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I read the following: enter image description here in John Fox's Applied Regression Analysis and Generalized Linear Models.

I don't understand this distinction.

In OLS, I have

$$E[y_i|X] = E[x_i^T\beta + \epsilon_i|X] = x_i^T\beta + E[\epsilon_i |X]= x_i^T\beta$$ Now suppose I have a log-linear model

$$\log(y) = X\beta + \epsilon$$

where transforming the data is similar to this case

enter image description here

enter image description here

where transforming the data leads to a linear relationship. source: https://kenbenoit.net/assets/courses/ME104/logmodels2.pdf

What do GLMs do that is different from this?

Stan Shunpike
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  • You seem to be asserting $z_i = \log(y_i) = x_i^\prime\beta + \epsilon_i = y_i$ which is impossible. When you correctly take logarithms you obtain $z_i = \log(x_i^\prime\beta + \epsilon_i)$ but that cannot be expressed in the form $x_i^\prime \gamma + \delta_i.$ – whuber Jan 23 '20 at 01:27
  • @whuber I tried to rephrase the question. Let me know if it's still nonsensical. – Stan Shunpike Jan 23 '20 at 01:46
  • Doesn't https://stats.stackexchange.com/questions/29325/ answer your question? Perhaps https://stats.stackexchange.com/questions/43930 adds a little something to that. Check out [our other highly-voted threads on GLM](https://stats.stackexchange.com/search?tab=votes&q=glm), too. – whuber Jan 23 '20 at 03:16
  • @whuber Your comments on one of the answers in the second link you posted touch on something I'm confused about. In OLS, the estimator $\hat{\beta}$ is derived from the data via a closed form solution. I don't understand why for GLM we are concerned with $E[y_i]$ instead of $y_i$. Do we treat each observation as a random variable? If so, why would we need to make that specific for each observation? If each $y_i$ is drawn from the same distribution, why wouldn't there just be one mean (i.e. the mean for that distribution)? – Stan Shunpike Jan 23 '20 at 19:03
  • In order: yes; because the observations are independent; yes. – whuber Jan 23 '20 at 19:35
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    @whuber ok thanks! i think you can close this as a duplicate of the second link you gave then. I can also delete it if thats preferred. – Stan Shunpike Jan 23 '20 at 20:20

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