# 2_05 - Factor Analytic Approach to Reliability Relationship...

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Factor Analytic Approach to Reliability -- Relationship to Coefficient α • If we assume that all the items have the same λ , i.e., the same factor loading or item discrimination power, () 2 2 22 2 2 X j T T XX X X mm λ σ λ σ ω= = = = σ σσ σ The estimate of σ 2 T is the average of all covariances: 2 ˆ (1 ) jk jk T σ σ= ∑∑ So, 2 2 . ) 1 jk jk T X m m ≠≠ σ σ σ = = σ Since , jk X j j σ− σ 2 1. 11 Xj j jj  σ  = −σ σ  * This is exactly what Coefficient alpha is!!

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• The result shows that Coefficient alpha is a special case of reliability ( ω ) from a one-factor model. Back to the One-Factor Model … jjj j X FE =µ +λ + For a specific item j , The error ( E j ) actually contains two parts 1. Random error 2. Error associated with model fit . Model fit = The degree how well the one-factor model fits the data • When the model fits the data perfectly, ω reflects only the random error, i.e., reliability of measure.
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2_05 - Factor Analytic Approach to Reliability Relationship...

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