Reliability-Constrained Latent Structure Models

Transcription

1 Reliability-Constrained Latent Structure Models Peter Westfall and Kevin Henning Texas Tech University

2 Latent Structure Models (LSMs) AKA "LISREL," or "structural equation models" Common in social science, often for survey data analysis Goal: etc. Model fuzzy concepts, "Love of humanity," "Adventurousness," Google: 232,000 hits for "LISREL"

3 Touted "Benefits" of LSMs over Regression LSMs allow you to estimate reliability of your measure of the fuzzy concept LSMs allow you to estimate relationships involving the fuzzy concepts LSMs eliminate measurement error bias

4 The Real Story LSMs cannot identify reliability. LSMs cannot identify relationships between fuzzy concepts. LSMs cannot correct for measurement error bias.

5 The Story Begins: The Single-Factor Model Y = Λη + ε Y a (p 1) vector of observable measures Λ an unknown (p 1) vector of constants η is a random, unobservable latent variable ε is a random (p 1) unobservable vector.

6 Assumptions for The Single-Factor Model Model: Y = Λη + ε 1. Var(η) =1 2. Cov(η, ε) =0 (1 p) 3. Cov(ε) =Ψ, sufficiently structured for identifiability (usually diagonal)

7 Implied Covariance Matrix Σ = Cov(Y )=ΛΛ 0 + Ψ = λ ψ 1 λ 1 λ 2 λ 1 λ p λ 2 λ 1 λ ψ 2 λ 2 λ p λ p λ 1 λ p λ 2 λ 2 p + ψ p

8 Example Five measures of "Job Salience": (4) Identity, (5) Autonomy. (1) Feedback, (2) Significance, (3) Variety,

9 Example, Continued Sample covariance matrix of standardized data (n =784)is S =

10 Example, Continued Estimated coefficients (ML): ˆΛ = , ˆΨ =

11 Example, Continued: Model Fit bσ = ˆΛˆΛ 0 + ˆΨ = , Goodness of fit = discrepancy from S to b Σ. RMR = root mean square difference. RMR=.020 here, "reasonably good."

12 Example, Continued: Use of Model Standard practice: The model "fits," so one can use it further. For example: assess reliability of F = 1 0 Y = Σ 5 i=1 Y i as a measure of η. Reliability = {Corr(F, η)} 2 = ρ 2. Note ρ ΛΛ 0 1 = 1 0 ΛΛ Ψ1. From the estimated model, bρ 2 =.845. By standard practice, bρ 2 >.7 is required for publication.

13 Example, Continued. The Crux If Ψ is allowed to be non-diagonal, any reliability ρ 2 is possible. = Rather than assume a diagonal Ψ and estimate ρ 2, we may as well assume a ρ 2 and estimate Ψ! Can ρ 2 be assumed? Yes: - If "job salience" is simply defined as F = Σ 5 i=1 Y i,thenρ 2 =1. - If η is defined otherwise, then ρ = Corr(F, η). - If the Y i "miss the mark" for the intended definition of "job salience," then ρ 2 <.845.

14 Why Errors Should Correlate Suppose η is measurable (Y i1,y i2,y i3,y i4,y i5,η i ), i =1,...,n are sampled Thecovariancematrixof(Y 1,Y 2,Y 3,Y 4,Y 5 ), partialing out η, iscomputed. This partial covariance matrix ˆΨ will be non-diagonal. = It is more sensible to assume Ψ is unstructured.

15 Standard LSMs, Parameters, and Identifiability Parallel: Y i = λη + ε i, i =1,...,p, Var(ε i ) ψ. Here θ =(λ, ψ). Congeneric: Y i = λ i η + ε i, i = 1,...,p, Var(ε i ) = ψ i. Here θ = (λ 1,...,λ p,ψ 1,...,ψ p ). Σ = Cov(Y )=Σ(θ). θ is identifiable if Σ(θ 1 ) 6= Σ(θ 2 ) whenever θ 1 6= θ 2.

