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The Relationship between Inattentiveness in the Classroom and Reading Achievement (Part A): Methodological Issues
The Relationship between Inattentiveness in the Classroom and Reading Achievement (Part A): Methodological Issues KENNETH J. ROWE, M.Sc.,
AND
KATHERINE S. ROWE, M.B., B.S., F.R.A.C.P.
Abstract. Conclusions drawn from recent reviews of research concerned with relationships between child behavior disorders and learning difficulties highlight major methodological and analytical problems that have plagued empirical research in the field. This paper (Part A) identifies and discusses some of these problems, including those associated with the design and use of psychometric instruments, cutoff scores, and statistical modeling techniques. The paper provides a contextual rationale for the choice of methods and statistical models used in a large explanatory study of the relationship between inattentiveness in the classroom and reading achievement, reported in Part B. J. Am. Acad. Child Adolesc. Psychiatry, 1992, 31, 2:349-356. Key Words: psychometric instruments, negative nomenclature, factor analysis, cutoff scores, statistical modeling techniques.
In educational theory, research, and practice, the notion of attentiveness has long been associated with the key operational construct of active learning time or its equivalents, time-on-task, engaged learning time, or perseverence. This notion derives from the theoretical work of Carroll (1963, 1984), Cooley and Lohnes (1976), and Bloom (1976) who argue that although students may differ in their aptitude for learning, the different amounts of time needed to achieve a given level of proficiency are a direct function of the amount of attention or effort invested by an individual in a learning task. Findings from research on student learning in classroom settings provide strong suport for this view, indicated that attentiveness is directly related to achievement outcomes (Fisher et aI., 1980; Keeves, 1986; Lahaderne, 1968). This work suggests that attentiveness (defined as purposeful activity showing a sustained attention span, perseverence, concentration, and not easily distracted) is a crucial variable associated with student behavior at home and at school, through which, the effects of learning experiences and attitudes are mediated positively to influence learning outcomes. Evidence from studies investigating the impact of maladaptive student behaviors provides strong support for the importance of inattentiveness as a major variable having negative effects on student achievement. These studies reflect a persistent concern of teachers, parents, and mental health professionals of the extent to which students' malAccepted May 20, 1991. Mr. Rowe is Senior Policy Officer-Research, School Programs Division, Ministry of Education and Training, Victoria, Australia. Dr. Rowe is Lecturer and Physician, Department of Pediatrics, The University of Melbourne, Royal Children's Hospital, Parkville, Victoria, Australia. The comments and suggestions made on an earlier version of this paper by Glenn Rowley, Phillip Holmes-Smith, Trevor Williams, and two anonymous reviewers were most helpful. Financial support was provided by the Commonwealth Resource Agreement. Reprint request to: Ken Rowe. Senior Policy Officer-Research. School Programs Division, Ministry ofEducation and Training, Level 8 Rialto Towers, Box 4367, GPO Melbourne. Victoria 3001, Australia. 0890-8567/92/31 02-Q349$03.0010© 1992 by the American Academy of Child and Adolescent Psychiatry. J. Am. A cad. Child Adolesc. Psychiatry, 31:2, March 1992
adaptive behaviors in the classroom (attention deficit/hyperactivity and conduct disorders/aggression) adversely affect learning outcomes in general and reading achievement in particular (Cairns, 1985; Kyriacou & Roe, 1988; McGee et aI., 1986; Rowe, 1988; Sturge, 1982; Wheldall and Merrett, 1988). Students whose behaviors are regarded as inattentive, disruptive, or maladjusted have been shown to be at risk of poor educational attainment (Cantwell and Baker, 1991; Davie et aI., 1972; Rutter, 1985; Szaday, 1989). Moreover, in addition to the consequences for an individual, behavior problems in the classroom diminish educational opportunities for other students and contribute to teacher stress (Brenner, et aI., 1985; Otto, 1986; Wearing, 1989). Although students' classroom behaviors have been found to be partly dependent on factors, such as ethnicity (Dunkin and Doenau, 1985), social background (Kahl, 1985), gender (Bank, 1985) as well as cognitive and affective characteristics (Debus, 1985; Sinclair, 1985), findings from a growing number of correlational studies indicate stronger direct associations between poor attention and reading difficulties both in general student populations and in identified learning-disabled groups (Dykman and Ackerman, 1991; Jorm et aI., 1986; Levy et aI., 1987; Maughan et aI., 1985; McGee et aI., 1987; McGee et aI., 1989; McKinney, 1989; Stanton et aI., 1990; Stevenson et aI., 1985). For example, in their longitudinal study in Dunedin, New Zealand, McGee and coworkers consistently found poor reading achievement to be strongly related to high ratings of inattention. McGee and Share (1988) estimate that 80% of their sample of ll-yearold children identified with attention deficit disorder with hyperactivity (ADDH), as defined by DSM-III, had learning disabilities in reading and written language skills. Resulting in part, from a variety of methodological and analytical limitations in these studies, however, both the direction and magnitude of "effect" relationships is not clear. From interest in the relationship between students' reading disabilities and problem behaviors, Rutter et ai. (1970) proposed four alternative "causal" hypotheses, namely: 1) problem behavior leads to reading difficulties; 2) reading disability produces behavior problems; 3) both problem behavior and reading disability are produced by some third factor; and 4) all of these hypotheses could be partly true.
