Randomized response ($q=(1-x)+(2x-1)p$)

Randomized response masks which of two complementary binary questions a person answered. Suppose the sensitive property has probability $p$, and independently the direct question is selected with known probability $x$, while the complementary question is selected otherwise. Truthful answers then have yes probability $q=xp+(1-x)(1-p)$. The investigator observes only the answer, preserving uncertainty about an individual's status while permitting population statistical estimation when $x\ne 1/2$.

For $n$ independent responses, $Y\sim \mathrm{Bin}(n,q)$, and the affine estimator $\tilde{p}=(Y/n-1+x)/(2x-1)$ is an unbiased estimator. Its variance is $q(1-q)/[n(2x-1{)}^2]$, by the mean and variance of a binomial distribution. Equivalently the variance is $p(1-p)/n+x(1-x)/[n(2x-1{)}^2]$, exposing the extra sampling uncertainty caused by the masking. At $x=1/2$, all $p$ produce $q=1/2$ and identifiability fails. At $x=0$ or $1$, there is no masking and the extra variance vanishes.

Because $0\le p\le 1$, maximum likelihood estimation gives $\hat{p}=\min (1,\max (0,\tilde{p}))$. This follows by maximizing the binomial likelihood over the interval $q\in [\min (x,1-x),\max (x,1-x)]$ and then inverting its monotone affine parametrization. Clipping can introduce bias: for $n=1$, $x=3/4$ and $p=0$, the clipped estimate has mean $1/4$.