How to Calculate Credible vs Confidence Interval

Understanding the difference between credible intervals and confidence intervals is crucial for statistical analysis. Both provide a range of values within which a population parameter is likely to fall, but they are based on different statistical philosophies and assumptions. This guide explains how to calculate each, their key differences, and when to use them.

What Are Credible and Confidence Intervals?

Interval estimation is a fundamental concept in statistics that provides a range of values within which a population parameter is likely to fall. There are two main types of intervals: confidence intervals and credible intervals.

Confidence Intervals

A confidence interval is a range of values that is likely to contain the true population parameter with a certain level of confidence. It is based on frequentist statistics, which treats parameters as fixed but unknown values.

Credible Intervals

A credible interval, also known as a Bayesian credible interval, is a range of values that is likely to contain the true population parameter based on the posterior distribution. It is based on Bayesian statistics, which treats parameters as random variables with a probability distribution.

Key Point: Confidence intervals are about the method of estimation, while credible intervals are about the parameter itself.

Key Differences Between Them

The main differences between confidence intervals and credible intervals lie in their statistical foundations and interpretations:

Aspect	Confidence Interval	Credible Interval
Statistical Philosophy	Frequentist	Bayesian
Interpretation	Probability that the interval contains the true parameter	Probability that the parameter falls within the interval
Assumptions	Parameters are fixed, unknown values	Parameters are random variables with a prior distribution
Calculation	Based on sample statistics and sampling distribution	Based on posterior distribution of the parameter

Understanding these differences is crucial for choosing the appropriate interval for your statistical analysis.

How to Calculate a Confidence Interval

Calculating a confidence interval involves several steps, including determining the sample size, calculating the sample mean and standard deviation, and using the appropriate formula based on the sample size and population standard deviation.

Steps to Calculate a Confidence Interval

Determine the sample size (n) and the sample mean (x̄).
Calculate the sample standard deviation (s) or use the population standard deviation (σ) if known.
Choose the desired confidence level (e.g., 95%).
Find the critical value (z or t) based on the confidence level and the sample size.
Calculate the margin of error (ME) using the formula: ME = critical value × (s/√n) for small samples or ME = critical value × (σ/√n) for large samples.
Calculate the confidence interval using the formula: x̄ ± ME.

Confidence Interval Formula:

For a 95% confidence interval with a known population standard deviation:

x̄ ± z*(σ/√n)

For a 95% confidence interval with an unknown population standard deviation:

x̄ ± t*(s/√n)

Example Calculation

Suppose you have a sample of 30 observations with a mean of 50 and a standard deviation of 10. To calculate a 95% confidence interval:

Determine the critical t-value for 29 degrees of freedom at 95% confidence: t = 2.045.
Calculate the margin of error: ME = 2.045 × (10/√30) ≈ 3.65.
Calculate the confidence interval: 50 ± 3.65 → (46.35, 53.65).

How to Calculate a Credible Interval

Calculating a credible interval involves several steps, including specifying the prior distribution, calculating the likelihood function, and combining them to form the posterior distribution.

Steps to Calculate a Credible Interval

Specify the prior distribution for the parameter of interest.
Calculate the likelihood function based on the observed data.
Combine the prior and likelihood to form the posterior distribution.
Choose the desired credible level (e.g., 95%).
Calculate the credible interval based on the posterior distribution.

Credible Interval Formula:

The credible interval is determined by the posterior distribution of the parameter. For a normal posterior distribution, the interval can be calculated as:

μ ± z*σ

where μ is the posterior mean, σ is the posterior standard deviation, and z is the critical value based on the credible level.

Example Calculation

Suppose you have a prior distribution N(50, 10) and observe a sample of 30 with a mean of 52 and standard deviation of 8. To calculate a 95% credible interval:

Combine the prior and likelihood to form the posterior distribution: N(51.25, 3.16).
Determine the critical z-value for 95% credible interval: z = 1.96.
Calculate the credible interval: 51.25 ± 1.96 × 3.16 → (45.06, 57.44).

When to Use Each Type

Choosing between a confidence interval and a credible interval depends on the statistical framework you are working within and the assumptions you are willing to make.

When to Use a Confidence Interval

When working within a frequentist statistical framework.
When you want to make inferences about the sampling process.
When you have a large sample size and can assume the sampling distribution is normal.

When to Use a Credible Interval

When working within a Bayesian statistical framework.
When you want to make inferences about the parameter itself.
When you have a small sample size and want to incorporate prior information.

Practical Tip: Consider the context of your analysis and the assumptions you are willing to make when choosing between a confidence interval and a credible interval.

FAQ

What is the difference between a confidence interval and a credible interval?

A confidence interval is based on frequentist statistics and provides a range of values that is likely to contain the true population parameter. A credible interval is based on Bayesian statistics and provides a range of values that is likely to contain the true population parameter based on the posterior distribution.

How do I choose between a confidence interval and a credible interval?

Choose a confidence interval if you are working within a frequentist statistical framework and want to make inferences about the sampling process. Choose a credible interval if you are working within a Bayesian statistical framework and want to make inferences about the parameter itself.

Can I use a confidence interval and a credible interval interchangeably?

No, you cannot use a confidence interval and a credible interval interchangeably because they are based on different statistical philosophies and have different interpretations. It is important to understand the differences between them before choosing the appropriate interval for your analysis.