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GLM assumes that the mean of some response depends linearly on a set of input variables. So given a particular input, we can know the associated mean. Does the model make any assumptions concerning the variance?

Nick Cox
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Mustafa Azzam
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  • Hi: the variance depends on the "family" which really means the distribution of the response. ( is the response logit, is it poisson, is it normal etc ). When trying to understand this , I remember John Fox's "companion to applied regression" helping a lot. I strongly suggest checking that out. He explains the purpose of the link function and the family chosen etc. – mlofton Nov 02 '19 at 04:15
  • see 1. https://stats.stackexchange.com/questions/228557/understanding-the-glm-coefficients-calculation/229712#229712 2. https://stats.stackexchange.com/questions/313785/what-is-variance-argument-in-quasi-generalized-linear-models/313812#313812 3. https://stats.stackexchange.com/questions/212430/what-are-the-error-distribution-and-link-functions-of-a-model-family-in-r/212433#212433 etc – Glen_b Nov 02 '19 at 06:08
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    @mlofton Logit is not a family; it’s a link function. The first sentence of the question also misses the role of the link function. – Nick Cox Nov 02 '19 at 08:51
  • Thanks Nick. I always confuse the two. Should I delete or fix it ? – mlofton Nov 03 '19 at 03:45
  • You can't edit a comment this long after posting it, but you can delete it and post a revised version. – Nick Cox Nov 03 '19 at 10:42

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For many families of distributions such as the Poisson distribution or Bernoulli/binomial distribution, the variance is already determined, if you have specified the mean (Poisson: variance=mean, Bernoulli: variance=mean*(1-mean)). For other distributions such as the normal distribution or negative binomial distribution (in mean rate + dispersion parameterization), there is an additional parameter (e.g. standard deviation, precision or variance for normal, dispersion or overdispersion parameter for negative binomial). Thus, for a GLM we typically specify the regression equation for the mean response transformed via the link function and - if necessary - also the additional parameter.

Björn
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