent [1 - (8.38 / 427.07) = .98] and 97 per cent [1 - (10.93/315.69) = .97] for 2-year-old boys and 2-year-old girls, respectively. Furthermore, we obtained very similar results for the other age groups (see Table 2); the unrestricted three-class model generally accounting for well over 90 per cent of the variance in the physical aggression data. Hence, on both accounts the unrestricted three-class model seems to provide an excellent fit to the physical aggression data; hence, by allowing for three homogeneous groups of children in the population we could account for the physical aggression data. And, therefore, even if it were possible to include an additional latent class to the model it would not significantly improve the fit of the model to the physical aggression data except for 6-year old boys.¹ Of course, if the three behaviour symptoms had been rated by the PMKs using a different Likert scale or if more than three behaviour symptoms had been used to assess physical aggression or if other behaviour symptoms had been used to assess physical aggression we may have obtained different results. No matter how interesting these questions may be they can not be answered using the physical aggression data from the NLSCY.

Table 4 contains the parameter estimates under the unrestricted three-class model. Figure 1, 2 and 3 display the estimates of the conditional behaviour symptom rating probabilities for two-year-old boys while Figure 4, 5 and 6 display the same estimates for two-year-old girls.

Inspection of the estimates of the conditional behaviour symptom rating probabilities, πj(k)|t, reveals a clear ordering among the three latent classes. First, the odds of being rated in rating category 1 (i.e., never or not true) tend to be much higher for children who are members of the first latent class than for those who belong to the second latent class. And, in turn, the odds of being rated in rating category 1 tend to be much higher for those who belong to the second latent class than for those who belong to the third latent class. Consider the odds of being rated in rating category 1 on the first behaviour symptom (i.e., Gets into many fights? —abecq6g) for boys. For instance, at two years of age, the odds were .94 to .06; that is, (.94 / .06) = 14.60 for boys who are members of the first latent class (see Table 3). Comparatively, the odds were only (.24 / .76) = .31 and (.08 / .92) = .08 for those who are members of the second and third latent class, respectively (see Table 3). Hence, the odds of being rated in rating category 1 were (14.60 / .31) = 46.62 times higher for 2-year-old boys who are members of the first latent class than for those who are members of second latent class. Similarly, the odds of being rated in rating category 1 were (.31 / .08) = 3.81 times higher for 2-year-old boys who are members of the second latent class than for those who are members of the third latent class. Second, the odds of being rated in rating category 2 (i.e., sometimes or somewhat true) tend to be much higher for children who are members of the second latent class than for those who belong to either the first or the third latent class. Consider the odds of being rated in rating category 2 on the first behaviour symptom (i.e., Gets into many fights?—abecq6g) for boys. For instance, at two years of age, the odds were (.72 / .28) = 2.58 for those who are members of the second latent class (see Table 3). Comparatively, the odds were only (.06 / .94) = .07 and (.34 / .66) = .51 for those who are members of the first and third latent class, respectively (see Table 3). Hence the odds of being rated in rating category 2 were (2.58 / .07) = 38.73 and (2.58 / .51) = 5.11 times higher for 2-year-old boys who are members of the second latent class than for those who belong to the first and third latent class, respectively. Third, the odds of being rated in rating category 3 (i.e., often or very true) tend to be higher for members of the third latent class than for those who belong to the second latent class. And, in turn, the odds of being rated in rating category 3 tend to be higher for those who belong to the second latent class than for those who belong to the first latent class. Consider the odds of being rated in rating category 3 on the first behaviour symptom (i.e., Gets into many fights?—abecq6g) for boys. For instance, at two years of age, the odds were (.59 / .41) = 1.43 for those who are members of the third latent class (see Table 3). Comparatively, the odds were only (.04 / .96) = .04 and (.002 / .998) = .002 for those who are members of the second and first latent class, respectively (see Table 3). Hence the odds of being rated in rating category 3 were (1.43 / .04) = 33.69 times higher for 2-year-old boys who are members of the third latent class than for those who belong to the second latent class. Similarly, the odds of being rated in rating category 3 were (.04 / .002) = 26.47 times higher for 2-year-old boys who are members of the second latent class than for those who are members of the first latent class. Thus, the estimated conditional behaviour symptom rating probabilities allow for a clear characterization of the latent classes under the unrestricted three-class model. A first latent class which we shall refer to as low-aggressive whose members do not tend to manifest the behaviour symptoms in question. A second latent class which we shall refer to as medium-aggressive whose members tend to manifest the behaviour symptoms in question but only occasionally. And, finally, a third latent class which we shall refer to as high-aggressive whose members tend to often manifest the behaviour symptoms in question.

