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I'm trying to calculate the Confidence Interval (CI) for binomial proportion. The point estimate of the proportion is 100%.

I first used the Bootstrapping and got a CI of [100%, 100%]. I was told it is not recommended to use bootstrapping CI in this case, and it's better to use the traditional method such as Wilson score CI. Using Wilson score method, I got a CI of [89%, 100%].

I'm wondering why Bootstrapping is not recommended if the observed proportion is too close to 100% and 0%? And when Bootstrapping is more preferred compared to traditional CI methods?

Thanks in advance.

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2 Answers 2

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100% proportion is an extreme and illustrative case of the limitations of the bootstrap assumption.

Consider the case of having only a single observation of a coin flip. The bootstrap CI of the coin's bias will always be 0% or 100% since the bootstrap assumes that all possible flips are like the one you've observed. This is obviously wrong.

Similarly, also in the case where the sample is larger than 1 but is all heads (or all tails), the bootstrap is unable to account for the probability of observing cases different than those you have observed since it uses the sample as a surrogate to the unknown population.

I'd use the bootstrap for CIs when: 1. the sample size is sufficiently large 2. the distribution is not very skewed and 3. no reasonable parametric CI is available.

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The bootstrap reuses samples from the data to simulate the sampling distribution of the statistic you're interested in.

Your samples is comprised entirely of ones. So resampling from your data will always yield a bootstrapped dataset comprised of ones. Thus, the bootstrapped statistics would be homogeneous and there would be no variation. No variation means no confidence interval because the variance is 0.

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