@ kjetil b halvorsen, thanks for the reply. We want to evaluate the relashionship between each explanatory variable with dependent variable (suppose: income with three levels) before considering all independent variables together and running the multiple regression.

thanks for the reply. Do you recommend to consider sigma.fo =~1 as pb(log(Alter_age)) is not significant?

thanks for your time. I am confused. The p-value of pb(log(Alter_age)) is not significant. And pb refers to the nonlinear trend which is smoothing term. Please correct me if I am not right? –

@ whuber, I made the question more clear here, stats.stackexchange.com/questions/601860/… so I would be so thankful if you share your thoughts on it.

thanks so much for the reply. I made the question more clear here, stats.stackexchange.com/questions/601860/… so I would be so thankful if you share your thouth with me regarding the question. I am using GAMLSS to calculate the centile curves based on LMS method including two independent variables. For the several dependent variables, the worm plots look fine but not the Q-statitics (as there are the Z-score >2)

@ whuber, Thank you very much for your response. We are more interested in residuals ranging between -3 and 3 and any improvement is primarily beyond +3 and -3 z-scores, is unimportant in clinical practice. Although it appears that these three models are the same, do you recommend that I choose one that is at least normal in the -3 to 3 range? or still normality assumption does not matter?

@Robert, thanks so much, Just for double checking : for calculating the ULN we need to use qBCCGo(0.05, mu = mu, sigma = sigma, nu = nu, lower.tail = FALSE, log.p = FALSE) which is which is equivalent to exp(log(mu)+log(1+1.645*nu*sigma)/nu).Is this right?

thank you so much for the reply. I need to calculate the LLN for each ID after applying the LMS method. For several Ids the value of sigma*abs(nu) is more than 0.27. So when I use qBCCGo function, I can calculate it but the answer would be NA as soon as I run the formula. So, which function should I use? The formula or qBCCGo?