Analysis of variance with Weibull or Gamma distributions

Question

I have been trying to find a method to analyse variance on Weibull and/or Gamma distributions but a Google search for

anovar  Weibull "gamma distribution"

yields nothing helpful. The data I have cannot be fitted to a normal distribution but fits a Weibull or Gamma distribution quite well.

Answer 1

Besides the issue of the misspelling of 'anova' that Stephan mentioned, I have a few points.

1) I'd also suggest inserting "|" (for "or") in your search, since otherwise it will be treated more like "and"

2) further, the word 'distribution' goes as much with Weibull as it does with gamma. So I suggest a search like so: anova weibull|gamma distribution.

3) further still, in the case of the gamma, an ANOVA-type model would normally be fitted using GLMs, so you may prefer to search on gamma GLM

4) Parametric Weibull models are often available under options relating to survival analysis in many statistics packages; while survival models often have censored data, they don't have to, so a Weibull model with grouping-factors as IVs ("ANOVA-like") models can often be fitted that way.

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The usual way to compare gamma means would be via a GLM.

This has the underlying assumption of equal shape parameters (much as ordinary ANOVA carries the assumption of equal variances).

This assumption can be assessed, for example, either visually (by looking at whether they seem to have similar shapes), or by finding MLEs of the shape parameters of the groups being compared. If the values are not too dissimilar then this approach should work fine.

[Similarly, a comparison of Weibull means might be achieved by treating the values as (uncensored) survival times in a survival model.]

On the other hand if it's not expected that the gammas are at least reasonably similar in shape, it's more complicated; one might try to form a confidence interval for the difference in means in any of several ways. In large samples, one might try bootstrapping, for example, or the distribution of a ratio of ML estimates of the mean (difference in logs) might be approximated or even assessed via simulation.

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For comparison purposes, the following can be done in R (the data set is built in):

summary(lm(weight~feed,chickwts)) # Linear regression model for one way model 

summary(glm(weight~feed,family=Gamma(link="identity"),chickwts)) # Gamma equivalent

summary(glm(weight~feed,family=Gamma(link="log"),chickwts)) # gamma with log-link

summary(survreg(Surv(weight)~feed,data=chickwts))  # Weibull model

anova(...) can be used in place of summary(...) in those calls to obtain other information.

The first model is an ordinary one-way anova type model.

The second is the equivalent using a gamma model (in this case with essentially identical parameter estimates to the first model but different standard errors).

The third model is also a gamma anova-type model but where the parameters describe effects on the log scale (a test of the model is still a test for differences in means on the original scale, though).

The fourth model is a Weibull model, which has log-scale parameter estimates (which can be compared with the third model).