2013 IAP. Chapter 2. Feedback Loops and Formative Measurement. Rex B. Kline

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1 Chapter 2 Reverse Arrow Dynamics Feedback Loops and Formative Measurement Rex B. Kline Life can only be understood backwards; but it must be lived forwards. Soren Kierkegaard (843; quoted in Watkin, 2004, para. 3) This chapter is about two types of special covariance structure models where some arrows (paths) point backwards compared with more standard models. The first kind is nonrecursive structural models with feedback loops where sets of variables are specified as causes and effects of each other in a cross-sectional design. An example of a feedback loop is the specification V V2 where V is presumed to affect V2 and vice-versa; that is, there is feedback. It is relatively easy to think of several real world causal processes, especially dynamic ones that may be based on cycles of mutual influence, including the relation between parental behavior and child behavior, rates of incarceration and crime rates, and violence on the part of protestors and police. In standard, or recursive, structural models Structural Equation Modeling: A Second Course (2nd ed.), pages Copyright 203 by Information Age Publishing All rights of reproduction in any form reserved. 39

2 40 R. B. KLINE estimated in cross-sectional designs, all presumed causal effects are specified as unidirectional only so that no two variables are causes of each other. Thus, it is impossible to estimate feedback effects when analyzing recursive structural models in cross-sectional designs. For example, a recursive model for such a design could include either the path V V2 or the path V2 V, but not both. The second type of special model considered in this chapter are formative measurement models also known as emergent variable systems where observed variables (indicators) are specified as causes of underlying factors (e.g., V F). The latter are referred to as latent composites because they are represented as the consequence of their indicators plus error variance. This directionality specification is reversed compared with standard, or reflective, measurement models where factors are conceptualized as latent variables (constructs) that affect observed scores (e.g., F V) plus measurement error. It is reflective measurement that is based on classical measurement theory, but there are certain research contexts where the assumption that causality flows from factors to indicator is untenable. The specification that measurement is formative instead of reflective may be an option in such cases. The specification of either nonrecursive structural models with feedback loops or formative measurement models potentially extends the range of hypotheses that can be tested in structural equation modeling (SEM). There are special considerations in the analysis of both kinds of models, however, and the failure to pay heed to these requirements may render the results meaningless. Thus, the main goal of this presentation is to help readers make informed decisions about the estimation of reciprocal causation in cross-sectional designs or the specification of formative measurement in SEM. Specifically, assumptions of both nonrecursive structural models with feedback loops and formative measurement models are explained, and an example of the analysis of each type of model just mentioned is described. Readers can download all syntax, data, and output files for these examples for three different SEM computer tools, EQS, LISREL, and Mplus, from a freely-accessible Web page. The computer syntax and output files are all plain-text (ASCII) files that require nothing more than a basic text editor, such as Notepad in Microsoft Windows, to view their contents. For readers who use programs other than EQS, LISREL, or Mplus, it is still worthwhile to view these files because () there are common principles about programming that apply across different SEM computer tools and (2) it can be helpful to view the same analysis from somewhat different perspectives. Contents of EQS, LISREL, and Mplus syntax files only for both examples are listed in chapter appendices.

3 Reverse Arrow Dynamics 4 Nonrecursive Models with Feedback Loops There are two different kinds of feedback loops that can be specified in a nonrecursive structural model, direct and indirect. The most common are direct feedback loops where just two variables are specified as reciprocally affecting one another, such as V V2. Indirect feedback loops involve three or more variables that make up a presumed unidirectional cycle of influence. In a path diagram, an indirect feedback loop with V, V2, and V3 would be represented as a triangle with paths that connect them in the order specified by the researcher. Shown without error terms or other variables in the model, an example of an indirect feedback loop with three variables is: V3 V Because each variable in the feedback loop just illustrated serves as a mediator for the other two variables (e.g., V2 mediates the effect of V on V3, and so on), feedback is thus indirect. A structural model with an indirect feedback loop is automatically nonrecursive. Both direct and indirect feedback effects are estimated among variables measured concurrently instead of at different times; that is, the design is cross-sectional, not longitudinal. Although there are theoretical models that would seem to map well onto indirect feedback loops for instance, Carson (982) described maladaptive self-fulfilling prophesies in psychopathology as an unbroken causal loop between social perception, behavioral enactment, and environmental reaction (p. 576) there are few reports of the estimation of models with indirect loops in the behavioral science literature. This is probably due to technical challenges in the analysis of such models. These same obstacles (elaborated later) also apply to models with direct feedback loops, but they are somewhat less vexing when there are only two variables involved in cycles of presumed mutual causation. Accordingly, only nonrecursive models with direct feedback loops are considered next. Estimating reciprocal causality in a cross-sectional design by analyzing a nonrecursive model with a feedback loop can be understood as a proxy for estimating such effects in a longitudinal design by analyzing a cross-lagged panel model (Wong & Law, 999). A feedback loop between V and V2 is presented in Figure 2.a without disturbances or other variables. Note that () Figure 2.a represents a fragment within a larger nonrecursive model and (2) it is not generally possible to estimate direct feedback without V2

