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STATISTICAL CORNER |
From the Division of Psychology in Education, Arizona State University, Tempe, Arizona.
Address correspondence and reprint requests to Samuel B. Green, PhD, Division of Psychology in Education, Box 870611, Arizona State University, Tempe, AZ 85287-0611. E-mail: samgreen{at}asu.edu
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMULTIVARIATE ANALYSIS OF...DISCRIMINANT ANALYSIS AND ITS...DIFFERENCES IN FACTOR MEANSCONCLUSIONSNOTESAPPENDIXREFERENCES |
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Key Words: structural equation modeling • MANOVA • structured means modeling • MIMIC models
Abbreviations: ANOVA = analysis of variance; MANOVA = multivariate analysis of variance; SEM = structural equation modeling; MIMIC = multiple-indicator, multiple-cause; RMSEA = root mean square error of approximation.
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INTRODUCTION |
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In all likelihood, researchers should consider, in most cases, multivariate methods that take into account not only the variances of the dependent variables, but also the covariances among the dependent variables. Multivariate techniques include MANOVA, discriminant analysis, and SEM analyses of factor means. Understanding the statistical models underlying the various group-comparison methods is crucial for making appropriate decisions. Accordingly, we present the statistical models for these methods using path diagrams within a structural equation modeling framework. With a better understanding of the statistical models associated with alternative methods, researchers may choose to use more frequently SEM analyses of factor means.
We developed a research scenario concerning coping with asthma to illustrate the various group comparison methods:
The purpose of the research is to assess whether adolescents with chronic asthma respond differently to their asthmatic attacks depending on whether they come from families with both parents present in the home (married parents) versus families in which the parents have gone through a divorce in the last year (recently divorced parents). Multiple measures are obtained on 100 asthmatic adolescents with married parents and 100 asthmatic adolescents with recently divorced parents. The focus of our example is on four scales from a self-report measure of how adolescents cope with asthmatic attacks. The four scales include two avoidant strategies—disengagement and denial—and two active coping strategies—problem-focused coping and seeking social support. Analyses are conducted to assess whether adolescents from the two types of families differ on their use of coping strategies. Other analyses could potentially be conducted to address process-oriented hypotheses (e.g., mediation).
The data created for this scenario are in the Appendix of this article.
In our presentation of statistical models, we first consider traditional multivariate techniques of MANOVA and discriminant analysis. We then present alternative SEM methods that assess differences in factor means. We illustrate all methods using our example data. We argue later in this article that our example implies a research question best addressed by assessing differences in latent means rather than MANOVA and discriminant analysis. Nevertheless, when presented with a general question of mean differences on a set of correlated measures, it is common for researchers to apply MANOVA (2,3). By presenting the results for all methods using the same data, we can demonstrate how the results obtained from MANOVA and discriminant analysis may differ markedly and can be less informative than those obtained from SEM-based tests of differences in latent means.
Two types of structural equation models are presented to analyze the difference in means: multiple-groups models and multiple-indicator, multiple-cause (MIMIC) models. The multiple-groups models may be conceptualized as analogous to ANOVA models, whereas MIMIC models may be thought to be analogous to regression models. For both multiple-groups and MIMIC models, we conduct analyses by comparing an initial model (MI) with a constrained model (MC). The constraints imposed on the initial model to produce the constrained model are that the population means are equal across groups, that is, the null hypothesis of interest. To the extent that the constrained model fits more poorly than the initial model, the constraints are judged to be inappropriate for the data set, and the null hypothesis that the population means are equal is rejected.
Our article is not a how-to guide for conducting tests of differences in multivariate means so the fine points of the analyses are not presented; rather, we provide references in our article to articles and chapters that discuss procedural details. For example, we do not illustrate preliminary analyses to the multivariate methods to assess violations of assumptions; however, we strongly encourage that these be conducted (4,5).
