Back Transform of SHASH Transformed Value

[#1]

Hello,

I am trying to calculate normal tolerance intervals for a population of data which follows a non-normal distribution. After using SHASH transformation to transform the data to normal, can tolerance intervals be calculated and back-transformed to the native distribution? If so, is there a readily available formula or script by which to execute this back transformation?

If this is not possible, are there other suitable transformations for completing this type of analysis? In the past I have used a Johnson Su transformation for a similar purpose.

Any help would be greatly appreciated.

[#2]

Check out this earlier Community Discussion Back transform of Student t distribution transformation 

[#3 · Accepted solution]

Hi @scappuccio : Presumably, you used SHASHTrans(x, gamma, delta, theta, sigma), to transform to a normal distribution. Could you then use SHASHInv(z, gamma, delta, theta, sigma) to back-transform? Or am I missing something?

[#4]

Hi @MRB3855,

Thank you for the reply. Please forgive my poor understanding of JMP and statistics in general. I used the drop down menus within the Analyze>Distributions window to transform my data set from its native distribution to SHASH. Is the SHASHInv function still applicable for returning a native value from a singular point (in this case, a tolerance interval calculated from the SHASH-transformed distribution) on the transformed distribution? 

[#5]

Hi @scappuccio : If your raw data has a SHASH distribution, then the raw data needs to be transformed to Normal before calculating tolerance intervals. You would do that via SHASHTrans(x, gamma, delta, theta, sigma). What function did you use to "transform" to SHASH? If you used "SHASH Distribution" or "SHASH Density", neither are transformations to (or from) SHASH.

Unless I'm misunderstanding, what you did appears to be the same thing that is going on in the link below. And please read the entire thread, as there is a lot going on here.

https://community.jmp.com/t5/Discussions/Back-transform-of-Student-t-distribution-transformation/m-p...

[#6]

Hi @MRB3855,

Thank you for linking to that thread - it appears I am trying to accomplish nearly the same thing as that user.

If I have used Continuous Fit>Fit SHASH and then Save Columns>Save Transformed, is this the same as using your suggested function on each point in my data set?  As you had mentioned in that thread my use-case is such that I am trying to meet the normality assumption of a traditional linear model.

If so, is it correct to calculate normal tolerance intervals on the new columns and use the SHASHInv(z, gamma, delta, theta, sigma) function to return the output to the native distribution? If not, is the script suggested by txnelson in the same thread the only option for returning a value?

[#7]

Hi @scappuccio : Yes, you are making the same mistake as that user...except that user is considering a t-distribution rather than SHASH distribution.  The formula should be in  the column properties so you can see which function is being used.

You can use the distribution platform, to see which distribution fits your data, but that platform cannot be used, as you describe, to transfer to/from any distribution.

If the distribution platform indicates your data has a SHASH distribution, then the way to transform it to a normal distribution is by the SHASHTrans(x, gamma, delta, theta, sigma) function. Create another column (let's call it Y in this example) and put the SHASHTrans(x, gamma, delta, theta, sigma) function in that column Y (x is your raw data).

Then, get a tolerance interval on Y using the distribution platform.

Then, save the tolerance interval to a JMP data set (right click the interval then click "Make into Data Table") . Then add three new columns each with the SHASHInv(z, gamma, delta, theta, sigma) function, where z is the lower TI bound, the mean, and the upper TI bound, respectively. Those three columns will result in a tolerance interval (and the median) for your raw data.   

[#8]

Hi @MRB3855,

Thank you again for the tutorial, this is extremely helpful. Where I am lost is that following your methodology seems to produce the same values as the method I method I described previously. I have attached an example file following both methods.

I have not confirmed this behavior over multiple data sets but could you please confirm whether these two paths should or should not produce the same results?

[#9 · Accepted solution]

Hi @scappuccio : It appears that there are more options available for SHASH distributions than other distributions. I stand corrected.

So, all you gotta do now is follow the steps as I outlined above.

[#10]

Hi @MRB3855,

That is great, thank you for looking into this. I appreciate you answering all my questions!