4 42 R. B. KLINE (a) Direct Feedback Loop V V2 (b) Panel Model Figure 2. Reciprocal causal effects between V and V2 represented with (a) a direct feedback loop based on a cross-sectional design and (b) cross-lagged effects based on a longitudinal design (panel model), both models shown without disturbances or other variables. other variables in the model due to identification. Variables V and V2 in Figure 2.a are measured concurrently in a cross-sectional design, which implies the absence of temporal precedence, or the measurement of a presumed cause before that of the presumed effect. Evidence for reciprocal causation is indicated when estimates of both direct effects in a feedback loop, or V V2 and V2 V in Figure 2.a, are of appreciable magnitude, given the research problem. Presented in Figure 2.b is a fragment from a cross-lagged panel model shown without disturbances or other variables and where V and V2 are each measured at different times in a longitudinal design. For example, V is measured at time and again at time 2 designated by, respectively, V and V 2 in Figure 2.b; variable V2 is likewise measured twice (V2 and V2 2 ). The possibility to collect additional measurements of V and V2 at later times is also represented in Figure 2.b. Presumed reciprocal causation is represented in the panel model of Figure 2.b by the cross-lagged direct effects between V and V2 measured at different times, specifically, V V2 2 and V2 V 2. Evidence for reciprocal causation is indicated when estimates of both cross-lagged direct effects in a panel model are of appreciable magnitude, again depending on the research area. 2 Although it is theoretically and mathematically possible to estimate reciprocal causation in cross-sectional designs when analyzing nonrecursive models with direct feedback loops, there is controversy about the adequacy of those estimates (Wong & Law, 999). The main reason is the absence of temporal precedence in cross-sectional designs. Causal effects are typically understood as taking place within a finite period, that is, there is a latency or lag between changes in causal variables and subsequent changes in outcome variables. With concurrent measurement there are no lags whatsoever, so in this sense the measurement occasions in cross-sectional designs are always wrong. This implies that the two-way paths that make up a direct V V2 V 2 V2 2

5 Reverse Arrow Dynamics 43 feedback loop, such as V V2 in Figure 2.a, represent an instantaneous cycling process, but in reality there is no such causal mechanism (Hunter & Gerbing, 982). 3 This may be true, but a feedback can still be viewed as a proxy or statistical model for a longitudinal process where causal effects occur within some definite latency. We do not expect statistical models to exactly mirror the inner workings of a complex reality. Instead, a statistical model is an approximation tool that helps researchers to structure their thinking (i.e., generate good ideas) in order to make sense of a phenomenon of interest (Humphreys, 2003). If the approximation is too coarse, then the model will be rejected. In contrast to model with feedback for cross-sectional data, definite latencies (lags) for presumed causal effects are explicitly represented in panel models by the measurement periods in a longitudinal design. For example, variables V and V2 2 in Figure 2.b represent the measurement of V at time and V2 at some later time 2. This temporal precedence in measurement is consistent with the interpretation that the path V V2 2 corresponds to a causal effect. But it is critical to correctly specify the lag interval when estimating cross-lagged direct effects in longitudinal designs. This is because even if V actually causes V2, the observed magnitude of the direct effect V V2 2 may be too low if the measurement interval is either too short (causal effects take time to materialize) or too long (temporary effects have dissipated). The same requirement for a correct measurement interval applies to the other cross-lagged direct effect in Figure 2.b, V2 V 2. The absence of time precedence in cross-sectional designs may not always be a liability in the estimation of reciprocal causation. Finkel (995) argued that the lag for some causal effects is so short that it would be impractical to measure them over time. An example is reciprocal effects of the moods of spouses on each other. Although the causal lags in this example are not zero, they may be so short as to be virtually synchronous. If so, then the assumption of instantaneous cycling for feedback loops in nonrecursive designs would not be indefensible. Indeed, it may even be more appropriate to estimate reciprocal effects with very short lags in a cross-sectional design even when longitudinal panel data are available (Wong & Law, 999). The true length of causal lags is not always known. In this case, longitudinal data collected according to some particular measurement schedule are not automatically superior to cross-sectional data. There is also the reality that longitudinal designs require more resources than cross-sectional designs. For many researchers, the estimation of reciprocal causation between variables measured simultaneously is the only viable alternative to a longitudinal design.

6 44 R. B. KLINE AU: There is no part (c) in Figure 2. Identification Requirements of Models with Feedback Loops Presented in Figure 2.2a is the most basic type of nonrecursive structural model with a direct feedback loop that is identified. This model includes observed variables only (i.e., it is path model), but the same basic configuration is required when variables in the structural model are latent each measured with multiple indicators (i.e., standard reflective measurement) and the whole model is a latent variable path model, not a measured variable path model (Kline, 200, chap. 6). The two-headed curved arrows that exit and re-enter the same variable ( ) in Figure 2.2a represent the variances of exogenous variables, which are generally free model parameters in SEM. The disturbances of the two variables that make up the feedback loop in Figure 2.2a, V3 and V3, are assumed to covary (D3 D4). This specification makes sense for two reasons: () if variables are presumed to mutually cause one another, then it is plausible that there are common omitted causes of both; (2) some of the errors in predicting one variable in a direct feedback loop, such as V3 in Figure 2.c, are due to the other vari- (a) All possible disturbance correlations V V2 V3 V3 (b) All possible disturbance correlations within recursively related blocks D3 D4 V V2 V3 V4 Figure 2.2 Two examples of nonrecursive path models with feedback loops. D3 D4 V5 V6 D5 D6

