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My analysis is about systematic changes over time. I have counted data (values range from 2 to 7) that follows a normal distribution (according to the histogram and kurtosis), but this answer recommends analyze count data with GLM (log link function). In my case, can I use simple linear regression? what would be a test to justify it? My sample size is big (~ 8,000)

thank you for your answers.

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MSS
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    In the olden days we regularly used simple linear regression routines for such data (well, when the counts were a bit higher than what you show) as software to perform a glm (with a log link function) wasn't so widely available. My suggestion is to follow @Bernhard 's advice but follow it with a simple linear regression to show your boss. You'll likely find that the residuals don't support the assumption of constant variance (although a square root transformation on the counts might remedy that problem). The point is show your supervisor the ugly characteristics of simple linear regression. – JimB Oct 23 '16 at 00:08
  • Thank you @Jim Baldwin, the mean = 3.81, variance = 1.15; variance-to-mean ratio = 0.30, the residual plots look almost the same. Do you think a likelihood ratio or anova test could be a good way to justify the use of Poisson? I ran it and the model with the lower value is the GLM (the best?) – MSS Oct 23 '16 at 02:47
  • small counts don't "follow a normal distribution". But in any case if you're examining the marginal distribution, you're not even checking the assumption that regression makes (which relates to the conditional distribution). If there's changes over time, you also have the potential issue of serial dependence. You can justify a simple linear regression without normality if you can demonstrate constant variance, linearity, etc; the assumption relates to the inference, not to fitting the regression (it impacts efficiency). Normality may not be the main issue (you'll lose a bit of power). – Glen_b Oct 23 '16 at 03:18

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Your count data does not follow a normal distribution, because it simply can not. Because it can not, simple linear regression is not the way to go. That being said, a GLM with poisson distribution is not all that difficult. It can be done by a beginner. So why not just try it? Can you think of any good reason, why not?

Bernhard
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  • My most powerful reason is that my supervisor wants the analysis like that, now I need to justify it. Any advice? Thanks for your answer @Bernhard! – MSS Oct 22 '16 at 23:54
  • I would be quite concerned if your supervisor wants you to use simple linear regression. It really isn't the way to analyse this type of data, and there's no good way to justify it really. I agree with the GLM approach using a poisson distribution – Conor Neilson Oct 23 '16 at 02:40
  • Thank you @Conor Neilson, can you please have a look at the question I just wrote in Jim Baldwin comment? – MSS Oct 23 '16 at 02:49
  • Finally, all models are wrong. Finally it is all about how useful a model is in your situation. The supervisor being your supervisor may be a really strong point and the world has seen worse than count data being tackled by a simple linear model. Without more information, you ask a real life question, not a statistics question. Ask yourself: Do.people die, if you do this wrong or is just your homework less elegant? – Bernhard Oct 23 '16 at 07:08
  • Thanks everyone! Indeed, my data violates the assumptions. – MSS Oct 26 '16 at 15:08