Table 4 Parameter Estimates Under the Unrestricted Three-class Model for Physical Aggression Latent class ( = 1, 2, 3)

Boys
	Low-aggressive ( = 1) Age (years)										Medium-aggressive ( = 2) Age (years)										High-aggressive ( = 3) Age (years)
	2	3	4	5	6	7	8	9	10	11	2	3	4	5	6	7	8	9	10	11	2	3	4	5	6	7	8	9
πt	.62	.69	.73	.77		.79	.77	.78	.82	.81	.34	.21	.24	.17		.18	.16	.20	.17	.17	.04	.10	.03	.06		.03	.06	.02
πA(1)|t	.94	.82	.76	.75		.76	.76	.78	.75	.78	.24	.13	.20	.07		.10	.07	.14	.24	.20	.08	.01	.07	.29		.00	.18	.02
πA(2)|t	.06	.16	.24	.23		.21	.23	.22	.23	.20	.72	.87	.67	.93		.82	.90	.76	.58	.68	.34	.42	.40	.00		.01	.30	.18
πA(3)|t	.00	.02	.01	.02		.03	.01	.00	.02	.02	.04	.01	.13	.00		.08	.04	.11	.18	.12	.59	.57	.53	.71		.99	.52	.80
πB(1)|t	.87	.73	.96	.97		.95	.99	.93	.98	.96	.38	.25	.14	.20		.21	.20	.30	.11	.31	.13	.22	.00	.00		.09	.15	.26
πB(2)|t	.13	.25	.03	.03		.05	.01	.07	.02	.04	.50	.64	.86	.77		.77	.75	.67	.89	.69	.20	.22	.33	.81		.68	.54	.07
πB(3)|t	.00	.02	.01	.00		.00	.00	.00	.00	.00	.11	.11	.00	.02		.01	.04	.02	.00	.00	.67	.56	.67	.19		.23	.32	.67
πC(1)|t	.77	.87	.92	.91		.91	.94	.98	.95	.97	.27	.22	.25	.23		.39	.39	.31	.44	.44	.00	.00	.10	.44		.00	.00	.00
πC(2)|t	.23	.13	.08	.08		.09	.06	.02	.05	.03	.69	.78	.74	.77		.54	.59	.66	.54	.56	.33	.71	.44	.45		.86	.90	.37
πC(3)|t	.00	.00	.00	.01		.00	.00	.00	.00	.00	.04	.00	.01	.00		.08	.03	.03	.03	.00	.67	.29	.46	.11		.14	.10	.63
Girls
	2	3	4	5	6	7	8	9	10	11	2	3	4	5	6	7	8	9	10	11	2	3	4	5	6	7	8	9
πt	.77	.65	.77	.81	.84	.87	.87	.90	.89	.90	.20	.34	.22	.19	.15	.12	.12	.10	.10	.10	.02	.01	.01	.00	.00	.00	.01	.00
πA(1)|t	.85	.86	.77	.83	.86	.79	.77	.82	.76	.76	.19	.20	.16	.20	.06	.24	.21	.15	.12	.18	.00	.05	.00	.00	.18	.00	.23	.76
πA(2)|t	.14	.13	.22	.15	.13	.20	.21	.16	.23	.21	.76	.77	.74	.72	.74	.62	.67	.74	.67	.62	.29	.23	.00	.00	.00	.00	.37	.00
πA(3)|t	.00	.00	.01	.02	.01	.01	.02	.02	.01	.03	.04	.03	.10	.08	.20	.14	.13	.11	.21	.20	.71	.72	1.0	1.0	.82	1.0	.41	.24
πB(1)|t	.82	.78	.96	.98	.99	.99	.98	.98	.98	.99	.40	.26	.40	.43	.47	.38	.25	.27	.23	.28	.04	.14	.00	.00	.18	.00	.19	.00
πB(2)|t	.14	.21	.04	.02	.01	.01	.01	.02	.01	.01	.54	.57	.58	.57	.54	.57	.75	.73	.77	.71	.13	.12	.34	.44	.18	.00	.00	.39
πB(3)|t	.03	.01	.00	.00	.00	.00	.01	.00	.00	.00	.06	.17	.02	.00	.00	.04	.00	.00	.00	.01	.83	.74	.66	.56	.64	1.0	.81	.61
πC(1)|t	.81	.89	.92	.96	.96	.97	.99	.97	.99	.98	.13	.33	.25	.39	.35	.28	.42	.44	.35	.46	.00	.02	.00	.00	.00	.00	.00	.00
πC(2)|t	.19	.10	.08	.04	.04	.03	.01	.03	.01	.02	.84	.67	.74	.60	.65	.72	.53	.56	.63	.54	.38	.17	.15	.41	.01	.25	.95	.51
πC(3)|t	.01	.00	.00	.00	.00	.00	.00	.00	.00	.00	.03	.00	.01	.00	.00	.00	.05	.00	.02	.00	.62	.81	.85	.59	.99	.75	.05	.49
Note: Behaviour symptom A and C refer to the first (i.e., Gets into many fights?—abecq6g) and third (i.e, Kicks, bites, hits other children?—abecq6nn) behaviour symptom, respectively. Behaviour symptom B refers to the second behaviour symptom (i.e, Recats with anger and fighting?—abecq6x—for 2-3 year-old children and Physically attacks people— abecq6aa—for 4-11 year-old children). πt refers to the probability of being a member of the -th latent class ( = 1, 2, 3). πj(k)| refers to the probability of a rating in category ( = 1, 2, 3) to behaviour symptom (j = A, B, C) given membership in latent class ( = 1, 2, 3). The latent class model reported here for the 11-year-old girls is a restricted three-class model wherein some restrictions were imposed on the conditional behaviour symptom rating probabilities such that πj(1)|1 = π j(3)|3, π j(2)|1 = π j(2)|3 and π j(3)|1 = π j(1)|3. The unrestricted three-class model did not fit the physical aggression data for 6-year-old boys.

