lecture12_s10 GM1 - General structural model Part 1: Power...

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General structural model – Part 1: Power of testing, mean-structure, etc. Psychology 588: Covariance structure and factor models Mar 10, 2010
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Estimation 2 • Fitting functions ( F ML , F GLS , F ULS ) of a general SE model have the same forms as those for path modeling of observed variable only and CFA, but the implied covariance matrix is differently defined, e.g., • Properties of the ML, GLS and ULS estimators hold essentially the same • Given a converged solution, all estimates must be substantively sensible --- Exercise: fit the model explained in p. 334 to the political democracy data (with and without the equal-loading constraints in nested modeling approach); which are given in the data directory (poldemcov.xls) ( ) () 1 ML ˆˆ log tr log F pq =+ + Σ S Σ S
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Power of chi-square tests 3 • Given a pair of nesting-nested models, we can set up H 0 and H a as follows: H 0 --- the constraints that make the only difference between the two models are correct, such that θ a = θ 0 , and θ b contains free parameters for both H 0 and H a H a --- • The constraints are often θ a = 0 , though not necessary; it equally holds for constraints at nonzero constants • If the nesting model (i.e., H a is true) has 0 df , the test is about goodness of fit of a hypothesized model [ ] ab ,, ′′ = θθ θ a0
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• Type-I error occurs only when H 0 is true while Type-II error (and hence power) is relevant only when H a is true ¾ Nominal vs. true Type-I error rate --- do we know true Type-I error rate in practice? And true power? • All chi-square tests so far assumed true H 0 • When H a is true, the chi-square value computed under H 0 does not follow the χ 2 distribution we use for null-hypothesis testing (which is called “central” χ 2 distribution); instead, it follows noncentral χ 2 distribution, which has one more parameter, noncentrality that depends on true values of θ a • Thus, calculating power of a chi-square test boils down to estimation of the noncentrality parameter
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• Noncentrality essentially defines how much chi-square values deviate from their incorrect expectation (= df ) due to the wrong assumption of true H 0 --- i.e., ¾ As in usual null hypothesis testing, smaller α leads to
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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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lecture12_s10 GM1 - General structural model Part 1: Power...

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