Advanced Applications of Structural Equation Modeling in ... - CiteSeerX

Advanced Applications of Structural Equation Modeling in Counseling Psychology Research Matthew P. Martens Richard F. Haase University at Albany, State University of New York Structural equation modeling (SEM) is a data-analytic technique that allows researchers to test complex theoretical models. Most published applications of SEM involve analyses of cross-sectional recursive (i.e., unidirectional) models, but it is possible for researchers to test more complex designs that involve variables observed at multiple points in time or variables implicated in reciprocal feedback loops (i.e., bidirectional models). Given SEM’s popularity among counseling psychology researchers, this article aims to introduce three SEM designs not often seen in the counseling psychology literature: cross-lagged panel analyses, latent growth curve modeling, and nonrecursive mediated model analysis. For each design, the authors provide a brief rationale regarding its purpose, procedures for specifying a model to test the design, and a worked illustration.

LONGITUDINAL ANALYSES AND NONRECURSIVE MODELS Structural equation modeling (SEM) is a data-analytic technique increasing in popularity during the past several years (Martens, 2005; Steiger, 2001; Weston & Gore, 2006 [TCP, special issue, part 1]; for other techniques in this special issue, see Helms, Henze, Sass, & Mifsud, 2006 [TCP, special issue, part 1]; Henson, 2006 [TCP, special issue, part 1]; Kahn, 2006 [TCP, special issue, part 1]; Quintana & Minami, 2006 [this issue]; Sherry, 2006 [TCP, special issue, part 1]; Worthington & Whittaker, 2006 [this issue]). We can attribute this, at least in part, to SEM’s flexibility in being able to answer a large number of research questions. Researchers can use SEM for purposes as varied as confirming the factor Matthew P. Martens and Richard F. Haase, Department of Educational and Counseling Psychology, University at Albany, State University of New York. Correspondence concerning this article should be addressed to Matthew P. Martens, Department of Educational and Counseling Psychology, ED220 University at Albany, State University of New York, 1400 Washington Ave., Albany, NY 12222; (518) 442-5039; [email protected] THE COUNSELING PSYCHOLOGIST, Vol. 34 No. 6, November 2006 878-911 DOI: 10.1177/0011000005283395 © 2006 by the Society of Counseling Psychology

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structure of a psychological assessment instrument, analyzing potential mediator and moderator effects, and specifying and testing individual changes during the course of a psychological intervention. In addition, when conducting an SEM analysis, the researcher can model observed variables, latent variables (i.e., the underlying, unobserved construct as measured by multiple observed variables), or some combination of the two. Regardless of the specific variables the researcher uses, SEM is a confirmatory technique where analyses typically involve testing at least one a priori, theoretical model, and unlike many other statistical techniques, when using SEM the researcher can test the entire theoretical model in one analysis. As part of the analysis, the researcher can test both the specific hypothesized relationships among his or her variables and the plausibility of the overall model (i.e., the fit of the model). Clearly, SEM has a number of benefits for the researcher interested in studying relatively complex theoretical models. The most straightforward types of SEM analyses commonly seen in the psychological literature are models that test cross-sectional, linear, recursive (i.e., unidirectional) relationships among continuously measured variables during a single period. For example, a model specifying that higher levels of vocational self-efficacy were associated with higher amounts of vocational exploration when both variables were measured at the same time, would be a recursive model measured at a single period. It is possible, however, to test more complex designs. Researchers now use SEM to test designs that include interactions and other nonlinear effects (e.g., Schumaker & Marcoulides, 1998), experimental manipulations (e.g., Russell, Kahn, Altmeir, & Spoth, 1998), and dichotomous variables (e.g., Glockner-Rist & Hoijtink, 2003). This article’s purpose is to describe and illustrate additional advanced uses of SEM that may be of particular interest to counseling psychologists. First, we discuss two types of longitudinal analyses, cross-lagged panel designs and latent growth curve (LGC) modeling. Second, we discuss how to use SEM to test nonrecursive models (i.e., bidirectional models that hypothesize reciprocal causation or feedback loops). We hope to present a nontechnical introduction to these topics that will allow the reader to become a more educated consumer of research that uses such models as well as to provide a foundation for using such designs in one’s own research. We assume that the reader has a basic working knowledge of SEM procedures and a general understanding of specific SEM practices. We suggest that readers without such knowledge consult one of several introductory texts on SEM (e.g., Bollen, 1989; Kline, 2004) and nontechnical reviews that address a number of specific SEM practices (e.g., MacCallum & Austin, 2000; Martens, 2005; McDonald & Ho, 2002; Quintana & Maxwell, 1999; Weston & Gore, 2006).

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SEM AND LONGITUDINAL ANALYSIS The general benefits of longitudinal analysis are well known (Menard, 1991). For example, longitudinal studies often allow the researcher to determine cause and effect in terms of change over time. There are additional benefits of longitudinal research that may not be related to definitively establishing causality itself but are often associated with such designs. Two of these benefits include determining the degree to which variables serve as temporal predictors of each other and testing individual rates of change over time. SEM is especially suited to answering research questions in these domains. SEM models are often referred to as causal models because they generally posit that one or more exogenous (or predictor or independent) variables are a cause of one or more endogenous (or outcome or dependent) variables in a model. We have adopted this terminology throughout the article to illustrate several SEM applications but want to be clear about what we call causality in the context of SEM analyses. We are not referring to the traditional, deterministic, scientific perspective of manipulating experimental variables and controlling for extraneous variables. We recognize that from this perspective, it is not possible to infer causality from nonexperimental or quasi-experimental designs. Nonetheless, researchers generally specify SEM models on the basis of a theoretical perspective that specifies a causal relationship (e.g., increased vocational self-efficacy causes greater vocational exploration). Therefore, when we are discussing causal structural equation models, it is important to recognize that we are doing so from a theoretical and philosophical, rather than design-based, perspective. It is also important to recognize that when using SEM in longitudinal analyses, the research can establish some but not all conditions necessary to determine causality (e.g., establishing an empirical relationship between two variables, determination of temporal precedence). What Causes What: Cross-Lagged Panel Designs Psychology researchers are often confronted with questions about potential causality between two related variables—that is, does Variable A cause Variable B, or vice versa? Or does reciprocal causation exist? Some examples of this question of potential causality include the relationship between physical and mental health, racial or ethnic minority group exposure and increased multicultural sensitivity, and motivation to learn and academic success. Cross-lagged panel designs are not intended to establish the causality of such relationships in the traditional framework of using an


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FIGURE 1. Cross-Lagged Panel Design Example

experimental design, controlling for extraneous variables, and so on. They can, however, provide information about the strength of the temporal relationship among the variables, which is necessary in establishing causality (Hill, 1965; Menard, 1991). To conduct a cross-lagged panel design, the researcher must have at least two variables measured on at least two occasions and then test the relationships between each of the variables at subsequent time points. By comparing these two cross-lagged relationships, the researcher is able to determine the variable that is a stronger temporal predictor of the other, which can be considered evidence that one variable is a more likely cause of the other. Cross-Lagged Panel Design Example Specifying a cross-lagged panel model. Figure 1 is an example of a model that involves a cross-lagged panel design. At this point, we are focusing only on the theoretical model itself; we discuss the nature and meaning of the parameter estimates later. In this model, the investigators (Kenny, Blustein, Haase, Jackson, & Perry, 2006) were interested in examining the relationship between

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two latent variables: career development (career) and school engagement (school). Each latent variable includes two measured indicator variables, all of which were measured at two different time points. Career development was measured by the Career Development Inventory (CDI; Super, Thompson, Lindeman, Jordaan, & Myers, 1971) and the Outcome Expectation Scale (McWhirter, Crothers, & Rasheed, 2000), whereas school engagement was measured by the Value and Belong subscales of the Identification with School Questionnaire (Voelkl, 1997). The notation pre (e.g., CDI Pre) represents baseline scores, whereas the notation post (e.g., CDI Post) represents followup scores after a career-exploration intervention. Specifying an SEM model that uses a cross-lagged panel design is relatively straightforward. The baseline latent variables (Career 1 and School 1 in Figure 1) are generally conceptualized as correlated with each other, as are the disturbance (or error) terms associated with the latent variables at later time points (D1 and D2). These disturbance terms, such as all error terms, indicate the amount of variability in the endogenous variables associated with unknown factors. Error terms of the same measured variable assessed on different occasions are also conceptualized as correlated with each other because of the assumption that factors contributing to measurement error in any specific variable will be consistent across measured occasions. For example, in Figure 1, the latent variables that represent the error associated with the CDI at baseline and follow-up, respectively, are modeled as being correlated with each other. In a cross-lagged panel design, the researcher is generally most interested in examining the strength of the direct paths between the latent variables. There are two main types of such paths: the paths within each latent variable and the paths between the latent variables. The autoregressive paths, or paths that link a latent variable measured later with the same variable measured earlier (e.g., the path between career at Time 1 and Time 2), provide information about the relative stability of the construct, with higher values indicating greater stability. The paths measured across latent variables (e.g., the path between career at Time 1 and school at Time 2) provide information about the degree to which one variable is a stronger temporal predictor of the other (e.g., Does a stronger relationship exist between baseline career development and follow-up school engagement, or vice versa?) and are used to draw conclusions about potential causality. Although more complex cross-lagged panel designs can be tested (e.g., Burkholder & Harlow, 2003), Figure 1 represents the procedures for specifying a relatively straightforward model. Model testing in a cross-lagged panel design. To illustrate a cross-lagged panel design, we used baseline and follow-up data from a study where the researchers (Kenny et al., 2006) provided a career-exploration intervention