 

 

 

 

 

 

Inspection of the estimates of the latent class probabilities, πt, reveals that a majority of children in the population were estimated to belong to the low-aggressive latent class. For instance, at two years of age, the estimated proportion of boys and girls belonging to the low-aggressive was 62 and 77 per cent, respectively (see Table 4). Comparatively, only a small but nonetheless significant percentage of the population of children were estimated to belong to the high-aggressive latent class. For instance, at two years of age, the estimated proportion of boys and girls belonging to the high-aggressive latent class was 4 and 2 per cent, respectively (see Table 4). Figure 7 and Figure 8 display the estimated latent class probabilities for boys and girls, respectively.

3.3 Predicting Latent Class Membership Under the Unrestricted Three-class Model

We are now in a position to infer from each child's response pattern his or her latent class membership (i.e., whether he or she belongs to the low-, medium- or high-aggressive latent class). As mentioned above, the assignment rule is based on the probability of a child being in latent class t (T = 1, 2, 3) given his or her observed response pattern. Each child was assigned to the latent class for which the probability of his or her response pattern is maximum.

Table 5 shows the predicted latent class for each observed response pattern. The percent of children correctly classified into the three latent classes is also reported in Table 3. Generally, over 90 per cent of children were correctly classified into the latent classes. This indicates an excellent ability to predict the child's latent class from his or her observed response pattern.

Table 6 presents the posterior conditional probability of membership for each latent class given children's predicted latent class membership. Inspection of Table 6 reveals that the posterior conditional probability of membership in the high-aggressive latent class for children who were predicted to belong to this latent class is very high. For instance, for two-year-old boys, the predicted value positive was estimated at .90. Hence, among 2-year-old boys who were predicted to belong to the high-aggressive latent class the odds of being high aggressive are .90 to .10; that is, 9.11 (i.e., the diagnostic information). Further inspection of Table 6 reveals that the posterior conditional probability of nonmembership in the high-aggressive latent class for children who were not predicted to belong to this latent class is very high. For instance, for two-year-old boys, the predicted value negative was estimated at .99 and .96 for children who were predicted to belong to the low- and medium-aggressive latent class, respectively. Hence, the odds of being high-aggressive are [(.90 / .10) / (.01 / .99)] = 15,213.33 and [(.90 / .10) / (.04 / .96)] = 225.73 times higher among 2-year-old boys who were predicted to belong to the high-aggressive than among those who were predicted to belong to the low- and medium-aggressive latent class, respectively. These results suggest that the assignment rule used to predict latent class membership permits the identification of a homogeneous group of high-aggressive children. That is, there are relatively few children who were predicted to belong to the high-aggressive latent class who actually do not manifest the characteristics of high-aggressive children (as defined by the conditional behaviour symptom rating probabilities presented in Table 4); and, in addition, there are relatively few children who were predicted not to belong to the high-aggressive latent class who actually manifest the characteristics of high-aggressive children.

 

 

Table 5 Predicted Latent Class Membership Under the Unrestricted Three-Class Model

Response pattern

Age (years)

