Convert lognormal distribution LS Means to the original scale

[LOLO12]

Hi All,  I have used this reference to convert the lognormal LS mean estimates to the original scale. The question is what is σ2 in the model outputs?  I'm using CS covariance structure 

E[Y]=exp⁡{μ}ω

Var[Y]=exp⁡{2μ}ω(ω−1)

ω=exp⁡{σ2}

https://documentation.sas.com/doc/en/statcdc/14.2/statug/statug_glimmix_syntax17.htm

[Ksharp]

You need to use sas built-in macro
%Margins
%NLMeans
%NLEstimate
to complete this task.

https://support.sas.com/kb/37/228.html
https://support.sas.com/kb/62/362.html

[LOLO12]

I just need to know how to find the Standard Deviation without the macro? 

[Rick_SAS]

I think you are asking where to find an estimate for the sigma^2 term in the formula among the PROC GLMMIX output.

If you add the SOLUTION option to the MODEL statement, you should get a ParameterEstimates table. The last row has an estimate of the scale parameter.

I believe that the estimate is for the sigma^2 term in the formulas.

I don't use PROC GLIMMIX very often, so perhaps a GLIMMIX expert can confirm?

[SteveDenham]

We have had luck using the standard error of the lsmean reported in GLIMMIX as the value to plug in for sigma^2 to give a standard error on the arithmetic scale. Logically this seems a bit off, but the values for the standard error (square root of the variance) are reasonable.

Wikipedia presents the following formulas. 

in which mu is the expected value (mean) on the log scale and sigma^2 is the variance on the log scale.

I know I must be missing something algebraic as to why the two sources use different representations. I definitely think @Rick_SAS 's statement about using the estimate of the scale parameter from the SOLUTION is helpful.

SteveDenham