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TABLE 1: Summary of Cross-Lagged Panel Design Models Model Autoregressive School → career Career→ school Fully cross-lagged

χ2

df

p

RMSEA

SRMR

CFI

18.37 18.37 12.88 12.68

12 11 11 10

.11 .07 .30 .24

.04 .04 .02 .03

.03 .03 .02 .02

1.00 0.99 1.00 1.00

NOTE: RMSEA = root mean square error of approximation; SRMR = standardized root meansquared residual; CFI = comparative fit index.

for 416 ninth-grade students. The researchers used the data to test the model depicted in Figure 1. Researchers often test at least four hypotheses in a cross-lagged panel design: (a) a baseline model with only the autoregressive effects, (b) a model with the autoregressive effects and one latent variable predicting the other at later time points, (c) a model with the autoregressive effects and the other latent variable predicting the former at later time points, and (d) a fully cross-lagged model with the autoregressive effects and both latent variables predicting each other at later time points. We tested these four models with these data and present the fit indices for each model in Table 1. We first tested the autoregressive model and found that it provided a strong fit to the data (i.e., a nonsignificant chi-square value)—χ2(12) = 18.37, p = .11. Next, we tested the School → Career model, where we estimated both the autoregressive paths and the path between School 1 and Career 2 (but not the path between Career 1 and School 2), and the Career → School model, where we estimated both the autoregressive paths and the path between Career 1 and School 2 (but not the path between School 1 and Career 2). In particular, we were interested in determining if one or both of these models provided a significantly better fit to the data than did the autoregressive model. Because the autoregressive model is nested within these models, one can conduct a chi-square difference test to determine if the model with freer parameters provides a significantly better fit to the data than does the more parsimonious model. Results indicated that the School → Career model did not provide a significantly better fit to the data than did the autoregressive model—χ2diff (1) = .00, p > .05—and in this model, the path from School 1 to Career 2 was not statistically significant (β = –.02, p = .90). In contrast, the Career → School model provided a significantly better fit to the data—χ2diff(1) = 5.49, p < .05—and in this model, the path from Career 1 to School 2 was statistically significant (β = .26, p = .01). Because the Career → School model provided the best fit of these first three models, we compared the fit of the fully cross-lagged design to this model to determine if it provided a significantly better fit to the data. For illustrative purposes, we present standardized parameter estimates for this

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model in Figure 1. Results indicated that the fully cross-lagged model did not provide a significantly better fit to the data than did the Career → School model—χ2diff(1) = 0.20 p > .05. In this model, the path from Career 1 to School 2 was statistically significant (β = .28, p = .01), but the path from School 1 to Career 2 was not (β = –.06, p > .05). Therefore, because the Career → School model was more parsimonious than the fully crosslagged model, because the path from Career 1 to School 2 was statistically significant in both models that estimated the parameter, and because the path from School 1 to Career 2 was not statistically significant in either model that estimated the parameter, we conclude that baseline career development’s causing postintervention school engagement is more likely than baseline school engagement’s causing postintervention career development. Cross-lagged panel design summary. Cross-lagged panel designs allow an investigator with longitudinal data to determine, across time, if Variable X is a more likely cause of Variable Y, or if Y is the potential cause of X. We have illustrated how to model and test a straightforward cross-lagged panel analysis, and we encourage the interested reader to consult sources that provide examples of more complex cross-lagged models, such as mediated cross-lagged models (e.g., Cole & Maxwell, 2003) and models that incorporate additional exogenous variables (e.g., Sher, Wood, Wood, & Raskin, 1996). We believe that cross-lagged panel designs can answer important research questions related to possible causality in counseling psychology research and encourage researchers to design studies that use the technique to answer such questions. Modeling Individual Change: LGC Models Another way to use SEM in longitudinal data analysis is in LGC modeling. Compared with traditional procedures for analyzing longitudinal data (e.g., repeated measures ANOVA), LGC procedures have several appealing aspects in assessing change over time, many of which are related to the technique’s flexibility.1 Two such aspects are (a) that LGC models allow the researcher, in one analysis, to determine if significant overall change occurs over time, if significant variability exists in individual change trajectories among one’s observations, and if a relationship exists between baseline values and change over time; and (b) that LGC models allow researchers to easily model and test both potential covariates of change (e.g., gender) and different change trajectories (e.g., linear vs. quadratic change) in the same analysis (e.g., Byrne & Crombie, 2003). Conducting an LGC analysis requires that the researcher have data from the same measure on at least three different occasions and that the time and


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spacing of the data collection are the same for all observations. These requirements are in addition to standard data requirements for any SEM analysis, such as multivariate normal data and adequate sample size. In most SEM analyses, the researcher specifies theoretical latent variables represented by a combination of several observed variables. For example, in Figure 1, the theoretical latent variable career is represented by two different observed measures of the construct. An LGC model, however, does not specify latent variables that represent some type of underlying theoretical construct that can be represented by observable measures. Instead, at least two other types of latent variables are determined: (a) an intercept parameter that represents each individual’s score or observation on the variable at Time 1 and (b) a growth parameter that represents each individual’s rate of change across the periods. Specifying an LGC model. Figure 2 contains a hypothetical example of a linear LGC model. Assume that a researcher is interested in assessing changes in perceived counseling competencies of counseling psychology graduate students during their first 4 years in a doctoral program and believes that linear growth in such competencies will occur. Four observed variables in the diagram indicate the four measurement occasions of perceived counseling competencies. PCC refers to a hypothetical perceived counseling competencies measure, and the postscript refers to the measurement occasion (baseline through 3-year follow-up). One can also see that we have included the two latent variables of the intercept and growth parameters, with the variable slope reflecting the linear growth parameter. Recall that the purpose of the intercept parameter is to represent each individual’s score at Time 1, which remains fixed across all measurement occasions. Therefore, all paths from the intercept latent variable to the measured PCC variables are constrained to a constant value (generally 1.0). The purpose of the slope variable is to model the impact of time on the measured variables. Because the measurement intervals in this hypothetical example were 1 year apart, the parameters are constrained in increments that represent this interval. Therefore, the parameter estimated at 0 refers to the baseline, 1 to the 1-year follow-up, 2 to the 2-year follow-up, and 3 to the 3-year follow-up. Zero is often selected for the earliest measurement occasion because constraining this parameter to 0 allows the intercept term to be interpreted as the value for the measured variable at its earliest occasion (Curran & Hussong, 2003). It is important to note that when conducting an LGC analysis, the parameters from the slope latent variable to the observed variables must correspond to the intervals between measurement occasions. For example, if data were collected at baseline as well as at 6-month and 3-year follow-ups, the slope loadings to the three measured variables would be 0, 0.5, and 3 to accurately represent these intervals. This requirement of

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FIGURE 2. LGC Model Testing for a Linear Effect

LGC analyses also highlights the necessity of having all measured observations occur at the same intervals in the study. Hypothetical LGC example from a mock data set. To illustrate an LGC analysis, we developed an example from mock data applied to the model in Figure 2. Assume that we scored the PCC measure on a scale from 1 to 10, with higher scores indicating greater perceived competencies, and that we obtained data from 300 counseling psychology graduate students measured at four different times: the beginnings of the 1st, 2nd, 3rd, and 4th years in their programs. It is important to note that in an LGC model, the researcher is generally interested in a different set of parameter estimates than in other types of SEM analyses. In most SEM analyses, the parameter estimates that the researcher is most interested in are various regression parameters or paths between variables. In an LGC analysis, however, the parameters of most interest are the means of the slope and the intercept latent variables, the


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TABLE 2: Parameter Estimates for Latent Growth Curve Linear Model (Figure 2) Variable Mean Intercept Slope Variance Intercept Slope Covariance Intercept-slope

Estimate

SE

z Value

3.03 0.83

0.09 0.03

34.48 27.67

1.87 0.10

0.20 0.03

9.36 3.59

0.25

0.06

4.57

NOTE: z scores greater than 1.96 are statistically significant (p < .05).