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In a review of the related research, McGee et aI. (1986) note: "All hypotheses have drawn support from the literature and the proposed mechanism underlying the relationship between reading disability and behavior disorder appear to be equally plausible" (p. 597). On the basis of a more recent and detailed review of the literature concerned specifically with the relationship between ADDH (or attention-deficit hyperactivty disorder in DSM-III-R) and failure to acquire literacy skills, McGee and Share (1988) conclude: "The evidence the authors have reviewed suggests that a substantial overlap exists between ADDH and learning difficulties and that, as yet, no unique pattern of cognitive or attentional deficits has been identified that can discriminate between these two types of disorder" (p. 322). (Fletcher et aI. 1991 make a similar observation.) Following Kinsbourne (1984), who argues that attentional problems are both "context" and "task" dependent, McGee and Share (1988) further conclude that "ADDH behaviors might best be considered as a disorder of conduct in the classroom, because the child with learning difficulties is excluded from much of the normal classroom activity" (p. 322). This view is consistent with the recent findings of Day and Peters (1989) who conclude that" ... learning-disabled children seem to be better characterized as 'inattentive in the classroom' ..." (p. 360). Despite the apparent simplicity of the Rutter et ai. (1970) hypotheses outlined above, the conclusions drawn from reviews of the related research highlight several methodological and analytical problems that have plagued empirical research in the field. This is especially the case for those attempts to address "causal" or "which comes first?" hypotheses (e.g., McGee and Share, 1988; McMichael, 1979). Because most of the evidence upon which such hypotheses are based derives from cross-sectional rather than longitudinal studies, it is difficult to determine either the nature or the direction of putative effect relationships. Moreover, findings from the few longitudinal studies that have been reported are not consistent (e.g., Chazan, 1985; Jorm et al., 1986; Maughan et aI., 1985; McGee et aI., 1986; Richman et aI., 1982; Stott, 1981). The purpose of this paper (Part A) is to provide a brief review of some major methodological and analytical problems endemic to research in child behavior disorders and learning difficulties. The rationale for this review is twofold: 1) to assist clinical and epidemiological researchers who may be struggling with such problems; and 2) to provide a substantive basis for presenting the methodology and findings from an explanatory study of the relationship between inattentiveness in the classroom and reading achievement for a sample of 5,000 students (age 5-14 years) drawn from a normal school population - reported in Part B (in this issue). Methodological Problems
Perhaps the most ubiquitous methodological problem faced by researchers is the attempt to identify the correlates unique to attentional problems on the one hand and learning difficulties on the other (McGee and Share, 1988). The lack of clear differentiation between the two domains on existing measures used to define them not only places the validity
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of diagnostic "categories" in question but imposes severe constraints on the interpretation of findings from related studies. The high level of overlap between learning difficulties and attentional problems is underscored by the lack of sensitivity and specificity attained by existing psychometric instruments and rating scales, suggesting that valid criteria for determining the existence of either disorder in childhood are, at best, dubious (Boyle and Jones, 1985; Hinshaw, 1987; Prior and Sanson, 1986; Ullmann et aI., 1985). Similarly, the consistent low levels of agreement between clinical diagnoses « 60%) is likewise too low to serve as a classification standard (Boyle and Jones, 1985; Rutter et aI., 1975). A related problem concerns the way in which diagnostic instruments are developed. Typical of such instruments is the use of items that focus exclusively on negative rather than positive behaviors (e.g., Achenback and Edelbrock, 1983; Conners, 1969, 1973; Quay and Peterson, unpublished manuscript, 1975; Rutter, 1967). Emphasis on negative nomenclature is at the expense of a more balanced assessment and increases the risk of a prejudicial searches for "pathology," regardless of its presence or absence; nor are such instruments independent of sociocultural differences (Yao et aI., 1988). For example, in a recent normative study of the Achenbach Child Behavior Checklist, Hensley (1988) found a consistent tendency by Australian parents to rate their child's behavior toward the negative ends of items, significantly more so than their North American counterparts. Sandoval (1977) has criticized the use ofrating scales exclusively employing negatively worded items on the grounds that they are highly susceptible to rater bias and response sets such as "reverse halo effects" or "reverse generosity errors" (Kerlinger, 1986; Sellitz et aI., 1976). In a comparative study of format effects in rating scales of hyperactivity, Sandoval (1981) has shown that for positively worded items, raters are more willing to use the extreme rating categories for a given item, thus increasing the dispersion and discrimination of the measurements. In contrast, an inspection of the marginal distributions for negatively worded items show that they tend to be highly skewed and leptokurtic. By far, the most popular means of defining and measuring emotional and behavioral disorders of childhood is via exploratory factor-analytic (FA) approaches to determine the underlying dimensionality of multiple-item rating inventories administered by parents, teachers, or clinicians. From Hinshaw's (1987) comprehensive review of the 60-FA studies published between 1970 and 1986, it is interesting to note that all used exploratory FA approaches, and that 56 (93%) used orthogonal methods of factor extraction and rotation (mostly principal components analysis or principal factor solutions, with varimax rotation). Such approaches are problematic on at least three grounds. First, in the case of exploratory (unrestricted) methods of FA, the factor solutions are arbitrary, data driven, hypothesis generating, and, invariably, result in theory conflation (1oreskog, 1981; Rowe, 1989). Second, orthogonal methods of factor extraction and rotation assume that the derived factors are uncorrelated or independent-by definition (Harman, 1976). Because all items are allowed to load on more than one factor, J. Am. Acad. Child Adolesc. Psychiatry, 31:2, March 1992
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the resulting correlated error variance alone is sufficient to yield shared variance across factors. Although the development of unidimensional scales is highly desirable from a measurement perspective, given the considerable literature concerning the nonindependent and overlapping dimensions of child behavior and psychopathology, orthogonal FA methods are difficult to justify either substantively or empirically. Third, and perhaps most serious of all, such methods are invariably applied to item responses measured on dichotomous or 3- to 5-point Likert-type ordinal scales and rely on the computation of Pearson product-moment interitem correlation matrices-estimated by default in most omnibus computer packages. What is overlooked in such instances is that the assumptions underlying Pearson product-moment correlations (i.e., normal distribution and homogeneity of variance) are always violated. As a consequence of using Pearson product-moment estimates for dichotomous or ordinal variables, instead of the consistently less biased tetrachoric correlations (dichotomous with dichotomous) or polychoric correlations (ordinal with ordinal), substantial bias in the interitem correlations and subsequent factor parameters is unwittingly introduced (Carroll, 1961; Joreskog and Sorborn, 1979, 1988). Failure to take account of the measurement and distributional properties of response variables in factor analysis amounts to what Hendrickson and Jones (1987) refer to as "an undisciplined romp through a correlation matrix" (p. 105). Given the almost universal application of these procedures, it could be argued that current claims to substantive knowledge about dimensions of child psychopathology may be little more than the products of methodological and statistical artifact (Scarr, 1985). An additional problem in applied research relates to the widespread use of rating inventories for the purposes of classification and diagnosis. Because of the inherent complexity of learning disabilities and behavioral disorders in childhood, Ullmann et aI. (1985) argue that the