SOC 681 – Causal Models with Directly Observed Variables James G. Anderson, Ph.D. Purdue University

Types of SEMs Regression Models Path Models Recursive Nonrecursive

Class Exercise: Example 7 SEMs with Directly Observed Variables Felson and Bohrnstedt’s study of 209 girls from 6th through 8th grade Variables Academic: Perceived academic ability Attract: Perceived attractiveness GPA: Grade point average Height: Deviation of height from the mean height Weight: Weight adjusted for height Rating: Rating of physical attractiveness

Assumptions Relations among variables in the model are linear, additive and causal. Curvilinear, multiplicative and interaction relations are excluded. Variables not included in the model but subsumed under the residuals are assumed to be not correlated with the model variables.

Assumptions Variables are measured on an interval scale. Variables are measured without error.

Objectives Estimate the effect parameters (i.e., path coefficients). These parameters indicate the direct effects of a variable hypothesized as a cause of a variable taken as an effect. Decompose the correlations between an exogenous and endogenous or two endogenous variables into direct and indirect effects. Determine the goodness of fit of the model to the data (i.e., how well the model reproduces the observed covariances/correlations among the observed variable).

AMOS Input ASCII SPSS Microsoft Excel Microsoft Access Microsoft FoxPro dBase Lotus

AMOS Output Path diagram Structural equations effect coefficients, standard errors, t-scores, R2 values Goodness of fit statistics Direct and Indirect Effects Modification Indices.

Model One

Decomposing the Effects of Variables on Achievement Direct Indirect Total Sex -.03 - FatherEd .17 Ethnic IndTrng .23* AStress -.17* ActMast .02 SelfCon .42*

Model Two

Goodness of Fit: Model 2 Chi-Square = 29.07 df = 15 p < 0.06 RMSEA = 0.086 GFI = 0.94 AGFI = 0.85 AIC = 67.82

Chi Square: 2 Best for models with N=75 to N=100 For N>100, chi square is almost always significant since the magnitude is affected by the sample size Chi square is also affected by the size of correlations in the model: the larger the correlations, the poorer the fit

Chi Square to df Ratio: 2/df There are no consistent standards for what is considered an acceptable model Some authors suggest a ratio of 2 to 1 In general, a lower chi square to df ratio indicates a better fitting model

Root Mean Square Error of Approximation (RMSEA) Value: [ (2/df-1)/(N-1) ] If 2 < df for the model, RMSEA is set to 0 Good models have values of < .05; values of > .10 indicate a poor fit.

GFI and AGFI (LISREL measures) Values close to .90 reflect a good fit. These indices are affected by sample size and can be large for poorly specified models. These are usually not the best measures to use.

Akaike Information Criterion (AIC) Value: 2 + k(k-1) - 2(df) where k= number of variables in the model A better fit is indicated when AIC is smaller Not standardized and not interpreted for a given model. For two models estimated from the same data, the model with the smaller AIC is preferred.

Model Building Standardized Residuals ACH – Ethnic = 3.93 Modification Index ACH – Ethnic = 10.05

Model Three

Goodness of Fit: Model 3 Chi-Square = 16.51 df = 14 p < 0.32 RMSEA = 0.037 GFI = 0.96 AGFI = 0.90 AIC = 59.87

Comparing Models Chi-Square Difference = 12.56 df Difference = 1 p < .0005 AIC Difference = 7.95

Difference in Chi Square Value: X2diff = X2 model 1 -X2 model 2 DFdiff = DF model 1 –DFmodel 2

Decomposing the Effects of Variables on Achievement Direct Indirect Total Sex - .09 FatherEd .- .06 Ethnic .29 .05 .34 IndTrng .25 .04 AStress -.14 -.03 -.17 ActMast .13 SelfCon .44

Class Exercise: Example 7 SEMs with Directly Observed Variables Attach the data for female subjects from the Felson and Bohrnstedt study (SPSS file Fels_fem.sav) Fit the non-recursive model Delete the non-significant path between Attract and Academic and refit the model Compare the chi square values and the AIC values for the two models

Class Exercise: Example 7 SEMs with Directly Observed Variables Felson and Bohrnstedt’s study of 209 girls from 6th through 8th grade Variables Academic: Perceived academic ability Attract: Perceived attractiveness GPA: Grade point average Height: Deviation of height from the mean height Weight: Weight adjusted for height Rating: Rating of physical attractiveness