variance of the slope and the intercept, and the covariance between the slope and the intercept. The chi-square statistic in the present example indicated an excellent model fit—χ2(5) = 7.05, p = .22—as did other fit indices (e.g., comparative fit index [CFI] = .99; standardized root mean-squared residual [SRMR] = .01; root mean square error of approximation [RMSEA] = .04). The mean of the intercept variable estimates the average Time 1 value of the measured variable (assuming that the parameter from the slope to the Time 1 measurement was set to 0). A statistically significant mean intercept indicates that the Time 1 value is significantly greater than 0. In the present example, the mean intercept estimate was 3.03 (p < .001), suggesting that the average baseline starting value was significantly greater than 0 (see Table 2 for all intercept and slope parameter estimates). In many instances, the mean intercept estimate is not that important to researchers, because they are most interested in the rate of change over time. The mean estimate of the slope provides this estimate of change. In the present example, this value was 0.83, which was statistically significant (p < .001). Therefore, these results indicate that, on average, the participants’ PCC scores increased by 0.83 units per year.2 Once the researcher has established the average Time 1 values and rates of change, he or she can examine the variances of the intercept and slope parameters to determine if significant individual differences exist in both the initial scores and the rates of change. If the variances are statistically significant, the researcher concludes that individual differences exist in terms of the average Time 1 value and average rate of change among the individuals in his or her sample. The ability to determine significant variability among one’s observations in terms of rate of change is one of the most important advantages of LGC analyses over more traditional longitudinal designs. For example, a researcher finding a large effect size in a longitudinal

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repeated measures analysis could conclude that significant change occurred over time but could not draw conclusions about any variability in rates of change among his or her observations. In contrast, in an LGC analysis, the researcher can identify both significant rates of change and variability among his or her observations in terms of change rates, and he or she can potentially model additional exogenous variables that might explain such variability. In the present example, the variance estimates for both the intercept (1.87) and the slope (0.10) were statistically significant (p < .001; see Table 2), indicating that significant variability existed in both Time 1 PCC scores and PCC rates of change among the participants. Furthermore, we can examine the covariance between the intercept and the slope to determine if a relationship existed between Time 1 PCC scores and rates of change. The estimated covariance between the intercept and the slope (0.25) was statistically significant (p < .001). Because the slope estimate was positive, we interpret the significant positive covariance as meaning that those with higher Time 1 perceived counseling competencies experienced a greater rate of change across the four periods. Therefore, based on our hypothetical example, we would conclude that psychology graduate students with higher baseline PCC scores experienced greater improvements during the course of their training than did those with lower baseline PCC scores. Modeling covariates of change. Because we found in the preceding analyses that significant interindividual differences existed regarding linear rates of change, the researcher may be interested in determining if variables exist that predict such differences. Therefore, he or she can model exogenous variables thought to predict differences in growth rates. For example, let us assume that the researcher believed that gender was related to Time 1 perceived counseling competencies and to rates of change over time. To test for this, the researcher would include an additional exogenous gender variable in the model, with parameters estimated between gender and both the intercept and the slope terms. The values for these parameters indicate whether a relationship existed between gender and Time 1 values and between gender and rates of change. Again using our mock data set, we found a standardized value of .17 (p < .01) for the gender-intercept parameter and of .03 (p = .70) for the gender-slope parameter. Assuming that females were coded 1 and males were coded 0 in our data, these results would indicate that women had significantly higher Time 1 PCC scores than men but that rates of growth across the four measurement occasions were similar for both men and women. Gender differences in growth rates (i.e., a statistically significant gender-slope parameter) would have been analogous to an interaction effect, where rates of change varied systematically between men and women. In this example, however, we found differences in only the baseline scores and not the rates of change.


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FIGURE 3. LGC Model Testing for a Quadratic Effect

Modeling exponential change. In the preceding example, we hypothesized that PCC scores would demonstrate a linear change over time. When the research problem focuses on a nonlinear effect—such as a quadratic, cubic, or other exponential model—LGC modeling can also be a useful tool. For example, a researcher might hypothesize that perceived counseling competencies improve at one rate during students’ first 2 years in a graduate program and then decelerate or accelerate to more modest or dramatic improvements during the remaining years. We present an example of a quadratic model in Figure 3. The latent variable slope2 represents the quadratic factor. To specify such a factor in an LGC analysis, the researcher must constrain the paths from the latent quadratic factor to the observed variables so that they represent the quadratic change over time. To do this, the researcher simply squares the corresponding parameters from the linear factor and uses those values for the quadratic parameter estimates (Kirk,

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1995). So, in our hypothetical example, to test for the linear effect, we set the slope parameters to 0, 1, 2, and 3 to correspond to the yearly interval measurement occasions. To test for a quadratic effect of time, then, we would set the corresponding quadratic parameters to 0, 1, 4, and 9, respectively. Because the LGC model testing for the linear effect is nested within the quadratic model, the researcher can use measures such as the chi-square difference test to determine if the quadratic model provides a significantly better fit than the linear model. In this case, the model testing the quadratic effect did not provide a significantly better fit—χ2diff (4) = 3.44, p > .05— so we concluded that the more parsimonious model (the linear LGC model) provided the best fit to the data. Summary of SEM and Longitudinal Models We hope that the preceding examples have provided the reader with a basic understanding of how to use SEM with two general types of longitudinal research questions. Although researchers can sometimes use other types of analyses—such as multiple regression for cross-lagged panel designs and hierarchical linear modeling for assessing individual growth, to answer such questions—we believe that SEM provides a relatively straightforward procedure with several methodological advantages when analyzing longitudinal data.

NONRECURSIVE MEDIATED STRUCTURAL EQUATION MODELS The Basic Ideas of Recursive Mediated Models Techniques for the analysis of mediated recursive path analytic models (Wright, 1934) are reasonably well known to psychologists, and numerous applications of such models have appeared in the counseling psychology literature (e.g., Lopez, Lent, Brown, & Gore, 1997; Nauta & Epperson, 2003). The essence of the idea of mediation is that some personal or environmental characteristic (say, X) affects an intervening variable (say, M), which, in turn, affects some relevant outcome variable (say, Y). The Variable M is said to mediate the relationship between X and Y, such that the correlation between X and Y vanishes when the relationship is adjusted for M.3 A popular class of such mediated models within counseling psychology is one that invokes self-efficacy as a variable that mediates the relationship between some personal or environmental characteristics (X) and some aspect of vocational behavior (Y) (Betz, 1999). Brown, George-Curran, and Smith


Utilization of Feelings

βM.X1 = .43 (p < .001) Self-Efficacy

βM.X2 = –.09 (n.s)

βY.X3 = –.30 (p < .001) Self-Control

βY.M = –.44 (p < .001)

Handling Relationships

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Vocational Exploration

FIGURE 4. Recursive Mediated Model of the Relationship Between Self-Efficacy and Vocational Exploration

(2003) studied the relationship between several aspects of emotional intelligence (i.e., handling of relationships, use of feelings, and self-control) and career decision-making self-efficacy (Betz, Klein, & Taylor, 1996), as well as vocational exploration and commitment (Blustein, Ellis, & Devenis, 1989). Although their hypotheses did not call for a mediated path-analytic solution, we can use their data to illustrate testing a mediated hypothesis in that we can empirically conform the data to the theoretical mandate that self-efficacy (M) mediates the relationship between emotional intelligence and vocational exploration. We present the fitted, single-indicator, path-analytic model based on the responses of 271 individuals in Figure 4.4 The primary exogenous variable of this model (use of feelings) is construed to influence the endogenous outcome variable (vocational exploration) by way of the endogenous mediator (self-efficacy). The reasons for the additional exogenous variables (handling of relationships and selfcontrol) are incidental to the current model and will be explained as we progress through the sequel. The zero-order correlations (Table 1 of Brown et al., 2003) between use of feelings and vocational exploration (r = –.32), between use of feelings and self-efficacy (r = .38), and between self-efficacy and vocational exploration (r = –.52) are all sizable and significantly different from 0. The justification for analyzing recursive mediated models

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(Baron & Kenny, 1986) requires that all these correlations be statistically significant; otherwise, there would be no possibility for mediation. There are several methods for testing the hypothesis of mediation (Shrout & Bolger, 2002). One is to test the indirect effect of the exogenous variable (feelings) on the endogenous outcome variable (exploration). To perform this test, we fitted the structural model implied in Figure 4 to the data by the method of maximum likelihood (LISREL 8.7; Jöreskog & Sörbom, 1993). To test the mediation hypothesis, we find that the indirect effect of use of feelings on exploration via self-efficacy—(.43)(–.44) = –.19—is significantly different from 0 (t = –5.32, p < .001). Hence, the conclusion follows that the ability to use feelings may be causally responsible for career decision-making selfefficacy. The level of self-efficacy is, in turn, responsible for the level of vocational exploration. Thus, high ability to use feelings ultimately leads to high levels of vocational exploration,5 and the process is mediated by increasing levels of self-efficacy for career decision-making tasks. In this sense, recursive mediated structural equation models are defined. Although not a necessity when testing for mediation, standard SEM statistics reveal that by many contemporary conventional criteria (with the exception of the RMSEA value), the Figure 4 model fits the data reasonably well: χ2(3) = 29.73, p < .001, CFI = .91, RMSEA = .18, SRMR = .08. Recursive structural equation models have seen frequent use in counseling psychology. For this article, we assume that researchers are generally familiar with the statistical solutions to such models (e.g., hierarchical regression, analysis of covariance structures) and with their advantages. Baron and Kenny (1986) and Frazier, Tix, and Barron (2004) discuss in general the mechanics of fitting single-indicator, mediated, path-analytic models by correlation and regression methods. Jöreskog and Sörbom (1993) and Kline (2004) discuss mediated, path-analytic models on more complex latent variables. Cole and Maxwell (2003), Holmbeck (1997), and Tomarken and Waller (2003) give interpretational and design advice about mediated models. Theoretical Limitations of Recursive Structural Equation Models It is a well-known limitation of statistical methods for assessing mediation in structural models that several alternative models might fit the data equally well (MacCallum, Wegener, Uchino, & Fabrigar, 1993). It is also quite possible that alternative theoretical models are equally plausible on logical grounds. In Figure 4, for example, it is plausible to reverse the roles of the mediator (selfefficacy) and the outcome (exploration) and to posit a quite different view of the process. It could be argued that as one has some success in the tasks of vocational exploration, however achieved, self-efficacy rises concomitantly—that is,


