{"id":361,"date":"2017-05-05T19:44:29","date_gmt":"2017-05-05T19:44:29","guid":{"rendered":"http:\/\/sites.warnercnr.colostate.edu\/gwhite\/?page_id=361"},"modified":"2017-05-05T19:44:29","modified_gmt":"2017-05-05T19:44:29","slug":"profile-likelihood-confidence-intervals","status":"publish","type":"page","link":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/profile-likelihood-confidence-intervals\/","title":{"rendered":"Profile Likelihood Confidence Intervals"},"content":{"rendered":"

Contents<\/a> – Index<\/a><\/span><\/p>\n

\n

Profile Likelihood Confidence Intervals<\/b><\/span><\/p>\n

The Run Window<\/a> includes a check box to request profile likelihood confidence intervals be computed for the real parameter<\/a> estimates.\u00a0 The default confidence intervals for real parameter estimates in the 0-1 interval are based on the standard error and the logit transformation.\u00a0 That is, a 95% confidence interval is computed on the logit estimate, and then these intervals are transformed to the real scale.\u00a0 Use of the logit transformation precludes confidence interval boundaries outside the 0-1 interval.\u00a0 However, problems are encountered with this approach when the parameter estimate is on the boundary, e.g., when a survival rate is estimated as 1, and\/or the standard error is estimated as zero.<\/span><\/p>\n

Profile likelihood confidence intervals are based on the log-likelihood function.\u00a0 For a single parameter, likelihood theory shows that the 2 points 1.92 units down from the maximum of the log-likelihood function provide a 95% confidence interval when there is no extrabinomial variation (i.e. c<\/i> = 1)..\u00a0 The value 1.92 is half of the chi-square value of 3.84 with 1 degree of freedom.\u00a0 Thus, the same confidence interval can be computed with the deviance by adding 3.84 to the minimum of the deviance function, where the deviance<\/a> is the log-likelihood multiplied by -2 minus the -2 log likelihood value of the saturated model<\/a>.<\/span><\/p>\n

MARK will compute profile likelihood confidence intervals with extrabinomial variation<\/a> assumed, i.e., c<\/i> > 1.\u00a0 To do so, the amount dropped below the maximum of the likelihood is c<\/span><\/i>-hat<\/a>*1.92, or the amount to increase above the minimum deviance is c<\/i>-hat*3.84.<\/span><\/p>\n

The method used in MARK to compute the profile likelihood lower bound for parameter i<\/i> is to minimize the following function:<\/span><\/p>\n

[-2 log likelihood of current parameter value\u00a0 –\u00a0 (-2 log likelihood of maximum likelihood estimates + c<\/i>-hat*3.84) ]**2 – (maximum likelihood estimate of parameter i\u00a0 –\u00a0<\/i> current parameter value of i<\/i>).<\/span><\/p>\n

The first portion of this expression finds the value of the deviance that is c<\/i>-hat*3.84 units larger than the deviance for the maximum likelihood parameter estimates.\u00a0 The second portion of the expression maximizes the difference between the maximum likelihood parameter estimate and the lower bound.<\/span><\/p>\n

For the upper bound, the function has to be modified maximize the difference between the maximum likelihood estimate and the upper bound:<\/span><\/p>\n

[-2 log likelihood of current parameter value\u00a0 –\u00a0 (-2 log likelihood of maximum likelihood estimates + c<\/i>-hat*3.84) ]**2 – (current parameter value of i\u00a0 –\u00a0 <\/i>maximum likelihood estimate of parameter i<\/i>).<\/span><\/p>\n

This procedure appears to be different from the typical approach to estimate profile likelihood intervals because in MARK, the parameters that are optimized are the beta estimates, but we are interested in the profile intervals for the real parameters.\u00a0 Thus, there is not always a 1 to 1 conversion from the beta estimates to the real estimates, e.g., trend models.\u00a0 The approach described above accommodates this additional complexity.<\/span><\/p>\n

Because of the additional optimization required to obtain the profile likelihood interval estimates, considerably more computer time is required to compute these intervals.<\/span><\/p>\n

