Poor sample measurement and the Central Limit Theorem

Question

I have a fairly basic question about the Central Limit Theorem. I understand it in principle, but I like to know specifically what happens when there is poor measurement on the samples.

Say for example I have a large population which I sample in order to determine the population mean. In sampling I make a measurement on each sample which will have one of 3 outcomes. However, my measurement mechanism is very poor, and I only have a 40% chance of making a good measurement - the rest of the the time I have a random chance of incorrectly getting one of the other values.

Can the Central Limit Theorem be applied in this case? Any pointers to further reading on this situation would be appreciated.

Answer 1

Assuming the values you're getting are numeric, so that it makes sense to talk about averages, and if the resulting contaminated distribution still satisfies the conditions for some version of the CLT, that CLT should still apply -- in that appropriately standardized sample means will still go to a standard normal (in the limit as $n\to\infty$).

However beware -- the distribution that you use to standardize it will be the contaminated one, not the 'good' one; in some cases that may not be useful.

Answer 2

You may want to look into the literature on randomized response. In this approach to asking confidential and sensitive survey questions, the interviewer asks the respondent to toss a coin and answer a sensitive question (e.g., "Do you use illicit drugs?") if heads come up, and an innocuous question (e.g., "Does your mother's birthday fall on an even-numbered day of the month?") if tails come up. That way, if the respondent says "Yes", the interviewer does not know whether they answered the mother question or the drug question, so it won't be as embarrassing or scary for the respondents to say "Yes". By working out the probabilities in a way similar to that in Dirk Horsten's answer, you can deduce the incidence of the sensitive behavior.

While there's a school of researchers following up and developing that technique, randomized response is rarely used in practice in large scale surveys, as both interviewers and respondents find it rather confusing, and lower literacy/numeracy respondents still don't understand how it protects their answers, and hence continue to underreport the sensitive behaviors.

Answer 3

CLT relates your mean on a sample to the parameters of your observed variable. In this case that observed variable is your biassed measurement, not your underlying actual variable.

Name the three possible outcomes x1, x2 and x3. Let r be your chance of correct measurement (in your case 0.4), so that both other outcomes have a chance of (1-r)/2 To estimate the distribution of your actual variable A, using your measured variable M, consider

P(M = xi) = sum over j (P(M = xi | A = xj) * P(A = xj))
= r * P(A = xi) + (1-r)/2 * sum for j not i P(A = xj)
= r * P(A = xi) + (1-r)/2 * (1 - P(A = xi))
= (1-r)/2 + (3r-1)/2 * P(A = xi)

In your case

P(M = xi) = 0.3 + 0.1 * P(A = xi) 
E(M) = 0.3 * mean(x1, x2, x3) + 0.1 * E(A)

As the standard deviation of M is less than ´max(x1, x2, x3) - min(x1, x2, x3)´ , the average of n samples will converge to have the above average and a standard deviation of less than

(max(x1, x2, x3) - min(x1, x2, x3)) / sqrt(n)

From the above we also know

P(A = xi) = 2/(3r-1) * (P(M = xi) + (r-1)/2)
= 2/(3r-1) * P(M = xi) + (r-1) / (3r-1)

in your case

P(A = xi) = 10 * P(M = xi) - 3

So if 40% of your measurements give a certain outcome, the actual variable will have a probability of 100% to have that value and if 30% of your measurements give a certain outcome, the actual variable will have a probability of 0% to have that value.