7 Reverse Arrow Dynamics 45 able in that loop, or V4 in the figure, and vice versa (Schaubroeck, 990). Although there is no requirement for correlated disturbances for variables involved in feedback loops, the presence of disturbance correlations in particular patterns in nonrecursive models helps to determine their identification status, a point elaborated next. The model in Figure 2.2a satisfies the two necessary identification requirements for any type of structural equation model: () all latent variables are assigned a metric (scale), and (2) the number of free model parameters is less than or equal to the number of observations (i.e., the model degrees of freedom, df M, are at least zero). In path models, disturbances can be viewed as latent (unmeasured) exogenous variables each of which requires a scale. The numerals () that appear in Figure 2.2a next to paths that point from disturbances to corresponding endogenous variables, such as D3 V3, are scaling constants that assign to the disturbances a scale related that of the unexplained variance of its endogenous variable. With four (4) observed variables in Figure 2.2a, there are a total of 4(5)/2 = 0 observations, which equals the number of variances and unique covariances among the four variables. The number of free parameters for the model in Figure 2.2a is 0, which includes the variances of four exogenous variables (of V, V2, D3, and D4), two covariances between pairs of exogenous variables (V V, D3 D4), and four direct effects, or V V3, V2 V4, V3 V4, and V4 V3. Because df M = 0 (i.e., 0 observations, 0 free parameters), the model in Figure 2.2a would, if actually identified, perfectly fit the data (i.e., the observed and predicted covariance matrices are identical), which means that no particular hypothesis would be tested in the analysis. It is usually only more complex nonrecursive models with positive degrees of freedom (df M > 0) which allow for the possibility of model-data discrepancies that are analyzed in actual studies. But the basic pattern of direct effects in Figure 2.2a from external variables (those not in the feedback loop, or V and V2) to variables involved in the feedback loop (V3, V4) respect both of the requirements for identifying certain kinds of nonrecursive models outlined next. If all latent variables are scaled and df M 0, then any recursive structural model is identified (e.g., Bollen, 989, pp ). This characteristic of recursive structural models simplifies their analysis. Cross-lagged panel models are typically recursive, but they can be nonrecursive depending on their pattern of disturbance covariances (if any). Unfortunately, the case for nonrecursive models is more complicated. There are algebraic means to determine whether a nonrecursive model is identified (Berry, 984), but these methods are practical only for very simple models. But there are two heuristics (rules) that involve determining whether a nonrecursive model

8 46 R. B. KLINE meets certain requirements for identification that are straightforward to apply (Kline, 200). These rules assume that df M 0 and all latent variables are properly scaled in a nonrecursive model. The first rule is for the order condition, which is a necessary requirement for identification. This means that satisfying the order condition does not guarantee identification, but failing to meet this condition says that the model is not identified. The second rule is for the rank condition, which is a sufficient requirement for identification, so a model that meets this requirement is in fact identified. The heuristics for the order condition and the rank condition apply to either () nonrecursive models with feedback loops and all possible disturbance covariances or (2) nonrecursive models with 2 pairs of feedback loops with unidirectional paths between the loops but all possible disturbance covariances within each loop. The latter models are referred to as block recursive by some authors even though the whole model is nonrecursive. Consider the two nonrecursive path models in Figure 2.2. For both models, df M 0 and every latent variable is scaled, but these facts are not sufficient to identify either model. The model of Figure 2.2a has an indirect feedback loop that involves V3 V4 and all possible disturbance correlations (), or D3 D4. The model of Figure 2.2b is a block recursive model with two direct feedback loops, one that involves V3 and V4 and another loop made up of V5 and V6. Each block of these variable pairs in Figure 2.2b contains all possible disturbance correlations () D3 D4 for the first block, D5 D6 for the second but the disturbances across the blocks are independent (e.g., D3 is uncorrelated with D5). The pattern of direct effects within each block in Figure 2.2b are nonrecursive (e.g., V3 V4), but effects between the blocks are unidirectional (recursive) (e.g., V3 V5). Thus, the two blocks of endogenous variables in the model of Figure 2.2b are recursively related to each other even though the whole model is nonrecursive. Order condition. The order condition is a counting rule applied to each variable in a feedback loop. If the order condition is not satisfied, the equation for that variable is under-identified, which implies that the whole model is not identified. One evaluates the order condition by tallying the number of variables in the structural model (except disturbances) that have direct effects on each variable in a feedback loop versus the number that do not; the latter are excluded variables. The order condition requires for models with all possible disturbance correlations that the number of variables excluded from the equation of each variable in a feedback loop exceeds the total number of endogenous in the whole model variables minus. For example, the model in Figure 2.2a with all possible disturbance correlations has two variables in a feedback loop, V3 and V4. These two variables are the only endogenous variables in the model, so the total number of endogenous variables is 2. Therefore, variables V3 and V4 in Figure 2.2a must each have a minimum of 2 = other variable excluded from each

