lecture14_s10 GM2 - General structural model Part 2: RAM,...

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General structural model – Part 2: RAM, special constraints and instrumental variables Psychology 588: Covariance structure and factor models March 31, 2010
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Alternative representation 2 • LISREL notation does not allow: ¾ Loadings from ξ to y or η to x , or from an indicator to another ¾ Covariances between different kinds of error terms (e.g., between δ and ε ) • These “unacceptable” parameters are a limitation of the LISREL notation, not genuine limitation of SEM • Reticular Action Model (RAM) provides an alternative, apparently a more general SEM representation
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LISREL model represented by RAM 3 , , ⎡⎤ ⎢⎥ ⎡ ⎤ == = ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣⎦ ε y x δ y ηζ y x η ζ ξ ξ ± ± ± , , y x ε δ = 00 Λ 0 Θ 0 I000 0 Θ 00 0 Λ B ΛΨ 0I00 Ψ 0 00 B Γ 0 Φ 00 0 0 ±± ± 11 ( ) () = −− Σθ Λ IB Ψ Λ ±±± =+ = η B y Λη ± ± ± ± ±
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LISREL & RAM really different? 4 • Either notation can represent almost all models --- compare Fig 9.1 (a) and (b), p. 398 ¾ All ξ are rewritten as η along with correlated ζ ¾ All ε are rewritten as ζ with y turning into η • RAM is more efficient in # of equations (2 vs. 3) and in kinds of parameters (6 vs. 15), while it requires much bigger parameter sets with most of them constrained to 0 or 1 • By RAM, the lesson is not what RAM can do more than LISREL, but what SEM can do as far as a model is identifiable • Most programs (e.g., AMOS) overcome the limitation of the LISREL notation
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Equality & inequality constraints 5 • Constraints at constant values and of equality are easy to impose in terms of numerical optimization, as readily available in all SEM programs • In contrast, parameter estimation is very difficult with constraints of inequality or a functional form --- such constraints are not widely available (EQS and COSAN seem to be exceptions) ¾ Inequality --- λ 12 > c , γ 12 > β 23 , γ 12 + β 23 –2 β 34 > c ¾ linear equality --- γ 12 + β 23 β 34 + c = 0 ¾ nonlinear equality --- 23 12 23 34 20 c γ ββ + −+ =
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This note was uploaded on 06/11/2010 for the course PSYC 588 taught by Professor Sunjinghong during the Spring '10 term at University of Illinois at Urbana–Champaign.

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lecture14_s10 GM2 - General structural model Part 2: RAM,...

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