Table 2:

Derived population correlations of different kinds of stacked long data when σ_X, σ_Y , X^PS, and Y ^PS follow a joint multivariate normal distribution.

Data Type	Derived population correlation	Involved parameters
Raw data (xit and yit)	${\rho }_{X,Y}=$ $\frac{\left({\mu }_{\sigma X}{\mu }_{\sigma Y}+{\sigma }_{\sigma X,\sigma Y}\right){\mu }_{\rho w}+{\sigma }_{\mu X}{\sigma }_{\mu Y}{\rho }_b}{\sqrt{\left({\mu }_{\sigma X}^2+{\sigma }_{\sigma X}^2+{\sigma }_{\mu X}^2\right)\left({\mu }_{\sigma Y}^2+{\sigma }_{\sigma Y}^2+{\sigma }_{\mu Y}^2\right)}}$	Average within-person correlation (µρw) Between-person correlation (ρb) Mean and SD of the person SDs (µσX , µσY , σσX , σσY) Covariance of the person SDs (σσX,σY) SD of the person means (σµX , σµY)
P-C data ($x_{it}^{PC}$ and $y_{it}^{PC}$)	${\rho }_{CX,CY}=$ $\frac{\left({\mu }_{\sigma X}{\mu }_{\sigma Y}+{\sigma }_{\sigma X,\sigma Y}\right){\mu }_{\rho w}}{\sqrt{\left({\mu }_{\sigma X}^2+{\sigma }_{\sigma X}^2\right)\left({\mu }_{\sigma Y}^2+{\sigma }_{\sigma Y}^2\right)}}$	Average within-person correlation (µρw) Mean and SD of the person SDs (µσX , µσY , σσX , σσY) Covariance of the person SDs (σσX,σY)
P-S data ($x_{it}^{PS}$ and $y_{it}^{PS}$)	${\rho }_{WX,WY}={\mu }_{\rho w}$	Average within-person correlation (µρw)