9 Reverse Arrow Dynamics 47 of their equations, which is here true: V is excluded from the equation for V4, and V2 is excluded from the equation for V3. Because there is variable excluded from the equation of each endogenous variable in Figure 2.2a, the order condition is satisfied. The order condition is evaluated separately for each feedback loop in a block recursive model. For example, variables V3 and V4 are involved in the first feedback loop in the block recursive model of Figure 2.2b. For the moment we ignore the second feedback loop comprised of V5 and V6. The total number of endogenous variables for the first feedback loop is 2, or V3 and V4. This means that at least 2 = variables must be excluded for each of V3 and V4, which is true here: V is omitted from the equation for V4, and V2 is the excluded from the equation for V3. But the total number of endogenous variables for the second feedback loop comprised of V5 and V6 in Figure 2.2b is 4, or V3 through V6. This is because the first feedback loop with V3 and V4 in Figure 2.2b is recursively related to the second feedback loop with V5 and V6. Therefore, the order condition requires a minimum of 4 = 3 excluded variables for each of V5 and V6, which is true here: V, V2, and V4 are omitted from the equation for V5, and V, V2, and V3 are excluded from the equation for V6. Thus, the block recursive model of Figure 2.2b satisfies the order condition. Rank condition. Because the order condition is only necessary, it is still uncertain whether the models in Figure 2.2 are actually identified. Evaluation of the sufficient rank condition, however, will provide the answer. The rank condition is usually described in the SEM literature in matrix terms (e.g., Bollen, 989, pp ). Berry (984) devised an algorithm for checking the rank condition that does not require extensive knowledge of matrix operations. A non-technical description follows. The rank condition can be viewed as a requirement that each variable in a feedback loop has a unique pattern of direct effects on it from variables outside the loop. Such a pattern of direct effects provides a statistical anchor so that the free parameters of variables involved in feedback loops, including V3 V4, V4 V3, Var(D3), Var(D4), and D3 for the model in Figure 2.2a where Var means variance, can be estimated distinctly from one another. Each of the two endogenous variables in Figure 2.2a has a unique pattern of direct effects on it from variables external to the feedback loop; that is, V V3 and V2 V4. If all direct effects on both endogenous variables in a direct feedback loop are from the same set of external variables, then the whole nonrecursive model would fail the rank condition. Because the analogy just described does not hold for nonrecursive structural models that do not have feedback loops (e.g., Kline, 200, pp. 06 D4

10 48 R. B. KLINE 08), a more formal means of evaluating the rank condition is needed. Such a method that does not require knowledge of matrix algebra is described in Kline (200). The application of this method to the model in Figure 2.2a is summarized in Appendix A of this chapter. The outcome of checking the rank condition is the conclusion that the model in Figure 2.2a is identified; specifically, it is just-identified because df M = 0. See Kline (200, p. 53) for a demonstration that the block recursive model in Figure 2.2b also meets the rank condition. Thus, the model is identified, specifically, it is also just-identified because df M = 0. See Rigdon (995), who described a graphical technique for evaluating whether nonrecursive models with direct feedback loops are identified and also Eusebi (2008), who described a graphical counterpart of the rank condition that requires knowledge of undirected, directed, and directed acyclic graphs from graphical models theory. There may be no sufficient conditions that are straightforward to apply in order to check the identification status of nonrecursive structural models with disturbance covariances (including none) that do not match the two patterns described earlier. There are some empirical checks, however, that can be conducted to evaluate the uniqueness of a converged solution for such models (see Bollen, 989, pp ; Kline, 200, p. 233). These tests are only necessary conditions for identification. That is, a solution that passes them is not guaranteed to be unique. Technical problems in the analysis are common even for nonrecursive models proven to be identified. For example, iterative estimation of models with feedback loops may fail to converge unless start values are quite accurate. This is especially true for the direct effects and disturbance variances and covariances of endogenous variables involved in feedback loops; see Kline (200, pp , 85) for an example of how to specify start values for a feedback loop. Even if estimation converges, the solution may be inadmissible, that is, it contains Heywood cases such as a negative variance estimate, an estimated correlation with an absolute value >.00, or other kinds of illogical result such as an estimated standard error that is so large that no interpretation is reasonable. Possible causes of Heywood cases include specification errors, nonidentification of the model, bad start values, the presence of outliers that distort the solution, or a combination of a small sample size and only two indicators when analyzing latent variable models (Chen, Bollen, Paxton, Curran, & Kirby, 200). Special Assumptions of Models with Feedback Loops Estimation of reciprocal effects by analyzing a nonrecursive model with a feedback loop is based on two special assumptions. These assumptions are required because data from a cross-sectional design give only a snapshot