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MULTIVARIATE ANALYSIS OF VARIANCE |
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MANOVA: Hypothesis and Statistical Models
MANOVA evaluates the null hypothesis that the population means on the dependent variables are equal for all groups. The null hypothesis is tested by comparing initial and constrained models, and these models can be represented using path diagrams. The path diagrams for the asthmatic adolescent example are presented in Figure 1. In these diagrams, like with other diagrams in this article, we present estimates only for parameters relevant to determining the means on the variables.
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For the initial model, the means for the measures are not constrained but allowed to differ between groups. As shown in the path diagram for the initial model, the coping scales (in the rectangles) are a function of two quantities: the means for these scales for the groups (the paths between 
and the coping scales) and the errors (the paths between the Es and the coping scales). The values on the paths between and the coping scales are the coping scale means for the asthmatics with married parents (group 0) and those with recently divorced parents (group 1). The errors for the coping scales have variances (double-headed arrows for each error) as well as covariance (the double-headed arrows between different errors). As dictated by the assumptions for MANOVA, the variances and covariances of the errors are equated across groups (not shown in the diagram).
The constrained model is comparable to the initial model except the means on the measures are constrained to be equal between groups. Accordingly, the coping scales are predicted on the basis of the grand means for the coping scales rather than the group means; that is, the values on the paths are the coping scale means for all 200 adolescents combined.
MANOVA Using Wilks’ Lambda and SEM
By comparing the fit of the two models, we assess whether the means should or should not be constrained to be equal between groups, that is, the difference between the two models. Within a traditional MANOVA framework, the relative fit of the two models is assessed using Wilks’ 
(6,7). To the extent Wilks’ is less than 1 and approaches 0, the initial model demonstrates improvement in fit, and the means on the dependent variables should be allowed to differ between groups. For our example, the Wilks’ of 0.96 is somewhat less than 1, suggesting some improvement in fit by allowing the group means to differ. Given the sample size, is Wilks’ sufficiently small to warrant rejecting the null hypothesis and to conclude that the means are different in the population? We can transform the Wilks’ to a 
2 statistic if we can assume the population distributions for errors are multivariate normally distributed for each group, the covariance matrices among the errors for these distributions are equal, and independent and random sampling.1 The 2 test is nonsignificant at the 0.05 level for our example (2 (4) = 8.60, p = .07). Accordingly, we cannot reject the hypothesis that the population means on the four coping scales are equivalent for asthmatics with married parents and asthmatics with recently divorced parents.
Based on the same models and assumptions (and, thus, the same path diagrams), the test of the null hypothesis that the population means are equal across groups can be conducted using an SEM multiple-groups approach. With SEM, lack of fit of a model may be assessed with a 2 statistic. For our example, the lack of fit was somewhat worse for the constrained model (2 (14) = 20.45, p = .13) than it was for the initial model (2 (10) = 11.45, p = .32). To assess the null hypothesis, a 2 difference statistic is computed: the 2 for the initial model is subtracted from the 2 for the constrained model, whereas the degrees of freedom (df) for the initial model is subtracted from the df for the constrained model. The 2 difference test is nonsignificant (Difference2 (4) = 8.60, p = .07).
The 2 values are the same to two decimal places for the traditional MANOVA and SEM approaches. We have found the traditional MANOVA test and the SEM test to yield similar results across data sets that we have analyzed. In addition, we would not expect the two tests to differ dramatically based on their statistical properties. However, we have seen no formal study investigating the comparability of the two approaches.
The SEM multiple-groups approach offers some advantage over traditional MANOVA in dealing with the restriction of equality of error variances and covariances and the assumption of normality. The 2 difference test of the hypothesis of equality of population means should not proceed if the initial model fits inadequately. The initial model fits poorly if the equality constraints of error variances and covariances are inconsistent with the data. In addition, the 2 test can be inflated if the scores on the dependent variables are nonnormally distributed. However, the requirement of equality of error variances and covariances and normality can be relaxed within an SEM approach.
