G89.2247 Lecture 9 1

G89.2247Lecture 9

• Measurement Error Models

• Bias due to measurement error

• Adjusting for bias with structural equation models

• Examples

• Alternative models that remind us of the limits of non-experimental data

G89.2247 Lecture 9 2

Measurement Error Models

• Suppose that T and V are random variables that are correlated TV in the population.For example, T might be current depressed mood and V

might be level of support sought in current day

• Suppose we cannot measure T and V directly, but instead have fallible measures X and YX = T + ex

Y = V + ey

• The reliability coefficients RXX and RYY tell us what proportion of the variance of X and of Y are due respectively to the variance of T and V

G89.2247 Lecture 9 3

Numerical Example

• Suppose Y1=V+E1Var(Y1) = 1.887Var(V) = 1.00R11 = .53

• We normally estimate the reliability by getting replicate measures of Y and looking at their correlation pattern.Test-retestInternal consistency

G89.2247 Lecture 9 4

Correlations of Fallible Scores

• Let's suppose E[X]=E[Y]=E[T]=E[V]=0Allows us to forget about means in Var() and Cov()

•

• Cov(X,Y) = E[XY]=E[(T+ex)(V+ey)] = E[TV]+E[Tey]+E[Vex]+E[exey] = E(TV) = TV

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),(

G89.2247 Lecture 9 5

Attenuation Results

• Correlations between fallible variables are smaller than correlations between the corresponding error free variables. If we can estimate RXX and RYY we can estimate the attenuation effect.

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G89.2247 Lecture 9 6

Example

• Suppose X=T+E1RXX = .53

• Suppose also Y = V+E2RYY = .39

• If Corr(F1,F2) = .58, thenCorr(X,Y) = .26 = .58*(.53*.39).5

• Notice that the square root of the product of the reliabilities can be thought as a geometric mean of the two reliabilities.

G89.2247 Lecture 9 7

Attenuation in Regression

• Suppose we were interested inV = B0 + B1T + rv

• But we only could observeY = b0 + b1X + ry

• What is the relation of b1 to B1?

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G89.2247 Lecture 9 8

Attenuation in Multiple Regression

• Measurement error produces bias in regression estimates in multiple regressionThe bias is not always positiveError attenuates correlations, but partial regression

coefficients are incompletely adjusted, often leading to estimates that are too large

• Correcting for bias using reliability estimates can be riskyReliability is often underestimated

G89.2247 Lecture 9 9

Numerical Example: Regressed Change

• Suppose there is a relation known to existT2 = .6*T1 + .1*X + r

• T might be distress and X might be level of environmental stress

• Suppose T1 and X are correlated .32

• Suppose, however, that T1 and T2 are measured with about 50% measurement error.Call the fallible measures Y1 and Y2 In a simulation with N=400 the estimates of the effects were

Y2 = .225*Y1 + .157*X + rThe effect of Y1 is too small and the effect of X is too big

G89.2247 Lecture 9 10

Adjusting Bias Through Latent Variables and SEM

• If it is possible to obtain several replicate measures of key variables, it may be possible to adjust for bias due measurement error.

• If the replicate measures were perfect replications ("parallel measures"), the error models for Ya, Yb, Yc would beYa = F1 + EaYb = F1 + EbYc = F1 + Ec

G89.2247 Lecture 9 11

Adjusting Bias Through Latent Variables and SEM

• A more flexible error model is the one-factor CFA model

Ya = F1 + Ea

Yb = bF1 + Eb

Yc = c F1 + Ec

• Where the lamdas are weights that adjust the possible scale or importance of each replicate measure

• Not only can this model be used to estimate the lamda weights, it allows us to envision F1 as a variable in a system of regression equations.

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G89.2247 Lecture 9 12

Estimating Latent Variable Models

• Inferences about latent variables are made by looking at the structure of the observed covariance matrixIf the latent variable model is correct, it creates a

pattern in the covariance matrixBy fitting the pattern to the observed covariance

matrix, we can obtain estimates

• For the CFA model Var(Y) = [Var(F)]' + Var(E)

G89.2247 Lecture 9 13

A simple covariance structure

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G89.2247 Lecture 9 14

Regressed Change SEM

This model fits covariances from five observed variables. There are 10 correlations between the five and seven paths to be estimated. This leaves 3 degrees of freedom.

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F1 F2