Extending reliability to intensive longitudinal data with the Kalman filter

Reliability is central to how researchers approach measurement in standard, group-based analyses of single-time-point data, yet this critical aspect is often overlooked in the analysis of repeated observations. Since its inception, reliability has been a between-person concept, but we redevelop this notion for within-person designs by proposing a new coefficient <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>κ</mml:mi></mml:mrow> <mml:annotation>$$ \kappa $$</mml:annotation></mml:semantics> </mml:math> of reliability for single-subject designs. This coefficient shares the same general definition of reliability as former coefficients-the ratio of the true score variance to the total variance-but applies to time-dependent within-person variability rather than independent between-person variability. Coefficient <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>κ</mml:mi></mml:mrow> <mml:annotation>$$ \kappa $$</mml:annotation></mml:semantics> </mml:math> begins with a latent variable time series model called a state space model, and is then extended to a state space model for multiple subjects with continuous or discrete variation across people. Using analytic methods, we derive coefficient <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>κ</mml:mi></mml:mrow> <mml:annotation>$$ \kappa $$</mml:annotation></mml:semantics> </mml:math> and prove its relations to other coefficients of reliability.