βM.X2 = –.04 (n.s)

βY.M = -.26 (p < .26)

Handling Relationships X2

.711 βY.X3 = –.37 (p < .001)

εY1

Self-Efficacy Y1 βM.Y = –.21 (p < .05)

Utilization of Feelings X1

.731

βM.X1 = .32 (p < .001)

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εY2

Vocational Exploration Y2

Self-Control X3

FIGURE 5. Nonrecursive Model of Self-Efficacy and VECS

increasing confidence could be a function of performance accomplishment and not vice versa (Bandura, 1977). Brown et al.’s (2003) data support such a view. The fit of this reverse-cause model with opposite causal directional arguments is nearly indistinguishable from the model in Figure 4. The standardized path coefficient leading from vocational exploration to self-efficacy (βSE.VECS = –.45, t = 9.08, p = .028) is nearly identical to that of the previous theoretical model from self-efficacy to VECS (βVECS.SE = –.44, t = 8.80, p = .032), and the effect is about equally significantly different from 0. These two diametrically opposed causal models appear to be equally plausible, both theoretically and statistically. A third alternative theoretical model is also quite plausible—that the variables of self-efficacy and vocational exploration are reciprocal—each providing one terminus of a cyclical process imbedded in a feedback loop. Such models are said to be nonrecursive. Nonrecursive Structural Equation Models With Reciprocal Feedback Loops In his seminal writings, Bandura (1977, 1978) argued that behavior, person or cognition, and environment all stand in reciprocal, deterministic relationship to one another, with feedback loops between each of the system’s three elements. Apply this argument to the Figure 5 model, ignoring the values of the path coefficients for the moment. A little reflection reveals that the

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pattern of the relationship between self-efficacy and vocational exploration may be reciprocal; as one gains some success at exploration, by whatever mechanism, one increases one’s confidence in performing the task at hand. Increased confidence in one’s ability to be successful then raises the probability that one will attempt further exploratory behaviors—a reciprocal, and cyclical, feedback loop that could continue almost indefinitely, or at least until the effect dissipates as is typical with many infinite series. Nonrecursive structural equation models—models that incorporate reciprocal feedback loops between variables—are far less well known, and far less frequently used, than are recursive mediated models, yet nonrecursive structural equation models represent processes that appear frequently in psychological theory. In the data-analytic realm, a considerable literature discusses the mechanics of the statistical solution to such models (Berry, 1984; Bollen, 1989; Hayduk, 1987; Kenny, 1979; Kline, 2004; Namboodiri, Carter, & Blalock, 1975). In the remainder of this section, we discuss the necessary details to statistically analyze problems involving reciprocal feedback loops. Models with feedback loops are considerably more difficult to implement than are recursive mediated models. All structural equation models face peculiar indeterminacies in their solutions, and many puzzling impediments can arise in the numerical analysis of SEM models, such as nonpositive definite covariance matrices, negative estimates of error variances (Heywood cases), inability to reach an iterative maximum likelihood solution, and other nefarious mathematical and numerical malfunctions. Researchers occasionally encounter these conditions in practice when fitting an SEM model by contemporary computer software (e.g., AMOS, LISREL, EQS, M-PLUS). An indepth discussion of these problematic issues is beyond the scope of the current article. The interested reader should be aware of the problems’ potential and consult the many available references that address methods to cope with such problems (e.g., Bentler & Chou, 1987; Wothke, 1993). Nonrecursive structural equation models are especially susceptible to these and other potentially problematic circumstances. In the remainder of this section, we discuss the necessary conditions to estimate the parameters of, and arrive at interpretable solutions to, models containing reciprocal feedback loops. In so doing, we illustrate the potential use of nonrecursive models in situations where the theory and substance of the problem involve cyclical processes. Necessary and Sufficient Conditions for Identifying Nonrecursive Models The identification problem in SEM models. Structural equation models are soluble to the extent that they are said to be identified. There are three different classes of identification in structural equation models: underidentified, just identified, and overidentified. Just-identified and overidentified


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models can be solved; underidentified structural equation models have no solutions. There are many reasons that a structural model may fail to be identified (Bentler & Chou, 1987; Bollen, 1989; MacCallum, 1995; Long, 1983; Wothke, 1993), but the fundamental problem for identifying structural equation models is the requirement that a model have sufficient data to estimate the number of parameters the model implies. The data of structural models typically consist of the variances and covariances associated with the model’s variables. Consider, for example, a partially mediated model in which a Variable X is said to affect a Mediator M, and the Mediator M is thought to affect an endogenous outcome Variable Y (i.e., X → M → Y). Consider also the fact that in addition to its indirect effect on Y through M, X is thought to have a direct effect on Y (i.e., X → Y). This model has three structural paths and three error variances (six parameters) to be estimated from the data.6 The structural (regression) equations that specify the model are X = εX M = βM.XX + εM Y = βY.MM + βY.XX + εY,

[1]

where X, M, and Y are variables in the model; βY.M, βM.X, and βY.X are the estimated structural parameters (path coefficients); and εX, εM, and εY are the errors associated with each equation. Models are said to be identified (and soluble) if the number of parameters to be estimated is equal to, or less than, the number of variances and covariances among the variables. For the mediated model of Equation 1, there are three variances (for X, M, and Y) and three covariances (X and M, X and Y, M and Y) available for fitting the model. For this model, the six parameters can be estimated from the six variances and covariances of the variables—the model is said to be just identified because the number of variances and covariances is equal to the number of parameters to be estimated. Now, consider the possibility of respecifying the Equation 1 model by setting the path from X → Y to 0—that is, by dropping the direct effect of X on Y. Dropping this path means that the solution to the resulting model requires that only five parameters be estimated from six variances and covariances. Models in which the number of variances and covariances is greater than the number of parameters to be estimated are soluble and are said to be overidentified. Finally, consider the possibility of altering the original Equation 1 model by retaining the three paths and the three error variances but adding a Y → M path to the model in addition to the M → Y path. This model now has four paths and three error variances to be estimated, yet the data still contain only six variances and covariances. Models in which the number of parameters to

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be estimated is greater than the number of available variances and covariances are said to be underidentified. These models have no solution. Underidentification is the central problem that faces the data analyst in solving nonrecursive models. In their recursive form (X → M → Y), analysts can easily structure a model to have sufficient variances and covariances such that it is overidentified and therefore soluble. When one adds a nonrecursive, cyclical feedback loop to the model, problems of underidentification, and insolubility, arise. Instrumental variables are the most usual method of coping with the underidentification problem in the fitting of nonrecursive models. Using instrumental variables also helps to meet two further conditions on which the solution of nonrecursive models depends. Both conditions are quite technical and beyond the scope of this article, but we provide an expository overview of the conditions and suggest further reading on the technical details. The order condition. The standard SEM solution to the problem of underidentification in structural models is to provide more data points (variances and covariances). In the literature of nonrecursive SEM, variables added to a model to ensure that it is estimable are called instrumental variables. In the Figure 5 model, for example, the exogenous variables labeled handling of relationships and self-control are instrumental variables, and their presence ensures that the model can be estimated. Instrumental variables, however, are constrained by certain restrictions. An instrumental variable can have a direct path to one of the endogenous variables involved in the feedback loop (e.g., vocational exploration) but at the same time must be excluded from having a direct path to the other endogenous variable in the model (e.g., self-efficacy). The rules for instrumental variables are more complicated as models grow in complexity. More complex nonrecursive models are beyond the scope of this article, but the reader may find more extensive discussion of the issues of identification of nonrecursive structural equation models in Berry (1984), Bollen (1989), Kenny (1979), Kline (2004), and Namboodiri et al. (1975). An additional point to keep in mind when choosing instruments is that the instrumental variables added to a model should ideally share as little common variance as possible with other instruments or with other exogenous variables in the system of equations. Too high a correlation between instruments or between instruments and other exogenous variables can induce severe multicollinearity (Cohen, Cohen, West, & Aiken, 2003) that can be troublesome when trying to arrive at an iterative analytic solution. The rank condition. A second required condition for identifying nonrecursive models is called the rank condition, which is both a necessary and a