11 Reverse Arrow Dynamics 49 of an ongoing dynamic process. One is that of equilibrium, which means that any changes in the system underlying a feedback relation have already manifested their effects and that the system is in a steady state. That is, the values of the estimates of the direct effects that make up the feedback loop do not depend upon the particular time point of data collection. Heise (975) described equilibrium this way: it means that a dynamic system has completed its cycles of response to a set of inputs and that the inputs do not vary over time. This implies that the causal process has basically dampened out and is not just beginning (Kenny, 979). The other assumption is that of stationarity, or the requirement that the underlying causal structure does not change over time. It is important to know that there is generally no statistical way in a cross-sectional design to verify whether the assumptions of equilibrium or stationarity are tenable. Instead, these assumptions must be argued substantively based on the researcher s knowledge of theory and empirical results in a particular area. However, these special assumptions are rarely acknowledged in the literature where feedback effects are estimated with cross-sectional data in SEM (Kaplan, Harik, & Hotchkiss, 200). This is unfortunate because results of two computer simulation studies summarized next indicate that violation of the special assumptions can lead to severely biased estimates of the direct effects in feedback loops:. Kaplan et al. (200) specified a true dynamic system that was perturbed at some earlier time and was headed toward but had not attained yet equilibrium. Computer-generated data sets simulated the results of cross-sectional studies conducted at different numbers of cycles before equilibrium was reached. Estimates of direct effects in feedback loops varied widely depending upon when the simulated data were collected. This variation was greatest for dynamic systems in oscillation where the signs of direct effects swung from positive to negative and back again as the system moved back toward stability. 2. Wong and Law (999) specified a cross-lagged panel model with reciprocal effects between two variables as the population (true) model and generated simulated cross-sectional samples in which nonrecursive models with a single feedback loop were analyzed under various conditions. They found that the adequacy of estimates from cross-sectional samples was better when (a) there was greater temporal stability in the cross-lagged effects (i.e., there is stationarity and equilibrium ); (b) the nonrecursive model had a disturbance covariance as opposed to uncorrelated disturbances; (c) the sample size was larger (e.g., N = 500) rather than smaller (e.g., N = 200); and (d) the relative magnitudes of direct effects from prior variables

12 50 R. B. KLINE on variables in the feedback loop (e.g., V V3 and V2 V4 in Figure 2.2a were approximately equal instead of disproportionate. Example Analysis of a Nonrecursive Model with a Feedback Loop Within a sample of 77 nurses out of 30 similar hospitals in Taiwan, H.-T. Chang, Chi, and Miao (2007) administered measures of occupational commitment (i.e., to the nursing profession) and organizational commitment (i.e., to the hospital that employs the nurse). Each commitment measure consisted of three scales, affective (degree of emotional attachment), continuance (perceived cost of leaving), and normative (feeling of obligation to stay). The affective, continuance, and normative aspects are part of a three-component theoretical and empirical model of commitment by Myer, Allen, and Smith (993). Results of some studies reviewed by H. T. Chang et al. (2007) indicate that commitment predicts turnover intentions concerning workers careers (occupational turnover intentions) and place of employment (organizational turnover intentions). That is, workers who report low levels of organizational commitment are more likely to seek jobs in different organizations but in the same field, and workers with low occupational commitment are more likely to change careers altogether. H.- T. Chang et. al (2007) also predicted that organizational turnover intention and occupational turnover intention are reciprocally related: plans to change one s career may prompt leaving a particular organization, and viceversa. Accordingly, H.-T. Chang et al. (2007) also administered measures of occupational turnover intention and of organization turnover intention to the nurses in their sample. Presented in Table 2. are the correlations, standard deviations, and internal consistency (Cronbach s alpha) score reliability coefficients for all eight measures analyzed by H.-T. Chang et al. (2007). H.T. Chang et al. (2007) hypothesized that () organizational turnover intention and occupational turnover intention are reciprocally related, that is, the intention to leave a place of employment affects the intention to remain in a profession, and vice versa; (2) the three components of organizational commitment (affective, continuance, normative) directly affect organizational turnover intention; and (3) the three components of occupational commitment directly affect occupational turnover intention. A nonrecursive path model that represents these hypotheses would consist of a direct feedback loop between organizational turnover intention and occupational turnover intention, direct effects of the three organizational commitment variables on organizational turnover intention, and direct effects of the three occupational commitment variables on occupational turnover intention. However, path analysis of a structural model of observed

13 Reverse Arrow Dynamics 5 TABLE 2.. Input Data (Correlations, Standard Deviations, Score Reliabilities) for Analysis of a Nonrecursive Model of Organizational and Occupational Commitment and Turnover Variable Organizational commitment. Affective Continuance Normative Occupational commitment 4. Affective Continuance Normative Turnover intention 7. Organizational Occupational SD Note: These data are from H.-T. Chang et al. (2007); N = 77. Values in the diagonal are internal consistency (Cronbach s alpha) score reliability coefficients. variables does not permit the explicit estimation of measurement error in either exogenous variables (commitment) or endogenous variables (turnover intention). Fortunately, the availability of score reliability coefficients for these data (Table 2.) allows specification of the latent variable model in Figure 2.3 with the features described next. Each observed variable in Figure 2.3 is represented as the single indicator of an underlying factor. The unstandardized loading of each indicator is fixed to.0 in order to scale the corresponding factor. Measurement error variance for each indicator is fixed to equal the product of the sample variance of that indicator, or s 2, and one minus the score reliability for that indicator, or r XX. The latter quantity estimates the proportion of observed variance due to measurement error. For instance, the score reliability coefficient for the affective organizational commitment measure is.82 and the sample variance is.04 2 =.086 (see Table 2.). The product (.82).086, or.947, is the amount of total observed variance due to measurement error. Accordingly, the error variance for the affective organizational commitment indicator is fixed to.947 for the model in Figure Error variances for the remaining 7 indicators in the figure were calculated in a similar way. So specified, estimation of the direct effects and disturbance variances and covariance for the model in Figure 2.3 explicitly controls for measurement error in all observed variables (see Kline, 200, pp , for additional examples). To save space, all possible covariances between

