PROC FMM WEIGHT Statement

[jl4443]

The WEIGHT statement documentation for PROC FMM (https://documentation.sas.com/doc/en/statug/15.2/statug_fmm_syntax15.htm) links to a page on Log-Likelihood Functions for Response Distributions (https://documentation.sas.com/doc/en/statug/15.2/statug_fmm_details07.htm) for further details. On that page, not all distributions indicate how the weight function is incorporated. For example, the Weibull distribution does not have a w_i term. How are weights applied to the weibull distribution? Additionally, it's not entirely clear to me how the weights are being applied more generally - would inverse probability of treatment weights be appropriately incorporated via the WEIGHT statement?

Appreciate any insights, thank you!

[Season]

For the first question, I think only the writer of the relevant documentation can give you a definite answer. It seems to me that the Weibull formula in the SAS documentation is wrong. I think w_iy_i's should replace the y_i's in the formula.

However, an easy way to see if the WEIGHT statement in PROC FMM works properly is to build one Weibull model with the WEIGHT statement and build another one without that statement and compare the results.

If you wish to have a definite answer as to whether PROC FMM maximized the log-likelihood function in exactly the same way you wanted it to be, then a more complicated approach is to compose custom log-likelihood function in SAS procedures like PROC NLMIXED, maximize the log-likelihood there and compare its result to the one generated by PROC FMM.

For the second question, I think whether a WEIGHT statement is specified is far more important than what the specified weights stand for. In other words, I think PROC FMM will give you the correct result as long as you have stored the weights in a variable and specified it in the WEIGHT statement, regardless of what the weights stand for and what they are supposed to do. Again, to see if PROC FMM gives you exactly what you wanted, you can also custom your log-likelihood function and maximize it in procedures like PROC NLMIXED and compare its result to the one obtained by PROC FMM.

[SAS_Rob]

It appears that the weights have been omitted from the formula in the documentation and it will need to be updated. 

The short answer as far as the formula is concerned is that FMM uses the weight as a multiplicative factor on the component log-likelihood.  That is, if you look at the formula for the Weibull log-likelihood that is in the FMM documentation, all you do is multiply that expression by ‘w_i’.  If you want to see what it looks like specifically, the GENSELECT documentation has the correct formulation.

SAS Help Center: Log-Likelihood Functions