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rgamma-distribution
viridius
viridius

Reputation: 497

Gamma equivalent to standard deviations

I have a gamma distribution fit to my data using libary(fitdistrplus). I need to determine a method for defining the range of x values that can be "reasonably" expected, analogous to using standard deviations with normal distributions.

For example, x values within two standard deviations from the mean could be considered to be the reasonable range of expected values from a normal distribution. Any suggestions for how to define a similar range of expected values based on the shape and rate parameters of a gamma distribution?

...maybe something like identifying the two values of x that between which contains 95% of the data?

Upvotes: 0

Views: 1989

Answers (2)

MC Kwit
MC Kwit

Reputation: 9

The mean expected value of a gamma is:

E[X] = k * theta

The variance is Var[X] = k * theta^2 where, k is shape and theta is scale.

But typically I would use 95% quantiles to indicate data spread.

Upvotes: 1

josliber
josliber

Reputation: 44330

Let's assume we have a random variable that is gamma distributed with shape alpha=2 and rate beta=3. We would expect this distribution to have mean 2/3 and standard deviation sqrt(2)/3, and indeed we see this in simulated data:

mean(rgamma(100000, 2, 3))
# [1] 0.6667945
sd(rgamma(100000, 2, 3))
# [1] 0.4710581
sqrt(2) / 3
# [1] 0.4714045

It would be pretty weird to define confidence ranges as [mean - gamma*sd, mean + gamma*sd]. To see why, consider if we selected gamma=2 in the example above. This would yield confidence range [-0.276, 1.609], but the gamma distribution can't even take on negative values, and 4.7% of data falls above 1.609. This is at the very least not a well balanced confidence interval.

A more natural choice might by to take the 0.025 and 0.975 percentiles of the distribution as a confidence range. We would expect 2.5% of data to fall below this range and 2.5% of data to fall above the range. We can use qgamma to determine that for our example parameters the confidence range would be [0.081, 1.857].

qgamma(c(0.025, 0.975), 2, 3)
# [1] 0.08073643 1.85721446

Upvotes: 1

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