For our example, the initial model fit well (2 (10) = 11.45, p = .32; CFI of 0.98, range, 0–1 with 1 being best fit) and root mean square error of approximation (RMSEA) of 0.03 (range, 0 to 
with 0 being best).2 Consequently, no further steps are necessary, and the fit of the initial model can be compared with the fit of the constrained model, as previously presented. However, if the fit were poor, we would be required to conduct additional analyses to deal with violation of the equality constraint of error variances and covariances or the normality assumption. Although the initial model fit well, we illustrate these additional analyses.
The initial and constrained models may be respecified such that some or all of the error variances and covariances are allowed to differ. When we permitted all error variances and covariances to differ for our example, the 2 difference test yielded a value similar to the one obtained when they were restricted to be equal: (Difference2 (4)= MI2 (4) – MC2 (0) = 8.59 – 0 = 8.59, p = .07). In addition, Satorra-Bentler rescaled 2 statistics could be computed to correct for a lack of normality based on the distributional characteristics of the sample data (5,8). When we allowed for nonnormal data (as well as unequal error variances and covariances), our 2 difference test based on the Satorra-Bentler 2 statistic was comparable to the previous one (Difference2 (4) = 8.59, p = .07).3 By conducting tests that have less restrictive assumptions like the Satorra-Bentler 2 statistic, we may feel more confident about our results.
Evaluation of Equality of Means for an Index
We previously stated the MANOVA null hypothesis very simplistically: the population means on the dependent variables are the same across groups. However, if this equality holds, the population means on any linear combination of dependent variables must also be equal for all groups. Consequently, the null hypothesis for MANOVA may be stated in a more general form: the population means on any linear combination of dependent variables are equal for all groups. The general form encompasses the more simplistic form in that any one dependent variable is a linear combination of the dependent variables: the focal dependent variable plus zero times each of the other dependent variables.
The statistical complexity introduced by a MANOVA over and above ANOVA is that MANOVA assesses equality of means of linear combination of dependent variables. In this context, two points are worth noting: a) a linear combination with more than one nonzero weight—which we refer to as an index—is created presumably to represent a concept that is broader than any one of the dependent variables making it up. Researchers need not compute a MANOVA if they are not potentially interested in these indices and instead might conduct ANOVAs on the separate dependent variables (10). They could control for type I error across these tests using a method such as Holm’s sequential Bonferroni approach (1). b) The null hypothesis for MANOVA does not specify any particular linear combination of variables, and therefore MANOVA considers differences in means on all possible linear combinations. If a researcher is interested in a particular linear combination, they should conduct an ANOVA on that linear combination to maximize power (11).
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DISCRIMINANT ANALYSIS AND ITS RELATIONSHIP TO MANOVA |
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The primary purpose of discriminant analysis is to define discriminant functions—linear combinations of scale scores—that differentiate groups. The first discriminant function maximizes differences between groups; each subsequent discriminant function maximizes differences between groups with the restriction that it is uncorrelated with all previous discriminant functions. The number of discriminant functions is equal to the number of groups minus one or the number of scales, whichever is smaller. To interpret the results, discriminant functions are named, and groups are described as differing on the basis of these functions. By interpreting group differences on the discriminant functions rather than individual scales, researchers are taking into account both the variances and covariances among the scales. For simplicity and consistency with our example, we restrict our focus to a two-group analysis, which yields a single discriminant function.
The null hypothesis that is evaluated with a two-group discriminant analysis is that the weights associated with the scales to form a discriminant function are all equal to zero in the population. This null hypothesis implies that the population means are equivalent between groups for all scales; that is, the null hypothesis for discriminant analysis is mathematically equivalent to the null hypothesis for MANOVA. In addition, the assumptions for discriminant analysis are identical to those for MANOVA. Accordingly, it is not surprising that the traditional tests for discriminant analysis and MANOVA are equivalent.
Next, we demonstrate the equivalence of the tests for MANOVA and two-group discriminant analysis with SEM MIMIC models. We cannot use multiple-groups models as we did in the previous section in that only MIMIC models have been presented in the literature for conducting discriminant analyses (15,16).4 After showing the equivalence of the tests, we discuss the interpretation of discriminant functions.
