Re: st: Choosing a family using glm

[Phil Schumm <pschumm@uchicago.edu>]

On Aug 24, 2010, at 12:10 PM, Laurie Molina wrote:
> I'm trying to fit a glm to get non negative fitted values.  I am  
> thinking to use a glm with a log link.  But i am  not sure about  
> wich family to use.  Is there any test i can perform to choose  
> between the normal and gamma distribution?


Everything Nick said is correct, of course -- I'll just expand a bit.   
WRT the distributional family, what is most important is that the  
variance function of the family (i.e., the way in which the variance  
changes WRT the mean) is consistent with your data.  For example, the  
variance function for the Normal distribution is V(mu) = 1 (where mu  
is E(Y) or the mean of Y), which corresponds to constant variance  
(i.e., this is why you look for homoscedasticity in residual plots  
after classical linear regression).  In contrast, the variance  
function for the gamma distribution is V(mu) = mu^2, which means that  
the variance increases with the square of the mean (i.e., constant  
coefficient of variation).  The easiest (and in any case  
indispensable) way to check if your variance function is plausible is  
to plot the standardized residuals versus the fitted values and verify  
that the amount of variation appears constant; in some cases it might  
be helpful to examine a plot of the absolute residuals versus the  
fitted values, together with the aid of -lowess-.


> My data is for the rent price of houses, so it is not count data and  
> therefore i think i should not use poisson.


Again, what's important is that you select a family whose variance  
function is consistent with your data.  For more information, see the  
book Generalized Linear Models by McCullagh and Nelder.


> To my understend in a clasical linear regression the asumption of  
> normality is in the distribution of the error term, but in glm the  
> asumption defined by the family selection is on the distribution of  
> the dependent variable. Isnt that a huge cost for using glm instead  
> of a clasical linear regression model?


You are laboring under a misunderstanding.  To say that the  
distribution of Y conditional on X is Normal with mean XB and variance  
sigma^2 is the same as saying that the distribution of the errors  
(i.e., Y - XB) is Normal with mean 0 and variance sigma^2.  And to  
emphasize the GLM approach, what is most important (if you're fitting  
a linear regression) is that the mean is XB and the variance is  
constant (i.e., that your assumptions about the first and second  
moments are correct).


-- Phil

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