common use of a single "cutoff" score on a rating scale to diagnose deviance is inappropriate and misleading (e.g., a score of 15 on Conners' 1973 Abbreviated Teacher Rating Scale to diagnose ADDH [Sprague et aI., 1974]). Although it is customary to select 2 standard deviations from the mean for these purposes, such selections are arbitrary and can be modified depending on whether one wishes to minimize false positives or false negatives (Loney and Milich, 1982). This approach is aptly illustrated by Szatmari et aI., (1989) in their review of 11 studies reporting prevalence rates of hyperactivity and ADDH. Four of these studies employed diagnostic' 'cutoff" scores of 1.0, 1.5, or 2.0 standard deviations from the mean, in the absence of substantive criteria for doing so, "resulting in the identification of different numbers and types of cases" (Szatmari et aI., 1989, p. 221). For example, reported prevalence rates for ADDH vary from less than 1% (Rutter et aI., 1970), 14.3% (Trites et aI., 1979) to as high as 20% (Shaywitz and Shaywitz, 1991), depending on the following: 1) the methods of data collection; 2) the sampling characteristics of the populations targeted; and 3) the arbitrary determination of deviance criteria. Variability in measurement and cutoff scores, together with sampling differences, leads to substantial differences in prevalence J. Am. Acad. Child Ado/esc. Psychiatry, 31:2, March 1992
estimates. In the context of predictive or explanatory research, there is little rational justification for identifying, a priori, a fixed proportion of the child population as ADDH, for example, particularly when such a dimension is more meaningfully viewed as a continuum, both in qualitative and quantitative terms. Despite the utility and obvious convenience of rating scales, especially for large-scale epidemilogical and survey research, the psychometric limitations endemic to their design, construction, and use seem to be largely unrecognized by most researchers. Of particular concern in many studies examining the relationship between student behavior and learning outcomes is the way in which major mediating variables of the "third kind" have been measured and treated (e.g., students' home background characteristics). In attempts to address Rutter et aI.'s (1970) Hypothesis 3 (mentioned earlier), the variables most often chosen to measure family background, for example, have typically not included direct measures of parental involvement or other family educational inputs. Whereas numerous studies have included surrogate measures of home background factors, the variables most often chosen are not measured directly. Instead, proxy measures, such as student self-report estimates of the number of books in the home and ordinal classifications of family social class or socioeconomic status are used (e.g., Davie et aI., 1972; Douglas, 1964; Fotheringham and Creal, 1980; Rutter et aI., 1979; Williams and Silva, 1985). Moreover, when such measures are used, it is mostly for group classification purposes (for subsequent comparative group analysis) or as a means of statistical control, despite the consistent finding that the cumulative effects of home background factors account for more than 60% of the variance in measures of student literacy performance (Rutter et aI., 1970; Thompson, 1985). The evidence for the mediating effects of home background influences appear to be particularly important. In a study of early reading achievement, the findings of Share et aI. (1983) indicate that the common practice of using a single index of socioeconomic status (SES) to measure home background severely underestimates the relationship between the home and educational achievement. Share et aI. show that whereas indices of socioeconomic status are associated positively with reading achievement, specific processes operating within the home (i.e., academic guidance, language models, levels of family literacy, parental participation and aspirations for the child) are more directly related to student achievement (Morgan and Lyon, 1979; Topping and Wolfendale, 1985; Winter, 1988). Quality home background influences are also important in the development of positive attitudes toward reading (Beach, 1985; Walberg and Tsai, 1985). When students are read to regularly by parents or other adults during their preschool years, such experiences are associated with subsequent positive attitudes toward reading, increased confidence and motivation to read, and are related to enhanced reading and writing skills (Bronfenbrenner, 1974; Spiegel, 1981; Stanovich, 1986). (For a review of the literature concerned with the mediating effects of students' home background and attitudinal factors on reading achievement, see Rowe, 1991). To merely examine the simple bivariate association between
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behavior and reading achievement and ignore the important mediating effects of student home background and attitude factors, in predictive or explanatory terms, is methodologically naive. Analytical Problems
Empirical research in this field has also been constrained by major analytical problems in both experimental and ex post facto designs. One of these lies in the uncritical use of first generation, univariate, and multivariate statistical techniques, based on the general linear model (Bibby, 1977; Rowe, 1989). A major shortcoming of many studies (particularly in the usual exploratory, data-driven mode) stems from the fact that hypotheses tested frequently attempt to link theoretical constructs that are not directly observable (i.e., those that specify relations among latent variables). Typically, researchers use either a single variable that is assumed to adequately measure its associated latent construct or "create" a latent variable by computing a "factor score" or "scale score" from a simple additive combination of two or more observed indicator variables, regardless of their individual or combined measurement properties. Such procedures increase the risk that variables so derived contain large amounts of measurement error, which when analysed via popular general linear model tehcniques, such as analysis of variance, multivariate analysis of variance, or regression (which assume that the observed variables are free from measurement error and that effects are unidirectional), yield coefficients that are often severely biased (Joreskog and Sorbom, 1979; Rowe, 1989). However, because many studies attempt to estimate' 'effect" relationships among latent constructs, the analytical methods used hitherto have been inadequate to provide answers to such questions. Moreover, these first generation techniques do not easily handle reciprocal effects among variables of interest. Such limitations suggest that traditional analytical techniques used in applied educational and psychiatric research require substantial revision. Specifically, it is timely for explanatory analyses to be undertaken that specify the directions and estimate the magnitudes of interdependent effects among critical variables. To this end, it is no longer sufficient merely to report a simple bivariate relationship between a given factor and a specified outcome (e.g., a correlation or regression coefficient). Rather, even at the risk of oversimplification, it is now necessary to develop explanatory models, based on substantive educational and clinical grounds. This approach to inquiry has been given special impetus from developments in the statistical theory of latent variables and the application of structural equation modeling (SEM) techniques (or covariance structure analysis) to data arising from epidemiological and psychosocial research questions (Bentler, 1980; Joreskog and Sorbom, 1979, 1989a; McDonald, 1978; Muthen, 1984). The use of such approaches, as illustrated in Part B, enables researchers to cast off the "shackles" imposed by mere exploratory "fishing expeditions" and to actually test substantive theory. The particular advantage of SEM procedures is that they employ confirmatory, model-testing tools that are based upon but
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go beyond conventional, data-driven, exploratory regression and FA techniques. To assist the reader who may be unfamiliar with SEM techniques, a relatively nontechnical overview of the LISREL model (Joreskog and Sorbom, 1989a) is provided in the Appendix to this paper. Concluding Comments
