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I have used the following negtaive binomial model:

p3<-glm(formula = Infected ~ log(1 + PopDensity) + log(1 + GDPPPP)+ LifeExpectancy,family=negative.binomial(1),maxit=10000,data=df)

How can I justify the following questions?

  1. What is the rationale behind the explanatory variables, log(1+Popdensity), log(1+GDPPPP)? Why do we need such unusual transformation for those variables?

  2. What is the problem if we use Popdensity and GDPPPP as covariates?

My intention was to use log transformation because I had some countries like India with high population density. Some countries had very low population density. I don't know how to justify this log transformation. I appreciate your suggestions! Thanks!

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    $\begingroup$ Does this answer your question? In linear regression, when is it appropriate to use the log of an independent variable instead of the actual values? Although that thread is in the context of linear regression, the issues are the same for any generalized linear model like your negative binomial. $\endgroup$
    – EdM
    Commented Jan 4, 2021 at 20:30
  • $\begingroup$ @Edm I saw that too. But I am not sure how can we incorporate the same idea for GLM! $\endgroup$
    – Simpson's Paradox
    Commented Jan 4, 2021 at 20:55
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    $\begingroup$ Whether in ordinary least squares (OLS) or GLM, you still are constructing a linear predictor: the sum of products of coefficients times corresponding (potentially transformed) predictors. With a GLM, you just can have something other than an identity link function between the linear predictor and the outcome; usually a log link for negative binomial. The rationales for predictor transformations are the same: you want transformations such that the linear predictor is linearly related to outcome (via the link function, when one is involved). $\endgroup$
    – EdM
    Commented Jan 4, 2021 at 21:27

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