We conducted three sets of taxometric analyses. The first set used combinations of indicators of inattention to examine whether attention problems are dimensional or categorical. The second set used examined the latent structure of hyperactivity/impulsivity. Finally, because it is possible for inattention and hyperactivity/impulsivity to each have a dimensional latent structure, yet for the unique combination of these two dimensions to be taxonic, we combined indicators of inattention and hyperactivity/impulsivity to examine the latent structure of ADHD.

The correlations among the primary and secondary indicators of ADHD are provided in . Because indicator skew can influence the shape of taxometric curves (e.g.,

Ruscio & Ruscio, 2002), the skew of each indicator is also provided in this table. All of the indicators were positively skewed except for the TOH and WJR, which were the only two variables in which higher scores indicated higher functioning. These two variables were reverse scored in the taxometric analyses.

| **Table 1**Correlations among primary and secondary indicators of ADHD ^{a}. |

Taxometric analyses require valid indicators and

Meehl (1995) recommended that they should demonstrate at least 1.25 standard deviation units (SDU) of separation between taxon and complement groups. Unfortunately, the SECCYD data set did not include formal diagnoses of ADHD. More importantly, there is no reason to presuppose that the diagnosis of ADHD would be isomorphic with a presumptive taxon; therefore, we used two strategies to estimate the validity of the various indicator sets. First, we presumed a taxon base rate of 7%, which is consistent with the population base rate of ADHD and also consistent with the participants’ scores on the DBQ (see below). Second, the MAMBAC and MAXEIG procedures provide base rate estimates based on the shape of the taxometric curves and these base rate estimates were used to compute indicator validities, by combining the indicators and using this composite to divide the sample according to the base rate. With the exception of a few individual indicators (see below), all of the indicator sets yielded validity coefficients above 1.25 SDUs ().

| **Table 2**Summary of the Taxometric Analyses Examining the Latent Structure of Inattention, Hyperactivity, and ADHD |

Ideally, the indicators for a taxometric analysis are correlated with one another in the full sample, but there is smaller covariance among the indicators within the taxon and complement groups (i.e., low nuisance covariance). also reports the average full sample and nuisance correlations for each set of indicators, once again presuming a 7% taxon base rate. The full sample correlations are .30 or greater for all but one of the reported sets of analyses. There was generally little evidence of nuisance covariance.^{1}

The indicators for the primary taxometric analyses for inattention, hyperactivity/impulsivity, and ADHD used the relevant DBQ items, because they map directly onto the DSM-IV-TR diagnostic criteria. The mother and teacher DBQs were analyzed separately and they were also combined across teacher and mother responses. If the teacher and mother both provided a report, these items were averaged; otherwise, the response provided was used. The rationale for averaging the mother and teacher responses was because children behave differently in different situations so these raters have access to separate types of information and because raters may interpret the same behavior differently (

Pelham, Fabiano, Massetti, 2005). Although average ratings are not the same as having independent reports as is required for comprehensive ADHD assessments (

Pelham et al., 2005), these average ratings were a way to use information from both the mother and teacher and can be considered a compromise between the “or” and the “and” rule for combining multi-informant data (

Gizer et al., 2008). In the interest of succinctness and because there is no rationale for assigning the separate mother or teacher ratings primacy, we provide detailed reports of the results from the average of the mother and teacher responses (and the corresponding graphs), but also provide the results from the separate analyses of the mother and teacher reports in .

Inattention

The average of the mother and teacher DBQ inattention items demonstrated strong validity: On the basis of the base rates yielded by the subsequent MAMBAC and MAXEIG analyses, the average degrees of separation for these nine indicators were 2.17 and 2.52^{2} SDUs, respectively. For the MAMBAC analyses, each item served as the output indicator for one graph, with the remaining 8 items summed to create the input indicator. All nine graphs exhibited a rising cusp on the right side of the graph that could either be indicative of a low baserate taxon or positively skewed indicators.^{3} The average of the nine curves was more similar to the dimensional comparison data than to the taxonic comparison data (, top graph), with a CCFI of .406 (). None of the nine individual MAXEIG curves displayed a clear peak consistent with taxonic structure. The average of the nine MAXEIG curves was more similar to the dimensional comparison data than to the taxonic comparison data (, middle graph), with a CCFI of .304. The L-Mode curve for the actual data was unimodal (CCFI = .379), unlike the L-Mode Curve for the simulated taxonic data, which was bimodal (, bottom graph). We conducted the same set of analyses separately for the mother and teacher DBQ forms. All of these results were consistent with a dimensional latent structure ().