MIMIC Models for MANOVA and Discriminant Analysis
In Figure 2, we show the initial and constrained MIMIC models for MANOVA for our example. MIMIC models are not specified for separate groups but rather for the total sample. With these models, a dummy variable is added to the data set. For our data set, we specified a dummy variable with 0 for asthmatics with married parents and 1 for asthmatics with recently divorced parents. The initial model indicates that the coping scales (V1 through V4) are a function of two quantities: the dummy variable (the paths between Vdummy and the coping scales) and error (the paths between the Es and the coping scales). The values on the paths between the dummy variable and the coping scales are the differences in scale means between the two asthmatic groups. The initial model must fit perfectly (i.e., it is a just-identified model) (2 (0) = 0, p = 1.00). The constrained model is identical to the initial model except it imposes on the model the constraint implied by the null hypothesis. Accordingly, the values on the paths between the dummy variable and the coping scales are constrained to 0 to indicate that the group means do not differ on the coping scales. The lack of fit is worse for the constrained model (2 (4) = 8.73, p = .07). To assess the null hypothesis, a 2 difference statistic is computed by subtracting the 2 for the initial model from the 2 for the constrained model (Difference2 (4) = 8.73, p = .07). The 2 statistic with the MIMIC models differs slightly from those for the traditional MANOVA and for the SEM multiple-groups approach.
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In Figure 3, we show the initial and constrained MIMIC models for discriminant analysis. The initial model indicates that a factor (F1) is a linear combination of the coping scales. The values on the arrows between the coping scales and the factor are the weights applied to the coping scales to create the factor. The factor is not a latent variable, but rather is directly computable on the basis of the coping scales and their weights. The dummy variable is a function of the linear combination of the coping scales (i.e., F1) plus error.5 This initial model must fit perfectly (i.e., a just-identified model) (2 (0) = 0, p = 1.00). The constrained model is identical to the initial model except it imposes on the model the constraint implied by the null hypothesis. The null hypothesis for a two-group discriminant analysis is that the weights associated with the scales to form the linear combination are all equal to zero. Accordingly, the values on the paths between the coping scales and the factor are constrained to 0, and the result is some lack of fit (2 (4) = 8.73, p = .07). To assess the null hypothesis, we compute a 2 difference statistic (Difference2 (4) = 8.73, p = .07). The 2 statistic for the MANOVA and discriminant analyses using the MIMIC model must be identical for all two-group cases.6
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Interpretation of Discriminant Functions
Discriminant functions are interpreted by examining the pattern (standardized) and structure coefficients (7,17).7 For our example, we determined these using the procedure CANDISC in SAS, but these coefficients are essentially identical to those that can be calculated using an SEM MIMIC model. The pattern coefficients are the standardized discriminant function weights, whereas the structure coefficients are the correlations between the scales and the discriminant functions. As shown on the left side of Table 1 (the remaining part of the table is discussed later), the discriminant function for our example appears to be a bipolar dimension with avoidant strategies (V1 and V2) on one pole and active coping strategies (V3 and V4) on the other pole. The group centroids (not presented) indicate that the adolescents with recently divorced parents tend to score, on average, higher on this bipolar dimension, implying that they are more likely to use avoidant strategies rather than active coping strategies.
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Application of MANOVA and Discriminant Analyses
MANOVA and discriminant analysis are appropriate analyses if researchers are interested in between-group differences on post hoc indices of dependent variables. For example, a researcher may be interested in comparing the physical health of two groups of 30- to 35-year-old women who have no chronic or acute medical disorders. One group might include women who have no children, whereas a second group might include women who have two or more children between the ages of 3 and 10 years of age. Measures of physical health might be blood pressure, body mass index, self-reported fatigue level, days with colds/flu in the last year, and number of trips for medical treatment in the last year. The researcher is not interested in defining a priori an index of physical health based on these measures but prefers the index to be determined empirically.
