Quantile regression with transformation of response

"A reference interval is a type of prediction interval and contains the middle 95% of measurements of some substance or item from a healthy population. Age specific reference intervals provide variable interval limits based on age. An Age-Specific Reference Interval gives the boundaries between which a typical measurement on an individual of a certain age from a population of individuals is expected to fall. The reference interval is often presented as percentiles of a healthy reference population, such as the 2.5th percentile and the 97.5th percentile".

I have a dependent variable and variable Age as the independent variable. I want to apply the nonparametirc quantile regression for 0.025 and 0.975 quantiles and I have this shape.

So I decided to use the power transformation method (calculated lambda=-0.37) for the dependent variable and work with the transformed variable.But I am not sure about the efficiency of this model to calculate the reference curve and the z score and predicted values. Should I remove the observations which are less than 0.5 in the second plot to have more smooth curve? What should I do when the nature of observation between 0 to one year old is completely different from 1 year to 19 years old?

Also this is the plot after using cubic root transformation.

• What is the problem, exactly? Your quantile regression appears to have succeeded. – whuber Jan 18 '18 at 15:45
• You need a better title. It's not informative. Something like "Quantile regression with transformation of response" would work better. – Nick Cox Jan 18 '18 at 16:18
• @whuber, thanks for the reply. It means you think I use the transformation method in a correct way and also the method for applying the quantile curves is chosen correctly? – shadi Jan 18 '18 at 17:16
• @NickCox, thanks for the reply, I changed the title as you requested. I used power normal transformation and the calculated lambda is -0.37. Based on your advice I used the cube root (please see the final plot above). The nature of observation between 0 to 1 year is different from the rest of the observations. So any advice? I want to make the growth chart at 0.025 and 0.95 quantiles. The data set includes the information of kids from birth to 19 years old. Do you think that I should apply two different quantile regression for 0-1 year old and 1 year to 19 years old? – shadi Jan 18 '18 at 17:18
• I don't know whether your regression is "correct": see Nick Cox's answer. I am only pointing out that your question is unclear because it asks how to apply quantile regression, you applied it, and you got a result that is consistent with the data. There's no information to determine whether that is meaningful, appropriate, or correct for your purposes. – whuber Jan 18 '18 at 17:33

• Regression here and elsewhere can be sensitive to granularity in the data. Your transformation is unsatisfactory as you've created outliers from your smallest values. At a guess you're using something like log$_{10}$ (response + 0.5), but that maps zeros to a value much less than the next lowest value. A transformation such as log$_{10}$ (response + 1) or cube root is likely to work better.