Controlling for Baseline

David A. Kenny

December 15, 2013

VariablesOutcome 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.

3

Equation

Y2 = a + Y1 + bX + e

X and Y1 as predictors

Comparable AnalysesAnalysis 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)

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) .

The estimated equals where 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 ReliabilityReliability or 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 by (- rXY12)/(1 - rXY1

2).

Williams & Hazer uses the reliability of Y1 so no adjustment is necessary.

Lord-Porter CorrectionReliability or Y1 controlling for X or must

be known.

Regress Y1 on X (and covariates) and compute the predicted score (P) and the residual (R)

Compute: Y1ʹ = P + R (is the reliability)

Regress

Y2 on Y1ʹ and X

Hardly ever done, but does not require a SEM program.

Williams & HazerError variance of for the T1 latent variable is

fixed to sY12(1- where is the known

reliability of Y1.

Latent VariableMultiple 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.

MoreTechnically 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.