In a comprehensive review of the vast reading research literature, Calfee and Drum (1986) note: "Literacy is the foundation for lifelong learning; thus its importance in practice and in research" (p. 843). However, to synthesize findings from the expanding body of research specifically related to relationships between students' inattentive behaviors and their learning outcomes in general and reading achievement in particular is difficult. Such is the case, not only because of the plethora of relationships that have been found, but also because of both the range and type of methodologies that have been employed. From exploratory work in this domain, mounds of significant associations have been identified. Although there is clearly no lack of empirical evidence, the problem remains one of explaining the observed relationships in terms of direction and magnitude of effects. Moreover, since little is known about the relative salience of home background, affective and behavioral factors affecting students' reading achievement, and the extent to which these factors are in turn modified and changed by achievement, it is not known which of these factors (or combination of factors) might best be targeted to maximize outcomes. Thus, the task confronting researchers is the estimation of effects among alterable variables (Bloom, 1980) that may be subsequently modified in ways that raise the level of student achievement and, for example, relieve the plight of the inattentive child with reading difficulties. Such information has crucial implications for the formulation and implementation of both educational and clinical intervention strategies. Appendix An Overview of Structural Equation Modeling
The analytic technique of structural equation modeling (SEM) derives from the recognition that many social and behavioral processes can be thought of as "causal" processes operating among unobserved constructs (latent variables). SEM techniques provide powerful explanatory methods of testing the "fit" of data to substantive theoretical models. Essentially, SEM is the outcome of combining two well-established approaches to model fitting: the structural approach of multiple regression analysis and the measurement approach of factor analysis. The term structural equation modeling is often used synonymously with covariance structure modeling (Joreskog, 1981; Scott Long, 1983), because the basic objective is to estimate and explain the extent of covariation among a number of unobserved latent constructs, each of whose measurement properties has been taken into account via confirmtory factor analysis of the related observed variables. l. Am. A cad. Child Adolesc. Psychiatry, 31 :2, March 1992
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The aim here is to highlight some of the major features of SEM, particularly as they relate to the application of LISREL 7 (Joreskog and Sorbom, 1989b) used in the explanatory study reported in Part B. For helpful and relatively nontechnical overviews of SEM, the reader is referred to the paper by Anderson (1987) and the chapter by Ecob and Cuttance (1987). Ecob and Cuttance also provide an outline of the underlying statistical assumptions of SEM techniques. Cuttance (1987) provides a more detailed discussion of issues associated with the application of SEM techniques, including estimation procedures. The formulation of a structural model begins with a verbal statement that makes explicit the hypothesized relations among a set of variables as well as the sequence of effects thought to exist among them. In fact, most research questions and theories in epidemiological and psychosocial research can be stated in terms of a series of linear structural equations and may also be presented diagrammatically, as shown in Figure 1. For simplicity, SEM equations are expressed in matrix notation where, by convention, observed variables are denoted by Latin (x and y) and latent variables by Greek symbols (~, xi and 11, eta). In general, and assuming linearity, the stuctural relations among the latent variables are given by: 11
= Bll + r~ + ~,
(1)
whi ch specifies the relationship between latent endogenous (dependent) variables (11) and latent exogenous (independent) variables (~). ~ denotes structural prediction errors; B is a coefficient matrix of effect relations among the latent endogenous variables (11), and r is a coefficient matrix of relations between exogenous variables (~). The key assumption underlying Equation 1 is that the observed variables are "indicators" of underlying latent (unobserved) variables that give rise to the observations, and the model states relations between the latent variables 11 and ~. The measurement models are expressed as confirmatory factor-analytic equations that relate the latent variables ~ and 11 to the observed variables x and y, respectively, as follows: x y
= Ax~ + 0, = Ayll + E,
(2)
(3)
w here x is a vector of observed independent variables and y is a vector of observed dependent variables. Ax is a matrix of factor loadings for the observed x variables on the latent
sumption that one or more pairs of error terms have nonzero correlations (Berry, 1984). To illustrate the essential features of these two types of SEM specification, simple diagrammatic illustrations of their forms are shown in Figure 1. From Figure 1, estimates for the measurement submodels (ellipses to rectangles) are given by the lambda coefficients (Ax> A y ), whereas for the structural submodel among the latent variables (ellipses to ellipses), the effect estimates are given by the gamma (y) and beta (~) coefficients. To assist interpretation, it is common to present completely standardized solutions (i.e., true path-analytic representations). Thus, each variable may be standardized to unit variance so that the A.xS and the AyS can be interpreted as standardized factor loadings for latent variables on observed variables, the structural coefficients (y and ~) among the latent variables (path arrows) can be interpreted as regression coefficients, and the covariances (two-way arrows) as correlations. It is important to note that, in general, SEM techniques assume that observed variables are quantitative variables measured, at least approximately, on an interval scale, and whose distributions are approximately multinormal. In most psychosocial research applications, however, the observed variables are typically nonnormal andlor of mixed scale types: categorical, ordinal (Likert-type rating scales), and continuous. Under such circumstances, the use of ordinary product-moment correlations is not recommended. Instead, tetrachoric (categorical with categorical), polychoric (ordinal with ordinal), and polyserial correlations (continuous with ordinal) should be computed, and the correct asymptotic covariance matrix of such correlations should be analyzed by the method of weighted least squares. Failure to do otherwise can lead to gross errors in correlation estimates, distorted structural equation parameter estimates, and incorrect goodness-of-fit measures and standard errors (Huba and Harlow, 1987; Joreskog and Sorbom, 1988, 1989b). In developing the weighted least squares method available in LISREL 7, Joreskog and Sorbom have extended Browne's (1984) "asymptotically distribution free best GLS estimators" method for obtaining an appropriate weight matrix, correct parameter estimates, standard errors, and a fit statistic. ''The weight matrix required for such an analysis is the inverse of the estimated asymptotic covariance matrix W of the polychoric and polyserial correlations" (Joreskog and Sorbom, 1989b, p. 193). Finally, SEM techniques allow researchers to test the "fit" of the substantive model to the data. For example, the LISREL 7 program provides three summary measures of the overall fit of the model: 1) the chi-square goodness-of-fit test statistic; 2) the goodness of fit index (GFI); and 3) the adjusted goodness of fit index (AGFI-which adjusts the GFI value for the degrees of freedom in the model). Values close to unity for the GFI and AGFI indices indicate that the model accounts for most of the joint variances and covariances among observed variables in the model. Unlike the X2 statistic, the GFI and AGFI indices are independent of sample size and are relatively robust against departures from normality. It should be noted that since the X2 statistic is a direct function of sample size, almost any model is likely to be rejected if the sample size is sufficiently large 353
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Arecursive structura1 equation model
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(N > 500). Moreover, the root mean square residual error is a measure of the average of the residual variances and covariances, when the observed and predicted covariance matrices are compared.
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analysis of covariance structures. Br. J. Math. Stat. Psychology, 31:59-72. McGee, R. & Share, D. L. (1988), Attention deficit disorder-hyperactivity and academic failure: which comes first and what should be treated? J. Am. Acad. Child Adolesc. Psychiatry, 27:318-325. - - Williams, S., Share, D. L., Anderson, J. & Silva, P. A. (1986), The relationship between specific reading retardation, general reading backwardness and behavioral problems in a large sample of Dunedin boys: a longitudinal study from five to eleven years. J.