1

2

3

4

6

7

8

9

10

11

M

F

M

F

M

F

M

F

M

F

M

F

M

F

M

F

M

F

M

F

1

1

1

L

L

L

L

L

L

L

L

L

L

L

L

L

L

L

L

L

L

L

2

1

1

L

L

L

L

L

L

L

L

L

L

L

L

L

L

L

L

L

L

L

3

1

1

M

L

L

L

L

L

L

L

L

L

L

L

L

M

L

L

L

L

L

1

2

1

L

L

L

L

L

L

L

L

L

L

L

L

M

L

L

M

L

L

L

2

2

1

M

L

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

3

2

1

M

M

L

M

M

M

H

M

M

M

M

M

M

M

M

M

M

M

M

1

3

1

M

L

L

L

L

L

H

—

—

M

M

M

L

M

—

L

L

L

M

2

3

1

M

L

M

M

L

M

M

—

—

M

M

M

L

M

—

L

L

L

M

3

3

1

M

L

L

M

H

M

H

—

—

M

M

M

L

M

—

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Note: The first digit of the response pattern refers to the observed rating on the first behaviour symptom (i.e., Gets into many fights?—abecq6g); the second digit refers to the rating on the second behaviour symptom (i.e., Reacts with anger and fighting?—abecq6x or Physically attacks people?—abecq6aa); and the third digit to the rating on the third behaviour symptom (i.e., Kicks, bites, hits other children?—abecq6nn). L, M and H refer to the low-, medium- and high-aggressive latent class, respectively. A dash refers to a cell with an expected value of zero for which the class to which these children should be assigned is undetermined. The latent class model reported here for the 11-year-old girls is a restricted three-class model wherein some restrictions were imposed on the conditional behaviour symptom rating probabilities such that πj(1)|1 = π j(3)|3, π j(2)|1 = π j(2)|3 and π j(3)|1 = π j(1)|3. The unrestricted three-class model did not fit the physical aggression data for 6-year-old boys.

 

Table 6 Posterior Conditional Probabilities of Membership to Each Latent Class for Children who were Predicted to Belong to the Low-, Medium and High-aggressive Latent Class [Latent class ( = 1, 2, 3)]

Boys
	Low-aggressive ( = 1) Age (years)										Medium-aggressive ( = 2) Age (years)										High-aggressive ( = 3) Age (years)
	2	3	4	5	6	7	8	9	10	11	2	3	4	5	6	7	8	9	10	11	2	3	4	5	6	7	8	9
Predicted
Low	.90	.97	.97	.98	—	.97	.98	.97	.97	.96	.10	.03	.03	.02	—	.03	.02	.04	.03	.05	.00	.00	.00	.01	—	.00	.00	.00
Medium	.10	.27	.14	.17	—	.22	.10	.10	.17	.17	.87	.57	.81	.65	—	.71	.67	.89	.81	.80	.04	.17	.05	.18	—	.08	.23	.02
High	.00	.00	.00	.00	—	.00	.00	.00	.00	.00	.10	.00	.02	.00	—	.24	.03	.08	.00	.00	.90	1.0	.98	1.0	—	.76	.97	.92
Girls
	Low-aggressive ( = 1) Age (years)										Medium-aggressive ( = 2) Age (years)										High-aggressive ( = 3) Age (years)
	2	3	4	5	6	7	8	9	10	11	2	3	4	5	6	7	8	9	10	11	2	3	4	5	6	7	8	9
Predicted
Low	.95	.83	.92	.91	.94	.95	.95	.95	.96	.95	.05	.17	.08	.09	.06	.05	.05	.05	.04	.05	.00	.00	.00	.00	.00	.00	.00	.00
Medium	.20	.09	.10	.05	.05	.07	.02	.08	.05	.08	.79	.90	.90	.95	.94	.93	.91	.91	.95	.93	.02	.01	.00	.00	.01	.00	.07	.01
High	.00	.00	.00	.00	.00	.00	.00	.00	.00	.00	.05	.16	.06	.02	.01	.32	.00	.00	.04	.32	.95	.84	.95	.98	.99	.68	1.0	1.0
Note: The unrestricted three-class model did not fit the physical aggression data for 6-year-old boys.

3.3.1 Testing the hypothesis of behaviour symptom interchangeability

As mentioned above, the traditional approach for making a formal diagnosis consists of using a cutoff point specified in terms of an arbitrary number of behaviour symptoms. This approach tacitly assumes that the behaviour symptoms are interchangeable and that it does not matter which behaviour symptoms the child displays as long as the number of symptoms exceeds the minimum specified by the cutoff. However, it may be that the behaviour symptoms differ in severity and that two children who manifest the same number of symptoms may nonetheless differ as to their level of the emotional or behavioural problem in question.

A formal test of the hypothesis that the three behaviour symptoms used to assess physical aggression are equally severe can be obtained by imposing equality constraints on the conditional behaviour symptom rating probabilities. That is, for any given latent class, the probability of being rated in any given rating category is constrained to be equal across the three behaviour symptoms. The goodness-of-fit statistics associated with this restricted three-class model are shown in Table 7.

The value of the L²associated with this restricted three-class model is 88.17 with 18 degrees of freedom (p < .00) and 70.58 with 19 degrees of freedom (p < .00), for 2-year-old boys and girls, respectively. Hence, the hypothesis of behaviour symptoms interchange-ability can be rejected for the 2-year-old children ( = .01). Further, the hypothesis of behaviour symptoms interchange-ability can be rejected for the other age groups as well (see Table 7). These results suggest that the behaviour symptoms differ in severity. Therefore, if one assumes that the unrestricted three-class model is appropriate for the data, adding up the ratings on the three behaviour symptoms may not constitute a proper way to compare individuals with respect to physical aggression. For instance, at two years of age, boys who would obtain a total score of 7 are, according to the assignment rule described above, either predicted to belong to the medium-or the high-aggressive latent class depending on their response patterns (see Table 5). Hence, children with the same total score do not necessarily constitute an homogeneous group of individuals with respect to physical aggression.

