4. Path Analysis. In the diagram: The technique of path analysis is originated by (American) geneticist Sewell Wright in early 1920.

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1 4. Path Analysis The technique of path analysis is originated by (American) geneticist Sewell Wright in early The relationships between variables are presented in a path diagram. The system of relationships involves two kinds of variables: indpendent or cause variables x 1,...,x q and dependent (or effect or response) variables y 1,...,y p. In the diagram: A straight, one-headed arrow represents a causal relationship between two variables. A curved, two-headed arrow represent correlation between the variables. Boxed variable in a diagram indicates an observed variable. Circled variable indicates unobserved variable. Error terms (which are unobserved) are sometimes indicated without frames. 1 2 In traditional path analysis variables are all directly observed. Given the correlations the path coefficients were solved (if possible) from appropriate equations. Example 4.1: A simple path diagram c x 1 x 2 x 3 a b d y ɛ There are causal links from x-variables to y, x 1 and x 2 are correlated, and x 3 is uncorrelated with other x-variables. Thus, the correlation between x 3 and y indicates directly the causal link from x 3 to y, i.e. d = r y,x3. The other relationships are solved as follows: r x1,x2 = c (1) r y,x1 = a + bc r y,x2 = b + ac Solving these gives, (2) a = ry,x 1 ry,x 2 rx 1,x 2 1 r 2 x 1,x 2 b = ry,x 2 ry,x 1 rx 1,x 2 1 r 2 x 1,x 2 c = r x1,x2 d = r y,x3 The (unobserved) error term is ɛ. 3 4

2 The coefficient in the above example are, in fact, standardized regression coefficients, called path coefficients (or beta coefficients). As seen, the graph leads to certain equations for the correlations between the variables. Thus this is one form of more general covariance structure analysis. Direct and Indirect Effects A direct (one headed) arrow from one variable to a response variable indicates a direct effect. The effect can go also via another variable. Such an effect is called indirect causal effect. 5 6 Example 4.2: Consider the following path diagram. The indirect effect are calculated by multiplying the path coefficients on the way. φ 21 x 1 x 2 x 3 γ 11 y 1 γ ζ1 12 γ 22 β 21 γ 13 y 2 ζ2 γ 23 The total effect is the sum of the direct and indirect effect. Example 4.3: In the above diagram the direct effect form x 2 to y 2 is γ 22 and the indirect effect γ 12 β 21. The total effect is γ 22 + γ 12 β 21. x 2, x 3 have in addition to the direct causal effects on both y 1 and y 2 also an indirect causal effect on y 2 via y 1 (which has also a direct effect on y 2. x 1 has only an indirect causal effect on y 2 (via y 1 ). 7 8

3 Example 4.4: Ambition and attainment. The following data is a sample of n = 767 twelfthgrade males. Model Specification: Suppose that the researchers have come up on the basis of theoretical considerations with following specification intl (x 1 ): Intelligence sibl (x 2 ): Number of siblings fedu (x 3 ): Father s education focc (x 4 ): Father s occupation grad (y 1 ): Grades eexp (y 2 ): Educational expectation occa (y 3 ): Occupational aspiration ========= x1 x2 x3 x4 y1 y2 y intl 1 sibl fedu focc grad eexp occa ========= 9 10 Identification: Altogether there are 7 (7 1)/2 = 21 correlations and 7 unit variances, thus altogether 28 variances and correlations in the correlation matrix. Estimation: LISREL gives the following estimates for the path coefficients. Thus, maximum 28 free parameters can be estimated. The above specification turns to be again saturated, because there are 4 3 = 12 causal links (coefficients) from the exogenous variables (x-variables) to the endogenous variables (y-variables), 3 links between y variables, 3 error variances, and 6 covariances between the x-variables, and 4 variances, thus altogether 28 free parameters to be estimated. In equation form the estimates are: 11 12

4 Testing and Modification: On the basis of the t-values, we find and many of the coefficient estimates are not statistically significant. Considering all those coefficients insignificant whose t-values are smaller (in absolute values) than 2, we get an over-identified model with estimates: We can use the modification indices to check which link(s) should not have been deleted from the model (View menu) and get The p-value of the chi-square goodness of fit statistic tells that the model does not fir in a satisfactory manner into the data. This suggests that the link focc eexp should have preserved. Thus, we reduced the model probably too much Returning this and re-estimating produces: We found above that for example, the direct effect from int occa was not statistically significant. However, intl grad, grad occa, intl eexp, and eexp occa are allhighly significant, so that the (positive) link seems through these. Now the model seems to fit the data and all the coefficient estimates are statistically significant

5 In summary the direct, indirect, and total effects are the following: Total Effects of X on Y =========== intl sibl fedu focc grad eexp occa ============ Indirect Effects of X on Y =========== intl sibl fedu focc grad eexp occa =========== Total Effects of Y on Y grad eexp occa grad eexp occa Indirect Effects of Y on Y grad eexp occa grad eexp occa Path models are recursive such that there are no feedback relations among the y-variables. Furthermore, the error terms are not allowed to correlate. These restriction imply that we can in fact estimate the path model as a series of single regression models. The direct and indirect effects can be solved simply by multiplying the relevant path coefficients. Remark 4.1: Traditionally path analysis has been applied to correlation matrices (standardized data), but covariance matrices (non-standardized raw data) apply equally well. 18 Next we move to the case, where the cause and effect variables are allowed to be latent variables (unobservable) that are observed via indicator variables. Latent variables related to x variables are denoted by ξ (xi) and those related to y variables by η (eta). Example 4.5: Example 4.2 in terms of latent variables. ξ 1 γ 11 φ 21 η 1 γ 12 ξ 2 γ 22 β 21 γ 13 η 2 ξ 3 γ 23 ζ1 ζ

6 Suppose we measure each of the latent variables by two indicator variables. The schematic setup of the full model is Compared to the traditional path model the general latent variable structural equation model is an extension in at least two important ways. δ 1 δ 2 δ 3 δ 4 δ 5 δ 6 x 1 x 2 x 3 x 4 x 5 x 6 ξ 1 γ 11 η φ 21 1 γ 12 ζ 1 ξ 2 ψ12 γ 22 β 21 ζ 2 η γ 13 2 ξ 3 γ 23 y 1 y 2 y 3 y 4 ɛ1 ɛ2 ɛ3 ɛ4 It allows for dealing with latent variables via measurement models and non-recursive relationships between the dependent variables (η -variables). Before introducing the general SEM, we consider some topics related to the measurement models