As illustrated in the physical health example, MANOVA and discriminant analysis ideally are applied when the following conditions are met:
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DIFFERENCES IN FACTOR MEANS |
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Evaluating Differences in Factor Means Using a Multiple-Groups Model
For simplicity, we divide into two parts the multiple-groups statistical models for assessing differences in factor means: a) the measurement structure that specifies the relationship between measures and underlying factors and b) the means structure that defines the means for factors and their measures. After defining these structures, we consider necessary equality constraints on model parameters and then illustrate SEM models using our asthmatic adolescent example. Finally, we consider testing differences in factor means using a MIMIC model.
Measurement Structure
A test of differences in factor means requires researchers to specify a measurement structure. In other words, researchers must specify that measured variables (i.e., dependent variables) are representations of a priori defined latent factors (i.e., constructs). By representations, we mean that constructs underlie the measures. In addition, the dependent variables are not assumed to be measured without error. Multiple-groups path diagrams of the initial and constrained models for testing differences in factor means are presented in Figure 4 for the asthmatic adolescent example. For these models, we hypothesize the construct avoidant coping (F1) to underlie disengagement (V1) and denial (V2), and the construct active coping (F2) to underlie problem-focused coping (V3) and seeking social support (V4). Because the measures are imperfect representations of the factors, measured variables are a function of not only the factors, but also errors, as indicated by the arrows in the diagram. In addition, because we postulate that avoidant and active coping strategies (F1 and F2) are correlated, we include a double-headed curved arrow in the diagram between F1 and F2 to estimate the covariance between the two factors. Differences in factor means should not be evaluated unless this a priori model appears appropriate based on model fit.
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Before discussing the remaining aspects of the initial and constrained models, we want to emphasize that these models are quite different from the ones discussed for MANOVA and discriminant analysis. The primary ways the statistical models for differences in factor means differ from the models for MANOVA/discriminant analyses are as follows:
Mean Structure
The focus of our analyses for the models in Figure 4 is on the factor means, that is, the arrows between and the factors. In particular, we want to evaluate whether the population factor means are equal across groups by comparing the initial model in which the factor means are allowed to vary with the constrained model in which the factor means are constrained to be equal between groups. Given the abstract nature of factors, we cannot determine the factor means for any one group in an absolute sense, but only relative to the means in other groups. Accordingly, we arbitrarily fix the means for the factors in group 0 to the value of 0. For the initial model, the factor means in group 1 are allowed to be estimated and thus vary from the factor means in group 0. In contrast, for the constrained model, the factor means in group 1 are fixed to 0 so that the factor means are the same across groups. By comparing the initial and constrained models, we assess whether constraining the factor means to be equivalent across groups is consistent with the data.
The part of the mean structure that we have not discussed yet is the intercept at the bottom of each of the models. Intercepts (i.e., the arrows from the intercept to the measured variables) are necessary to allow for different means for the various measures of a factor. To understand this better, let us consider the means for V3 and V4 for group 0 in the initial model. These means can be determined by tracing the paths between and the Vs and between the intercept and the Vs. For V3 and V4,
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Because the factor means in group 0 are set to zero, the means for V3 and V4 would have to be the same if we failed to include intercepts. By including intercepts, we allow these measured variables to have different means:
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Necessary Equality Constraints
For both the initial and constrained models, we have constrained all factor loadings (the paths between the factors and the measured variables) and all intercepts to be equivalent across groups. These constraints may be appropriate for some data sets and inappropriate for others. In particular, the factor loadings for a measured variable would differ between groups if the degree that the measured variable is saturated with the factor differs. Also, the intercepts for a measured variable would differ between groups if the mean of a measure is affected by sources other than the factor specified in the model and the effect of these sources differs between groups (i.e., the measure as an indicator of the factor is biased with respect to a particular group). To estimate and test differences in factor means, all loadings and intercepts do not have to be constrained to be equal across groups; only the loadings and intercepts for one variable must be constrained for each factor (23). This makes sense intuitively in that it would be difficult to determine differences in factor means—the focus of the model comparisons—if all measures that assess that factor have different relationships with it across groups. As discussed in the literature, it can be problematic when fitting these models to decide what factor loadings and intercepts should and should not be constrained to be equal across groups (22,24).
