G Lecture 81 Comparing Measurement Models across Groups Reducing Bias with Hybrid Models Setting the Scale of Latent Variables Thinking about Hybrid Model Fit Recap of Measurement Model Issues

G Lecture 82 Strategies for Comparing Groups Are all paths exactly the same? Are all paths except the residual variances the same? Are residual variances and factor variances different? Are loadings the same, but factor correlations, variances and residual variances different? Are correlations the same, but variances and loadings different?

G Lecture 83 Payoff of Measurement Models in SEM Bias Reduction Measures often are contaminated by noise  Transient changes in subjects, random variation in rater selection, instrument failures, and so on. Noise in explanatory variables leads to biased structural effects Noise in covariates or control variables leads to underadjusted (biased) structural effects of variables that are measured perfectly Hybrid models (when properly specified) reduce or eliminate bias

G Lecture 84 Psychometric details of bias Suppose we have the simple structural model Y = b 0 + b 1 X + r.  b 0 and b 1 can be estimated without bias if X is measured without error Suppose X * =X+e is a contaminated version of X  Suppose e~N(0,   ), Then E(X * )=X If we estimate Y = b * 0 + b * 1 X * + r * b * 1 = R XX b 1, where R XX is the reliability coefficient for X *

G Lecture 85 Example of Bias: Single Predictor I simulated a data set with N=1000 with  X 2 = 24 .5*X 1 + r  I then created added a version X 1A of X (by adding error variance equal to the variance of X). Several noisy versions can be created. These have R XX of.50 See SPSS listing  Compare regressions of error free variables with noisy variables

G Lecture 86 Example of Bias: Single Predictor Simulated (true) model: X 2 = *X 1 Estimate of true model X 2 = *X 1 Estimate of model when variables have reliability of.5 X 2A = *X 1A Note that coefficient is about.5 less than true value

G Lecture 87 Example of Bias: Two Predictors Measurement error of an independent variable in multiple regression has two effects  Its own coefficient is biased  It incompletely adjusts other variables for its effect Adjustment is an important reason for multiple regression  We may want to adjust for selection effects by including measures of social class, IQ, depressed affect and so on

G Lecture 88 Example of Bias continued: Two Predictors Simulated (true) model: Y = *X 1 +.6*X 2 Estimate of true model Y = *X *X 2 Estimate of model when variables have reliability of.5 Y = *X *X 2 Note that the bias is not a simple function of R XX

G Lecture 89 Hybrid Model: Taking Measurement Error into Account

G Lecture 810 Setting the Scale of Latent Variables The latent variables can be scaled to have variance 1 They can also be scaled to have units like the original indicators The indicators themselves do not have to be in the same units, so long as the units are linearly related  E.G. inches, mm, cm

G Lecture 811 Setting the Scale, Continued In simple confirmatory factor analysis we tend to set the variance of latent variables to 1.0 In hybrid models, we tend to set the scale of the latent variables to the units that have meaning for the structural model  Choosing the "best" indicator to have loading set to 1.0  Standardized versions of the analysis are always available as well

G Lecture 812 Fit of Hybrid Models Hybrid models may not fit for two reasons  Measurement part  Structural part Kline and others recommend a two step model fit exercise  Test fit of CFA with no structural model  Impose additional constraints due to structural model  Don't claim validity for structural model that arises from a good fitting measurement model

G Lecture 813 How Fitted Variances and Covariances are Represented in EQS, AMOS, LISREL Consider nine variable system with three latent variables EQS LISREL E5 V1 V2 V3 V4 V5 V6 V7 V8 V9 F1 F2 F3 E1 E9 E8 E7 E6 E4 E3 E2 D3 D2 X1 X2 X3 Y1 Y2 Y3 Y4 Y5 Y6              

G Lecture 814 Structural Equation Forms

G Lecture 815 Matrix Algebra Version of LISREL X =  X  Y =  Y   = 

G Lecture 816 Variances of Exogenous Variables (in LISREL) The variances of the predicted (endogenous) variables are calculated from the structural models The variances of the exogenous variables, however, must be specified (defined):  Var(    Var(    Var(   Var 

G Lecture 817 Variances of Endogenous Variables Can be Expressed as Functions of Parameters Var(X) = Var(  X  ) =  X  X T   Var(Y) = Var(  Y  ) =  Y [Var(  )]  Y T   Var(  ) = Var    )] =   Var(  )[   ] T =   [  T  [   ] T

G Lecture 818 Voila! The Fitted Variance/Covariance Matrix Can Be Written Once the form of the model is specified, and the parameters indicated, we can begin to fit the variance covariance matrix of the data.

G Lecture 819 Issues in Measurement Models Model identification Number of factors Second order factor models? Scaling of latent variables Influence of parts of model on overall fit Naming of latent variables Reification of latent variables Items, Item parcels Overly simple factor models