Dispersion parameter
Topic of Introduction to generalised linear models, part of Generalised linear models
Dispersion parameter
| Type | Topic |
| Taught in | Introduction to generalised linear models |
| Part | Generalised linear models |
| Questions using it | 8 |
The parameter \phi that scales the variance of a GLM, \operatorname{Var}(Y)=\phi V(\mu). It is \sigma^2 for the normal and 1/\alpha for a gamma with shape \alpha; it is fixed at 1 for the Poisson and binomial unless a quasi-likelihood frees it.
In Introduction to generalised linear models
The times themselves are continuous and strictly positive, so they are not normal. A gamma distribution is a reasonable start: it is supported on (0,\infty) and it extends the exponential. A gamma law has a second parameter, a dispersion, which plays the role that \sigma^2 plays for the normal and can be chosen separately from the mean. In the shape-and-rate parameterisation, a \mathrm{Gamma}(\alpha,\alpha/\mu) random variable has mean \mu and variance \mu^2/\alpha. The model is the mean and the variance: Y_i\mid {\boldsymbol{X}}_i\stackrel{\mathrm{ind}}{\sim} \mathrm{Gamma}\!\left(\mu_i=\exp({\boldsymbol{X}}_i^\top\boldsymbol{\beta}),\ \mathrm{Var}=\phi\mu_i^2\right). Here \phi is the dispersion. The variance grows with the square of the mean, which is a stronger mean–variance relationship than the Poisson’s \mathrm{Var}=\mu. It is the same shape you see in the data: times near zero have little room to vary, and large mean times can vary a lot. Written out in full, this is shape 1/\phi and rate 1/(\phi\mu), because then the mean is \mu and the variance is \phi\mu^2.
Used in
- Taught in Introduction to generalised linear models
- Also used in Standard errors, and models for counts · Underdispersion, positive responses, and quasi-likelihood · Categorical responses · Paired comparisons, and how far a fit is from the data · Marginal models, and two responses at once
- Questions that use it (8) — Q47 (answer) · Q54 · Q55 (answer) · Q57 (answer) · Question 19 (answer) · C23 (answer) · C28 · C29 (answer)
- All 9 questions on Introduction to generalised linear models
See also
Variance · Generalised linear model · Quasi-score and quasi-likelihood · Exponential family · Overdispersion · Scaled deviance · Mean and variance of an exponential family · Canonical link