Chapter 3 Structure of psychometric models The general
- Slides: 10
Chapter 3: Structure of psychometric models The general measurement model: Three components: 1. Latent constructs and their relationships. 2. Observed measures. 3. Response model: Mapping latent vari ables to observed measures.
Chapter 3: Structure of psychometric models The general measurement model: Variables q Latent abilities (eta) Ø Multivariate normal distributed random variables. q Hidden responses (xi) Ø Represent hidden continu ous response processes. q Error variables (epsilon) Ø Represent unmeasured influences. q Observed variables Y
Chapter 3: Structure of psychometric models The general measurement model: q Circles: Latent variables q Rectangles: Observed variables q Triangles: Fixed values and not random variables Ø Either constants (e. g. 1) or parameters. q Greek letters outside circles: Parameters Ø Fixed parameters are represented by numbers
Chapter 3: Structure of psychometric models The general measurement model: q Covariance arcs Ø Unmodeled covariances between variables Ø Exist only between latent abilities and, possibly, between errors. q Arrows Ø Represent linear causal influences (details later). q Red sigmoid curves Ø Represent response functions ICC = Item characteristic curves.
Chapter 3: Structure of psychometric models The general measurement model: Parameters q Variance of and covariances between latent abilities: (phi) q Mean parameters of latent abilities (mu) q Variances of and covari ances between error variables: (theta) q Regression weights: Ø Loading coefficients: (lambda) Ø Item difficulties: (beta) q Discrimination parameters of response functions (alpha)
Chapter 3: Structure of psychometric models The general measurement model: Linear equations q The hidden responses are linear combinations of: Ø Latent abilities Ø Errors Ø Items difficulties q The linear equations: (more explanations later)
Chapter 3: Structure of psychometric models The general measurement model: Response functions: q Logistic function: q Normal ogive: q No response function: CTT
Chapter 3: Structure of psychometric models The general measurement model: Response functions: The problem of indeterminacy q Response function: The participant selects response category c 1 if the hidden re sponse j of that person j surpasses a certain threshold j, other wise she selectsresponse category c 2. q The problem of indeterminacy concerning the source of randomness: Ø Random thresholds Ø Random error associated with hidden responses
Chapter 3: Structure of psychometric models The general measurement model: Response functions: The problem of indeterminacy q Random thresholds: q Random error in the hidden response:
Chapter 3: Structure of psychometric models The general measurement model: Response functions: The problem of indeterminacy q Random error in the hidden response: q Conclusion: Location of the random component cannot be determined uniquely.