Randomized Response and Estimation

In survey sampling, directly asking sensitive questions often leads to evasive answers or untruthful responses. The randomized response technique introduces a probabilistic privacy mechanism: a respondent answers truthfully with some probability r, and otherwise gives a random response (such as flipping a fair coin). This plausible deniability protects individual privacy while still allowing the aggregate prevalence to be estimated.

Worked Example

Suppose the true population prevalence is π = 1/4 and the truthful response probability is r = 1/2.

1. Calculate the probability of a "Yes" report (θ):
   θ = r × π + (1 - r) / 2
   θ = (1/2) × (1/4) + (1 - 1/2) / 2 = 1/8 + 1/4 = 3/8.
2. With n = 8 respondents, the expected number of "Yes" responses is nθ = 8 × (3/8) = 3.
3. Suppose we observe y = 4 "Yes" responses. The unbiased moment estimator for the prevalence is:
   π̂ = (y/n - (1 - r)/2) / r
   π̂ = (4/8 - 1/4) / (1/2) = (1/4) / (1/2) = 1/2.
4. The true-law variance of this estimator is:
   Var(π̂) = θ(1 - θ) / (nr²)
   Var(π̂) = (3/8)(5/8) / (8 × (1/2)²) = (15/64) / 2 = 15/128 ≈ 0.1171875.

The experiment is available here.

Experiment

Explore the trade-off between privacy and estimation variance:

1. Set Prevalence quarters to 1.
2. Set Truth quarters to 2.
3. Set Respondents to 8.
4. Adjust the Observed yes count and note how the Moment Estimate changes. Notice that it can exceed 1 or fall below 0, which is normal for this unbiased estimator.
5. Change Truth quarters to 0 to observe that the prevalence becomes unidentifiable and the estimator unavailable.

The experiment is available here.

Limitations

This module considers a specific unrelated-question design equivalent to a fair coin flip for the random response. Other designs exist (e.g., forced "Yes", forced "No", or using a different known-probability event). The moment estimator provided is unbiased but may yield estimates outside the meaningful [0, 1] range in small samples; these should not be silently truncated or interpreted as posterior probabilities.

References

Below is a reference to learn more about the survey-sampling methodology applied here.

• Penn State STAT506 survey-sampling notes (Section 12.3)