Table 7 Pearson and Likelihood-ratio Chi-square Statistics for a Restricted Three-class Model with Equality Restrictions on the Conditional Behaviour Symptoms Rating Probabilities Across Behaviour Symptoms

Age (in years)	Pearson chi-square (X2)		Likelihood-ratio chi-square (L2)		Degrees of freedom		a
	M	F	M	F	M	F	M	F
2	91.81	67.35	88.17	70.58	18	19b	.00	.00
3	127.95	103.74	138.61	108.95	18	18	.00	.00
4	178.48	195.37	173.75	189.79	18	19b	.00	.00
5	229.79	175.82	209.52	172.81	18	19b	.00	.00
6	232.41	226.22	226.52	220.96	18	19b	.00	.00
7	188.89	229.39	190.59	224.97	18	19b	.00	.00
8	246.19	304.90	254.44	300.93	19b	19b	.00	.00
9	188.78	207.71	190.91	203.87	18	18	.00	.00
10	270.89	338.07	259.48	336.16	19b	19b	.00	.00
11	215.17	314.94	202.21	307.91	19b	19b	.00	.00
Note: a Significance level associated with the likelihood-ratio chi-square statistic. b Strictly speaking there are 18 degrees of freedom for this model, but a terminal solution was obtained in which one or more conditional behaviour symptom rating probabilities converged to either 0 or 1. Therefore, a new solution was estimated in which these parameters were set a priori equal to that value. This procedure is described and utilized in Clogg (1979) and Goodman (1974a, 1974b). This allows us to assume that L2 has nonetheless a large sample 2 distribution (i.e., L2 is asymptotically distributed as a chi-square).

3.4 Comparing the Prevalence of Physical Aggression Across Age Groups and Across Sexes

3.4.1 Investigating the relationships of the three behaviour symptoms with sex and age

Figures 9 and 10 display the percentage of children who received ratings in the various response categories for the first behaviour symptom (i.e., Gets into may fights?—abecq6g), for boys and girls, respectively. In addition, Figures 11 and 12 present the same information for the second behaviour symptom (i.e., Reacts with anger and fighting?—abecq6x—or Physically attacks people?—abecq6aa), for boys and girls, respectively. And, in addition, Figures 13 and 14 show the same information for the third behaviour symptom (i.e., Kicks, bites, hits other children?—abecq6nn), for boys and girls, respectively. The extent to which the PMKs' ratings on any behaviour symptom vary as a function of children's age and sex can be investigated using log-linear models (Bishop, Fienberg & Holland, 1975). First, we considered the independence model.

 

 

 

 

 

This model assumes that the variables are statistically independent. This model yields a L of 12,044.02 with 59 degrees of freedom for the first behaviour symptom (i.e., Gets into many fights—abecq6g). Similarly, the L² associated with this model for the third behaviour symptom (i.e., Kicks, bites, hits other children?—abecq6nn) is 20,462.69 with 59 degrees of freedom. Hence, not surprisingly, there seems to exist a very strong relationships among these variables. Second, we considered a model with first- and second-order effects only. This model yields a L²of 59.67 with 18 degrees of freedom ( < .00) for the first behaviour symptom (i.e., Gets into may fights—abecq6g). Similarly, the L² associated with this model for the third behaviour symptom (i.e., Kicks, bites, hits other children?—abecq6nn) is 59.67 with 18 degrees of freedom ( < .00). Thus, a model with only first- and second-order effects does not seem adequate to represent the relationships among these variables. In other words, a third-order effect seems necessary to account for the relationships among these variables suggesting that the extent to which the PMKs' ratings vary as a function of the child's sex depends on his or her age and vice versa. In contrast, a model with only first- and second-order effects yields a L²of 1.63 with 2 degrees of freedom ( < .44) for the second behaviour symptom (i.e., Reacts with anger and fighting?—abecq6x) for 2- and 3-year-old children. Moreover, this model represents a decrease of 2045.14 with 9 degrees of freedom from the independence model (L² = 1.63 - 2,046.78 = 2,045.14, = 11 - 2 = 9, < .00). Among the various second-order effects only the age by behaviour symptom effect reached significance ( = .01) (X² = 48.62, = 2, < .00) suggesting that the PMKs' ratings on this behaviour symptom vary as a function of the child's age only. Similarly, a model with only first- and second-order effects yields a L² of 23.27 with 14 degrees of freedom ( < .06) for the second behaviour symptom (i.e., Physically attacks people?—abecq6aa) for 4- to 11-year-old children. Further, this model represents a decrease of 17,735.63 with 33 degrees of freedom from the independence model (L² = 23.27 - 17,758.90 = 17,735.63, = 47 - 14 = 33, < .00). Within this model both the age by behaviour symptom (X² = 100.54, = 14, < .00) and the sex by behaviour symptom (X² = 239.12, = 2, p < .00) effects have reached significance suggesting that the PMKs' ratings on this behaviour symptom vary as a function of both the child's age and sex. All in all, the PMKs' ratings on the three behaviour symptoms seem to vary as a function of both the child's age and sex. The extent to which these observed differences in the PMKs' ratings imply differences in the prevalence of physical aggression between boys and girls and/or across the different age groups will be investigated next.

