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Recently I’m studying modelling and prediction with R, here is a full module after data cleaning. I’m wondering the Z distribution and P(|Z|) in my coefficients table.

pic: the full module after data cleaning and data exploration I suppose each parameter has t distribution because we are using Sample Variance instead of Variance!

In my opinion, we’d like to use Z distribution, t distribution, chi square distribution and F distribution to evaluate our prediction module, but I’m not very clear when to use which distribution and what conclusions we can get from them.

And here in the quasipoisson module, we have t distribution and P(|t|), but why? pic: same data but with quasipoisson distribution

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As the coefficients table suggests (z value), the GLM model treats the parameter estimates (scaled by their standard errors) as following (based on asymptotic distribution) a standard normal or Z distribution. The t-distribution only applies when we assume (some function of) the data follow a normal distribution, too. So in linear regression, when the outcome is assumed to have a normal distribution, the parameters then follow a t distribution. But for (non-linear) GLMs this is explicitly not assumed.

In brief, then, the t distribution doesn't really enter the picture.

The Chi-squared distribution can be used to test differences between models as a whole, for example to compare the 'null' model and its deviance to your fitted model with its residual deviance. here are some more details about that test and its rationale.

The F distribution is simply the distribution of the ratio of two (independent) Chi-squared random variables but it doesn't necessarily have an obvious application in evaluating your model.

It's just as important to look at your data and try to check or test the model assumptions - a standard move for Poisson GLM is to also fit a negative binomial and compare the fits. Consider too whether any of your predictors have interactions or are highly correlated.

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  • $\begingroup$ Thank you sooo much! That’s really a big help for me. $\endgroup$ – Jeff Luo Nov 14 '19 at 6:05
  • $\begingroup$ And how it’s possible to compute Z distribution of each parameters? I mean we even have no clue about the the mean and the variance for each parameter, which are essential in computing the Z value and P(|Z|). Instead we only have one sample and its sample mean and sample variance from our training set(abalone1.df). $\endgroup$ – Jeff Luo Nov 14 '19 at 6:24
  • $\begingroup$ "we even have no clue about the the mean and the variance for each parameter" The mean is in the "Estimate" column, the square-root of the variance is in the "Std. Error" column. $\endgroup$ – Roland Nov 14 '19 at 11:23

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