Child Psychol. Psychiatry, 27:597-610. - - - - Silva, P. A. (1987), A comparison of boys and girls with teacher-identified problems of inattention. J. Am. Acad. Child Ad-
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Psycho/., 49:226-238. Morgan, R. & Lyon, E. (1979), 'Paired Reading'-a preliminary report on a technique for parental tuition of reading-retarded children. J. Child Psycho/. Psychiatry, 20:151-160. Muthen, B. O. (1984), A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators. Psychometrika, 49:115-132. Otto, R. (1986), Teachers under Stress. Melbourne: Hill of Content. Prior, M. & Sanson, A. (1986), Attention deficit disorder with hyperactivity: a critique. J. Child Psycho/. Psychiatry, 27:307-319. Richman, N., Stevenson, J. & Graham, P. J. (1982), Pre-school to School: A Behavioral Study. London: Academic Press. Rowe, K. J. (1989), The commensurability of the general linear model in the context of educational and psychosocial research. Australian
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Educ. Psycho/., 61:19-35. Rowe, K. S. (1988), Synthetic food colourings and 'hyperactivity': a double blind crossover study. Aust. Paediatr. J., 24:143-147. Rutter, M. (1967), A children's behaviour questionnaire for completion by teachers: preliminary findings. J. Child PsychoI. Psychiatry, 8:1-11. - - (1985), Family and school influences on behavioural development. J. Child Psycho/. Psychiatry, 26:349-368. - - Tizard, J. & Whitmore, K. (1970), Education Health and Behaviour. London: Longmans. - - Shaffer, D. & Shepherd, M. (1975), A Multiaxial Classification ofChild Psychiatric Disorders: An elevation ofa Proposa/. Geneva: World Health Organization. - - Maughan, B., Mortimer, P., Ouston, J. & Smith, A. (1979),
Fifteen Thousand Hours: Secondary Schools and their Effects on Children. Somerset: Open Books. Sandoval, J. (1977), The measurement of the hyperactive syndrome in children. Review of Educational Research, 47:293-318. - - (1981), Format effects in two teacher rating scales of hyperacti vity. J. Abnorm. Child Psycho/., 9:203-218. Scarr, S. (1985), Constructing psychology: making facts and fables for our times. Am. Psycho/., 40:499-512. Scott Long, J. (1983), Covariance structure models: an introduction to LlSREL. Sage University Paper Series on Quantitative Applications in the Social Sciences, series no. 07-034. Beverly Hills and London: Sage Publications. Sellitz, C., Wrightsman, L. S. & Cook, S. W. (1976), Research Meth-
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J. Am. Acad. Child Adolesc. Psychiatry, 31:2, March 1992
AND
KATHERINE S. ROWE, M.B., B.S., F.R.A.C.P.
Abstract. Conclusions drawn from recent reviews of research concerned with relationships between child behavior disorders and learning difficulties highlight major methodological and analytical problems that have plagued empirical research in the field. This paper (Part A) identifies and discusses some of these problems, including those associated with the design and use of psychometric instruments, cutoff scores, and statistical modeling techniques. The paper provides a contextual rationale for the choice of methods and statistical models used in a large explanatory study of the relationship between inattentiveness in the classroom and reading achievement, reported in Part B. J. Am. Acad. Child Adolesc. Psychiatry, 1992, 31, 2:349-356. Key Words: psychometric instruments, negative nomenclature, factor analysis, cutoff scores, statistical modeling techniques.
In educational theory, research, and practice, the notion of attentiveness has long been associated with the key operational construct of active learning time or its equivalents, time-on-task, engaged learning time, or perseverence. This notion derives from the theoretical work of Carroll (1963, 1984), Cooley and Lohnes (1976), and Bloom (1976) who argue that although students may differ in their aptitude for learning, the different amounts of time needed to achieve a given level of proficiency are a direct function of the amount of attention or effort invested by an individual in a learning task. Findings from research on student learning in classroom settings provide strong suport for this view, indicated that attentiveness is directly related to achievement outcomes (Fisher et aI., 1980; Keeves, 1986; Lahaderne, 1968). This work suggests that attentiveness (defined as purposeful activity showing a sustained attention span, perseverence, concentration, and not easily distracted) is a crucial variable associated with student behavior at home and at school, through which, the effects of learning experiences and attitudes are mediated positively to influence learning outcomes. Evidence from studies investigating the impact of maladaptive student behaviors provides strong support for the importance of inattentiveness as a major variable having negative effects on student achievement. These studies reflect a persistent concern of teachers, parents, and mental health professionals of the extent to which students' malAccepted May 20, 1991. Mr. Rowe is Senior Policy Officer-Research, School Programs Division, Ministry of Education and Training, Victoria, Australia. Dr. Rowe is Lecturer and Physician, Department of Pediatrics, The University of Melbourne, Royal Children's Hospital, Parkville, Victoria, Australia. The comments and suggestions made on an earlier version of this paper by Glenn Rowley, Phillip Holmes-Smith, Trevor Williams, and two anonymous reviewers were most helpful. Financial support was provided by the Commonwealth Resource Agreement. Reprint request to: Ken Rowe. Senior Policy Officer-Research. School Programs Division, Ministry ofEducation and Training, Level 8 Rialto Towers, Box 4367, GPO Melbourne. Victoria 3001, Australia. 0890-8567/92/31 02-Q349$03.0010© 1992 by the American Academy of Child and Adolescent Psychiatry. J. Am. A cad. Child Adolesc. Psychiatry, 31:2, March 1992
adaptive behaviors in the classroom (attention deficit/hyperactivity and conduct disorders/aggression) adversely affect learning outcomes in general and reading achievement in particular (Cairns, 1985; Kyriacou & Roe, 1988; McGee et aI., 1986; Rowe, 1988; Sturge, 1982; Wheldall and Merrett, 1988). Students whose behaviors are regarded as inattentive, disruptive, or maladjusted have been shown to be at risk of poor educational attainment (Cantwell and Baker, 1991; Davie et aI., 1972; Rutter, 1985; Szaday, 1989). Moreover, in addition to the consequences for an individual, behavior problems in the classroom diminish educational opportunities for other students and contribute to teacher stress (Brenner, et aI., 1985; Otto, 1986; Wearing, 1989). Although students' classroom behaviors have been found to be partly dependent on factors, such as ethnicity (Dunkin and Doenau, 1985), social background (Kahl, 1985), gender (Bank, 1985) as well as cognitive and affective characteristics (Debus, 1985; Sinclair, 1985), findings from a growing number of correlational studies indicate stronger direct associations between poor attention and reading difficulties both in general student populations and in identified learning-disabled groups (Dykman and Ackerman, 1991; Jorm et aI., 1986; Levy et aI., 1987; Maughan et aI., 1985; McGee et aI., 1987; McGee et aI., 1989; McKinney, 1989; Stanton et aI., 1990; Stevenson et aI., 1985). For example, in their longitudinal study in Dunedin, New Zealand, McGee and coworkers consistently found poor reading achievement to be strongly related to high ratings of inattention. McGee and Share (1988) estimate that 80% of their sample of ll-yearold children identified with attention deficit disorder with hyperactivity (ADDH), as defined by DSM-III, had learning disabilities in reading and written language skills. Resulting in part, from a variety of methodological and analytical limitations in these studies, however, both the direction and magnitude of "effect" relationships is not clear. From interest in the relationship between students' reading disabilities and problem behaviors, Rutter et ai. (1970) proposed four alternative "causal" hypotheses, namely: 1) problem behavior leads to reading difficulties; 2) reading disability produces behavior problems; 3) both problem behavior and reading disability are produced by some third factor; and 4) all of these hypotheses could be partly true.
