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I used the gamlss to fit the model. Bottom right panel: Normal QQPlot of the quantile residuals. The plots suggest a slight left skewness of the quantile residuals. My questions is should this be consider als a issue or this can be taken als ok? enter image description here

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I prefer a negative binomial (NBI or NBII in gamlss) to a quasi-Poisson.

The negative binomial is a proper distribution and NBI and NBII have 2 different variance-mean relationships. For NBI the variance = mu + sigma*(mu^2) For NBII the variance = mu + sigma*mu

The mu and sigma parameters of NBI (or NBII) can each be modelled using explanatory variables in the gamlss R package.

In gamlss the normalised (randomised) quantile residuals are calculated so that if the model is correct, then the true residuals have a standard normal distribution.

Packages for gam models usually use deviance residuals, which I do not like since they do not have a normal distribution, so it is difficult to check them.

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The normalised quantile residuals are positively skewed, i.e skewed to the right, (i.e. a heavier right tail).

Whether this is an important issue depends on the purpose of your analysis. For example if you are estimating centiles (especially in the tails) then it is an important issue.

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  • $\begingroup$ Hi Robert, thanks for the answer, that was really hepful! The model was fitted to detect the effects of temperature on disease counts. The covariates are meoteorologische variable like Temperature, Rainfall... Y variable is the disease counts. The purpose of the analysis was to plot the exposure-responde relationtionship between temperature and disease counts. In this Fall wurde the QQ-plot say the exposure-responde relationtionship would not be correct? wenn positively skewed, how should I ajust the modell? Still struggling with Statistik, any help would be much appreciated, Thanks a lot! $\endgroup$
    – Thomas
    Commented Dec 16, 2021 at 7:41
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A possible adjustment to the model would be to change to a response distribution which is more flexible, (allowing for a higher positive skewness).

I don't know what distribution you are currently using.

However, since the disease counts is discrete, possible discrete flexible distributions include:

NBI and NBII, different parameterisations of the (2 parameter) negative binomial distribution

PIG, 2 parameter Poisson-inverse Gaussian distribution

SICHEL, 3 parameter Sichel distribution

DEL, 3 parameter Delaporte distribution

BNB, 3 parameter beta negative binomial distribution

and also zero-adjusted and zero-inflated versions of the above distributions.

For a full list of distributions available in gamlss, see

‘Distributions for Modeling Location, Scale, and Shape: Using GAMLSS in R’ R. A. Rigby, M. D. Stasinopoulos, G. Z. Heller and F. De Bastiani. Chapman and Hall/CRC, Boca Raton, 2019

Paperback version (2021) www.routledge.com//9780367278847

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  • $\begingroup$ Thanks a lot! currently I am using the quasi-Poisson distribution. Maybe I should condiser Negative binomial distribution. The daily count number is arond 5, which is really small. Maybe this is one of the reason that the normalised quantile residuals can be easily skewed? But I also heard that while applying GAM it is not that important that the Error term are normally distributed? $\endgroup$
    – Thomas
    Commented Dec 19, 2021 at 21:46

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