Example for Multiple-Groups Model
For our example, the initial model fits the data extremely well (2 (6) = 2.17, p = .90, CFI of 1.0, and an RMSEA of 0.00). Accordingly, no changes in model were deemed necessary. If fit were inadequate, we could have considered a variety of changes, including allowing differences between groups for the factor loadings on measures, allowing differences between groups for intercepts of measures or, more basically, altering the relationships between factors and measures. See Thompson and Green (22) for a detailed discussion concerning decisions about model specification for testing differences in factor means.
Given the adequacy of the initial model, its fit may be compared with the fit of the constrained model: (Difference2 (2) = 8.03, p = .02). These results suggest the hypothesis that the two factor means are equivalent across the two groups in the population can be rejected. Tests of differences in means for individual factors were significant: (Difference2 (1) = 4.75, p = .03) for F1 and (Difference2 (1) = 5.84, p = .02) for F2. It may be unnecessary to evaluate whether the means on the two factors are simultaneously equal across groups before testing differences in means on the separate factors as we have done. The test of the omnibus hypothesis may be considered unnecessary in that we planned a priori to assess mean differences on each of these two factors.
We computed effect size (ES) statistics for each factor mean by dividing the difference in factor means for the groups by the within-group standard deviation combined across groups (25): (ESF1 = .39) and (ESF2 = –.39). Based on Cohen’s guidelines for size of effects for measured variables, these effects would be classified as small to medium (26). However, because the computed effects for our example are on factors that exclude measurement error, we judge these effects to be relatively small. Given the results, we conclude that adolescents who have recently divorced parents show some tendency to use more avoidant strategies and less problem-focused coping strategies than adolescents who have married parents.
Evaluating Differences in Factor Means Using a MIMIC Model
Factor mean differences could also be conducted using a MIMIC model as shown in Figure 5. The results for this approach are similar to those for the multiple-groups analysis. The initial model specifies that the avoidant and active coping factors are affected by the parental status variable, a dummy-coded variable. This model fits extremely well (2 (3) = 0.96, p = .81, CFI of 1.0, and an RMSEA of 0.00). Accordingly, the initial model can now be compared with the constrained model, which fixes the effects of the dummy-coded variable on the factors to zero. The differential fit of the model indicates the null hypothesis that the factor means are equal to zero in the population should be rejected (Difference2(2) = 8.25, p = .02). The differences in means for both factors were statistically different for the two groups: (2 (1) = 4.95, p = .03) for F1 and (2 (1) = 5.94, p = .01) for F2. The effect size statistics were ESF1 = .40 and ESF2 = –.39.
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A disadvantage of the MIMIC approach is that it is not as flexible as the multiple-groups method. If the fit of the initial MIMIC model is inadequate, we are more restricted in how we might modify the model. For example, the MIMIC model does not allow for differences in factor loadings across groups. Consequently, the MIMIC approach might have to be abandoned for the multiple-groups approach. On the other hand, the simplicity of the MIMIC approach offers some advantages. First, given both the MIMIC and multiple-groups approaches are applicable, the MIMIC approach is likely to be slightly more powerful (3). Second, it is rather straightforward to adapt the MIMIC approach for more complex designs such as analysis of covariance and factorial analysis of variance.
Multiple Testing in the Assessment of Differences in Factor Means
A few questions are likely to arise when assessing differences of factor means using either the multiple-groups or the MIMIC approach: a) Is it necessary to control for type I error when conducting tests on multiple factors (for our example, avoidant and active-coping factors)? Tests involving differences in factor means should not be conducted unless the model relating the factors to the measures is meaningful and supported empirically. Thus, the results of the SEM analyses must be consistent with the conclusion that factors represent conceptually distinct constructs. In addition, researchers are likely to have focused research questions about these constructs. Under these conditions, we would want to maximize power and not control for type I error across factors. However, control for type I error may be advisable across factors if a factor model is developed post hoc to obtain adequate fit, especially if the number of factors is large. b) With more than two groups, is it necessary to control for type I error when conducting pairwise comparisons for any one factor? Yes, in that the same logic applies for multiple comparisons in this context as it does for multiple comparisons in ANOVA. The modified Bonferroni methods proposed by Shaffer (27) are simple but relatively powerful approaches for controlling for type I error across pairwise comparisons. c) Finally, in addition to tests of factor means, is it required to assess differences in means on the separate measures to determine which measures are most sensitive to group distinctions? If the structural equation model is appropriate, testing of individual measures is unnecessary. The most sensitive measure is the one most saturated by the factor as assessed by the standardized factor loadings and the standardized paths between the errors and the measures.