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In a review of the related research, McGee et aI. (1986) note: "All hypotheses have drawn support from the literature and the proposed mechanism underlying the relationship between reading disability and behavior disorder appear to be equally plausible" (p. 597). On the basis of a more recent and detailed review of the literature concerned specifically with the relationship between ADDH (or attention-deficit hyperactivty disorder in DSM-III-R) and failure to acquire literacy skills, McGee and Share (1988) conclude: "The evidence the authors have reviewed suggests that a substantial overlap exists between ADDH and learning difficulties and that, as yet, no unique pattern of cognitive or attentional deficits has been identified that can discriminate between these two types of disorder" (p. 322). (Fletcher et aI. 1991 make a similar observation.) Following Kinsbourne (1984), who argues that attentional problems are both "context" and "task" dependent, McGee and Share (1988) further conclude that "ADDH behaviors might best be considered as a disorder of conduct in the classroom, because the child with learning difficulties is excluded from much of the normal classroom activity" (p. 322). This view is consistent with the recent findings of Day and Peters (1989) who conclude that" ... learning-disabled children seem to be better characterized as 'inattentive in the classroom' ..." (p. 360). Despite the apparent simplicity of the Rutter et ai. (1970) hypotheses outlined above, the conclusions drawn from reviews of the related research highlight several methodological and analytical problems that have plagued empirical research in the field. This is especially the case for those attempts to address "causal" or "which comes first?" hypotheses (e.g., McGee and Share, 1988; McMichael, 1979). Because most of the evidence upon which such hypotheses are based derives from cross-sectional rather than longitudinal studies, it is difficult to determine either the nature or the direction of putative effect relationships. Moreover, findings from the few longitudinal studies that have been reported are not consistent (e.g., Chazan, 1985; Jorm et al., 1986; Maughan et aI., 1985; McGee et aI., 1986; Richman et aI., 1982; Stott, 1981). The purpose of this paper (Part A) is to provide a brief review of some major methodological and analytical problems endemic to research in child behavior disorders and learning difficulties. The rationale for this review is twofold: 1) to assist clinical and epidemiological researchers who may be struggling with such problems; and 2) to provide a substantive basis for presenting the methodology and findings from an explanatory study of the relationship between inattentiveness in the classroom and reading achievement for a sample of 5,000 students (age 5-14 years) drawn from a normal school population - reported in Part B (in this issue). Methodological Problems
Perhaps the most ubiquitous methodological problem faced by researchers is the attempt to identify the correlates unique to attentional problems on the one hand and learning difficulties on the other (McGee and Share, 1988). The lack of clear differentiation between the two domains on existing measures used to define them not only places the validity
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of diagnostic "categories" in question but imposes severe constraints on the interpretation of findings from related studies. The high level of overlap between learning difficulties and attentional problems is underscored by the lack of sensitivity and specificity attained by existing psychometric instruments and rating scales, suggesting that valid criteria for determining the existence of either disorder in childhood are, at best, dubious (Boyle and Jones, 1985; Hinshaw, 1987; Prior and Sanson, 1986; Ullmann et aI., 1985). Similarly, the consistent low levels of agreement between clinical diagnoses « 60%) is likewise too low to serve as a classification standard (Boyle and Jones, 1985; Rutter et aI., 1975). A related problem concerns the way in which diagnostic instruments are developed. Typical of such instruments is the use of items that focus exclusively on negative rather than positive behaviors (e.g., Achenback and Edelbrock, 1983; Conners, 1969, 1973; Quay and Peterson, unpublished manuscript, 1975; Rutter, 1967). Emphasis on negative nomenclature is at the expense of a more balanced assessment and increases the risk of a prejudicial searches for "pathology," regardless of its presence or absence; nor are such instruments independent of sociocultural differences (Yao et aI., 1988). For example, in a recent normative study of the Achenbach Child Behavior Checklist, Hensley (1988) found a consistent tendency by Australian parents to rate their child's behavior toward the negative ends of items, significantly more so than their North American counterparts. Sandoval (1977) has criticized the use ofrating scales exclusively employing negatively worded items on the grounds that they are highly susceptible to rater bias and response sets such as "reverse halo effects" or "reverse generosity errors" (Kerlinger, 1986; Sellitz et aI., 1976). In a comparative study of format effects in rating scales of hyperactivity, Sandoval (1981) has shown that for positively worded items, raters are more willing to use the extreme rating categories for a given item, thus increasing the dispersion and discrimination of the measurements. In contrast, an inspection of the marginal distributions for negatively worded items show that they tend to be highly skewed and leptokurtic. By far, the most popular means of defining and measuring emotional and behavioral disorders of childhood is via exploratory factor-analytic (FA) approaches to determine the underlying dimensionality of multiple-item rating inventories administered by parents, teachers, or clinicians. From Hinshaw's (1987) comprehensive review of the 60-FA studies published between 1970 and 1986, it is interesting to note that all used exploratory FA approaches, and that 56 (93%) used orthogonal methods of factor extraction and rotation (mostly principal components analysis or principal factor solutions, with varimax rotation). Such approaches are problematic on at least three grounds. First, in the case of exploratory (unrestricted) methods of FA, the factor solutions are arbitrary, data driven, hypothesis generating, and, invariably, result in theory conflation (1oreskog, 1981; Rowe, 1989). Second, orthogonal methods of factor extraction and rotation assume that the derived factors are uncorrelated or independent-by definition (Harman, 1976). Because all items are allowed to load on more than one factor, J. Am. Acad. Child Adolesc. Psychiatry, 31:2, March 1992
INATIENTIVENESS AND READING ACHIEVEMENT (PART A)
the resulting correlated error variance alone is sufficient to yield shared variance across factors. Although the development of unidimensional scales is highly desirable from a measurement perspective, given the considerable literature concerning the nonindependent and overlapping dimensions of child behavior and psychopathology, orthogonal FA methods are difficult to justify either substantively or empirically. Third, and perhaps most serious of all, such methods are invariably applied to item responses measured on dichotomous or 3- to 5-point Likert-type ordinal scales and rely on the computation of Pearson product-moment interitem correlation matrices-estimated by default in most omnibus computer packages. What is overlooked in such instances is that the assumptions underlying Pearson product-moment correlations (i.e., normal distribution and homogeneity of variance) are always violated. As a consequence of using Pearson product-moment estimates for dichotomous or ordinal variables, instead of the consistently less biased tetrachoric correlations (dichotomous with dichotomous) or polychoric correlations (ordinal with ordinal), substantial bias in the interitem correlations and subsequent factor parameters is unwittingly introduced (Carroll, 1961; Joreskog and Sorborn, 1979, 1988). Failure to take account of the measurement and distributional properties of response variables in factor analysis amounts to what Hendrickson and Jones (1987) refer to as "an undisciplined romp through a correlation matrix" (p. 105). Given the almost universal application of these procedures, it could be argued that current claims to substantive knowledge about dimensions of child psychopathology may be little more than the products of methodological and statistical artifact (Scarr, 1985). An additional problem in applied research relates to the widespread use of rating inventories for the purposes of classification