3.4.2 Comparing the prevalence of physical aggression in children from 2 to 11 years of age

As mentioned above, the traditional approach for making a formal diagnosis uses the same cutoff score irrespective of the sex and/or the age of the child. This approach tacitly assumes that any observed difference between children who differ in terms of their age and/or sex is due to a true difference in the prevalence of the emotional or behavioural problem in question. However, it may that the observed differences between children of different age and/or sex are due to the fact that the behaviour symptoms are not necessarily functioning the same way depending on children's sex and/or age. For example, the propensity to manifest the three behaviour symptoms used to assess physical aggression may be higher in boys than in girls because these behaviours are more socially acceptable for the former than for the latter (see Peppler & Slaby, 1994). Therefore, the prevalence of physical aggression may or may not differ between boys and girls once we take into account the factors that affect the way these behaviour symptoms function in the two groups. Similarly, the prevalence of physical aggression may or may not differ across age groups once we take into account the factors that affect the way the behaviour symptoms function in different age groups.

A formal test of the hypothesis that the prevalence of physical aggression is the same across age different groups for children of the same sex can be obtained by imposing equality restrictions on the latent class probabilities. That is, for any given sex, the latent class probabilities are constrained to be equal across age groups. The goodness-of-fit statistics associated with this restricted three-class model are shown in Table 8.

First, we considered a restricted three-class model which assumes that the prevalence of physical aggression is the same between 2 and 11 years of age among girls. The value of the L² associated with this restricted three-class model is 173.75 with 131 degrees of freedom (p < .01). Hence, the hypothesis of homogeneity in the estimated latent class probabilities between 2 and 11 years of age among girls is rejected. Similarly, a restricted three-class model which assumes that the prevalence of physical aggression is the same between 3 and 11 years of age among girls is also rejected (see Table 8). In contrast, a restricted three-class model which assumes that the prevalence of physical aggression is the same between 4 and 11 years of age among girls can not be rejected (see Table 8). However, this model represents an increase of 37.39 in L² with a corresponding increase of 14 in the degrees of freedom from the unrestricted model (L² = 119.45 - 82.05 = 37.39, = 127 - 113 = 14, < .00). Since this increase in L² is large compared to the increase the degrees of freedom this suggests that the hypothesis of homogeneity in the estimated latent class probabilities between 4 and 11 years of age among girls is too restrictive. This has lead us to consider a restricted three-class model which assumes that the prevalence of physical aggression is the same between 5 and 11 years of age among girls. The value of the L² associated with this model is 105.12 with 125 degrees of freedom ( < .90). Moreover, this model does not represent a significant decrement in fit from the unrestricted three-class model (L² = 105.12 -82.05 = 23. 06, = 125 - 113 = 12, < .03). And, moreover, other restricted three-class models that are nested within this model also fit the physical aggression data for girls and, in addition, they do not represent a significant decrement in fit over the unrestricted three-class model (see Table 8). These results seem to suggest that the prevalence of physical aggression may not vary among girls between the age of 5 and 11 years once we take into account that the three behaviour symptoms may function differently depending on the age of the child. Second, we considered a restricted three-class model which assumes that the prevalence of physical aggression is the same between 2 and 4 years of age among girls. This model yields a L² of 94.20 with 117 degrees of freedom (p < .94). Furthermore, this model does not represent a significant decrement in fit from the unrestricted three-class model (L² = 94.20 - 82.05 = 12.14, = 117 - 113 = 4, < .02). And, furthermore, other restricted three-class models that are nested within this model also fit the physical aggression data for girls and, in addition, they do not represent a significant decrement in fit over the unrestricted three-class model (see Table 8). These results indicate that the prevalence of physical aggression among girls do not seem to vary between 2 and 4 years of age.

