As is known, The class of generalized linear models is an extension of traditional linear models that allows the response probability distribution to be any member of an exponential family of distributions. When the response variable is continuous and it is not appropriate to assume the response distrubution as normal distribution, especially when it is left-skewed distribution (most of value fall toward larger scale) , how to specify an appropriate ditribution for the model? (Inverse Gaussian distribution and Gamma distribution seem to be appropriate for right-skewed response.) Dose it matters to choose different type of response distibution? Can I choose the model type based on the goodness of fit?
It depends on the range of your response variable. If the lower bound truly is zero, then Gamma usually is a good choice, because it can accomodate extra variance (overdispersion towards an exponential distribution).
However, oftenresponses cannot truly become zero, but have a natural limit somewhat higher, such as with reaction times. In such a case you can use the exponentially-modified Gaussian distribution. Albeit I am aware of only one regression engine that supports the latter, the R package brms.
With leave-one-out cross validation or information criteria you can compare the predictive accuracy of different response distributions. The brms package gives you LOO and the Widely Appicable Information Criterion (WAIC).
Read more about the topic in my online book (especially chapter 7.5):
https://www.researchgate.net/project/Book-New-statistics-for-the-design-researcher
Martin Schmettow ，thank for your answer. I have read the related charpters in your online book and they are very helpful. I have two more questions.
Gamma distribution may be appropriate for the right-skewed response variables with positive scale value, like the reaction times (Yes, the exponentially-modified Gaussian distribution is more suitable according your book but most softwares do not support it) in your book. Dose it still appropriate for left-skewed response variables？
Can I use AIC (or QIC in GEE) to compare the goodness of fit of models which based on different assumptions of distribution or different link functions?
I think you have it backwards. As Martin suggests it is the properties of the DV that tell you what type of Regression to use a good reference is given here:
https://www.amazon.com/Fundamentals-Biostatistics-Bernard-Rosner/dp/130526892X/ref=sr_1_fkmr1_1?keywords=rosen+fundamentals+of+biostatistics&qid=1560724075&s=books&sr=1-1-fkmr1
Best, D. Booth
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