16 Equivalent Models Researcher A likes the model Y = Λη + ε, with Λ = Λ(θ), Ψ = Ψ(θ), andσ = Σ(θ) =ΛΛ 0 + Ψ. Researcher B likes the model Y = Λ η + ε, with Λ = r Σ1 (1 0 Σ1) 1/2, Ψ = Σ r 2 Σ11 0 Σ 1 0,forB schoiceofr [0, 1]. Σ1

17 Equivalent Models, Different Reliabilities Models cannot be distinguished since Σ(θ) =ΛΛ 0 + Ψ = Λ Λ 0 + Ψ. However: With A s model, Reliability= 1 0 ΛΛ ΛΛ Ψ1. With B s model, Reliability= 1 0 Λ Λ Λ Λ Ψ 1 = r2.

18 Demo Using PROC CALIS of SAS/STAT. Shows how to specify any reliability you want; the model fits identically.

19 Measurement Error Bias Suppose Y = βx + ε X = X + δ Then in the model Y = β X + ε, β = Var(X) Var(X)+Var(δ) β. The biasing factor Var(X) Var(X)+Var(δ) equals the reliability ({Corr(X, X )} 2 ).

20 The Two-Factor Model " X Y # = " #" Λx 0 η1 0 Λ y η 2 # + " ε1 ε 2 # X (p 1 1)and Y (p 2 1)are random vectors of observable measures Λ i are unknown (p i 1) vectors of constants η 1,η 2 are scalar latent variables ε i is a random (p i 1) vector of residuals.

21 Assumptions for the LSM Var(η i )=1, Cov(η 1,η 2 )=φ Cov(η i, ε j )=0 Cov(ε i )=Ψ i,sufficiently structured Cov(ε 1, ε 2 )=0 Thus Σ = Cov(Y )= " Λx Λ 0 x + Ψ 1 φλ x Λ 0 # y φλ y Λ 0 x Λ y Λ 0 y + Ψ 2

22 Using LSMs to "Correct" for Measurement Error Bias: Standard Practice Estimate θ by minimizing discrepancy between S to Σ(ˆθ). If the models "fits", then ˆφ is assumed free of measurement error bias Large ˆφ (say >.3) are needed for publication. Note: φ might be estimated as Corr(1 0 Y, 1 0 X); thisispresumed"toolow" because of measurement error in 1 0 Y and 1 0 X.

23 Models Assume parallel models. " X Y # Researcher A likes the model " #" # " λx 1 0 η1 ε1 = + 0 λ y 1 ε 2 η 2 # for which the implied reliabilities are ρ 2 X and ρ2 Y. Researcher B thinks the reliabilities are rx 2 and r2 Y, and uses the model " # " #" # " # X (rx /ρ = x )λ x 1 0 η 1 ε Y 0 (r x /ρ x )λ x 1 η ε 2

24 Equivalent Models, Different Reliabilities, Different Interfactor Correlation The models are equally supported by the data since Σ(θ) =ΛΛ 0 + Ψ = Λ Λ 0 + Ψ. However, in B s model, Reliabilities are r 2 X and r2 Y Interfactor correlation is φ = ρ xρ y r x r y φ. = All interfactor correlations in the range [ρ x ρ y φ, 1] are equally supported.

25 SAS Demo: "Correcting" Measurement Error Bias Same five measures of "Job Salience". Seven measures of "Job Satisfaction": (1) with supervisor, (2) with careerfuture, (3) with compensation, (4) with workload, (5) with company identification, (6) with kind of work, and (7) overall.

26 Figure 1: Path Diagram

27 Effect of Preselected Reliability on Interfactor Correlation Interfactor correlation φ as a function of reliability r 2.

28 The Bad News 1. LSMs cannot identify reliability, 2. LSMs cannot identify factor loadings, and 3. LSMs cannot identify interfactor correlations (nor path coefficients). Diametrically opposed theories cannot be distinguished using data.

29 And Now the Bad News (with recommendations) Don t use LSMs: Use path diagrams for theory development only; not for data analysis. Use manifest measures for data analysis. IfyoumustuseLSMs: state the assumptions, state that the assumptions cannot be tested, and state that all conclusions rely on the assumptions, and that your conclusions will change dramatically if your assumptions are wrong.