James G. Anderson, Ph.D. Purdue University Model Building James G. Anderson, Ph.D. Purdue University

Causal Relationships Imply: An association between the variables Lack of spuriousness of the relation Evidence to support the direction of causality

Research Designs Experimental Designs Spuriousness Randomization Direction of Causality Manipulation

Research Designs (2) Cross Sectional Data Specification Direction of Causality Not dealt with in recursive models Nonrecursive Models Impose restrictive constraints to identify model

Research Designs (3) Panel Designs Evaluate direction of causality Estimate the stability of endogenous variables over time

Research Designs (4) Latent Variable Models Disattentuated relations can be estimated by taking random error into account in the measures Conceptualize constructs as made up of multiple indicators The effects of fixing or constraining parameters can be examined Better represent complex social-psychological processes

Recursive Model Includes directly observed variables Presumes no measurement error Involves one-way causal relations The model hypothesizes that family socialization affects achievement both directly and indirectly through achievement values and self-concept

Parameter Estimates: Recursive Model Effects Parameter Z Ethnicity on Achievement Values 0.36 4.22 Ethnicity on Self-concept 0.21 2.07   Sex on Independence Training 0.31 3.30 Father's Ed. On Independence Training 0.30 3.23 Father's Ed. On Stress on Achievement 0.17 1.68 Father's Ed. On Achievement (t1) 0.29 3.73

Parameter Estimates: Recursive Model (2) Effects Parameter Z Independence Training on Achievement Values 0.29 3.38 Independence Training on Achievement (t1) 0.24 3.17   Stress on Achievement on Achievement Values -0.24 -2.9 Stress on Achievement on Achievement (t1) -0.19 -2.61 Achievement Values on Self Concept 0.20 1.93 Self-Concept on Achievement (t1) 0.45 6.07

Goodness of Fit: Recursive Model Chi square: 10.93 df: 12 Significance: NS Chi Square (null): 149.39 df: 20 Significance: Sig Δf : 0.93 Adjusted Goodness of Fit Index: 0.92 Root Mean Square Residual: 0.05

Nonrecursive Model Includes directly observed variables Assumes no measurement error Assumes reciprocal causation between Achievement and Self-Concept of Ability

Parameter Estimates: Nonrecursive Model Effects Parameter Z Ethnicity on Achievement Values 0.36 4.22 Ethnicity on Self-concept 0.26 1.72   Sex on Independence Training 0.31 3.30 Father's Ed. On Independence Training 0.30 3.23 Father's Ed. On Stress on Achievement 0.17 1.68 Father's Ed. On Achievement (t1) 0.21 2.18

Parameter Estimates: Nonrecursive Model (2) Effects Parameter Z Independence Training on Achievement Values 0.29 3.38 Independence Training on Achievement (t1) 0.21 2.61   Stress on Achievement on Achievement Values -0.24 -2.90 Stress on Achievement on Achievement (t1) -0.17 -2.12 Achievement Values on Self Concept 0.18 1.35 Self-Concept on Achievement (t1) 0.86 2.89 Achievement (t1) on Self-concept -0.14 -0.13

Goodness of Fit: Nonrecursive Model Chi square: 8.32 df: 10 Significance: NS Chi Square (null): 149.39 df: 20 Δf : 0.94 Adjusted Goodness of Fit Index: 0.93 Root Mean Square Residual: 0.04

Incremental Fit Indices Consider hierarchically nested models: Mk, Mt, Mo where: Mk is the most restricted model Mo is the null model The models can then be evaluated relative to each other The Normed Fit Index: Δf = Xk2 – Xt2 Xo2 0 < Δf < 1

Models 1 and 2 The null model involves only: Variances and covariances of exogenous variables Variances of the endogenous variables No structural relations among the endogenous and exogenous variables

Chi Square Difference Test To compare nested models, if M1 can be obtained from M2 by constraining one or more parameters of M2: Χ2 = Χ12 – Χ22 df = df1 – df2

