lecture4_s10 observed

lecture4_s10 observed - SEM with observed variables:...

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SEM with observed variables: parameterization and identification Psychology 588: Covariance structure and factor models Feb 3, 2010
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Limitations of SEM as a causal modeling 2 • If an SEM model reflects the reality, the data will be consistent with the model, given that measurement errors are tolerable, all assumptions made are tenable, etc.; but the reverse is not true (e.g., see Fig. 3.9, p. 70) • Theory- vs. data-driven • Cause vs. effect indicators ¾ simultaneous reciprocal relation (e.g., financial health and stock price of companies) --- really concurrent? • To be cautious about goodness (mostly badness) of fit testing
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Fundamental equation with observed variables 3 () ( ) 1 =+ =− + + y IB ζ By Γ Γ x x ζ • Dependent variables y are modeled as linear combinations of (a subset of) y and x as Hypothesized explanation for the covariances of y unexplained (but allowed to exist) in the y variation • From the measurement perspective, y and x may be considered as “single-indicator latent variables” with no measurement errors Demographic variables are often considered so (e.g., sex, age)
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Fundamental equation with observed variables 4 • Alternatively, y is modeled as linear combinations of a partitioned vector of y , x and ζ as [] ⎡⎤ ⎢⎥ == =+ + ⎣ ⎦ y yB y B Γ Ix B y Γ x ζ ζ ± ± • This is essentially the approach taken by RAM (reticular action model), though RAM incorporates all latent variables, ξ and η --- to be discussed later
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Two conditions for recursive models 5 B should be a lower triangular matrix, and so no feedback loop of causal paths • All error terms ζ are not correlated with one another Correlated errors themselves don’t result in a feedback loop; instead, they lead to inconsistent estimates due to errors correlated with explanatory variables ¾ Note: any exogenous variables (including error terms) are allowed to be inter-correlated while no endogenous variables are allowed so (instead, correlated through exogenous vars) x1 y1 zeta1 y2 zeta2 1 1
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Implied covariance matrix 6 • Basic hypothesis with ideal measurement: Σ = Σ ( θ ) --- population covariance matrix Σ is a function of free model parameters θ • Basic hypothesis in reality: --- sample covariance matrix S is a function of estimates of model parameters • Discrepancy in the data ( Σ vs. S ) is due to sampling errors while the discrepancy in the parameters ( θ vs. ) is due to not only sampling errors but also any violated assumptions made for a particular way of finding the estimates (e.g., ML) () ˆ = S Σθ ˆ θ ˆ θ
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Implied covariance matrix 7 () [] ( ) ()() ( ) 1 11 1 1 E EE −− ⎛⎡ ⎤ ′′ == + ⎜⎟ ⎢⎥ ⎝⎣ ⎦ ⎡⎤ = ⎣⎦ ⎡′ −+ = y Σθ yx y IB Γ x ζ x yy xy xx ΓΦΓ Ψ ΓΦ ΦΓ Φ
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lecture4_s10 observed - SEM with observed variables:...

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