and diagnosis. Because of the inherent complexity of learning disabilities and behavioral disorders in childhood, Ullmann et aI. (1985) argue that the common use of a single "cutoff" score on a rating scale to diagnose deviance is inappropriate and misleading (e.g., a score of 15 on Conners' 1973 Abbreviated Teacher Rating Scale to diagnose ADDH [Sprague et aI., 1974]). Although it is customary to select 2 standard deviations from the mean for these purposes, such selections are arbitrary and can be modified depending on whether one wishes to minimize false positives or false negatives (Loney and Milich, 1982). This approach is aptly illustrated by Szatmari et aI., (1989) in their review of 11 studies reporting prevalence rates of hyperactivity and ADDH. Four of these studies employed diagnostic' 'cutoff" scores of 1.0, 1.5, or 2.0 standard deviations from the mean, in the absence of substantive criteria for doing so, "resulting in the identification of different numbers and types of cases" (Szatmari et aI., 1989, p. 221). For example, reported prevalence rates for ADDH vary from less than 1% (Rutter et aI., 1970), 14.3% (Trites et aI., 1979) to as high as 20% (Shaywitz and Shaywitz, 1991), depending on the following: 1) the methods of data collection; 2) the sampling characteristics of the populations targeted; and 3) the arbitrary determination of deviance criteria. Variability in measurement and cutoff scores, together with sampling differences, leads to substantial differences in prevalence J. Am. Acad. Child Ado/esc. Psychiatry, 31:2, March 1992
estimates. In the context of predictive or explanatory research, there is little rational justification for identifying, a priori, a fixed proportion of the child population as ADDH, for example, particularly when such a dimension is more meaningfully viewed as a continuum, both in qualitative and quantitative terms. Despite the utility and obvious convenience of rating scales, especially for large-scale epidemilogical and survey research, the psychometric limitations endemic to their design, construction, and use seem to be largely unrecognized by most researchers. Of particular concern in many studies examining the relationship between student behavior and learning outcomes is the way in which major mediating variables of the "third kind" have been measured and treated (e.g., students' home background characteristics). In attempts to address Rutter et aI.'s (1970) Hypothesis 3 (mentioned earlier), the variables most often chosen to measure family background, for example, have typically not included direct measures of parental involvement or other family educational inputs. Whereas numerous studies have included surrogate measures of home background factors, the variables most often chosen are not measured directly. Instead, proxy measures, such as student self-report estimates of the number of books in the home and ordinal classifications of family social class or socioeconomic status are used (e.g., Davie et aI., 1972; Douglas, 1964; Fotheringham and Creal, 1980; Rutter et aI., 1979; Williams and Silva, 1985). Moreover, when such measures are used, it is mostly for group classification purposes (for subsequent comparative group analysis) or as a means of statistical control, despite the consistent finding that the cumulative effects of home background factors account for more than 60% of the variance in measures of student literacy performance (Rutter et aI., 1970; Thompson, 1985). The evidence for the mediating effects of home background influences appear to be particularly important. In a study of early reading achievement, the findings of Share et aI. (1983) indicate that the common practice of using a single index of socioeconomic status (SES) to measure home background severely underestimates the relationship between the home and educational achievement. Share et aI. show that whereas indices of socioeconomic status are associated positively with reading achievement, specific processes operating within the home (i.e., academic guidance, language models, levels of family literacy, parental participation and aspirations for the child) are more directly related to student achievement (Morgan and Lyon, 1979; Topping and Wolfendale, 1985; Winter, 1988). Quality home background influences are also important in the development of positive attitudes toward reading (Beach, 1985; Walberg and Tsai, 1985). When students are read to regularly by parents or other adults during their preschool years, such experiences are associated with subsequent positive attitudes toward reading, increased confidence and motivation to read, and are related to enhanced reading and writing skills (Bronfenbrenner, 1974; Spiegel, 1981; Stanovich, 1986). (For a review of the literature concerned with the mediating effects of students' home background and attitudinal factors on reading achievement, see Rowe, 1991). To merely examine the simple bivariate association between
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behavior and reading achievement and ignore the important mediating effects of student home background and attitude factors, in predictive or explanatory terms, is methodologically naive. Analytical Problems
Empirical research in this field has also been constrained by major analytical problems in both experimental and ex post facto designs. One of these lies in the uncritical use of first generation, univariate, and multivariate statistical techniques, based on the general linear model (Bibby, 1977; Rowe, 1989). A major shortcoming of many studies (particularly in the usual exploratory, data-driven mode) stems from the fact that hypotheses tested frequently attempt to link theoretical constructs that are not directly observable (i.e., those that specify relations among latent variables). Typically, researchers use either a single variable that is assumed to adequately measure its associated latent construct or "create" a latent variable by computing a "factor score" or "scale score" from a simple additive combination of two or more observed indicator variables, regardless of their individual or combined measurement properties. Such procedures increase the risk that variables so derived contain large amounts of measurement error, which when analysed via popular general linear model tehcniques, such as analysis of variance, multivariate analysis of variance, or regression (which assume that the observed variables are free from measurement error and that effects are unidirectional), yield coefficients that are often severely biased (Joreskog and Sorbom, 1979; Rowe, 1989). However, because many studies attempt to estimate' 'effect" relationships among latent constructs, the analytical methods used hitherto have been inadequate to provide answers to such questions. Moreover, these first generation techniques do not easily handle reciprocal effects among variables of interest. Such limitations suggest that traditional analytical techniques used in applied educational and psychiatric research require substantial revision. Specifically, it is timely for explanatory analyses to be undertaken that specify the directions and estimate the magnitudes of interdependent effects among critical variables. To this end, it is no longer sufficient merely to report a simple bivariate relationship between a given factor and a specified outcome (e.g., a correlation or regression coefficient). Rather, even at the risk of oversimplification, it is now necessary to develop explanatory models, based on substantive educational and clinical grounds. This approach to inquiry has been given special impetus from developments in the statistical theory of latent variables and the application of structural equation modeling (SEM) techniques (or covariance structure analysis) to data arising from epidemiological and psychosocial research questions (Bentler, 1980; Joreskog and Sorbom, 1979, 1989a; McDonald, 1978; Muthen, 1984). The use of such approaches, as illustrated in Part B, enables researchers to cast off the "shackles" imposed by mere exploratory "fishing expeditions" and to actually test substantive theory. The particular advantage of SEM procedures is that they employ confirmatory, model-testing tools that are based upon but
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go beyond conventional, data-driven, exploratory regression and FA techniques. To assist the reader who may be unfamiliar with SEM techniques, a relatively nontechnical overview of the LISREL model (Joreskog and Sorbom, 1989a) is provided in the Appendix to this paper. Concluding Comments
