Random Response

Surveys

What do we use surveys for in our field?

## Non-Sampling Error

  • Differences b/w estimate and true population quantities that do not arise because you didn’t observe the whole population

Non-Sampling Error

Examples??

  • Sensitive topics
  • detection errors

Dealing with non-response

  • Prevent it through the design (best solution!)
  • Ignore it (common and may lead to bias)
  • Take representative subsample of non-respondents
  • Use a model / make assumptions

Random Response Survey

How can we deal with non-response and “wrong” responses?


Warner, S. L. (1965). Randomized response: A survey technique for eliminating evasive answer bias. Journal of the American statistical association, 60, 63-69.

Random Response Survey

Combine probability math and a survey technique to ask your question indirectly

  • First: you need a mechanism to receive an answer w/o knowing which question is being asked
  • Second: you need to know the probability of the question
  • Third: record the answers, frequency of yes and no’s

Random Response Survey

  • Start with identical note cards with a fraction (\(\theta\); this has to be known) marked with \(A\) and the remaining \((1- \theta)\) marked with \(B\).

  • If \(A\), answer the question, e.g.,

    • have you ever cheated on an exam

  • If \(B\), answer the question, e.g.,

    • have you never cheated on an exam

  • A sample of people are chosen; each is asked to randomly draw a card and state ‘yes’ if they agree with the letter on the card or ‘no’ otherwise.

  • The interviewer does not ever see the card – just records ‘yes’ or ‘no’.

Random Response Estimator

Random Response Estimator

\[ \frac{n_{\text{yes}}}{n_{\text{total}}} = \theta p+(1-\theta)\times(1-p) \]

Random Response Estimator

\[ \hat{p} = \frac{1}{2\theta-1} \frac{n_{\text{yes}}}{n_{\text{total}}}-\frac{1-\theta}{2\theta-1}\text{where }\theta\ne0.5 \]

\[ \hat{\sigma}^2_p = \frac{1}{n_{\text{total}}}\left( \frac{1}{16(\theta-0.5)^2}- (p-0.5)^2\right) \]

Lots of variations

  • flip a coin to pick the question
  • Q1) primary, Q2) whether your phone number ends with an odd number

Test it?

What’s our question?

Evaluate the Estimator

Let’s consider how the sample size (\(n\)) effects the sampling distribution of \(\hat{p}\)

Also, consider how our choice of \(\theta\) effect the sampling distribution of \(\hat{p}\)