G Lecture 61 G SEM Lecture 6 An Example Measures of Fit Complex nonrecursive models How can we tell if a model is identified? Direct and Indirect Effects Testing Indirect Effects

G Lecture 62 An Example from JPSP: Effects of Resource Loss /Gain JPSP September 1999 Vol. 77, No. 3, Resource Loss, Resource Gain, and Depressive Symptoms A 10-Year Model Charles J. Holahan, Rudolf H. Moos, Carole K. Holahan, Ruth C. Cronkite This study examined a broadened conceptualization of the stress and coping process that incorporated a more dynamic approach to understanding the role of psychosocial resources in 326 adults studied over a 10-year period. Resource loss across 10 years was significantly associated with an increase in depressive symptoms, whereas resource gain across 10 years was significantly associated with a decrease in depressive symptoms. In addition, change in the preponderance of negative over positive events across 10 years was inversely associated with change in resources during the period. Finally, in an integrative structural equation model, the association between change in life events and depressive symptoms at follow-up was completely mediated through resource change.

G Lecture 63 The Model

G Lecture 64 The Description of Fit Test of hypothesized model. The results of the LISREL test of the hypothesized mediational model are presented graphically in Figure 2, which includes standardized estimates of parameters in the measurement and structural models. The  and  represent unique variance in the observed x and y variables, respectively. The unlabeled arrows pointing to the three endogenous latent variables show the residual (unaccounted-for) variance for each of these variables. The model provided a good fit to the data: overall  2 (16, N = 195) = 25.62, p >.05; goodness-of-fit index =.97; adjusted goodness-of-fit index =.92; normed fit index =.95; nonnormed fit index =.97. All parameter estimates in the measurement model and all hypothesized parameter estimates in the structural model were significant at the.01 level. Change in excess negative events was inversely related to change in psychosocial resources; in turn, change in psychosocial resources was inversely associated with depressive symptoms at follow-up. These effects control for the influence of initial depressive symptoms on all three endogenous variables. Thus, as we predicted, an increase in excess negative events showed an indirect relationship to an increase in depressive symptoms at follow-up, mediated by a decline in psychosocial resources. Initial depressive symptoms were not significantly related to changes in either events or resources.

G Lecture 65 A Fit Measure Worth Considering: RMSEA  Root Mean Square Error of Approximation where F is the minimized fitting function Look for values less than.05  In example

G Lecture 66 Error of Approximation vs. Errors of Estimation Browne, M. W. and Cudeck, R. (1993) Alternative ways of assessing model fit. In Bollen, K.A. & Long, J.S. (eds) Testing structural equation models. Newbury Park, CA: Sage. When measurement models are considered along with structural models, almost all SEM models are misspecified to some extent  We should ask about that extent  We should distinguish between how close the model approximates the true covariance structure and how fuzzy is our estimate of the model.  They find support for RMSEA as an index

G Lecture 67 Revisiting the Holahan et al Structural Model X1 Y2 Y1 Y3   

G Lecture 68 Thinking about Correlated Residuals Holahan assume that the residuals are uncorrelated (  is diagonal)  If there are response biases or other unmeasured variables at work the residuals would be correlated.  With uncorrelated residuals the model is recursive Easily shown to be identified  With completely correlated residuals, the model would not be identified.

G Lecture 69 Illustration of Underidentified Nonrecursive Model Kline (and Bollen) talk about several rules that hint at the identification problem  Count of parameters/covariance; "Order Rule"; "Rank Rule" X1 Y2 Y1 Y3   

G Lecture 610 Count of Parameters and Variance/Covariance Elements With 1+3=4 variables there are 4*3/2=6 covariances and 4 variances The model has five structural paths, three correlation paths, four variance estimates  There are two too many parameters  These calculations do not tell us which parameters need to be constrained The Counting rule is necessary but not sufficient to guarantee identification

G Lecture 611 The Order Condition When one is interested in models with correlated residuals (  nondiagonal) Suppose we have p endogenous (Y) variables For each endogenous variable  The number of excluded potential explanatory must be be greater or equal to (p-1)  We count both exogenous and endogenous explanatory variables This condition alerts us to equations that need to have more excluded explanatory variables.

G Lecture 612 The Order Condition (Bollen’s Matrix Version) Suppose there are p endogenous variables and q exogenous variables  is the pxp matrix of paths linking endogenous variables  is the pxq matrix of paths linking endogenous variables to exogenous variables [  is a px(p+q) matrix summarizing all the possible explanatory paths C = [(  is a px(p+q) matrix that is useful in counting for the order condition  Each row of C should have ≥(p-1) zeros

G Lecture 613 Example of Order Condition Row 1 has two zeros, but rows two and three only have one zero each. The order condition is a necessary but not sufficient condition for identification.

G Lecture 614 Another Identification Check: The Rank Condition Consider the C= [( . For example Form three new smaller matrices, indexed by row  Retain columns that have zero’s in index row. E.g.  See if Rank(C i )=(p-1). This holds for C 1 but not C 2,C 3

G Lecture 615 SPSS can be used to help check Rank COMMENT this illustrates some SPSS matrix manipulations. MATRIX. COMPUTE A={1, 1, 1; 2, 4, 8; 3, 9, 27}. PRINT A. COMPUTE B={1, 2, 3; 2, 3, 4; 3, 4, 5}. PRINT B. COMPUTE RANKA=RANK(A). PRINT RANKA. COMPUTE RANKB=RANK(B). PRINT RANKB. COMPUTE INVA=INV(A). PRINT INVA. COMPUTE D=INVA*B. PRINT D. COMPUTE RANKD=RANK(D). PRINT RANKD. COMPUTE E=A*D. PRINT E. END MATRIX.

G Lecture 616 Checking an alternative just-identified model for Holahan X1 Y2 Y1 Y3   

G Lecture 617 Problems can still arise Inferences about correlated residuals come from relation of X to Y’s  But X1 and Y1 are barely related  See Handout

G Lecture 618 Model as graphed can be shown to be identified with simulation This simulation makes V1 =>V2 connection strong See handout

G Lecture 619 Indirect Effects in Path Models Indirect effects of one variable on another are the effects that go through mediators. For recursive models we calculate indirect effects by forming products of mediating effects. X1 Y2 Y1 Y3   

G Lecture 620 Indirect Effects in Nonrecursive Models In nonrecursive models there is infinite regress The indirect effect can be estimated if the system is in equilibrium  Bollen shows that this is true if B k →0 as k →   This happens when absolute value of largest eigenvalue of B is <1 Y2 Y1 Y3   

G Lecture 621 Nonrecursive Equilibrium Models Exogenous Variables Endogenous Variables Direct Effects  Indirect Effects      Total Effects     

G Lecture 622 Testing Indirect Effects LISREL, EQS, AMOS all compute large sample standard errors for indirect effects Baron and Kenny recommend using this standard error to test the indirect test using usual normal theory. Call the estimate I  Reject H 0 : indirect effect=0 if |[I/se(I)]|>1.96  However, test assumes that the estimate is normally distributed  Often the sampling distribution is skewed. ^ ^^

G Lecture 623 A Bootstrap Confidence Interval for Indirect Effect Amos provides a convenient method for estimating sampling variability of the estimates of indirect effects. The use of the bootstrap is described in Shrout & Bolger (2002) Psychological Methods In the example, the indirect effect of Y1(V2) on Y3(V4) is.208 (.035) according to EQS.