Table 1.

Brief Derivations of the Variance of One or Both Items with a Correlated Residual

	Single Item with Correlated Residual	Both Items with Correlated Residuals
Indicators	I2 = T1 + s2 + e2 I3 = T1 + s3 + e3 I4 = T1 + s4 + x + e4	I3 = T1 + s3 + e3 I4 = T1 + x + s4 + e4 I5 = T1 + x + s5 + e5
Parcel	$\frac{1}{3}\left[I_2+I_3+I_4\right]$	$\frac{1}{3}\left[I_3+I_4+I_5\right]$
Variance of the parcel	$var\left(\frac{1}{3}\left[3T_1+s_2+s_3+s_4+x+e_2+e_3+e_4\right]\right)$	$var\left(\frac{1}{3}\left[3T_1+2x+s_3+s_4+s_5+e_3+e_4+e_5\right]\right)$
Restated as a covariance	$\frac{1}{9}cov\left(\begin{array}{l}3T_1+s_2+s_3+s_4+x+e_2+e_3+e_4,\hfill \\ 3T_1+s_2+s_3+s_4+x+e_2+e_3+e_4\hfill \end{array}\right)$	$\frac{1}{9}cov\left(\begin{array}{l}3T_1+2x+s_3+s_4+s_5+e_3+e_4+e_5,\hfill \\ 3T_1+2x+s_3+s_4+s_5+e_3+e_4+e_5\hfill \end{array}\right)$
Expanded with zero terms removed	$\frac{1}{9}\left[\begin{array}{l}cov\left(3T_1,3T_1\right)+cov\left(s_2,s_2\right)+cov\left(s_3,s_3\right)\hfill \\ +cov\left(s_4,s_4\right)+cov\left(x,x\right)+cov\left(e_2,e_2\right)\hfill \\ +cov\left(e_3,e_3\right)+cov\left(e_4,e_4\right)\hfill \end{array}\right]$	$\frac{1}{9}\left[\begin{array}{l}cov\left(3T_1,3T_1\right)+cov\left(2x,2x\right)\hfill \\ +cov\left(s_3,s_3\right)+cov\left(s_4,s_4\right)+cov\left(s_5,s_5\right)\hfill \\ +cov\left(e_3,e_3\right)+cov\left(e_4,e_4\right)+cov\left(e_5,e_5\right)\hfill \end{array}\right]$
Restated as variances	$\begin{array}{l}\frac{1}{9}\left[\begin{array}{l}9var\left(T_1\right)+var(s_2)+var\left(s_3\right)+var\left(s_4\right)\hfill \\ +var\left(x\right)+var\left(e_2\right)+var\left(e_3\right)+var\left(e_4\right)\hfill \end{array}\right]\\ var\left(T_1\right)+\frac{1}{9}\left[\begin{array}{l}var\left(s_2\right)+var\left(s_3\right)+var\left(s_4\right)+var(x)\hfill \\ +var\left(e_2\right)+var\left(e_3\right)+var\left(e_4\right)\hfill \end{array}\right]\end{array}$	$\begin{array}{l}\frac{1}{9}\left[\begin{array}{l}9var\left(T_1\right)+4var(x)+var\left(s_3\right)+var\left(s_4\right)\hfill \\ +var\left(s_5\right)+var\left(e_3\right)+var\left(e_4\right)+var\left(e_5\right)\hfill \end{array}\right]\\ var\left(T_1\right)+\frac{4}{9}var(x)+\frac{1}{9}\left[\begin{array}{l}var\left(s_3\right)+var\left(s_4\right)+var\left(s_5\right)\hfill \\ +var\left(e_3\right)+var\left(e_4\right)+var\left(e_5\right)\hfill \end{array}\right]\end{array}$

Note. Figure 2 shows the item-level model that describes these items, and all components of the equations. T₁ is not expanded into C + U₁ here, for simplicity.