Next, we used indicators that operationalized inattention using diverse methods. The four indicators for these analyses were (a) observer rated inattention problems (the inattention ratings averaged across both the mother and teacher DBQ, the TRF, and the CBCL), (b) academic achievement (average Broad Reading and Math scores on the WJ-R), (c) omissions on the CPT, and (d) TOH. Because these diverse operationalizations do not share method variance, they did not cohere as strongly as indicators all derived from the same methodology. As a result, the average degrees of separation for these four indicators, based on the base rates derived from the subsequent MAMBAC and MAXEIG analyses, were 1.41 and 1.52 SDUs, respectively, which is adequate but less than ideal.

Two of the MAMBAC graphs exhibited a rising cusp on the right side of the graph that could be indicative of a low base-rate taxon or positively skewed indicators, whereas the other two graphs had right side peaks suggestive of a taxon. However, the average of the four curves was more similar to the dimensional comparison data than to the taxonic comparison data, (CCFI= .429, ). The MAXEIG analyses were consistently dimensional. The four MAXEIG curves were flat and the average MAXEIG curve was more similar to the dimensional comparison data than to the taxonic comparison data, (CCFI = .328; ). The L-Mode graph was unimodal and more similar to the dimensional comparison data (CCFI = .273). Finally, because the CPT omissions yielded the lowest indicator validity, we repeated the MAMBAC and MAXEIG analyses with the three remaining indicators. These analyses yielded better average indicator validities (1.59 for MAMBAC; 1.74 for MAXEIG), and dimensional results for MAMBAC (CCFI = .287) and MAXEIG (CCFI = .348). Across almost all of the analyses using the various combinations of inattention indicators, there was strong evidence that inattention problems have a dimensional latent structure.

Hyperactivity/Impulsivity

The average of the mother and teacher DBQ hyperactivity/impulsivity items also yielded acceptable validity coefficients: On the basis of the base rates yielded by the subsequent MAMBAC and MAXEIG analyses, the average degrees of separation for these nine indicators were 1.93 and 2.31 SDUs, respectively. All nine MAMBAC curves exhibited a rising cusp on the right side of the graph. Overall, the average MAMBAC curve was closer to the simulated dimensional data than to the simulated taxonic data (CCFI = .426; , top graph). Only two of the nine MAXEIG curves displayed a clear peak consistent with a taxonic structure. The average of the nine MAXEIG curves was clearly more similar to the dimensional comparison data than to the taxonic comparison data (, middle graph), CCFI = .272. The L-Mode curve for the actual data was unimodal, (, bottom graph), and the CCFI (.376) was more consistent with a dimensional latent structure. We also ran separate sets of taxometric analyses for the mother and teacher forms of the DBQ. These analyses also yielded consistently dimensional results ().

Next, we used indicators that operationalized hyperactivity/impulsivity using diverse methods. Specifically, the three indicators for these analyses were (a) observer rated hyperactivity/impulsivity problems (the hyperactivity/impulsivity ratings averaged across both the mother and teacher DBQ, the TRF, and the CBCL), (b) classroom disruptive behavior as rated by an observer, and (c) commission errors on the CPT. The average degrees of separation for these three indicators, based on the base rates derived from the subsequent MAMBAC and MAXEIG analyses, were 1.41 and 2.23 SDUs, respectively.

Two of the MAMBAC graphs exhibited a rising cusp on the right side of the graph, whereas the other graph had a right side peak consistent with a taxon. However, the average of the three curves was much more similar to the dimensional comparison data than to the taxonic comparison data, CCFI = .285 (). The average MAXEIG curve was clearly more similar to the dimensional comparison data than to the taxonic comparison data, CCFI = .378 (). Overall, it appears that hyperactivity/impulsivity has a dimensional latent structure.

ADHD

We used multiple sets of indicators to examine whether an ADHD taxon emerged from the combination of inattention and hyperactivity/impulsivity problems. The first analyses used the total DBQ inattention score and total DBQ hyperactivity/impulsivity score (each averaged across mother and teacher reports when both were available) as the two indicators. Because there were only two indicators for this analysis, we were limited to MAMBAC. Both curves had a rising cusp on the right. The average of the two curves was considerably more similar to the curve from the simulated dimensional data than the curve produced by the simulated taxonic data (CCFI = .370). Separate MAMBAC analyses with just the mother DBQ and teacher DBQ reports also yielded dimensional results ().