In a comprehensive review of the vast reading research literature, Calfee and Drum (1986) note: "Literacy is the foundation for lifelong learning; thus its importance in practice and in research" (p. 843). However, to synthesize findings from the expanding body of research specifically related to relationships between students' inattentive behaviors and their learning outcomes in general and reading achievement in particular is difficult. Such is the case, not only because of the plethora of relationships that have been found, but also because of both the range and type of methodologies that have been employed. From exploratory work in this domain, mounds of significant associations have been identified. Although there is clearly no lack of empirical evidence, the problem remains one of explaining the observed relationships in terms of direction and magnitude of effects. Moreover, since little is known about the relative salience of home background, affective and behavioral factors affecting students' reading achievement, and the extent to which these factors are in turn modified and changed by achievement, it is not known which of these factors (or combination of factors) might best be targeted to maximize outcomes. Thus, the task confronting researchers is the estimation of effects among alterable variables (Bloom, 1980) that may be subsequently modified in ways that raise the level of student achievement and, for example, relieve the plight of the inattentive child with reading difficulties. Such information has crucial implications for the formulation and implementation of both educational and clinical intervention strategies. Appendix An Overview of Structural Equation Modeling
The analytic technique of structural equation modeling (SEM) derives from the recognition that many social and behavioral processes can be thought of as "causal" processes operating among unobserved constructs (latent variables). SEM techniques provide powerful explanatory methods of testing the "fit" of data to substantive theoretical models. Essentially, SEM is the outcome of combining two well-established approaches to model fitting: the structural approach of multiple regression analysis and the measurement approach of factor analysis. The term structural equation modeling is often used synonymously with covariance structure modeling (Joreskog, 1981; Scott Long, 1983), because the basic objective is to estimate and explain the extent of covariation among a number of unobserved latent constructs, each of whose measurement properties has been taken into account via confirmtory factor analysis of the related observed variables. l. Am. A cad. Child Adolesc. Psychiatry, 31 :2, March 1992
INATIENTIVENESS AND READING ACHIEVEMENT (PART A)
The aim here is to highlight some of the major features of SEM, particularly as they relate to the application of LISREL 7 (Joreskog and Sorbom, 1989b) used in the explanatory study reported in Part B. For helpful and relatively nontechnical overviews of SEM, the reader is referred to the paper by Anderson (1987) and the chapter by Ecob and Cuttance (1987). Ecob and Cuttance also provide an outline of the underlying statistical assumptions of SEM techniques. Cuttance (1987) provides a more detailed discussion of issues associated with the application of SEM techniques, including estimation procedures. The formulation of a structural model begins with a verbal statement that makes explicit the hypothesized relations among a set of variables as well as the sequence of effects thought to exist among them. In fact, most research questions and theories in epidemiological and psychosocial research can be stated in terms of a series of linear structural equations and may also be presented diagrammatically, as shown in Figure 1. For simplicity, SEM equations are expressed in matrix notation where, by convention, observed variables are denoted by Latin (x and y) and latent variables by Greek symbols (~, xi and 11, eta). In general, and assuming linearity, the stuctural relations among the latent variables are given by: 11
= Bll + r~ + ~,
(1)
whi ch specifies the relationship between latent endogenous (dependent) variables (11) and latent exogenous (independent) variables (~). ~ denotes structural prediction errors; B is a coefficient matrix of effect relations among the latent endogenous variables (11), and r is a coefficient matrix of relations between exogenous variables (~). The key assumption underlying Equation 1 is that the observed variables are "indicators" of underlying latent (unobserved) variables that give rise to the observations, and the model states relations between the latent variables 11 and ~. The measurement models are expressed as confirmatory factor-analytic equations that relate the latent variables ~ and 11 to the observed variables x and y, respectively, as follows: x y
= Ax~ + 0, = Ayll + E,
(2)
(3)
w here x is a vector of observed independent variables and y is a vector of observed dependent variables. Ax is a matrix of factor loadings for the observed x variables on the latent
sumption that one or more pairs of error terms have nonzero correlations (Berry, 1984). To illustrate the essential features of these two types of SEM specification, simple diagrammatic illustrations of their forms are shown in Figure 1. From Figure 1, estimates for the measurement submodels (ellipses to rectangles) are given by the lambda coefficients (Ax> A y ), whereas for the structural submodel among the latent variables (ellipses to ellipses), the effect estimates are given by the gamma (y) and beta (~) coefficients. To assist interpretation, it is common to present completely standardized solutions (i.e., true path-analytic representations). Thus, each variable may be standardized to unit variance so that the A.xS and the AyS can be interpreted as standardized factor loadings for latent variables on observed variables, the structural coefficients (y and ~) among the latent variables (path arrows) can be interpreted as regression coefficients, and the covariances (two-way arrows) as correlations. It is important to note that, in general, SEM techniques assume that observed variables are quantitative variables measured, at least approximately, on an interval scale, and whose distributions are approximately multinormal. In most psychosocial research applications, however, the observed variables are typically nonnormal andlor of mixed scale types: categorical, ordinal (Likert-type rating scales), and continuous. Under such circumstances, the use of ordinary product-moment correlations is not recommended. Instead, tetrachoric (categorical with categorical), polychoric (ordinal with ordinal), and polyserial correlations (continuous with ordinal) should be computed, and the correct asymptotic covariance matrix of such correlations should be analyzed by the method of weighted least squares. Failure to do otherwise can lead to gross errors in correlation estimates, distorted structural equation parameter estimates, and incorrect goodness-of-fit measures and standard errors (Huba and Harlow, 1987; Joreskog and Sorbom, 1988, 1989b). In developing the weighted least squares method available in LISREL 7, Joreskog and Sorbom have extended Browne's (1984) "asymptotically distribution free best GLS estimators" method for obtaining an appropriate weight matrix, correct parameter estimates, standard errors, and a fit statistic. ''The weight matrix required for such an analysis is the inverse of the estimated asymptotic covariance matrix W of the polychoric and polyserial correlations" (Joreskog and Sorbom, 1989b, p. 193). Finally, SEM techniques allow researchers to test the "fit" of the substantive model to the data. For example, the LISREL 7 program provides three summary measures of the overall fit of the model: 1) the chi-square goodness-of-fit test statistic; 2) the goodness of fit index (GFI); and 3) the adjusted goodness of fit index (AGFI-which adjusts the GFI value for the degrees of freedom in the model). Values close to unity for the GFI and AGFI indices indicate that the model accounts for most of the joint variances and covariances among observed variables in the model. Unlike the X2 statistic, the GFI and AGFI indices are independent of sample size and are relatively robust against departures from normality. It should be noted that since the X2 statistic is a direct function of sample size, almost any model is likely to be rejected if the sample size is sufficiently large 353
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Arecursive structura1 equation model
~1
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FIG. 1. Diagrammatic representation of structural equation models.
(N > 500). Moreover, the root mean square residual error is a measure of the average of the residual variances and covariances, when the observed and predicted covariance matrices are compared.
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