soc 681 – causal models with directly observed variables

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 7SEMs 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

GPA

HEIGHT

WEIGHT

RATING

ACADEMIC

ATTRACT

e1

e2

1

1

GPA

HEIGHT

WEIGHT

RATING

ACADEMIC

ATTRACT

e1

e2

1

1

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

Variables Direct Indirect Total

Sex -.03 - -.03

FatherEd .17 - .17

Ethnic .17 - .17

IndTrng .23* - .23*

AStress -.17* - -.17*

ActMast .02 - .02

SelfCon .42* - .42*

Model Two

Goodness of Fit: Model 2

Chi-Square = 29.07df = 15

p < 0.06 Chi-Square/df = 1.8 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 ResidualsACH – Ethnic = 3.93

Modification IndexACH – Ethnic = 10.05

Model Three

Goodness of Fit: Model 3

Chi-Square = 16.51df = 14

p < 0.32 Chi-Square/df = 1.08 RMSEA = 0.037 GFI = 0.96 AGFI = 0.90 AIC = 59.87

Comparing Models

Chi-Square Difference = 12.56df 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

Variables Direct Indirect Total

Sex - .09 .09

FatherEd .- .06 .06

Ethnic .29 .05 .34

IndTrng .25 .04 .29

AStress -.14 -.03 -.17

ActMast - .13 .13

SelfCon .44 - .44

Class Exercise: Example 7SEMs 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 7SEMs 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

GPA

HEIGHT

WEIGHT

RATING

ACADEMIC

ATTRACT

e1

e2

1

1

GPA

HEIGHT

WEIGHT

RATING

ACADEMIC

ATTRACT

e1

e2

1

1