Table 8 Pearson and Likelihood-ratio Chi-Square Statistics for Some Multi-Group Latent Class Models for Comparing the Prevalence of Physical Aggression Across Age Groups

Model	Pearson chi-square (X2)		Likelihood-ratio chi-square (L2)		Degrees of freedom		a		% of variance
	M	F	M	F	M	F	M	F	M	F
Independence	17,700.40	38,451.44	4,827.02	4,334.80	180	200	.00	.00
Unrestricted three-class 2-to 11-year-olds	80.97	72.72	92.85	82.05	85b	113c	26	.99	98	98
RTC 2- to 11-year-olds	139.70	159.64	152.05	173.75	101	131	.00	.01	97	96
RTC 3- to 11-year-olds	120.70	146.25	132.46	160.11	99	129	.01	.03	97	96
RTC 4- to 11-year-olds	111.84	107.78	119.73	119.45	97	127	.06	.67	98	97
RTC 5- to 11-year-olds	106.62	94.59	114.32	105.12	95	125	.09	.90	98	98
RTC 6- to 11-year-olds		86.40		96.28		123		.96		98
RTC 7- to 11-year-olds	95.14	81.03	103.72	91.30	93	121	.21	.98	98	98
RTC 8- to 11-year-olds	91.45	75.89	103.27	86.36	91	119	.18	.99	98	98
RTC 9- to 11-year-olds	84.58	74.30	95.78	84.93	89	117	.29	.99	98	98
RTC 10- to 11-year-olds	81.19	74.13	93.09	84.09	87	115	.31	.99	98	98
RTC 2- to 10-year-olds	135.25	154.50	148.78	169.40	99	129	.00	.01	97	96
RTC 3- to 10-year-olds	116.98	141.88	130.06	156.51	97	127	.01	.04	97	96
RTC 4- to 10-year-olds	109.78	105.47	118.22	117.88	95	125	.05	.66	98	97
RTC 5- to 10-year-olds	104.81	93.32	113.22	104.42	93	123	.08	.89	98	98
RTC 6- to 10-year-olds		85.51		95.67		121		.96		98
RTC 7- to 10-year-olds	94.36	80.05	103.14	90.21	91	119	.18	.98	98	98
RTC 8- to 10-year-olds	90.69	74.55	102.66	84.35	89	117	.15	.99	98	98
RTC 9- to 10-year-olds	84.35	72.67	95.55	82.93	87	115	.25	.99	98	98
RTC 2- to 9-year-olds	108.97	142.71	124.89	156.42	97	127	.03	.04	97	96
RTC 3- to 9-year-olds	93.81	130.60	110.26	143.68	95	125	.14	.12	98	97
RTC 4- to 9-year-olds	90.04	98.01	104.26	109.51	93	123	.20	.80	98	97
RTC 5- to 9-year-olds	81.83	88.98	98.38	98.92	91	121	.28	.93	98	98
RTC 6- to 9-year-olds		83.37		92.58		119		.97		98
RTC 7- to 9-year-olds	79.79	79.76	95.72	88.93	89	117	.29	.98	98	98
RTC 8- to 9-year-olds	82.82	74.40	94.94	84.24	87	115	.26	.99	98	98
RTC 2- to 8-year-olds	103.80	132.36	120.65	144.91	95	125	.04	.11	98	97
RTC 3- to 8-year-olds	90.16	121.48	107.64	133.77	93	123	.14	.24	98	97
RTC 4- to 8-year-olds	84.67	91.61	103.07	102.90	91	121	.18	.88	98	98
RTC 5- to 8-year-olds	77.48	84.47	95.87	94.64	89	119	.29	.95	98	98
RTC 6- to 8-year-olds		80.11		89.86		117		.97		98
RTC 7- to 8-year-olds	78.42	78.20	95.12	87.66	87	115	.26	.97	98	98
RTC 2- to 7-year-olds	100.66	124.45	117.52	133.89	93	123	.04	.24	98	97
RTC 3- to 7-year-olds	88.71	111.44	106.15	121.63	91	121	.13	.47	98	97
RTC 4- to 7-year-olds	87.17	85.51	101.69	95.25	89	119	.17	.95	98	98
RTC 5- to 7-year-olds	77.82	79.53	95.59	87.23	87	117	.25	.98	98	98
RTC 6- to 7-year-olds		76.92		84.50		115		.99		98
RTC 2- to 6-year-olds		107.31		111.94		121		.71		97
RTC 3- to 6-year-olds		98.83		108.52		119		.74		97
RTC 4- to 6-year-olds		78.10		88.09		117		.98		98
RTC 5- to 6-year-olds		74.68		83.70		115		.99		98
RTC 2- to 5-year-olds	101.57	95.72	112.68	104.72	91	119	.06	.82	98	98
RTC 3- to 5-year-olds	90.69	88.00	102.34	96.76	89	117	.16	.91	98	98
RTC 4- to 5-year-olds	89.05	75.00	99.15	84.25	87	115	.18	.99	98	98
RTC 2- to 4-year-olds	92.53	84.02	103.61	94.20	89	117	.14	.94	98	98
RTC 3- to 4-year-olds	85.30	80.81	97.46	89.89	87	115	.21	.96	98	98
RTC 2- to 3-year-olds	85.09	77.95	97.25	88.32	87	115	.21	.97	98	98
Note: The unrestricted three-class model did not fit the physical aggression data for 6-year-old boys. RTC=Restricted three-class model with equal latent class probabilities across groups. a Significance level associated with the likelihood-ratio chi-square statistic. b Strictly speaking there are 54 degrees of freedom for this model, but a terminal solution was obtained in which one or more conditional behaviour symptom rating probabilities converged to either 0 or 1. Therefore, a new solution was estimated in which these parameters were set a priori equal to that value. This procedure is described and utilised in Clogg (1979) and Goodman (1974a, 1974b). This allows us to assume that L2 has nonetheless a large sample c2 distribution (i.e., L2 is asymptotically distributed as chi-square). c Strictly speaking there are 60 degrees of freedom for this model, but a terminal solution was obtained in which one or more conditional behaviour symptom rating probabilities converged to either 0 or 1. Therefore, a new solution was estimated in which these parameters were set a priori equal to that value (see justification above).