14 52 R. B. KLINE.947 E aoc.288 E coc.2446 E noc.603 E apc.764 E cpc.90 E npc affective org commit continuance org commit normative org commit affective occ commit continuance occ commit normative occ commit Affective Org Commit Continuance Org Commit Normative Org Commit Affective Occ Commit Continuance Occ Commit Normative Occ Commit org turnover intent occupational turnover the 6 exogenous factors (5 altogether) are not shown in Figure 2.3, but they are specified in computer syntax for this analysis. With eight observed variables, there are a total 8(9)/2, or 36 observations available to estimate the model s 32 free parameters, which include 8 variances (of 6 exogenous factors and 2 disturbances), 6 covariances ( disturbance covariance and 5 exogenous factor covariances), and 8 direct effects, so df M = 4. Each latent variable in the model of Figure 2.3 is assigned a scale, but we still do not know if the model is identified. Now looking only at the structural part of the model (i.e., the pattern of direct effects among Org Turnover Intent Occ Turnover Inent.2744 E ort D OrT D OcT.2700 E oct Figure 2.3 Initial latent variable nonrecursive model of the relation of organizational/occupational commitment to organizational/occupational turnover intention. Covariances among exogenous factors are omitted to save space. Unstandardized error variances for single indicators are fixed to the values indicated.

15 Reverse Arrow Dynamics 53 the 8 factors), there are two endogenous factors, so a minimum of 2 = exogenous factor must be excluded from each equation for Occupational Turnover Intention and for Organization Turnover Intention, which is true (see Figure 2.3). For example, the affective, continuance normative, and organizational commitment factors are all excluded from the equation for the Occupational Turnover Intention factor. Thus, the model in Figure 2.3 meets the necessary order condition. It can also be shown by applying the method described in Appendix A to the structural model in Figure 2.3 that the sufficient order condition is also satisfied. This observation plus other features of the model (e.g., each observed variable loads on a single factor, measurement error variances are fixed to equal the values shown in Figure 2.3) indicate that the whole model is identified, specifically, it is overidentified because df M > 0. The model in Figure 2.3 was fitted to the covariance matrix based on the correlations and standard deviations in Table 2. using maximum likelihood (ML) estimation in EQS, LISREL, and Mplus. The EQS program issued warnings about singular (nonpositive definite) parameter matrices in early iterations, but then recovered to generate a converged and admissible solution. Both LISREL and Mplus generated converged and admissible solutions without incident. Values of fit statistics and parameter estimates are very similar across the three programs, so results from EQS only are summarized here. Values of the model chi-square, Bentler Comparative Fit Index (CFI), Jöreskog-Sörbom Goodness of Fit Index (GFI), standardized root mean square residual (SRMR), and Steiger-Lind root mean square error of approximation (RMSEA) with its 90% confidence interval calculated are listed next: 2 χ (4) = 9.77, p =.057 M CFI =.999;GFI =.987;SRMR =.08 RMSEA =.086 (0.60) The model just passes the chi-square test at the.05 level, and values of the CFI, GFI, and SRMR are generally favorable. In contrast, the RMSEA results are poor, specifically, the upper bound of its confidence interval (.60) suggests poor fit within the limits of sampling error. Although none of the absolute correlation residuals (calculated in EQS) exceeded.0, there were several statistically significant standardized residuals (calculated by LISREL and Mplus), including the residual for the association of the continuance organizational commitment and occupational turnover intention variables. A respecification consistent with this result is to add to the initial model in Figure 2.3 a direct effect from continuance organizational commitment to occupational turnover intention.

16 54 R. B. KLINE The model in Figure 2.3 was respecified by adding the direct effect just mentioned. All analyses in EQS, LISREL, and Mplus of the respecified model converged to admissible solutions that were all very similar. Listed in Appendices B D is, respectively, EQS, LISREL, and Mplus syntax for this analysis. Values of selected fit statistics calculated by EQS for the respecified model are reported next: 2 χ (3) =.84, p =.848 M CFI=.000;GFI =.999;SRMR =.005 RMSEA = 0(0.07) The results just listed are all favorable. Also, the largest absolute correlation is.02, and there are no statistically significant standardized residuals for the revised model. The improvement in fit of the revised model relative to that of the initial model is also statistically significant, or 2 χ D() = = 8.363, p =.004. Based on all these results concerning model fit (Kline, 200, chap. 8), the respecified nonrecursive model with a direct effect from continuance organizational commitment to occupational turnover intention is retained. Reported in Table 2.2 are the ML estimates of the direct effects and the disturbance variances and covariance for the revised model. The best predictor of organizational turnover intention is normative organizational commitment. The standardized path coefficient is.662, so a stronger sense of obligation to stay within the organization predicts lower levels of the intention to leave that organization. In contrast, the best predictor of occupational turnover intention is continuance occupational commitment, for which the standardized path coefficient is.650. That is, higher estimation of the costs associated with leaving a discipline predicts a lower level of intention to leave that discipline. The second strongest predictor of occupational turnover intention is continuance organizational commitment and, surprisingly, the standardized path coefficient for this predictor is positive, or.564. That is, higher perceived costs of leaving an organization predicts a higher level of intention to leave the discipline. This result is an example of a suppression effect because although the Pearson correlation between continuance organizational commitment and occupational turnover intention is about zero (.02; see Table 2.), the standardized weight for the former is positive (.564) once other predictors are held constant. See Maasen and Bakker (200) and Kline (200, pp , 60 72) for more information about suppression effects in structural models.

