I log transformed my dependent variable, can I use GLM normal distribution with LOG link function? I have a question concerning Generalized Linear Models (GLM).My dependent variable (DV) is continuous and not normal. So I log transformed it (still not normal but improved it).
I want to relate the DV with two categorical variables and one continuous covariable. For this I want to conduct a GLM (I am using SPSS) but I am unsure how to decide on the distribution and function to choose.
I have conducted Levene's nonparametric test and I have homogeneity of variances so I am inclined to use the normal distribution.
I have read that for linear regression the data does not need to be normal, the residuals do. So, I have printed the standardized Pearson residuals and predicted values for linear predictor from each GLM individually (GLM normal identity function and normal log function). I have conducted normality tests (histogram and Shapiro-Wilk) and plotted residuals against predicted values (to check for randomness and variance) for both individually. Residuals from identity function are not normal but residuals from log function are normal. I am inclined to choose normal with log link function because the Pearson residuals are normally distributed.
So my questions are:  


*

*Can I use GLM normal distribution with LOG link function on a DV that has already been log transformed?  

*Is the variance homogeneity test sufficient to justify using normal distribution?  

*Is the residual checking procedure correct to justify choosing the link function  model?  


Image of the DV distribution on the left and residuals from the GLM normal with log link function on the right.

 A: 
Can I use GLM normal distribution with LOG link function on a DV that has already been log transformed?

Yes; if the assumptions are satisfied on that scale

Is the variance homogeneity test sufficient to justify using normal distribution?

Why would equality of variance imply normality?

Is the residual checking procedure correct to justify choosing the link function model?

You should beware of using both histograms and goodness of fit tests to check the suitability of your assumptions:
1) Beware using the histogram for assessing normality. (Also see here)
In short, depending on something as simple as a small change in your choice of binwidth, or even just the location of the bin boundary, it's possible to get quite different impresssions of the shape of the data:

That's two histograms of the same data set. Using several different binwidths can be useful in seeing whether the impression is sensitive to that. 
2) Beware using goodness of fit tests for concluding that the assumption of normality is reasonable. Formal hypothesis tests don't really answer the right question.
e.g. see the links under item 2. here

About the variance, that was mentioned in some papers using similar datasets "because distributions had homogeneous variances a GLM with a Gaussian distribution was used". If this is not correct, how can I justify or decide the distribution? 

In normal circumstances, the question isn't 'are my errors (or conditional distributions) normal?' - they won't be, we don't even need to check. A more relevant question is 'how badly does the degree of non-normality that's present impact my inferences?"
I suggest a kernel density estimate or normal QQplot (plot of residuals vs normal scores). If the distribution looks reasonably normal, you have little to worry about. In fact, even when it's clearly non-normal it still may not matter very much, depending on what you want to do (normal prediction intervals really will rely on normality, for example, but many other things will tend to work at large sample sizes)
Funnily enough, at large samples, normality becomes generally less and less crucial (apart from PIs as mentioned above), but your ability to reject normality becomes greater and greater.
Edit: the point about equality of variance  is that really can impact your inferences, even at large sample sizes. But you probably shouldn't assess that by hypothesis tests either. Getting the variance assumption wrong is an issue whatever your assumed distribution.

I read that scaled deviance should be around N-p for the model for a good fit right? 

When you fit a normal model it has a scale parameter, in which case your scaled deviance will be about N-p even if your distribution isn't normal.

in your opinion the normal distribution with log link is a good choice

In the continued absence of knowing what you're measuring or what you're using the inference for, I still can't judge whether to suggest another distribution for the GLM, nor how important normality might be to your inferences. 
However, if your other assumptions are also reasonable (linearity and equality of variance should at least be checked and potential sources of dependence considered), then in most circumstances I'd be very comfortable doing things like using CIs and performing tests on coefficients or contrasts - there's only a very slight impression of skewness in those residuals, which, even if it's a real effect, should have no substantive impact on those kinds of inference. 
In short, you should be fine.
(While another distribution and link function might do a little better in terms of fit, only in restricted circumstances would they be likely to also make more sense.) 