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sufficient condition for model identification. The technical description of models of deficient rank is best understood by recourse to matrix algebraic methods, which we do not pursue here. Suffice it to say that in multiequation systems, estimating parameters of the model requires solving a set of simultaneous equations. A set of simultaneous equations will be of deficient rank if one of the equations can be expressed as a linear combination of another equation or set of equations. Not satisfying the rank condition leaves one in a predicament where the equations cannot be solved (Johnston, 1972). Adding instrumental variables to the design will often help provide the data necessary to meet the rank condition. As a practical matter, it is important to recognize that every endogenous variable involved in a reciprocal feedback loop must have its own instrumental variables. Ensuring this condition will often satisfy both the order and the rank conditions and make the model estimable and its results interpretable. Those who desire a more detailed discussion of both rank and order condition diagnosis should refer to presentations in Berry (1984) and Kline (2004). As a practical matter, data analysts often take a trial-and-error approach to meeting the order and rank conditions necessary to solve nonrecursive models. When both the order and the rank conditions are satisfied and sufficient data points are available to estimate the model, one can proceed to the next stage of the analysis. Some Tactics for Choosing Instrumental Variables It is often difficult to choose exogenous instrumental variables that will allow a nonrecursive model to be identified and hence to be numerically soluble. It is obviously necessary for an investigator to choose instruments in the design phase of a study prior to collecting data. The problem is a challenging one, but a few suggestions can prove useful in choosing instruments to solve the identification problem in a nonrecursive model. Even making judicious choices in advance may not be completely satisfactory, and some trail and error may well be necessary (Berry, 1984; Schaubroeck, 1990), but some rules of thumb can ease the process. First, the researcher can identify the model by restricting certain paths to 0—adding an instrumental variable to the model that is directed to one endogenous variable and that has no paths to other endogenous variables can identify a model. In SEM terminology, this is the equivalent of setting the path from that instrument to another endogenous variable equal to 0, which eliminates the instrument from the second variable’s equation and helps to meet the order condition. Second, although not emphasized here, there is a disturbance term associated with each endogenous variable in a nonrecursive model. Allowing these disturbances to correlate can sometimes identify a model, although this is recommended only if such a path were theoretically defensible.

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Third, imposing an equality constraint on the reciprocal paths can sometimes identify a model. Such a constraint renders the paths equal and requires estimating only one, rather than two, parameters. Although this is a technically feasible solution, one should be sure that the path coefficients in the population are in fact equal to avoid serious misspecifications by implementing this tactic. These methods for avoiding underidentified, insoluble, nonrecursive models are technical solutions. Although the analyst must make some form of numeric accommodation just to reach a solution, it is extremely important to keep in mind that he or she should base choosing instruments on sound theoretical grounds such that the instrument is defensible in the context of the explanatory theory. Furthermore, all the conditions that plague ordinary least squares (OLS) regression analysis also apply in this context— excessive multicollinearity among instruments or between instruments and other exogenous variables, low relationships between instrument and endogenous variables that can lead to suppression effects, and exaggeration of the standard errors estimated within the model. The best guidance for the data analyst is to satisfy the theoretical requirements of specifying the model first and, after doing so, to attend to the technical, numeric tactics to get the model to resolve. Equilibrium Assumptions and the Problem of Correlated Errors Brief mention should be made of two other conditions that must be assumed in the analysis and interpretation of cross-sectional, nonrecursive, mediated models: equilibrium and correlated errors. The equilibrium assumption. When we use cross-sectional, single-pointin-time data to fit a structural equation model to observations that were presumably generated by a theoretical process that has taken place across time (i.e., Y1 → Y2 → Y1 → . . . → Y2), it is important to recognize that there is a strong assumption being made about the stage of the process being studied. We must assume that the reciprocal process has run its course and that the cyclical effect has actually dissipated at the time of measurement. In addition, we must assume that at the time our measurements were taken, the system is in equilibrium, which is to say that the estimates of the effect of Y1 → Y2 and Y2 → Y1 are not dependent on any particular time point in the relationship cycle (i.e., the cycle has exhausted itself and the looping has stopped; Kenny, 1979; Kline, 2004). Correlated errors and the failure of OLS. One of the most popular methods of analyzing single-indicator, recursive, mediated models is by applying OLS regression analysis, often hierarchically (Cohen et al., 2003;


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Frazier et al., 2004). Nonrecursive mediated models, however, present problems for OLS regression techniques. Specifying a nonrecursive mediated model requires the definition of reciprocal paths between the endogenous variables, and consequently, each endogenous variable serves as both an independent and a dependent variable in the structural equations. In the Figure 5 model, self-efficacy is both a hypothesized outcome of vocational exploration and a putative cause of vocational exploration. The converse is also true. The two structural regression equations of the previous section (Equation 1) illustrate the problem. If Y1 did not appear in the second equation of this set, and if Y2 did not appear in the first equation of the set, there would be little problem with the estimation phase of fitting the model to the data. However, because of the reciprocal nature of the model (Y1 ¿ Y2), the errors of either equation are correlated with the exogenous variables of the equations—a situation expressly forbidden in the solution of OLS. Furthermore, the errors across equations are correlated (i.e., the errors are not independent), also an inadmissible condition under OLS assumptions. Failing these conditions, regression techniques will produce biased estimates of the parameters. It therefore follows that for any nonrecursive model, the OLS assumptions cannot be met, because of the very nature of the structural equations. One method for solving nonrecursive mediated models that suffer from the problems described earlier was popularized in the econometric literature—the technique of two-stage least squares regression analysis. Namboodiri et al. (1975) and James and Singh (1978) discussed this early procedure. A more contemporary solution to these computational difficulties is found in the full-information, maximum-likelihood solution implemented in most SEM software (e.g., LISREL, AMOS, EQS, M-PLUS). The SEM solution is preferable for two important reasons: (a) full-information, maximum-likelihood SEM software can handle myriad solutions to problems involving correlated errors of the structural equations and (b) SEM software is capable of specifying and estimating models with latent variables (Bollen, 1989; Jöreskog & Sörbom, 1993; Kline, 2004). Two-stage least squares cannot address either of these cases. In the following sections, we illustrate an SEM solution to the nonrecursive model in Figure 5. The Fitted Nonrecursive Model We fitted the nonrecursive, mediated, path-analytic model in Figure 5 by the method of maximum likelihood with LISREL 8.7 and with initial start values of the model estimated by two-stage least squares (Jöreskog & Sörbom, 1993). Figure 5 displays the path coefficients and their levels of statistical significance, as denoted by the ratio of the parameter estimate

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divided by its standard error. By some criteria, the model fits the data only modestly well, and the RMSEA value is above values typically associated with a well-fitting model: χ2(2) = 27.57, p < .001, CFI = .92, RMSEA = .21, SRMR = 07. However, in the hypothesis-testing application of the nonrecursive model illustrated here, we are not especially concerned about the overall level of model fit. The magnitude of the individual path coefficients is of greater interest in the analysis of this model. Note that the reciprocal path coefficients between self-efficacy and vocational exploration and vice versa are of the same sign and nearly equal (–.21 and –.26). The path from self-efficacy to VECS is significantly different from 0 (p’s < .01), and the path from VECS to self-efficacy differs significantly from 0 (p < .01). The analysis suggests that the nonrecursive model is tenable in the sense that the reciprocal paths are approximately equal and that both are significantly different from 0. Imposing an equality constraint on the two reciprocal path coefficients and reestimating the model yielded a model fit of χ2(3) = 28.19. The chi-square difference test, obtained from contrasting the constrained model and the freely estimated model, is not statistically significant: χ2diff(1) = .62, p > .80. The nonrecursive analysis of these data yields a fairly unambiguous interpretation: There is evidence within these data that the relationship between self-efficacy and vocational exploration may well be reciprocal in nature. As vocational exploration increases because of increasing self-efficacy, increases in self-efficacy would follow in turn. The cycle would be repeated throughout the feedback loop until damping down. One must be mindful of the fact that the data analyzed here are cross-sectional, and consequently, it is not possible to determine the order of the loop. The analysis suggests only that the causal influence is bidirectional and reciprocal. Tracking the indirect effects of the reciprocal feedback loop. Feedback loops can cycle through themselves numerous times, with each cycle in the loop multiplicatively affecting the next cycle. That is, Y1 affects Y2, which, in turn, affects Y1, and so on through a theoretically infinite number of cycles of the reciprocal loop. Research has shown (Kenny, 1979; Namboodiri et al., 1975) that the loop defines an infinite series but usually damps down after several iterations, provided that the coefficient estimates are less than one. For the current model, we note that a one-unit increase in Y1 leads to a βY2.Y1 = –.21 decrease in Y2, and one-unit increase in Y2 would be followed by a βY1.Y2 = –.26 decrease in Y1.7 By the third or fourth cycle through the loop, the constantly changing effect will become quite small (if the coefficients are less than one) because the limit of Y2 in this infinite series (Kenny, 1979) is


