Table 1.

Identifiability and bounds on the causal risk difference CRD_I(0). 3000 households of size 2 with individual 1 randomized to vaccine or control 1:1. In the 1500 households with Z_i = (0, 0), individual 1 became infected in 1000, and individual 2 became infected in 400 of those, so p₀ = 0.4. In the 1500 households with Z_i = (1, 0), individual 1 became infected in 500, and individual 2 became infected in 100 of those, so p₁ = 0.2. The 500 transmission units where individual 1 was randomized to vaccine and became infected must be in the doomed stratum. The 500 transmission units where individual 1 was randomized to control and was not infected must be in the immune stratum. Thus, ignoring sampling variability, the proportions in the three principal strata π_i, π_p, and π_d all equal 1/3, and ρ = (1/3)/(2/3) = 0.5. The number of households with Y_i1 = 1, Z_i1 = 0 in which Y_i2 = 1 is 400. It is not known which of these 400 households are in the doomed or protected stratum. The number assumed in the doomed stratum under different selection assumptions is denoted by x (eSection 2.1). If individual 1 does not become infected, then Y_i2(z_i1, z_i2) is undefined and denoted by *.

Basic Principal Stratum		Potential infection outcomes Yi1(zi)		Observed transmission outcomes $Y_{i2}^{\mathit{obs}}=1$		Selection assumptions Yi2(zi)
	Prop.	(0,0)	(1,0)	(0,0)	(1,0)	(0,0)	(0,0)	(0,0)
						no selection bias	HH lower bound	HH upper bound
Immune	πi	$\begin{array}{c}0\\ 500\end{array}$	$\begin{array}{c}0\\ 500\end{array}$	*	*
Protected	πp	$\begin{array}{c}1\\ 500\end{array}$	$\begin{array}{c}0\\ 500\end{array}$	$\begin{array}{c}\uparrow ?\\ 400\\ \downarrow ?\end{array}$	*	200	0	400
Doomed	πd	$\begin{array}{c}1\\ 500\end{array}$	$\begin{array}{c}1\\ 500\end{array}$		100	x = 200	400	0
					CRDI (0)	−0.2	−0.6	0.2