Identity Link and Real Confidence Intervals

[tgarrison]

I just ran a very simple CJS in MARK placing the identity link on both Phi and p. I've also removed the intercept from the design matrices so that parameters aren't estimated as offsets from the intercept. When looking at the output file I'm finding that the Beta estimates and real estimates are identical as expected, the Beta and real standard errors are identical as expected, but the Beta and real confidence intervals don't align. What's going on here ? These confidence intervals should align when placing no constraint on the parameters (i.e. using the identity link and removing the intercept from the design matrix), correct ?

From bottom of Mark Output file:

IDENTITY Link Function Parameters of { Phi(~-1 + time)p(~-1 + time) }
95% Confidence Interval
Parameter Beta Standard Error Lower Upper
------------------------- -------------- -------------- -------------- --------------
1:Phi:time1 0.5857530 0.0904862 0.4084000 0.7631059
2:Phi:time2 0.7128867 0.1138726 0.4896964 0.9360771
3:Phi:time3 0.7778674 0.1265650 0.5297999 1.0259348
4:p:time2 0.0149880 0.0029750 0.0091569 0.0208191
5:p:time3 0.4451827 0.0202556 0.4054817 0.4848837
6:p:time4 0.0613498 0.0108518 0.0400802 0.0826194
7:p:time5 0.2097905 0.0340483 0.1430559 0.2765251


Real Function Parameters of { Phi(~-1 + time)p(~-1 + time) }
95% Confidence Interval
Parameter Estimate Standard Error Lower Upper
-------------------------- -------------- -------------- -------------- --------------
1:Phi g1 c1 a0 t1 0.5857530 0.0904862 0.4050480 0.7459915
2:Phi g1 c1 a1 t2 0.7128867 0.1138726 0.4548747 0.8807850
3:Phi g1 c1 a2 t3 0.7778674 0.1265650 0.4545312 0.9363710
4:p g1 c1 a1 t2 0.0149880 0.0029750 0.0101470 0.0220869
5:p g1 c1 a2 t3 0.4451827 0.0202556 0.4059122 0.4851499
6:p g1 c1 a3 t4 0.0613498 0.0108518 0.0432227 0.0863927
7:p g1 c1 a4 t5 0.2097905 0.0340483 0.1507486 0.2842177
8:Phi g1 c1 a3 t4 1.0000000 0.0000000 1.0000000 1.0000000 Fixed

[gwhite]

The beta parameters always just get a +/- 2 SE CI. The real parameters get a CI based on the type of parameter: logit for parameters 0-1, log for parameters >0, etc.

[tgarrison]

Thanks for the response, but with an identity function, f(x) = x, so the real parameter CI shouldn't need any adjustment ??

[gwhite]

It is not the link function used to obtain the estimates, but the range of the parameter values that determine the type of CI used.

[tgarrison]

Interesting, thanks for the clarification. I doubt many people setup models as shown in this example (identity link with no intercept design matrix) and so it's likely not a big deal, but I don't think it's very clear that if you setup the model as I described, and went to "Output > Specific Model Output > Parameter Estimates > Real Estimates" that you would actually get parameter estimates transformed from the logit scale (which isn't how the model was setup) to the real scale.

[cooch]

Interesting, thanks for the clarification. I doubt many people setup models as shown in this example (identity link with no intercept design matrix) and so it's likely not a big deal, but I don't think it's very clear that if you setup the model as I described, and went to "Output > Specific Model Output > Parameter Estimates > Real Estimates" that you would actually get parameter estimates transformed from the logit scale (which isn't how the model was setup) to the real scale.

Fair point -- but the motivation for calculating the CI on a transformed scale (say, logiit) is to guarantee the CI is bounded [0,1]. This is especially important for parameters estimated near the boundaries of that interval. This is mentioned in a couple of places scattered throughout the MARK book.