. Author manuscript; available in PMC: 2025 Mar 1.

Published in final edited form as: Psychophysiology. 2024 Jan 16;61(3):e14522. doi: 10.1111/psyp.14522

Table 1.

Generalizability Theory Estimates of Internal Consistency and Test-Retest Reliability

	Generalizability Coefficient ( $E ρ^{2}$ )	Dependability Coefficient ( $ϕ$ )
Single Session
CE: Internal Consistency	$\frac{σ_{p}^{2}}{σ_{p}^{2} + \frac{σ_{p i, e}^{2}}{n_{i}}}$	$\frac{σ_{p}^{2}}{σ_{p}^{2} + \frac{σ_{p i, e}^{2}}{n_{i}^{'}} + \frac{σ_{i}^{2}}{n_{i}^{'}}}$
Multiple Sessions
CE: Internal Consistency	$\frac{σ_{p}^{2} + \frac{σ_{p o}^{2}}{n_{o}^{'}}}{σ_{p}^{2} + \frac{σ_{p i}^{2}}{n_{i}^{'}} + \frac{σ_{p o}^{2}}{n_{o}^{'}} + \frac{σ_{p i o, e}^{2}}{n_{i}^{'} n_{o}^{'}}}$	$\frac{σ_{p}^{2} + \frac{σ_{p o}^{2}}{n_{o}^{'}}}{σ_{p}^{2} + \frac{σ_{p i}^{2}}{n_{i}^{'}} + \frac{σ_{p o}^{2}}{n_{o}^{'}} + \frac{σ_{p i o, e}^{2}}{n_{i}^{'} n_{o}^{'}} + \frac{σ_{i}^{2}}{n_{i}^{'}} + \frac{σ_{i o}^{2}}{n_{i}^{'} n_{o}^{'}}}$
CS: Test-Retest Reliability	$\frac{σ_{p}^{2} + \frac{σ_{p i}^{2}}{n_{i}^{'}}}{σ_{p}^{2} + \frac{σ_{p i}^{2}}{n_{i}^{'}} + \frac{σ_{p o}^{2}}{n_{o}^{'}} + \frac{σ_{p i o, e}^{2}}{n_{i}^{'} n_{o}^{'}}}$	$\frac{σ_{p}^{2} + \frac{σ_{p i}^{2}}{n_{i}}}{σ_{p}^{2} + \frac{σ_{p i}^{2}}{n_{i}^{'}} + \frac{σ_{p o}^{2}}{n_{o}^{'}} + \frac{σ_{p i o, e}^{2}}{n_{i}^{'} n_{o}^{'}} + \frac{σ_{o}^{2}}{n_{o}^{'}} + \frac{σ_{i o}^{2}}{n_{i}^{'} n_{o}^{'}}}$
CES	$\frac{σ_{p}^{2}}{σ_{p}^{2} + \frac{σ_{p i}^{2}}{n_{i}^{'}} + \frac{σ_{p o}^{2}}{n_{o}^{'}} + \frac{σ_{p i o, e}^{2}}{n_{i}^{'} n_{o}^{'}}}$	$\frac{σ_{p}^{2}}{σ_{p}^{2} + \frac{σ_{p i}^{2}}{n_{i}^{'}} + \frac{σ_{p o}^{2}}{n_{o}^{'}} + \frac{σ_{p i o, e}^{2}}{n_{i}^{'} n_{o}^{'}} + \frac{σ_{i}^{2}}{n_{i}^{'}} + \frac{σ_{o}^{2}}{n_{o}^{'}} + \frac{σ_{i o}^{2}}{n_{i}^{'} n_{o}^{'}}}$

Note: For single session data, internal consistency estimated using the ratio of universe score variance ( $σ_{p}^{2}$ ), item variance ( $σ_{i}^{2}$ ), error variance ( $σ_{p i, e}^{2}$ ), and the number of trials ( $n_{i}^{'}$ ). Multiple session data considers the universe score variance ( $σ_{p}^{2}$ ), item variance ( $σ_{i}^{2}$ ), occasion variance ( $σ_{o}^{2}$ ), transient error variance ( $σ_{p o}^{2}$ ), specific-facet trial score variance ( $σ_{p i}^{2}$ ), specific-facet occasion score variance ( $σ_{p o}^{2}$ ), error variance ( $σ_{p i o, e}^{2}$ ), the number of trials ( $n_{i}^{'}$ ), and the number of occasions ( $n_{o}^{'}$ ). These formulas and their derivations can be found elsewhere (see Clayson et al., 2021a; Clayson et al., 2021c; Clayson et al., 2021d; Clayson & Miller, 2017a; Vispoel et al., 2018a). CE = coefficient of equivalence; CS = coefficient of stability; CES = coefficient of equivalence and stability.