Using MANOVA/Discriminant Analysis as a Substitute for Differences in Factor Means
The results from the asthmatic adolescent example make it clear that problems can arise if MANOVA/discriminant analysis is applied when tests of differences in factor means are appropriate. As shown in Table 1, based on the results for differences in factor means, separate conclusions should be made about mean differences in avoidant coping and active coping. In contrast, based on the discriminant analysis results, a single conclusion is made about mean differences on a bipolar dimension with avoidant coping on one pole and active coping on the other pole. With discriminant analysis, the number of dimensions (i.e., discriminant functions) is limited by the number of groups minus one or the number of measured variables, whichever is smaller. The results of discriminant analysis and factor mean differences are likely to differ in many applications. See Cole et al. (2) and Green and Thompson (16) for a more detailed presentation of theses issues.
In addition, the MANOVA/discriminant analysis was nonsignificant, whereas the SEM analysis of factor mean differences was significant. Hancock et al. (3) present results from a Monte Carlo study that suggest that applying incorrectly MANOVA for SEM analysis of factor mean differences results in a loss of power.
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CONCLUSIONS |
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If a series of measures are believed to form an index, then MANOVA/discriminant analysis is an appropriate method. In this case, the measures, in combination, are not thought to have underlying constructs, but rather come together to produce some meaningful conglomerate index. An example might be physical health. On the other hand, if the measures have underlying constructs, then it is important to represent these constructs formally in the statistical model so they can be assessed. Structural equation modeling is a flexible methodology that can be used to assess all these models and other models, too, but is particularly useful for assessing factor mean differences.
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APPENDIX |
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NOTES |
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2 test can be used as the test statistic for multivariate analysis of variance (MANOVA). The MANOVA is also nonsignificant using an F statistic (F(4,195) = 2.19, p = .07). 
22 tests, global and local fit indices, and assessment of parameter estimates may all be used to evaluate model fit. In our article, we choose not to discuss in any detail the complexities associated with model fit in that it would detract from our goal of demonstrating the usefulness of an SEM framework to make decisions about multivariate analyses of differences in means.
3In general, Satorra-Bentler rescaled 2 statistics cannot be subtracted to assess differences in fit between initial and constrained models (9). However, they can be subtracted in this case in that the initial model is a just-identified model.
4Fan (15) demonstrated the equivalence between canonical correlation and analyses using SEM MIMIC models. Their results can be applied to discriminant analysis in that it is a subcase of canonical correlation. These issues are discussed in detail by Thompson and Green (16).
5The weights assigned to the coping scales to create the index (F1) are computed to predict as accurately as possible the dummy variable. Accordingly, the weight between the index and the dummy variable can be arbitrarily set to 1.
6We are unfamiliar with any literature that would extend this work to a discriminant analysis with more than two groups. Fan’s (15) demonstration of the equivalence of canonical correlational analysis and analyses using SEM MIMIC was in terms of weights determined to form the functions and not the significance tests.
7Both total-sample and within-group statistics can be reported for interpretation of the discriminant function. These statistics are reported by some statistical packages such the SAS CANDISC program. The total-sample statistics are ones that are comparable to those reported by standard error of mean programs. It should be noted that some researchers suggest that statistics based on the total sample confound two sources: the within-group and between-group statistics. Conveniently, the total-sample and within-group statistics are almost identical for our example.
Received for publication September 29, 2005; revision received February 23, 2006.
DOI:10.1097/01.psy.0000237859.06467.ab
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