Chi Square Difference Test (3) XR2 = 10.93 XNR2 = 8.32 Xo2 = 149.39 For the recursive model: Δf = 149.39 – 10.93 = 0.92 149.39 For the nonrecursive model: Δf = 149.39 – 8.32 = 0.93 To compare the two models: Δf = 10.93 – 8.32 = 0.017

Longitudinal Model Includes directly observed variables Assumes no measurement error Includes Achievement measures at two points in time Assume time lagged effects of Achievement on Self-concept of Ability and Self-concept on Achievement

Parameter Estimates: Longitudinal Model Effects Parameter Z Ethnicity on Achievement Values 0.32 3.75 Ethnicity on Self-concept 0.18 1.84   Sex on Independence Training 0.31 3.3 Father's Ed. On Independence Training 0.3 3.23 Father's Ed. On Stress on Achievement 0.17 1.68 Father's Ed. On Achievement (t0) 0.33 3.43 Father's Ed. On Achievement (t1) 0.14 1.87

Parameter Estimates: Longitudinal Model (2) Effects Parameter Z Independence Training on Achievement Values 0.26 3.02 Independence Training on Achievement (t0) 0.13 1.42 Independence Training on Achievement (t1) 0.17 2.94   Stress on Achievement on Achievement Values -0.20 -2.29 Stress on Achievement on Achievement (t1) -0.30 -3.27 Achievement Values on Self Concept 0.11 1.07 Self-Concept on Achievement (t1) 0.27 4.79 Achievement (t0) on Achievement Values 0.20 2.22 Achievement (t0) on Self-Concept 2.73 Achievement (t0) on Achievement (t1) 0.60 3.42

Goodness of Fit: Longitudinal Model Chi square: 23.21 df: 14 Significance: NS Chi Square (null): 264.44 df: 27 Δf : 0.91 Adjusted Goodness of Fit Index: 0.86 Root Mean Square Residual: 0.05

Latent Variable Model Assumes that a latent variable, control, underlies the two observed variables: Achievement values and Self-concept of Ability Both measures of Achievement are adjusted for unreliability It is assumed that the errors in the measurement of Achievement at Time 1 and Time 2 are correlated.

Parameter Estimates: Latent Variable Model Effects Parameter Z Ethnicity on Control 0.52 4.28   Sex on Independence Training 0.31 3.30 Father's Ed. On Independence Training 0.30 3.23 Father's Ed. On Stress on Achievement 0.17 1.68 Father's Ed. On Achievement (t0) 0.34 3.43

Parameter Estimates: Latent Variable Model (2) Effects Parameter Z Independence Training on Control 0.37 3.56 Independence Training on Achievement (t0) 0.14 1.43   Stress on Achievement on Achievement (t0) -0.32 -3.33 Control on Achievement (t1) 0.59 3.86 Achievement (t0) on Control 0.61 4.06 Achievement (t0) on Achievement (t1) 0.23 1.19

Goodness of Fit: Latent Variable Model Chi square: 22.27 df: 17 Significance: NS Chi Square (null): 255.37 df: 28 Δf : 0.91 Adjusted Goodness of Fit Index: 0.88 Root Mean Square Residual: 0.06

Model 4 The null model involves only: Variances and covariances of exogenous variables Variances of exogenous variables No structural relations among the exogenous and endogenous variables

Model 4 (2) Factor loadings equal to 1 XLV2 = 22.27 X02 = 255.37

Results Several variables have larger effects than previous models: Ethnicity Control Lagged effects of Family Socialization: Ind. Training Ach (t0) Control Ach (t1) Stress on Ach.

Results (2) Stability coefficients β73 is much smaller than earlier estimates: Achievement (t0) Achievement (t1) Lagged effects of Achievement on Control is larger than its effect on (Y4) Achievement Values (Y5) Self-Concept than in the previous model.

Results (3) Control has a stronger effect on Achievement (t1) than Self-Concept in the previous model.

Inferences Control has a direct effect (0.59): Control Ach(t1) Ethnicity has an indirect effect (0.31): Ethnicity Control Ach(t1) Father’s education has an indirect effect (0.26): Father’s Ed Ach Train. Ach(t0) Ach(t1) Control

Inferences (2) Achievement has a lagged feedback effect through Control