Classical Test Theory (CTT), also known as True Score Theory, is the oldest and simplest psychometric framework. Its central axiom — that every observed score is the sum of a true score and random error — has been the foundation of psychological testing for over a century.
E(X) = T (expected observed score equals true score)
ρ(T,E) = 0 (true score and error are uncorrelated)
ρ(E₁,E₂) = 0 (errors across forms are uncorrelated)
Reliability
Reliability is defined as the ratio of true score variance to observed score variance: ρₓₓ = σ²_T / σ²_X. Since true scores are unobservable, reliability must be estimated through methods such as test-retest correlation, parallel forms, split-half methods, or internal consistency measures like Cronbach's alpha. Reliability ranges from 0 (all error) to 1 (no error), with values above 0.80 considered adequate for most research purposes.
Limitations
CTT has well-known limitations: its statistics are sample-dependent and test-dependent, meaning that item difficulty and discrimination values change with the sample, and person scores depend on which items were administered. Item Response Theory was developed specifically to overcome these limitations by modeling the relationship between latent ability and item responses directly.