17 Reverse Arrow Dynamics 55 TABLE 2.2. Maximum Likelihood Estimates for a Nonrecursive Model of Organizational and Occupational Commitment and Turnover Intention Parameter Unstandardized SE Standardized Direct effects AOC OrgTO COC OrgTO NOC OrgTO.028 ** OccTO OrgTO APC OccTO.727 ** CPC OccTO.395 * NPC OccTO OrgTO OccTO.28 * COC OccTO.969 * Disturbance variances and covariance OrgTO.774 ** OccTO OrgTO OccTO Note: AOC, affective organizational commitment; COC, continuance organizational commitment; NOC, normative organizational commitment; APC, affective occupational commitment; CPC, continuance occupational commitment; NPC, normative occupational commitment; OrgTO, organizational turnover intention; OccTO, occupational turnover intention. Standardized estimates for disturbance variances are calculated as minus the Bentler-Raykov corrected R 2. * p <.05; ** p <.0. As expected, the direct effects of organizational turnover intention and occupational turnover intention on each other are positive. Of the two, the standardized magnitude of the effect of organizational turnover on occupational turnover (.259) is stronger than the effect in the other direction (.040). If there is reciprocal causation between these two variables, then it is stronger in one direction than in the other. The EQS program prints values of the Bentler-Raykov corrected R 2, which controls for model-implied correlations between predictors and disturbances for variables in feedback loops (Kline, 200, pp ). These corrected R 2 values are listed next: Organizational Turnover Intention factor,.54; Occupational Turnover Intention factor,.764. The prediction of occupational turnover intention is thus somewhat better than the prediction of organizational turnover intention by all prior variables of each endogenous factor in Figure 2.3. Formative Measurement Models A reflective measurement model for a single latent variable F with three indicators V V3 is presented in Figure 2.4a. This single-factor model is

18 56 R. B. KLINE (a) L M block V E V2 F E2 V3 E3 (c) Single-indicator specification V E E2 E3 F ( r ) s 2 ( r 22 ) s 2 2 ( r 33 ) s 3 2 V2 F2 F4 D V3 F3 (b) M L block (d) MIMIC Factor E2 V V2 V3 identified but has no degrees of freedom (df M = 0), but with 4 effect indicators it would be possible to test the fit of a single-factor reflective measurement model to the data in confirmatory factor analysis (CFA) because df M 0 in this case. The general goal of CFA is to explain (model) the observed covariances among the indicators. Specifically, CFA minimizes the differences between the observed covariances among the indicators and those predicted by a reflective measurement model. Because the direct effects in V F V2 F D D Figure 2.4 Directionalities of effects between indicators and a (a) latent variable and (b) latent composite. (c) Single-indicator specification of cause indicators. (d) A MIMIC factor. L, latent; M, manifest; MIMIC, multiple indicators and multiple causes. V3 E3

19 Reverse Arrow Dynamics 57 Figure 2.4a point from F to V V3, the latter are referred as reflective indicators or effect indicators. Grace and Bollen (2008) used the term L M block (latent to manifest) to describe the association between factors and their effect indicators in reflective measurement models. Measurement error in reflective measurement models is represented at the indicator level as represented by the terms E E3 in Figure 2.4a. Reflective measurement is based on the domain sampling model (Nunnally & Bernstein, 994, chap. 6). From this perspective, a set of effect indicators of the same factor should be internally consistent, which means that their intercorrelations should be positive and at least moderately high in magnitude; otherwise, estimates of the reliability of factor measurement (Hancock & Mueller, 200) may be low. Sometimes intercorrelations are both positive and negative among indicators with a mix of positively- versus negatively-word items. For example, the item My health is good is positively worded, but the item I am often worry about my health is negatively worded. Participants who respond true to the first item may respond false to the second, and vice versa. In this case, the technique of reverse coding could be used to change the scoring of negatively-worded items to match that of positively worded items (or vice-versa) so that intercorrelations among recoded items are all positive. The domain sampling model also assumes that effect indicators of the same construct with equal score reliabilities are interchangeable. This implies that indicators can be substituted for one another without appreciably affecting construct measurement. The assumption that indicators are caused by factors is not always appropriate. Some indicators are viewed as cause indicators or formative indicators that affect a factor instead of the reverse. Consider this example by Bollen and Lennox (99): The variables income, education, and occupation, are used to measure socioeconomic status (SES). In a reflective measurement model, these observed variables would be specified as effect indicators that are caused by an underlying SES factor. But we usually think of SES as the outcome of these variables (and others), not vice-versa. For example, a change in any one of these indicators, such as a salary increase, may affect SES. From the perspective of formative measurement, SES is a composite or index variable that is caused by its indicators. If it is assumed that SES is also caused by other, unmeasured variables besides income, education, and occupation, then SES would be conceptualized as a latent composite, which in a formative measurement model has a disturbance. Cause indicators are not generally interchangeable. This is because removal of a cause indicator is akin to removing a part of the underlying factor (Bollen & Lennox, 99). Cause indicators may have any pattern of intercorrelations, including ones that are basically zero. This is because composites reflect the contribution of multiple dimensions through its cause indicators. That is, composites are not unidimensional, unlike latent variables in

