Controlling for Baseline David A. Kenny

Variables Outcome variable Y Baseline measure: Y1 Follow-up measure: Y2 Causal variable X measured at time 1 The question is how to measure the effect of X on Y2 and control for Y1.

Equation Y2 = a + bY1 + bX + e X and Y1 as predictors

Comparable Analyses Analysis of Covariance with Y1 as a covariate. Control for Y1, but make the outcome change or Y2 – Y1 Residualized change (or gain) score analysis Regress Y2 on Y1 and compute residuals Regress X on Y1 and compute residuals (usually not done) Regress the first residual on the second Reduce df by 1. (usually not done)

b

Measurement Error in Y1 Assume Y1 mediation of the Z (the assignment variable) to Y2. The measurement error in Y1 attenuates (pushes it toward zero) b. The estimated b equals br where r is the reliability of the pretest (controlling for X). Not enough of the “gap” is subtracted.

Solutions to the Measurement Error Problem Known Reliability Solutions Lord-Porter Correction Williams & Hazer Strategy Reliability Estimation Solution Latent Variable Analysis with Multiple Indicators

Known Reliability Reliability or r must be known. Perhaps use an internal consistency estimate. Lord-Porter uses the reliability of Y1 controlling for X not the reliability of Y1. Must adjust r by (r - rXY12)/(1 - rXY12). Williams & Hazer uses the reliability of Y1 so no adjustment is necessary.

Lord-Porter Correction Reliability or Y1 controlling for X or r must be known. Regress Y1 on X (and covariates) and compute the predicted score (P) and the residual (R) Compute: Y1ʹ = P + rR (r is the reliability) Regress Y2 on Y1ʹ and X Hardly ever done, but does not require a SEM program.

Williams & Hazer Error variance of for the T1 latent variable is fixed to sY12(1- r) where r is the known reliability of Y1. b

Latent Variable Multiple indicators of latent Y at each time Set up a Latent Variable Model Test to see that the loadings are the same at each time. To be safely identified, need at least 3 indicators at each time. Correlate errors of the same indicator at different times.

b

More Technically need only a Time 1 latent variable and no Time 2 latent variable. If latent variables at two times, do not necessarily need temporally invariant loadings though you do with CSA.