Table 4.

Brief Derivations of the Variances and Covariances of Facet-Representative and Domain-Representative Parcels

	Facet-representative Parcel	Domain-representative Parcel
Indicators	N1 = T + S + e1 N2 = T + S + e2 N3 = T + S + e3	N1 = T + N + e1 V1 = T + V + e4 S7 = T + S + e7
Parcel	$\left(\frac{1}{3}\right)I_1+I_2+I_3$	$\left(\frac{1}{3}\right)I_1+I_4+I_7$
Variance of the parcel	$var\left(\frac{1}{3}\left[3T+3S+e_1+e_2+e_3\right]\right)$	$var\left(\frac{1}{3}\left[3T+S+V+N+e_1+e_4+e_7\right]\right)$
Restated as a covariance	$\frac{1}{9}cov\left(\begin{array}{l}3T+3S+e_1+e_2+e_3,\hfill \\ 3T+3S+e_1+e_2+e_3\hfill \end{array}\right)$	$\frac{1}{9}cov\left(\begin{array}{l}3T+S+V+N+e_1+e_4+e_7,\hfill \\ 3T+S+V+N+e_1+e_4+e_7\hfill \end{array}\right)$
Expanded with zero terms removed	$\frac{1}{9}\left[\begin{array}{l}cov\left(3T,3T\right)+cov(3S,3S)+cov\left(e_1,e_1\right)\hfill \\ +cov\left(e_2,e_2\right)+cov\left(e_3,e_3\right)\hfill \end{array}\right]$	$\frac{1}{9}\left[\begin{array}{l}cov\left(3T,3T\right)+cov(S,S)+cov(V,V)+cov(N,N)\hfill \\ +cov\left(e_1,e_1\right)+cov\left(e_4,e_4\right)+cov\left(e_7,e_7\right)\hfill \end{array}\right]$
Restated as variances	$\begin{array}{l}\frac{1}{9}\left[9var(T)+9var(S)+var\left(e_1\right)+var\left(e_2\right)+var\left(e_3\right)\right]\\ var(T)+var(S)+\frac{1}{9}\left[var\left(e_1\right)+var\left(e_2\right)+var\left(e_3\right)\right]\end{array}$	$\begin{array}{l}\frac{1}{9}\left[\begin{array}{l}9var(T)+var(S)+var(V)+var(N)\hfill \\ +var\left(e_1\right)+var\left(e_4\right)+var\left(e_7\right)\hfill \end{array}\right]\\ var(T)+\frac{1}{9}\left[\begin{array}{l}var\left(S\right)+var(V)+var(N)+\hfill \\ var\left(e_1\right)+var\left(e_4\right)+var\left(e_7\right)\hfill \end{array}\right]\end{array}$
Covariance between two parcels	$cov(\frac{1}{3}\left[3T+3S+e_1+e_2+e_3\right],\frac{1}{3}\left[3T+3V+e_4+e_5+e_6\right])$	$\begin{array}{l}cov(\frac{1}{3}\left[3T+S+V+N+e_1+e_2+e_3\right],\\ \frac{1}{3}\left[3T+S+V+N+e_4+e_5+e_6\right])\end{array}$
Expanded with zero terms removed	$\frac{1}{9}cov(3T,3T)$	$\frac{1}{9}\left[cov(3T,3T)+cov(S,S)+cov(V,V)+cov(N,N)\right]$
Restated as variances	var(T)	$var(T)+\frac{1}{9}\left[var(S)+var(V)+var(N)\right]$

Note. Figure 3 depicts the item-level and parcel-level models that describes facet-representative parceling. Figure 4 depicts the item-level and parcel-level models that describe domain-representative parceling.