Separate factor analyses of the ADHD items from the TRF and the mother report CBCL, yielded inattention and hyperactivity/impulsivity factors for the TRF, but only a single combined ADHD factor for the CBCL. Based on the subsequent MAMBAC (2.02 SDUs) and MAXEIG (2.42 SDUs) analyses, these three scales had good indicator validity. This set of indicators also yielded results that were consistent with a dimensional latent structure (MAMBAC CCFI = .236; MAXEIG CCFI = .291).

Finally, we used indicators that operationalized ADHD using diverse methods. The five indicators for these analyses were (a) observer rated inattention problems (the inattention ratings averaged across both the mother and teacher DBQ, the TRF, and the CBCL), (b) observer rated hyperactivity/impulsivity problems (the hyperactivity/impulsivity ratings averaged across both the mother and teacher DBQ, the TRF, and the CBCL), (c) academic achievement (average Broad Reading and Math scores on the WJ-R), (d) omission errors on the CPT,^{4} and (e) TOH. The average degrees of separation for these five indicators, based on the base rates derived from the subsequent MAMBAC and MAXEIG analyses, were 1.42 and 1.57 SDUs, respectively, which is adequate.

One of the five MAMBAC curves had a clear peak, but the other four were ambiguous (two had a rising cusp and two had multiple peaks). The average of the five curves was more similar to the dimensional comparison data than to the taxonic comparison data, (CCFI = .359; , top graph). All five MAXEIG curves were flat and the average MAXEIG curve was more similar to the dimensional comparison data than to the taxonic comparison data (CCFI = .418; , middle graph). The L-Mode graph was unimodal and more similar to the dimensional comparison data (CCFI = .278; , bottom graph; ). Finally, because the CPT omissions yielded the lowest validity scores, we ran the analyses with just the other four ADHD indicators. These analyses yielded greater separation (1.57 and 1.79 SDUs), and the results remained dimensional (see ).

Comparison of Dimensional and Dichotomous Models

Examination of features associated with ADHD provides a further test of the relative merits of taxonic versus dimensional models of the construct (

Watson, 2006). To examine the utility of these alternative models, we computed correlations between dimensional and categorical ADHD scores (based on two sets of scores) and numerous associated features. First, because the DBQ items map directly onto the DSM-IV-TR criteria, DBQ case assignments were considered positive if (a) mother and teacher reports both met the six-symptom cut off for either inattention or hyperactivity/impulsivity symptoms, or (b) either the mother or teacher report met the criteria for both inattention and hyperactivity/impulsivity. These criteria yielded a 7% rate of ADHD, consistent with population base rates.

^{5} The second set of scores combined the data from the inattention and hyperactivity/impulsivity measures from the DBQ-mother, DBQ-teacher, TRF, and CBCL (total of eight scales converted to

*z*-scores and averaged). Composite case assignments were made, splitting the sample into two groups matching the putative taxon base rate across all of the analyses that examined the latent structure of ADHD (26.7%). Based on Monte Carlo simulations

Ruscio (2009a) concluded that case assignment based on the estimated taxon base rate is at least as accurate as other methods such as using Bayes’ theorem. The DBQ and Composite continuous scores were the average scores of the respective measures.

The correlations among the dimensional scores and the associated features were consistently higher than the correlations among the dichotomous scores and the associated features, (). In fact, the DBQ-based case assignments accounted for 7.6% of the variance (on average) in the associated features, whereas the continuous DBQ score accounted for 17.4% of the variance in the associated features. Likewise, the composite case assignments accounted for an average of 6.5% of the variance in these variables, compared to 18.8% of the variance accounted for when treating the composite scores continuously. Overall, treating ADHD continuously accounted for 2.6 times as much variance as treating ADHD categorically. Finally, we compared the differences in the correlations between the associated features and the DBQ continuous and case assignment scores, as well as the differences in the correlations between the associated features and the composite continuous and case assignment scores.

Steiger’s (1980) method for comparing two dependent correlations (i.e., those sharing one common variable—the associated feature) was used. These results () indicated that correlations in which ADHD was treated continuously were significantly larger for 42 of 48 comparisons.

| **Table 3**Correlations comparing the dimensional and dichotomous variables with outcome variables. |