Calculating Real Parameter SEs from Betas - MLogit

[adamd]

I am trying to calculate SEs of real parameter estimates using Beta output from MARK. I am using multistrata modeling to estimate the effects of certain management actions on movement of cormorants among colonies. I know that I need to use the delta method to estimate variances. I've read Appendix B of Gentle Introduction and the documentation from Colorado State's FW663 class on the subject. I have replicated SE estimates for apparent survival and resight probability, that were modeled using the logit link. I can not figure out how to calculate SE for movement parameters that were estimated using the multinomial logit link. I have calculated the partial derivative of the mlogit equation in Mathematica D, and tried to calculate variance as:

var(Y) = DV transposeD, where V is the variance-covariance matrix.

Estimating SEs for movement rates based upon Beta estimates is an intermediate step in calculating certain effect sizes. I have included the effect of oiling cormorant eggs on movement as the proportion of the colony that was treated (in the design matrix). So the size of the effect of oiling eggs would the difference between movement rates with egg oiling and movement rates without egg oiling. Where each of these effects and their SEs are estimated from the Beta equation.

In an ideal world, we would have been able to apply a single new management option each year, so that we could simply calculate effect sizes as differences in real parameter estimates from one year to the next. Of course, this is assuming there is no annual variation. Unfortunately, I can not estimate the effect sizes in this manner.

Any help would be greatly appreciated.

[gwhite]

Adam:
I compute the variance of the real parameters from the mlogit beta parameters using numerical methods, not analytical methods like most of the other simpler link functions.
My advice is to re-run the models that you want variances from, but use a simpler link function for the parameters, and start the models at the solution obtained from the mlogit analysis. That is, suppose you replace the mlogit link with a logit link. Then, compute the starting value of the logit beta parameter as log(theta/(1-theta)) where theta is the real parameter estimate from the mlogit analysis.
Using this approach, you will still have valid parameter estimates, but can now manipulate them much more easily.
Gary

[adamd]

Gary:

Thanks for this piece of advice. At first, this seemed simple; however, when I tried to apply this I became confused. My data included 6 strata and 6 years, therefore I have 210 real parameter estimates. My best model includes 23 parameters (betas). When I provide starting values for parameter estimates, I only need to input 23 values. So I don't understand which of the real parameter estimates should be transformed into starting values.

I can estimate the effect size for each of the paramers. I transformed effect sizes and used these as starting values when I substituted the logit link for the Mlogit link. Incidentally, I got exactly the same estimates (beta and real) as when I didn't provide starting values and substituted link functions. So the process of estimating effect sizes didn't get me closer to the estimates from the model with Mlogit links.

When comparing estimates from the model with Logit to the model with Mlogit links, three real parameter estimates were quite a bit different. These differences were about 0.03 increases in movement rates or 22-28% greater - the other 210 were very similar. Is this what I am trying to do? Get close and use the SE estimates for my calculations of SE of effect sizes?

Another alternative is to use beta estimates from the model that included the Mlogit links as a starting point for the model with only Logit links. This does not make much sense, since these link functions should provide different beta estimates.

Any clarification that you can provide would be greatly appreciated!

[adamd]

I made an error in my last post.

When I attempted to calculate effect sizes for each beta parameter and transform these using the function log (theta/(1-theta)), where theta was the effect size, I got vastly different results compared to my model using the MLogit link.

When I set the starting values equal to the betas from the MLogit link and substituted the Logit link for the MLogit link, I got very similar results. This was when three real parameter estimates for movement differed by 0.03.

In either case, I was not exactly following the advice that Dr. White provided. I am still confused.

[gwhite]

Adam:
From your response, I suspect you are using one or more covariates to model the psi values? If so, you will not be able to do the simple trick I previously described, because the covariates affect ALL of the real parameters in the MLOGIT link set. Given that you were within 0.03, you probably were doing what I said, but didn't understand that now the covariate will have a different effect with the MLOGIT link than it now will have with the logit link.
Gary

[adamd]

Gary:

Ok, to calculate SEs from Beta estimates, I should not use the short cut method because I am using covariates. These are not individual covariates, but are covariates that apply to the strata where cormorants move away from for 3 covariates and where cormorants move to for another covariate.

Would I then estimate variances for movement parameters (modeled with Mlogit link) as the product of:

the vector of partial derivatives of the Multinomial transform function (vector D)

the portion of the variance-covariance matrix for movement, and

the transpose of vector D?

Second, if this is correct, would you expect these estimates of variance to include relatively more rounding error than estimates of variance calculated using the derivative of the logit transform function? I think you might because error propogation should be greater for the Mlogit function, since for each estimate you must include error for all other moves that an animal could make.

My SE estimates for parameters that I modeled with the Logit Link are accurate to 7 or 8 decimal places. My SE estimates for movement parameters are accurate to 2 to 3 decimal places in most cases and in a few cases only to 1 decimal place. Is this expected or a sign that I messed up?

Thank you for the help you have already provided and any additional suggestions that you might have!