Randomized response is a tool used in survey research to increase subject privacy and mitigate potential bias, particularly when our inquiries of interest pertain sensitive or controversial behavior or beliefs. For example, researchers may be interested in measuring participation in violence, engagement in illicit activities, or support for the opposition in authoritarian contexts.

Let us consider a specific type of randomized response design in which respondents are asked to use a randomization device, such as a die, whose outcome is unobserved to the enumerator. Respondents are then asked to answer “Yes” if the die shows numbers “1”, “2”, “3”, or “4” and to answer truthfully if the die shows “5”, or “6.” Only the respondent knows what they truly answered, so their anonymity is protected. The researcher can never know the true answer provided by any particular individual, but can back out an estimate of the rate of true responses from what she knows about the probabilities that underly the randomization mechanism.

Say, for example, we are interested in measuring the rate of intimate partner violence (IPV) in a given locality. Our design compares estimates from direct questions and questions using the randomized response principle.

## Design Declaration

**M**odel:Respondents are expected to answer the sensitive question with a probability \(p\) (a known quantity). \(L_i\) is a latent binary response to the sensitive question that equals 0 for respondent \(i\) if he or she does not engage in IPV and 1 if he or she does. This is defined in our design as the prevalence rate. \(Y_i\) is the response observed by researchers from the randomized approach. \(D_i\) is the response to the direct question of whether respondent engages in IPV. We expect respondents who engage in IPV to be inclined to misreport if asked to answer directly (we define this as the withholding rate).

**I**nquiry:Our estimand is the rate of intimate partner violence in the studied locality, or the population mean of L. Formally, we expect \(Pr(Y_i = 1) = p + (1-p)Pr(L_i = 1)\).

**D**ata strategy:We collect survey data on a representative sample of 1,000 individuals and respondents are assigned to respond “Yes” with probability \(p = \frac{4}{6}\) and truthfully with probability \(1 - p = \frac{1}{3}\).

**A**nswer strategy:We estimate our population rate of IPV in two ways: firstly by averaging the responses observed via direct question (\(\hat{\bar{L}} = \bar{D}\)), and secondly via randomized response: \(\hat{\bar{L}} = \frac{\bar{Y} - p}{1 - p}\).

```
N <- 1000
prob_forced_yes <- 0.6
prevalence_rate <- 0.1
withholding_rate <- 0.5
population <- declare_population(N = N, sensitive_trait = draw_binary(prob = prevalence_rate,
N = N), withholder = draw_binary(prob = sensitive_trait *
withholding_rate, N = N), direct_answer = sensitive_trait -
withholder)
potential_outcomes <- declare_potential_outcomes(Y_Z_Yes = 1,
Y_Z_Truth = sensitive_trait)
estimand <- declare_estimand(true_rate = mean(sensitive_trait))
assignment <- declare_assignment(prob = prob_forced_yes,
conditions = c("Truth", "Yes"))
estimator_randomized_response <- declare_estimator(handler = tidy_estimator(function(data) with(data,
data.frame(estimate = (mean(Y) - prob_forced_yes)/(1 -
prob_forced_yes)))), estimand = estimand, label = "Forced Randomized Response")
estimator_direct_question <- declare_estimator(handler = tidy_estimator(function(data) with(data,
data.frame(estimate = mean(direct_answer)))), estimand = estimand,
label = "Direct Question")
randomized_response_design <- population + assignment + potential_outcomes +
estimand + declare_reveal(Y, Z) + estimator_randomized_response +
estimator_direct_question
randomized_response_design <- set_diagnosands(design = randomized_response_design,
diagnosands = declare_diagnosands(select = bias))
```

## Takeaways

`diagnosis <- diagnose_design(randomized_response_design)`

Design Label | Estimand Label | Estimator Label | N Sims | Bias |
---|---|---|---|---|

randomized_response_design | true_rate | Direct Question | 500 | -0.05 |

(0.00) | ||||

randomized_response_design | true_rate | Forced Randomized Response | 500 | -0.00 |

(0.00) |

Our diagnosis of the design indicates that randomized response yields an unbiased estimate of the true rate of IPV in the study sample and offer a better alternative to direct questions involving sensitive or controversial inquiries.

## References

Blair, Graeme, Kosuke Imai, and Yang-Yang Zhou. 2015. “Design and Analysis of the Randomized Response Technique.” *Journal of the American Statistical Association* 110 (511): 1304–19.

Warner, Stanley L. 1965. “Randomized Response: A Survey Technique for Eliminating Evasive Answer Bias.” *Journal of the American Statistical Association* 60 (309): 63–69.