Y2 = lim

βY2 ⋅ Y1 1 − βY2 ⋅ Y1 βY1 ⋅ Y2

901

,

which, for the current example, would lead to a total change in Y2 of –.22. A similar process operates for Y1 in the reciprocal loop, with its limit defined by Y1 = lim

1 1 − βY2 ⋅ Y1 βY1 ⋅ Y2

,

which leads to a total effect of Y1 in the limit equal to –1.06. More complicated nonrecursive models with more than two endogenous variables are possible, and the rules for ascertaining their effects are similar to those just described, albeit more tedious (Namboodiri et al., 1975). Nonrecursive Models Involving Latent Variables The typical recursive model. To this point, we have considered nonrecursive models that contain only single indicators of the involved concepts. It is a straightforward matter to generalize the process of fitting nonrecursive models with latent variables. Latent variables can be defined by the presence of more than one measured indicator variable. Consider the recursive mediated model (Wang, 2004) that is embedded in the Figure 6 diagram.8 In this model, extraversion (X1) is the exogenous predictor for career decision-making self-efficacy (Y1), which mediates the relationship between extraversion and commitment to career choices (CCCP, Y2). Neuroticism (X2) is also an exogenous predictor explaining CCCP.9 The latent variable of CCCP is operationally defined by the measured indicators of VECS (Blustein et al., 1989), career indecision (Osipow, Carney, Winer, Yanico, & Koschier, 1987), and vocational identity (Holland, Daiger, & Power, 1980). CCCP was scaled to the units of the VECS (with high scores therefore indicating lack of progress), and the model was fitted to the data of 185 respondents by the method of maximum likelihood (LISREL 8.7). The fitted path coefficients of a model similar to that of Figure 6 (but that has no path from CCCP to self-efficacy) conform closely to the requirements for inferring mediation set forth by Baron and Kenny (1986): (a) The direct path from extraversion to self-efficacy is significantly different from 0, (b) the direct path from self-efficacy to progress toward CCCP is significantly different form 0, and (c) the direct path from extraversion to CCCP is set to 0.10 In these data, convincing evidence exists that the effect of extraversion on CCCP is mediated through self-efficacy. But alternative, competing theoretical models are logically possible and equally compelling.

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.88 .223 **

ε1

Self Efficacy

Extraversion

–.339**

–.337**

–.351**

.89** Career Commitment Progress

.60 ** Neuroticism

.57

–.85**

VECS MVS

.86** CDS

ε2

FIGURE 6. Nonrecursive Model of Self-Efficacy and the Latent Variable Career Choice Progress

The nonrecursive model of self-efficacy and progress in CCCP. For the variables in this example, the nonrecursive hypothesis would suggest that an increase in progress toward CCCP (as documented by increased exploration, less career indecision, and greater vocational identity) may well have a reciprocal, causal effect on self-efficacy. If self-efficacy were raised in this fashion, it would, in turn, raise the probability that one would engage in further exploration with a concomitant positive result and so forth through several cycles of the loop. As a practical matter, the loop would probably exhaust itself after several iterations, and the process would damp down to a state of stable equilibrium (Kenny, 1979). We present the fitted nonrecursive model in Figure 6. This nonrecursive model fits the data reasonably well, especially in terms of the CFI value, χ2(7) = 26.40, p = .0004, CFI = .97, SRMR = .09, RMSEA = .12. Following Schaubroeck’s (1990) recommendations, we tested the fit of this nonrecursive model against the strictly recursive model by a chi-squared difference test and found it to be not significantly different from the more parsimonious recursive model: χ2diff(1) = .92, p = .45. Hence, the nonrecursive model is as good of an account of the data as the recursive model. However, there is additional


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evidence in the fit of the nonrecursive model that is compelling: both the path from self-efficacy to career progress and the path from career progress to self-efficacy are significantly different from 0 (t = –2.24 and –2.16, respectively; ps < .05). Examination of the values of the reciprocal path coefficients reveals that they are nearly identical (–.339 and –.337, respectively). To test the proposition that the path coefficients of this nonrecursive model are equal, we placed an equality constraint on the reciprocal paths of the Figure 6 model and refitted it—χ2(1) = 26.27, p = .0009. The chi-square difference test between the two models differing only on the fixed versus free parameter estimates is χ2diff(1) = .13, p = .85. The freely estimated path coefficients do not differ from one another. All of the evidence points to the facts that the model is nonrecursive, both paths differ significantly from 0, and the paths are equally strong in both directions. These data clearly suggest that self-efficacy and progress in career choice commitment (identity, exploration, and decision) stand in reciprocal relationship to one another. The data contradict the notion that the causal path is unidirectional. Clearly, we must alter the theory of self-efficacy’s role in determining career choice to accommodate these findings. Some Prospects and Problems of Nonrecursive, Mediated, SEM Models The potential value of considering nonrecursive relationships. Throughout this section, there has been an underlying assumption that nonrecursive analytic procedures may offer a wider array of methodological possibilities than are typically considered in the counseling psychology literature. We certainly intended to convey that impression. Many models that enjoy currency in counseling psychology appear somewhat limited in potential because of the assumption that causality flows in only one direction. The numerous articles addressing the mediating role of self-efficacy in the psychological literature during the past 20 years, for example, have been almost exclusively of a static, cross-sectional nature. Notable exceptions to this trend are Nauta and Epperson (2003) and Lapan, Shaughnessy, and Boggs (1996), who colleted data across time, such that the causal direction of a model might be more readily tested. But the sine qua non of testing causal structure—experimental manipulation—is lacking in virtually all the published research on mediating conditions in counseling psychology. This inherent obstacle raises again the specter of the many cautions of SEM analysis of which we must be ever mindful. Many ways for things to go wrong: Some important cautions. Users of SEM analysis, and its supporting software, are well aware of the myriad

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difficulties they can encounter on the journey from data to fitted model. Because SEM numerical solutions are not based on mathematically closed forms but instead are iterative approximations, analytic solutions can often go wrong. Bentler and Chou (1987), Kline (2004), and Wothke (1993) discuss in detail the many problems generally encountered in fitting SEM models. Models may not converge, because of failure of the data to be normally distributed, failure to have sufficient variances and covariances to estimate the model’s parameters, and failure to avoid deficient rank matrices due to multicollinearity and other problems. More vexingly, nonrecursive models add an additional level of potential problems, including insufficient instruments to ensure an identified model (i.e., failure to satisfy the order and rank conditions), excessive multicollinearity among the instruments and the exogenous variables, and the possible need to allow for correlated errors of the structural equations that define the source of variation in the endogenous variables (Schaubroeck, 1990). Moreover, meeting the assumption that the reciprocal feedback loop of the model has stabilized and reached equilibrium at the time of measurement is critical but cannot be diagnosed statistically. Despite all these difficulties, we believe that the benefits of testing nonrecursive models, which are often theoretically necessary to give a full account of a psychological process or mechanism, far outweigh the burdens of solving the technical analytic problems.

SUMMARY This article’s purpose was to describe and illustrate advanced uses of SEM that we believe particularly useful to counseling psychology researchers. We described two longitudinal approaches, one of which (LGC modeling) provides investigators with an alternative strategy for investigating rates of change over time, and another (cross-lagged panel analysis) that provides the investigator with information about the possible causal relationship among sets of variables. We also described the approach for testing nonrecursive models, where two variables are thought to be reciprocal causes of each other. Together, we believe that these three methods have promise in advancing research within our field, especially in terms of understanding potential causal relationships among variables. Throughout this tutorial, we have made frequent reference to the notion of causality. The study of cause-and-effect relationships is deeply embedded within the long-lived, controversial, contentious, and often torturous history of the philosophy of science. Yet issues of causality are crucial to the advance of science in any discipline. It is clear that statistical analyses alone, SEM included, cannot decide issues of causality. The statistical


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methodology discussed in this article provides a necessary, but not sufficient, condition for interpreting causal relationships among constructs (for more on construct validation, see Hoyt, Warbasse, & Chu, 2006 [this issue]). We have observed that most, if not all, applications of SEM methodology in the social sciences are intended to implement a fundamental belief on the part of the investigator that a causal process underlies the specification of the model. This belief is usually supported by the philosophical ideas of the British empiricists represented by David Hume (1740) and John Stuart Mill (Copi, 1982). Hume laid out the fundamental conditions for inferring cause-and-effect relationships from observational data. He argued that for an Event A to be considered the cause of an Event B, one must be able to assert that A preceded B in time (temporal precedence), that A and B are correlated (constant conjunction), and that A and B are reasonably close in space (contiguity). Mill completed the algorithm by adding a last condition: that every alternative explanation be eliminated, leaving only the presumed A → B connection intact. Longitudinal designs applied to nonexperimental, observational data, as discussed in the first section, can help to reduce the ambiguities of causal inference about change by documenting constant conjunction (correlation or association) and by introducing an explicit time dimension into the study’s design and thus resolving issues of temporal precedence. In addition to the control of temporal precedence, the experimental manipulation of exogenous variables in designed studies can additionally strengthen inferences about causality in SEM models (Russell et al., 1998). Introducing experimentally manipulated factors with random assignment of participants to conditions meets Mill’s requirement for eliminating alternative explanations. Even in the absence of experimental manipulation, however, the possibility of drawing causal inferences from observational data seems to us to have promise—the investigator must, of course, be far more active in controlling and eliminating the third-variable explanations that inevitably lurk in the background. This process is limited only by the creativity and resources available to any research program. In theory, one can never control for all possible confounds (one cannot control what one cannot see), but as a practical matter, it is possible to eliminate the most obvious theoretically compelling and measurable alternative explanations. Despite the inherent uncertainties in this approach, the good-enough principle of approaching deterministic, population, cause-and-effect relationships from observational (correlational) data has gained considerable currency in contemporary philosophical and mathematical circles (Holland, 1986; Papineau, 2001; Pearl, 2000; Rosenbaum, 1984; Suppes, 1970) under the rubric of stochastic causality. Even in the face of observational data that

