Table 2.

Brief derivations of the variance of two types of parcels with cross-loadings

	Cross-loading with correlated factors	Cross-loading with uncorrelated factors
Indicators	I2 = T1 + s2 + e2 = U1 + C + s2 + e2 I3 = T1 + s3 + e3 = U1 + C + s3 + e3 I6 = T1 + T2 + s6 + e6 = U1 + C + U2 + C + s6 + e6	I1 = T1 + U3 + s1 + e1 = U1 + C + U3 +s1 + e1 I2 = T1 + s2 + e2 = U1 + C + s2 + e2 I3 = T1 + s3 + e3 = U1 + C + s3 + e3
Parcel	$\left(\frac{1}{3}\right)I_2+I_3+I_6$	$\left(\frac{1}{3}\right)I_1+I_2+I_3$
Variance of the parcel	$var\left(\frac{1}{3}\left[3U_1+4C+U_2+s_2+s_3+s_6+e_2+e_3+e_6\right]\right)$	$var\left(\frac{1}{3}\left[3U_1+3C+U_3+s_1+s_2+s_3+e_1+e_2+e_3\right]\right)$
Restated as a covariance	$\frac{1}{9}cov\left(\begin{array}{l}3U_1+4C+U_2+s_2+s_3+s_6+e_2+e_3+e_6,\hfill \\ 3U_1+4C+U_2+s_2+s_3+s_6+e_2+e_3+e_6\hfill \end{array}\right)$	$\frac{1}{9}cov\left(\begin{array}{l}3U_1+3C+U_3+s_1+s_2+s_3+e_1+e_2+e_3,\hfill \\ 3U_1+3C+U_3+s_1+s_2+s_3+e_1+e_2+e_3\hfill \end{array}\right)$
Expanded with zero terms removed	$\frac{1}{9}\left[\begin{array}{l}cov\left(3U_1,3U_1\right)+cov(4C,4C)+cov\left(U_2,U_2\right)\hfill \\ +cov\left(s_2,s_2\right)+cov\left(s_3,s_3\right)+cov\left(s_6,s_6\right)\hfill \\ +cov\left(e_2,e_2\right)+cov\left(e_3,e_3\right)+cov\left(e_6,e_6\right)\hfill \end{array}\right]$	$\frac{1}{9}\left[\begin{array}{l}cov\left(3U_1,3U_1\right)+cov(3C,3C)+cov\left(U_3,U_3\right)\hfill \\ +cov\left(s_1,s_1\right)+cov\left(s_2,s_2\right)+cov\left(s_3,s_3\right)\hfill \\ +cov\left(e_1,e_1\right)+cov\left(e_2,e_2\right)+cov\left(e_3,e_3\right)\hfill \end{array}\right]$
Restated as variances	$\begin{array}{l}\frac{1}{9}\left[\begin{array}{l}9var\left(U_1\right)+16var(C)+var\left(U_2\right)+var\left(s_2\right)\hfill \\ +var\left(s_3\right)+var\left(s_6\right)+var\left(e_2\right)+var\left(e_3\right)+var\left(e_6\right)\hfill \end{array}\right]\\ var\left(U_1\right)+\frac{16}{9}var(C)\\ +\frac{1}{9}\left[\begin{array}{l}var\left(U_2\right)+var\left(s_2\right)+var\left(s_3\right)+var\left(s_6\right)\hfill \\ +var\left(e_2\right)+var\left(e_3\right)+var\left(e_6\right)\hfill \end{array}\right]\end{array}$	$\begin{array}{l}\frac{1}{9}\left[\begin{array}{l}9var\left(U_1\right)+9var(C)+var\left(U_2\right)+var\left(s_1\right)\hfill \\ +var\left(s_2\right)+var\left(s_3\right)+var\left(e_1\right)+var\left(e_2\right)+var\left(e_3\right)\hfill \end{array}\right]\\ var\left(U_1\right)+var(C)\\ +\frac{1}{9}\left[\begin{array}{l}var\left(U_3\right)+var\left(s_1\right)+var\left(s_2\right)+var\left(s_3\right)\hfill \\ +var\left(e_1\right)+var\left(e_2\right)+var\left(e_3\right)\hfill \end{array}\right]\end{array}$

Note. Figure 2 shows the model that describes these items, and all components of the equations.