Calculate Negative Binomial in R

The negative binomial distribution is a probability distribution that models the number of trials needed to achieve a given number of successes in repeated, independent Bernoulli trials. This guide explains how to calculate the negative binomial distribution in R, including practical examples and interpretation.

What is the Negative Binomial Distribution?

The negative binomial distribution describes the number of failures before a specified number of successes (r) occur in a series of independent Bernoulli trials. It's often used in quality control, reliability engineering, and other fields where the number of trials until a certain number of successes is important.

Probability Mass Function:

P(X = k) = C(k + r - 1, r - 1) * p^r * (1 - p)^k

Where:

k = number of failures
r = number of successes
p = probability of success on an individual trial
C(n, k) = binomial coefficient

The negative binomial distribution is different from the binomial distribution in that it models the number of trials until a fixed number of successes, rather than the number of successes in a fixed number of trials.

Calculating Negative Binomial in R

R provides several functions to work with the negative binomial distribution:

dnbinom() - Probability mass function
pnbinom() - Cumulative distribution function
qnbinom() - Quantile function
rnbinom() - Random number generation

Basic Syntax

To calculate probabilities for the negative binomial distribution:

dnbinom(x, size, prob)

x - number of failures
size - number of successes
prob - probability of success

For example, to calculate the probability of 3 failures before 2 successes with a success probability of 0.5:

dnbinom(3, size = 2, prob = 0.5)

Example Code

Here's a complete example of working with the negative binomial distribution in R:

# Calculate probability of 3 failures before 2 successes
# with success probability 0.5
prob <- dnbinom(3, size = 2, prob = 0.5)
print(prob)

# Calculate cumulative probability of 3 or fewer failures
cumulative_prob <- pnbinom(3, size = 2, prob = 0.5)
print(cumulative_prob)

# Find the number of failures needed for 90% cumulative probability
quantile <- qnbinom(0.9, size = 2, prob = 0.5)
print(quantile)

# Generate 10 random numbers from negative binomial distribution
random_numbers <- rnbinom(10, size = 2, prob = 0.5)
print(random_numbers)

Example Calculation

Let's calculate the probability of getting exactly 4 failures before 3 successes in a series of independent trials where each trial has a 30% chance of success.

Given:

Number of failures (k) = 4
Number of successes (r) = 3
Probability of success (p) = 0.3

Using the negative binomial formula:

P(X = 4) = C(4 + 3 - 1, 3 - 1) * 0.3³ * (1 - 0.3)⁴

P(X = 4) = C(6, 2) * 0.027 * 0.0081

P(X = 4) = 15 * 0.0002187

P(X = 4) ≈ 0.00328

In R, this calculation would be:

dnbinom(4, size = 3, prob = 0.3)

Result: ≈ 0.00328

This means there's approximately a 0.33% chance of getting exactly 4 failures before 3 successes in this scenario.

Interpreting Results

When working with the negative binomial distribution, consider these key points:

Overdispersion: The negative binomial distribution accounts for overdispersion in data, where the variance exceeds the mean.
Modeling: It's often used in generalized linear models (GLMs) when the response variable has overdispersed count data.
Parameters: The size parameter (r) and probability parameter (p) together determine the shape of the distribution.

Common applications include:

Quality control in manufacturing
Reliability engineering
Biological and ecological studies
Financial risk modeling

Note: The negative binomial distribution is different from the binomial distribution. While the binomial distribution models the number of successes in a fixed number of trials, the negative binomial models the number of trials until a fixed number of successes.

FAQ

What is the difference between binomial and negative binomial distributions?: The binomial distribution models the number of successes in a fixed number of trials, while the negative binomial models the number of trials until a fixed number of successes occur.
When should I use the negative binomial distribution?: Use the negative binomial when you're interested in the number of trials until a certain number of successes, especially when the data shows overdispersion (variance > mean).
How do I interpret the size parameter in R's negative binomial functions?: The size parameter in R's negative binomial functions corresponds to the number of successes (r) in the distribution's definition. It's not the same as the number of trials.
Can the negative binomial distribution have a probability of 1?: No, the probability parameter (p) must be between 0 and 1, exclusive. A probability of 1 would imply certain success, which isn't meaningful in the negative binomial context.