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cannot be experimentally manipulated, we believe that applications of the techniques discussed in this article can increase the potential to address causal and directional hypotheses in counseling psychology. Anecdotally, we should remember that in 1959, the U.S. Surgeon General’s Report on Smoking and Cancer (Cornfield et al., 1959) was based exclusively on a summary of nonexperimental, observational data. In 2005, there seems to be little doubt left as to the deterministic relationship between smoking and cancer. In that sense, the cumulative inference of cause-and-effect relationships in the social sciences can indeed be approximated and strengthened by meeting a series of increasingly stringent conditions (Hill, 1965). Complex SEM models as discussed in this article, most often applied to observational data, can readily be seen to support such an inferential framework. On a more pragmatic level, the SEM models in this article can have direct and immediate implications for counseling psychology research topics. With the appropriate longitudinal data, counseling psychology researchers can use cross-lagged models to answer questions about the temporal relationship between variables. For example, is the relationship between multicultural competencies measured at Time 1 and counseling competencies measured at Time 2 stronger than the relationship between Time 1 counseling competencies and Time 2 multicultural competences? As a second example, is the relationship between the working alliance measured at Time 1 and the psychotherapy outcome measured at Time 2 stronger than the relationship between Time 1 psychotherapy outcome and Time 2 working alliance? Similarly, and again with the appropriate data, counseling psychology researchers could use LGC models to address any question that involves hypothesized change over time. For example, is there a discernable pattern of changes in self-efficacy as a career counselor that take place during the course of graduate training? What happens during the course of counselor training in a counselor’s ability to confront racism? What are the observable longitudinal consequences of interventions designed to affect vocational exploration, or to affect changes in multicultural sensitivity, among high school students? Finally, counseling psychology researchers could use nonrecursive models to estimate mediated relationships that have the clear possibility to be reciprocal in nature. The many published studies of the mediating role of self-efficacy are a case in point. In addition to the example in this article (the relationship between self-efficacy and vocational exploration), counseling psychologists could use such models to explore relationships such as that between research interests, research mentoring, or program research training environments and research productivity or to study potentially reciprocal models of the effect of social support on willingness to seek therapy. We believe that the longitudinal and nonrecursive models in this article can help counseling psychology researchers answer novel questions that are important to our field.


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NOTES 1. Latent growth curve (LGC) modeling via structural equation modeling and hierarchical linear modeling (HLM) share several common features in terms of modeling and testing individual change over time. Comparing the LGC and HLM approaches goes beyond the scope of this article. We encourage the interested reader to consult sources on HLM and trajectory modeling (e.g., Raudenbush & Bryk, 2002) and sources that compare the two methods (e.g., MacCallum, Kim, Malarkey, & Kiecolt-Glaser, 1997; Willett & Sayer, 1994). 2. Because, for the present example, we are using mock data, we are not making any interpretations or drawing any conclusions regarding the practical (rather than statistical) significance of our findings. In analyses with actual data, we would encourage researchers to discuss their findings in terms of practical meaning. 3. Statistically, the difference between a confound and a mediator is indistinguishable. A confounding variable, when controlled by partial correlation, makes the correlation between X and Y vanish. This is the basis of the third variable problem in research design (Cohen et al., 2003; Shadish, Cook, & Campbell, 2002). Conversely, a mediator, when controlled, also causes the correlation between X and Y to vanish. The distinction between mediator and confound can be made only on logical and theoretical grounds (MacKinnon, Krull, & Lockwood, 2000). 4. Brown et al. (2003) did not perform their analysis as a mediated model. Their concern was directed toward accounting for variability in the two dependent variables as a function of four predictors and selected interactions with gender. We use their data here simply because the characteristics meet the requirements of mediated self-efficacy theory and can be modeled perfectly in recursive and nonrecursive forms of a structural equation model. Professor Brown was kind enough to provide additional information regarding their data such that the analyses could be performed. Any errors of omission and interpretation are therefore ours and not theirs. 5. Blustein et al. (1989) scaled their vocational exploration measure (the VECS) in such a way as to have low scores indicate greater amounts of VECS and commitment than high scores. Hence, a negative correlation between the vocational exploration and self-efficacy implies a positive relationship between self-efficacy and exploration. 6. Readers can find many examples of such models in counseling psychology. Some recent examples include Hollingsworth and Fassinger (2002), Lopez et al. (1997), and Lapan et al. (1996). 7. The VECS is scaled such that low scores indicate higher levels of exploration. 8. We depend on the diagram of the full nonrecursive model here to save space. If the path from commitment to career choices (CCCP) to self-efficacy were deleted, the recursive model with one latent variable would be portrayed. 9. The full model is formulated as in Figure 6 to meet the ultimate requirements for identification in a later discussion of the nonrecursive model. Neuroticism and extraversion each act as instruments. 10. A simple model with a direct path from extraversion to CCCP showed the effect to be statistically significant. Augmenting the path model by including a direct path from extraversion to CCCP shows the effect to be nonsignificant and to nearly vanish in magnitude (–.013, p > .05).

REFERENCES Bandura, A. (1977). Self-efficacy: Toward a unifying theory of behavioral change. Psychological Review, 81, 191-215.

908


Bandura, A. (1978). The self system in reciprocal determination. American Psychologist, 34, 344-358. Baron, R. M., & Kenny, D. A. (1986). The moderator-mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations. Journal of Personality and Social Psychology, 51, 1173-1182. Bentler, P. M., & Chou, C. (1987). Practical issues in structural modeling. Sociological Methods & Research, 16, 78-117. Berry, W. L. (1984). Nonrecursive causal models. Thousand Oaks, CA: Sage. Betz, N. E. (1999). Getting clients to act on their interests. Self-efficacy as a mediator of the implementation of vocational interests. In M. L. Savickas & A. R. Spokane (Eds.), Vocational interests (pp. 327-344). Palo Alto, CA: Davies-Black. Betz, N. E., Klein, K. L., & Taylor, K. M. (1996). Evaluation of a short form of the Career Decision-Making Self-Efficacy Scale. Journal of Career Assessment, 4, 47-57. Blustein, D. L., Ellis, M. V., & Devenis, L. E. (1989). The development and validation of a two-dimensional model of the commitment to career choices process. Journal of Vocational Behavior, 35, 342-378. Bollen, K. A. (1989). Structural equations with latent variables. New York: John Wiley. Brown, C., George-Curran, R., & Smith, M. (2003). The role of emotional intelligence in the career commitment and decision-making process. Journal of Career Assessment, 11, 379-392. Burkholder, G. J., & Harlow, L. L. (2003). An illustration of a longitudinal cross-lagged design for larger structural equation models. Structural Equation Modeling, 10, 465-486. Byrne, B. M., & Crombie, G. (2003). Modeling and testing change: An introduction to the latent growth curve model. Understanding Statistics, 2, 177-203. Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2003). Applied correlation/regression analysis for the behavioral sciences. Mahwah, NJ: Lawrence Erlbaum. Cole, D. A., & Maxwell, S. E. (2003). Testing mediational models with longitudinal data: Questions and tips in the use of structural equation modeling. Journal of Abnormal Psychology, 112, 558-577. Copi, I. (1982). Introduction to logic. New York: Macmillan. Cornfield, J., Haenszel, W., Hammond, E. C., Lilienfeld, A. M., Shimkin, M. B., & Wynder, E. L. (1959). Smoking and lung cancer: Recent evidence and a discussion of some questions. Journal of the National Cancer Institute, 22, 173-202. Curran, P. J., & Hussong, A. M. (2003). The use of latent trajectory models in psychopathology research. Journal of Abnormal Psychology, 112, 526-544. Frazier, P. A., Tix, A. P., & Barron, K. E. (2004). Testing moderator and mediator effects in counseling psychology research. Journal of Counseling Psychology, 51, 115-134. Glockner-Rist, A., & Hoijtink, H. (2003). The best of both worlds: Factor analysis of dichotomous data using item response theory and structural equation modeling. Structural Equation Modeling, 10, 544-565. Hayduk, L. A. (1987). Structural equation modeling with LISREL: Essentials and advances. Baltimore: Johns Hopkins University Press. Helms, J. E., Henze, K. T., Sass, T. L., & Mifsud, V. A. (2006). Treating Cronbach’s alpha reliability coefficients as data in counseling research. The Counseling Psychologist, 34, 630-660. Henson, R. K. (2006). Effect-size measures and meta-analytic thinking in counseling psychology research. The Counseling Psychologist, 34, 601-629. Hill, A. B. (1965). The environment and disease: Association or causation? Proceedings of the Royal Society of Medicine, 58, 295-300. Holland, P. W. (1986). Statistics and causal inference. Journal of the American Statistical Association, 81, 945-960. Holland, J. L., Daiger, D. C., & Power, P. G. (1980). Some diagnostic scales for research in decision-making and personality: Identity, information, and barriers. Journal of Personality and Social Psychology, 39, 1191-1200.


