1. G89.2247 Lect 21 G89.2247 Lecture 2 Regression as paths and covariance structure Alternative “saturated” path models Using matrix notation to write linear models Multivariate Expectations Mediation

2. G89.2247 Lect 22 Question: Does exposure to childhood foster care (X) lead to adverse outcomes (Y) ? Example of purported "causal model" X Y Y = B 0 + B 1 X + e Regression approach  B 0 and B 1 can be estimated using OLS  Estimates depend on sample standard deviations of Y and X, sample means, and covariance between Y and X B 1 = S XY /S 2 X B 0 = M Y -B 1 M X  Correlation, r XY = S XY /S X S Y, can be used to estimate the variance of the residual, e, V(e). S 2 e = S 2 Y (1-r 2 XY ) = S 2 Y - S 2 XY /S 2 X e B1B1

3. G89.2247 Lect 23 A Covariance Structure Approach If we have data on Y and X we can compute a covariance matrix This estimates the population covariance structure,    Y can itself be expressed as B 2 1  2 X +  2 e  Three statistics in the sample covariance matrix are available to estimate three population parameters

4. G89.2247 Lect 24 Covariance Structure Approach, Continued A structural model that has the same number of parameters as unique elements in the covariance matrix is "saturated". Saturated models always fit the sample covariance matrix.

5. G89.2247 Lect 25 Another saturated model: Two explanatory variables The first model is likely not to yield an unbiased estimate of foster care because of selection factors (Isolation failure). Suppose we have a measure of family disorganization (Z) that is known to have an independent effect on Y and also to be related to who is assigned to foster care (X) Y X Z e    XZ

6. G89.2247 Lect 26 Covariance Structure Expression The model: Y=b 0 +b 1 X+b 2 Z+e  If we assume E(X)=E(Z)=E(Y)=0  and V(X) = V(Z) = V(Y) = 1  then b 0 =0 and  's are standardized The parameters can be expressed  When sample correlations are substituted, these expressions give the OLS estimates of the regression coefficients.

7. G89.2247 Lect 27 Covariance Structure: 2 Explanatory Variables In the standardized case the covariance structure is: Each correlation is accounted by two components, one direct and one indirect There are three regression parameters and three covariances.

8. G89.2247 Lect 28 The more general covariance matrix for two IV multiple regression If we do not assume variances of unity the regression model implies

9. G89.2247 Lect 29 More Math Review for SEM Matrix notation is useful

10. G89.2247 Lect 210 A Matrix Derivation of OLS Regression OLS regression estimates make the sum of squared residuals as small as possible.  If Model is Then we choose B so that e'e is minimized. The minimum will occur when the residual vector is orthogonal to the regression plane  In that case, X'e = 0

11. G89.2247 Lect 211 When will X'e = 0? When e is the residual from an OLS fit.

12. G89.2247 Lect 212 Multivariate Expectations There are simple multivariate generalizations of the expectation facts:  E(X+k) = E(X)+k =  x +k  E(k*X) = k*E(X) = k*  x  V(X+k) = V(X) =  x 2  V(k*X) = k 2 *V(X) = k 2 *  x 2 Let X T =[X 1 X 2 X 3 X 4 ],  T =[         ] and let k be scalar value  E(k*X) = k*E(X) = k*   E(X+k* 1 ) = { E(X) + k* 1} =  + k* 1

13. G89.2247 Lect 213 Multivariate Expectations In the multivariate case Var(X) is a matrix  V(X)=E[(X-  ) (X-  ) T ]

14. G89.2247 Lect 214 Multivariate Expectations The multivariate generalizations of  V(X+k) = V(X) =  x 2  V(k*X) = k 2 *V(X) = k 2 *  x 2 Are:  Var( X + k* 1 ) =   Var(k* X ) = k 2  Let c T = [c 1 c 2 c 3 c 4 ]; c T X is a linear combination of the X's.  Var( c T X) = c T  c This is a scalar value If this positive for all values of c then  is positive definite

15. G89.2247 Lect 215 Semi Partial Regression Adjustment The multiple regression coefficients are estimated taking all variables into account  The model assumes that for fixed X, Z has an effect of magnitude  Z.  Sometimes people say "controlling for X" The model explicitly notes that Z has two kinds of association with Y  A direct association through  Z (X fixed)  An indirect association through X (magnitude  X  XZ )

16. G89.2247 Lect 216 Pondering Model 1: Simple Multiple Regression The semi-partial regression coefficients are often different from the bivariate correlations  Adjustment effects  Suppression effects Randomization makes  XZ = 0 in probability. Y X Z e    XZ

17. G89.2247 Lect 217 Mathematically Equivalent Saturated Models Two variations of the first model suggest that the correlation between X and Z can itself be represented structurally. Y X Z eYeY   eZeZ  Y X Z eYeY   eXeX 

18. G89.2247 Lect 218 Representation of Covariance Matrix Both models imply the same correlation structure The interpretation, however, is very different.

19. G89.2247 Lect 219 Model 2: X leads to Z and Y X is assumed to be causally prior to Z.  The association between X and Z is due to X effects. Z partially mediates the overall effect of X on Y  X has a direct effect  1 on Y  X has an indirect effect      on Y through Z  Part of the bivariate association between Z and Y is spurious (due to common cause X) Y X Z eYeY   eZeZ 

20. G89.2247 Lect 220 Model 3: Z leads to X and Y Z is assumed to be causally prior to X.  The association between X and Z is due to Z effects. X partially mediates the overall effect of Z on Y  Z has a direct effect  2 on Y  Z has an indirect effect      on Y through X  Part of the bivariate association between X and Y is spurious (due to common cause Z) Y X Z eYeY   eXeX 

21. G89.2247 Lect 221 Choosing between models Often authors claim a model is good because it fits to data (sample covariance matrix)  All of these models fit the same (perfectly!) Logic and theory must establish causal order There are other possibilities besides 2 and 3  In some instances, X and Z are dynamic variables that are simultaneously affecting each other  In other instances both X and Z are outcomes of an additional variable, not shown.

22. G89.2247 Lect 222 Mediation: A theory approach Sometimes it is possible to argue on theoretical grounds that  Z is prior to X and Y  X is prior to Y  The effect of Z on Y is completely accounted for by the indirect path through X. This is an example of total mediation If   is fixed to zero, then Model 3 is no longer saturated.  Question of fit becomes informative  Total mediation requires strong theory

23. G89.2247 Lect 223 A Flawed Example Someone might try to argue for total mediation of family disorganization on low self-esteem through placement in foster care Baron and Kenny(1986) criteria might be met  Z is significantly related to Y  Z is significantly related to X  When Y is regressed on Z and X,   is significant but   is not significant. Statistical significance is a function of sample size. Logic suggests that children not assigned to foster care who live in a disorganized family may suffer directly.

24. G89.2247 Lect 224 A More Compelling Example of Complete Mediation If Z is an experimentally manipulated variable such as a prime X is a measured process variable Y is an outcome logically subsequent to X  It should make sense that X affects Y for all levels of Z  E.g. Chen and Bargh (1997) Are participants who have been subliminally primed with negative stereotype words more likely to have partners who interact with them in a hostile manner?