A Survey Design for a Sensitive Binary Variable Correlated with Another Nonsensitive Binary Variable

Table 3

Cell probabilities, observed and unobservable counts for the main questionnaire.

Category	𝑊=1	𝑊=2	𝑊=3	Total
I: {𝑋=0,𝑌=0}	𝑝1𝜃1	𝑝2𝜃1	𝑝3𝜃1	𝜃1(𝑍1)
II: {𝑋=1,𝑌=0}				𝜃2(𝑍2)
III: {𝑋=1,𝑌=1}				𝜃3(𝑍3)
IV: {𝑋=0,𝑌=1}		𝜃4(𝑛4)		𝜃4(𝑛4)
Total	𝑝1𝜃1(𝑛1)	𝑝2𝜃1+𝜃2(𝑛2)	𝑝3𝜃1+𝜃3(𝑛3)	1(𝑛)

Note: 𝑛=∑⁴_𝑖=1𝑛_𝑖, 𝑍₁=𝑛−(𝑍₂+𝑍₃)−𝑛₄, where (𝑍₂,𝑍₃) are unobservable.