909

Hollingsworth, M. A., & Fassinger, R. E. (2002). The role of faculty mentors in the research training of counseling psychology doctoral students. Journal of Counseling Psychology, 49, 324-330. Holmbeck, G. N. (1997). Toward terminological, conceptual, and statistical clarity in the study of mediators: Examples from child-clinical and pediatric psychology literatures. Journal of Consulting and Clinical Psychology, 65, 599-610. Hoyt, W. T., Warbasse, R. E., & Chu, E. Y. (2006). Construct validation in counseling psychology research. The Counseling Psychologist, 34, 769-805. Hume, D. (1740). A treatise of human nature. Oxford, UK: Clarendon. James, L. R., & Singh, K. (1978). An introduction to the logic, assumptions, and basic analytic procedures of two-state least squares. Psychological Bulletin, 85, 1104-1122. Johnston, J. (1972). Econometric methods. New York: McGraw-Hill. Jöreskog, K. G., & Sörbom, D. (1993). LISREL 8: User’s reference guide. Chicago: Scientific Software International. Kahn, J. H. (2006). Factor analysis in counseling psychology research, training, and practice: Principles, advances, and applications. The Counseling Psychologist, 34, 684-718. Kenny, D. A. (1979). Correlation and causality. New York: John Wiley. Kenny, M. E., Blustein, D. L., Haase, R. F., Jackson, J., & Perry, J. C. (2006). Setting the stage: Career development and the student engagement process. Journal of Counseling Psychology, 53, 272-279. Kirk, R. E. (1995). Experimental design: Procedures for behavioral sciences (3rd ed.). Belmont, CA: Wadsworth. Kline, R. B. (2004). Principles and practice of structural equation modeling. New York: Guilford. Lapan, R. T., Shaughnessy, P., & Boggs, K. (1996). Efficacy expectations and vocational interests as mediators between sex and choice of math/science college majors: A longitudinal study. Journal of Vocational Behavior, 49, 277-291. Long, J. S. (1983). Covariance structure models: An introduction to LISREL. Beverly Hills, CA: Sage. Lopez, F. G., Lent, R. W., Brown, S. D., & Gore, P. A. (1997). Role of social-cognitive expectations in high school students’ mathematics-related interest and performance. Journal of Counseling Psychology, 44, 44-52. MacCallum, R. C. (1995). Model specification: Procedures, strategies, and related issues. In R. H. Hoyle (Ed.), Structural equation modeling: Concepts, issues, and applications. Thousand Oaks, CA: Sage. MacCallum, R. C., & Austin, J. T. (2000). Applications of structural equation modeling in psychological research. Annual Review of Psychology, 51, 201-226. MacCallum, R. C., Kim, C., Malarkey, W., & Kiecolt-Glaser, J. (1997). Studying multivariate change using multilevel models and latent curve models. Multivariate Behavioral Research, 32, 215-253. MacCallum, R., Wegener, D., Uchino, B., & Fabrigar, L. (1993). The problem of equivalent models in applications of covariance structure analysis. Psychological Bulletin, 114, 185-199. MacKinnon, D. P., Krull, J. L., & Lockwood, C. M. (2000). Equivalence of the mediation, confounding, and suppression effect. Prevention Science, 1, 173-181. Martens, M. P. (2005). The use of structural equation modeling in counseling psychology research. The Counseling Psychologist, 33, 269-298. McDonald, R. P., & Ho, M. R. (2002). Principles and practice in reporting structural equation analyses. Psychological Methods, 7, 64-82. McWhirter, E. H., Crothers, M., & Rasheed, S. (2000). The effects of high school career education on social-cognitive variables. Journal of Counseling Psychology, 47, 330-341. Menard, S. (1991). Longitudinal research. Newbury Park, CA: Sage.

910


Namboodiri, N. K., Carter, L. F., & Blalock, H. M. (1975). Applied multivariate analysis and experimental designs. New York: McGraw-Hill. Nauta, M. M., & Epperson, D. L. (2003). A longitudinal examination of the social-cognitive model applied to high school girls’ choice of nontraditional college majors and aspirations. Journal of Counseling Psychology, 50, 448-457. Osipow, S. H., Carney, C. G., Winer, J. L., Yanico, B. J., & Koschier, M. (1987). The Career Decision Scale. Odessa, FL: Psychological Assessment Resources. Papineau, D. (2001). Metaphysics over methodology—or, why infidelity provides no grounds to divorce causes from probabilities. In M. C. Gaolavotti, P. Suppes, & D. Costantini (Eds.), Stochastic causality (pp. 15-38). Palo Alto, CA: Stanford University Center for the Study of Language and Information. Pearl, J. (2000). Causality. Cambridge, UK: Cambridge University Press. Quintana, S. M., & Maxwell, S. E. (1999). Implications of recent developments in structural equation modeling for counseling psychology. The Counseling Psychologist, 27, 485-527. Quintana, S. M., & Minami, T. (2006). Guidelines for meta-analyses of counseling psychology research. The Counseling Psychologist, 34, PAGES. Raudenbush, S. W., & Bryk, A. S. (2002). Hierarchical linear models: Applications and data analysis methods (2nd ed.). Newbury Park, CA: Sage. Rosenbaum, P. R. (1984). From association to causation in observational studies: The role of tests of strongly ignorable treatment assignment. Journal of the American Statistical Association, 79, 41-48. Russell, D. W., Kahn, J. H., Altmeir, E. M., & Spoth, R. (1998). Analyzing data form experimental studies: A latent variable structural equation modeling approach. Journal of Counseling Psychology, 45, 18-29. Schaubroeck, J. (1990). Investigating reciprocal causation in organizational behavior research. Journal of Organizational Behavior, 11, 17-28. Schumaker, R. E., & Marcoulides, G. A. (Eds.). (1998). Interaction and nonlinear effects in structural equation modeling. Mahwah, NJ: Lawrence Erlbaum. Shadish, W. R., Cook, T. D., & Campbell, D. T. (2002). Experimental and quasi-experimental designs for generalized causal inference. Boston: Houghton Mifflin. Sher, K. J., Wood, M. D., Wood, P. K., & Raskin, G. (1996). Alcohol outcome expectancies and alcohol use: A latent variable cross-lagged panel study. Journal of Abnormal Psychology, 105, 561-574. Sherry, A. (2006). Discriminant analysis in counseling psychology research. The Counseling Psychologist, 34, 661-683. Shrout, P. E., & Bolger, N. (2002). Mediation in experimental and nonexperimental studies: New procedures and recommendations. Psychological Methods, 7, 422-445. Steiger, J. H. (2001). Driving fast in reverse: The relationship between software development, theory, and education in structural equation modeling. Journal of the American Statistical Association, 96, 331-338. Super, D. E., Thompson, A. S., Lindeman, R. H., Jordaan, J. P., & Myers, R. A. (1971). The Career Development Inventory. Palo Alto, CA: Consulting Psychologists Press. Suppes, P. C. (1970). A probabilistic theory of causality. Amsterdam, the Netherlands: North Holland. Tomarken, A. J., & Waller, N. G. (2003). Potential problems with “well fitting” models. Journal of Abnormal Psychology, 112, 578-598. Voelkl, K. E. (1997). Identification with school. American Journal of Education, 105, 294-318.


911

Wang, N. (2004). The relations of neuroticism and extraversion to progress in commitment to career choice: A model of mediating mechanisms through career decision-making self-efficacy. Unpublished doctoral dissertation, University at Albany, State University of New York. Weston, R., & Gore, P. A., Jr. (2006). A brief guide to structural equation modeling. The Counseling Psychologist, 34, 719-751. Willett, J. B., & Sayer, A. G. (1994). Using covariance structure analysis to detect correlates and predictors of individual change over time. Psychological Bulletin, 116, 363-381. Worthington, R. L., & Whittaker, T. A. (2006). Scale development research: A content analysis and recommendations for best practices. The Counseling Psychologist, 34, 806-838. Wothke, W. (1993). Nonpositive definite matrices in structural modeling. In K. A. Bollen & J. S Long (Eds.), Testing structural equation models (pp. 256-293). Newbury Park, CA: Sage. Wright, S. (1934). The method of path coefficients. Annals of Mathematical Statistics, 5, 161-215.

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