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# 1_22 - Reliability of a Composite 1 Spearman-Brown formula...

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1 Reliability of a Composite 1. Spearman-Brown formula • Assume k parallel composite; C = X 1 + X 2 + …. + X k - Reliability is; 2 ' 2 . C T CC C σ ρ = σ - Equal true scores = equal observed score means. - Equal error variance = equal observed score variance. - There are k variance terms and k ( k – 1) covariance terms. • The true score variance of the composite; 1 2 2 2 2 2 . C k i j i j T T T T TT T T i j σ = σ + σ + + σ + ρ σ σ ∑∑ " 2 2 2 2 2 2 2 2 2 ( 1) . C i i i i i T T T i T T T k k k k k k k σ = σ + σ = σ + σ σ = σ Since 1 2 2 2 2 1.00 , k i j T T T TT σ = σ = = σ ρ = " • The observed variance of the composite; [ ] 2 2 2 2 1 2 2 2 2 2 ' ( 1) 1 ( 1) 1 ( 1) . C k ij i j i j i ij i i ij i ii k k k k k k k σ = σ + σ + + σ + ρ σ σ = σ + ρ σ = σ + ρ = σ + ρ ∑∑ "

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2 • Then, [ ] [ ] [ ] 2 2 2 ' 2 2 ' 2 2 ' ' ' 1 ( 1) 1 ( 1) . 1 ( 1) C i i T T CC C i ii T ii i ii ii k k k k k k k σ σ ρ = = σ σ + ρ σ = × + ρ σ ρ = + ρ (Spearman-Brown formula) • The Spearman-Brown formula shows that the reliability of a composite is a function of ; - the reliability of a component, and - the number of components. 2. Cronbach’s Coefficient Alpha • It does not have to assume that the components are parallel. • It estimates the lower bound of the reliability of the composite.
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