We have obtained very similar results for boys. First, a restricted three-class model which assumes that the prevalence of physical aggression is the same among boys between 5 and 11 years of age yields a L² of 114.32 with 95 degrees of freedom ( < .09). Moreover, this model does not represent a significant decrement in fit from the unrestricted three-class model (L² = 114.32 - 92.85 = 21. 48, = 95 - 85 = 10, < .02). And, moreover, other restricted three-class models that are nested within this model also fit the physical aggression data for boys; and, in addition, they do not represent a significant decrement in fit over the unrestricted three-class model (see Table 8). Second, we considered a restricted three-class model which assumes that the prevalence of physical aggression is the same between 2 and 4 years of age among boys. This model yields a L² of 103.61 with 89 degrees of freedom ( < .14). Furthermore, this model represents an increase of only 10.76 in L² with an increase of 4 degrees of freedom from the unrestricted three-class model (L² = 103.61 - 92.85 = 10.76, = 89 - 85 = 4, < .03). And, furthermore, other restricted three-class models that are nested within this model also fit the physical aggression data for boys; and, in addition, they do not represent a significant decrement in fit over the unrestricted three-class model (see Table 8). These results indicate that the prevalence of physical aggression among boys do not seem to vary between 5 and 11 years of age and between 2 and 4 years of age.

3.4.3 Comparing the prevalence of physical aggression between boys and girls

We are now in a position to compare the prevalence of physical aggression between boys and girls. First, we considered a restricted three-class model which assumes that the prevalence of physical aggression is the same between boys and girls from 2 to 4 years of age. The value of the L² associated with this model is 90.73 with 60 degrees of freedom (p < .01). Hence, the hypothesis of homogeneity in the latent class probabilities between boys and girls from 2 to 4 years of age can not be rejected. Moreover, this model represents an increase of only 8.44 in L²with an increase of 2 degrees of freedom over the restricted three-class model which assumes that the prevalence of physical aggression is the same between 2 and 4 years of age for each sex (L² = 90. 73 - 82.29 = 8.44, = 60 -58 = 2, < .02 ). These results suggest that the prevalence of physical aggression may be the same between boys and girls for the period ranging from 2 to 4 years of age. Second, we considered a restricted three-class model which assumes that the prevalence of physical aggression is the same between boys and girls for the period ranging from 5 to 11 years of age. The L² associated with this model is 176.09 with 167 degrees of freedom (p < .30). However, this model represents a significant decrement of fit over the restricted three-class model which assumes that the prevalence of physical aggression is the same between 5 and 1 years of age within each sex (L² = 176.09 - 143.71 = 32.38, = 167 - 165 = 2, < .00 ). These results seem to suggest that the prevalence of physical aggression is not the same between boys and girls from 5 and 11 years of age.

Figure 15 displays the latent class probabilities estimated under these restricted three-class models. Between 2 and 4 years of age, the proportion of Canadian children who belong to the high-aggressive latent class was estimated at 3.5 per cent, the same for boys and girls. Between 5 and 11 years of age the proportion of Canadian children who belong to this latent class was estimated at 3.3 and .6 per cent, for boys and girls, respectively. Note that the odds of belonging to the high-aggressive latent class were [(.033 / .967) / (.006 / .994)] = 5.34 times higher for boys than for girls between 5 and 11 years of age. Further, note that the odds of belonging to the high-aggressive latent class were [(.035 / .965) / (.006 / .994)] = 5.75 times higher for girls between 2 and 4 years of age than for those between 5 and 11 years of age. In comparison, the odds of belonging to the high-aggressive latent class were practically the same between 2 and 11 years of age for boys [(.035 / .965) / (.033 / .967)] = 1.08.

• ¹Note that with only three trichotomous behaviour symptoms a four-class model is not identifiable (for more details on latent class model identifiability see McCutcheon, 1987).