20 58 R. B. KLINE reflective measurement models. There are many examples of the analysis of composites in economics and business research (Diamantopoulos, Riefler, & Roth, 2005). Presented in Figure 2.4b is a formative measurement model for a single latent composite F with three cause indicators V V3. It depicts an M L block (manifest to latent) because F is assumed to be caused by its indicators. With no disturbance, F in Figure 2.4b would be just a linear combination of its cause indicators and thus not latent. 5 To scale the latent composite, the unstandardized direct effect of one of its cause indicators, V, is fixed to.0. Cause indicators in formative measurement models are exogenous variables that are free to vary and covary in any pattern. Also, they have no measurement errors in the standard representation of formative measurement. Thus, it is assumed for the model in Figure 2.4b that cause indicators V V3 have perfect score reliabilities (i.e., r XX =.00). The assumption of perfect score reliability is unrealistic for most observed variables. Consequently, measurement error in cause indicators is manifested in the disturbance term of the latent composite (D in Figure 2.4b). This means that measurement error in formative measurement is represented at the construct level, not at the indicator level as in reflective measurement. In other words, unreliability in formative indicators tends to increase the disturbance variance of the corresponding latent composite. Unlike a reflective measurement model, a formative measurement model does not explain the variances and covariances of the indicators. Instead, the formative indicators are specified as predictors of their corresponding latent composites, and the computer is asked to estimate regression coefficients for the indicators and the proportions of explained variance in the composites, not the indicators. Note that the formative measurement model in Figure 2.4b is not identified. In order to estimate its parameters, it would be necessary to embed it in a larger model. Identification requirements of formative measurement models are considered later. There is an alternative to assuming that cause indicators of latent composites have perfect score reliabilities as in Figure 2.4b. This alternative is illustrated by the model in Figure 2.4c where. each cause indicator is specified as the single indicator of an underlying exogenous factor, such as F V; 2. the measurement error variance of each indicator is fixed to equal the product of one minus the score reliability and the observed variance, such as ( r 2 2 ) s where r is a reliability coefficient and s is the sample variance for indicator V; 3. factors F F3 are free to vary and covary; and 4. factors F F3 have direct effects on the latent composite F4.

21 Reverse Arrow Dynamics 59 The observed variables V V3 in Figure 2.4c are no longer technically cause indicators (they are effect indicators), but the corresponding factors F F3 are the cause indicators of the latent composite F4. Estimation of the disturbance term D in Figure 2.4c would control for measurement error in the cause indicators V V3, if this model were embedded in a larger structural equation model and corresponding identification requirements were met. Specifically, measurement error in the cause indicators V V3 of Figure 2.4c is estimated separately from the disturbance variance for the latent composite. This means that the disturbance in Figure 2.4c reflects only unexplained variance in the latent composite F4. In contrast, the disturbance of the latent composite F in Figure 2.4b reflects both unexplained variance and measurement error in V V3. There is a kind of compromise between specifying that the indicators are either all effect indicators or all causal indicators. It is achieved by specifying a MIMIC (multiple indicators and multiple causes) factor with both effect indicators and cause indicators. A MIMIC factor with a single cause indicator V and two effect indicators V2 V3 is presented in Figure 2.4d. A MIMIC factor is endogenous, which explains why F has a disturbance. Note in the figure that effect indicators V2 and V3 have measurement errors (E2, E3), but cause indicator V does not. This means that measurement error in V2 and V3 is represented at the indicator level, but error in V is manifested at the construct level; specifically, in the disturbance of the MIMIC factor. The single-mimic-factor model in Figure 2.4d with 3 indicators is just-identified (df M = 0). With more than 3 indicators, a measurement model with a single MIMIC factor and a single cause indicator is an equivalent version of a single-factor reflective measurement model where all the indicators are effect indicators (Kline, 200, pp ). There are many examples in the SEM literature of the analysis MIMIC factors. For example, Hershberger (994) described a MIMIC depression factor with indicators that represented various behaviors. Some of these indicators, such as crying and feeling depressed, were specified as effect indicators because they are symptoms of depression. However, another indicator, feeling lonely, was specified as a cause indicator. This is because feeling lonely may cause depression rather than vice-versa. Formative measurement and the analysis of composites is better known in areas such as economics, commerce, management, and biology than in psychology or education. For example, Grace (2006, chap. 6) and Grace and Bollen (2008) described the analysis of composites in the environmental sciences. There is a relatively recent special issue about formative measurement in the Journal of Business Research (Diamantopoulos, 2008). Jarvis, MacKenzie, and Podsakoff (2003) and others advised researchers in the consumer research area and the rest of us, too not to automatically specify factors with effect indicators only because doing so may re-