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1.Comparing Different Confidence Interval Methods[Original Blog]
- The frequentist approach is perhaps the most commonly used method for constructing CIs. It relies on the concept of repeated sampling: if we were to collect an infinite number of samples from the same population, the resulting CIs would cover the true parameter value (e.g., conversion rate) with a certain probability (usually 95%).
- Formula: For a proportion (e.g., conversion rate), the CI is calculated as:
$$\text{CI} = \hat{p} \pm z \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$
Where:
- $$\hat{p}$$ is the sample proportion.
- $$z$$ is the critical value from the standard normal distribution (e.g., 1.96 for a 95% CI).
- $$n$$ is the sample size.
- Example: Suppose we have a sample of 500 users, and 80 of them converted. The 95% CI for the conversion rate would be [0.144, 0.176].
2. Bayesian Approach:
- Bayesian CIs incorporate prior knowledge (prior distribution) and update it based on observed data. Unlike frequentist CIs, Bayesian intervals provide a probability distribution for the parameter.
- Formula: The Bayesian CI is derived from the posterior distribution, which combines the likelihood function and the prior distribution.
- Example: Using a Beta(2, 10) prior, we observe 80 conversions out of 500 users. The 95% Bayesian CI might be [0.139, 0.175].
- The bootstrap method is non-parametric and resamples from the observed data to estimate the sampling distribution of a statistic (e.g., conversion rate). It provides an empirical CI.
- Procedure:
- Randomly sample with replacement from the original data to create multiple bootstrap samples.
- Calculate the statistic (e.g., conversion rate) for each bootstrap sample.
- Construct the CI from the percentile-based distribution of these statistics.
- Example: After bootstrapping, we obtain a 95% CI of [0.142, 0.178].
- An improvement over the standard frequentist CI, the Agresti-Coull interval adjusts for small sample sizes by adding pseudo-counts to the numerator and denominator.
- Formula:
$$\text{CI} = \hat{p} + z \cdot \sqrt{\frac{\hat{p}(1-\hat{p}) + \frac{z^2}{4}}{n + z^2}}$$
- Example: With the same data, the Agresti-Coull 95% CI becomes [0.141, 0.177].
- The Wilson score interval balances precision and coverage. It accounts for both sample size and proportion.
- Formula:
$$\text{CI} = \frac{\hat{p} + \frac{z^2}{2n} \pm z \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n} + \frac{z^2}{4n^2}}}{1 + \frac{z^2}{n}}$$
- Example: The 95% Wilson score interval for our data is [0.141, 0.177].
In summary, each CI method has its merits and assumptions. Consider the context, sample size, and your prior beliefs when choosing an appropriate method. Remember that CIs provide a range of plausible values, not a single point estimate. By comparing and contrasting these methods, we can make informed decisions in statistical inference.
Comparing Different Confidence Interval Methods - Conversion Confidence Interval Understanding Conversion Confidence Intervals: A Practical Guide