The following is an example comparing the default confidence intervals with the profile likelihood confidence intervals.\u00a0 The example is from the North Park sage grouse data distributed with Program MARK in the NPMALES.DBF file.\u00a0 First is the default output for the 30 survival estimates from the {S(a*T) r(a*T)} model.<\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 North Park Male Sage Grouse Band Recovery Data<\/span>
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Real Function Parameters of {S(a*T) r(a*T)}<\/span>
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 95% Confidence Interval<\/span>
\nParameter\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Estimate\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Standard Error\u00a0\u00a0\u00a0\u00a0\u00a0 Lower\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Upper<\/span>
\n————————-\u00a0 ————–\u00a0 ————–\u00a0 ————–\u00a0 ————–<\/span>
\n\u00a0\u00a0\u00a0 1:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.5212251\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0463568\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4306641\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.6104122\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0\u00a0 2:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.5017107\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0406542\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4226535\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.5806825\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0\u00a0 3:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4821911\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0351372\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4140880\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.5509623\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0\u00a0 4:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4627257\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0300246\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4046607\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.5218196\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0\u00a0 5:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4433733\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0256337\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3938728\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4940243\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0\u00a0 6:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4241912\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0224038\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3809848\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4685882\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0\u00a0 7:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4052348\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0208173\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3651713\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4466016\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0\u00a0 8:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3865568\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0211150\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3460756\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4286692\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0\u00a0 9:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3682068\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0230520\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3242870\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4144267\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 10:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3502306\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0260782\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3009809\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4028942\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 11:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3326698\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0296737\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.2772552\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3931355\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 12:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3155620\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0334756\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.2538794\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3845078\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 13:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.2989398\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0372553\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.2313377\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3766175\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 14:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.2828314\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0408711\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.2099224\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3692287\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 15:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.2672600\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0442350\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.1898019\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3621974\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 16:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.8083942\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0596942\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.6646855\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.8997973\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 17:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.7757114\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0608475\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.6353812\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.8728417\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 18:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.7392520\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0609946\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.6039334\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.8405448\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 19:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.6991644\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0602202\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.5700751\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.8028948\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 20:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.6557833\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0587967\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.5334585\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.7604383\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 21:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.6096393\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0571881\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4936989\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.7143893\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 22:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.5614463\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0559738\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4505298\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.6665446\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 23:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.5120654\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0556664\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4041007\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.6189161\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 24:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4624480\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0564830\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3552982\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.5731824\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 25:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4135633\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0582287\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3057997\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.5302933\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 26:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3663247\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0603937\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.2577011\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4904807\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 27:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3215234\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0623774\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.2129511\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4535516\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 28:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.2797822\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0636788\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.1729395\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4191781\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 29:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.2415309\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0639865\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.1383693\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3870549\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 30:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.2070061\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0631856\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.1093399\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3569485\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n

Next is the output for the same model, but with profile likelihood intervals computed for the 30 survival estimates. The value of c<\/i>-hat used to compute the profile likelihood confidence intervals is reported in the output,\u00a0<\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 North Park Male Sage Grouse Band Recovery Data<\/span>
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Real Function Parameters of {S(a*T) r(a*T)}<\/span>
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 95% Confidence Interval<\/span>
\nParameter\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Estimate\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Standard Error\u00a0\u00a0\u00a0\u00a0\u00a0 Lower\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Upper<\/span>
\n————————-\u00a0 ————–\u00a0 ————–\u00a0 ————–\u00a0 ————–<\/span>
\n\u00a0\u00a0\u00a0 1:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.5212251\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0463568\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4305389\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.6107000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0\u00a0 2:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.5017107\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0406542\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4226006\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.5809796\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0\u00a0 3:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4821911\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0351372\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4140988\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.5512675\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0\u00a0 4:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4627257\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0300246\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4047206\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.5221246\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0\u00a0 5:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4433733\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0256337\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3939565\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4943047\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0\u00a0 6:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4241912\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0224038\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3810508\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4687985\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0\u00a0 7:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4052348\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0208173\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3651681\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4467037\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0\u00a0 8:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3865568\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0211150\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3459763\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4286955\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0\u00a0 9:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3682068\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0230520\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3241126\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4144641\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 10:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3502306\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0260782\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3007706\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4030047\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 11:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3326698\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0296737\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.2770367\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3933405\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 12:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3155620\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0334756\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.2536662\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3848095\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 13:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.2989398\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0372553\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.2311351\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3770121\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 14:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.2828314\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0408711\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.2097321\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3697119\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 15:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.2672600\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0442350\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.1896241\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3627656\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 16:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.8083942\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0596942\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.6637455\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.8966843\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 17:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.7757114\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0608475\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.6360563\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.8699084\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 18:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.7392520\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0609946\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.6062872\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.8382203\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 19:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.6991644\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0602202\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.5740674\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.8017697\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 20:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.6557833\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0587967\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.5388831\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.7611976\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 21:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.6096393\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0571881\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.5001267\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.7176146\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 22:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.5614463\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0559738\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4573229\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.6724149\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 23:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.5120654\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0556664\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4105786\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.6270169\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 24:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4624480\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0564830\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3609972\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.5826686\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 25:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4135633\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0582287\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3106001\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.5403518\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 26:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3663247\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0603937\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.2617290\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.5007203\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 27:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3215234\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0623774\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.2164036\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4640376\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 28:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.2797822\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0636788\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.1759763\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.4301850\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 29:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.2415309\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0639865\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.1410853\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3988211\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span>
\n\u00a0\u00a0 30:S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.2070061\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0631856\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.1117793\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.3695837\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n

The difference between the 2 sets of intervals is basically negligible when c-hat = 1.\u00a0\u00a0<\/span><\/p>\n

To view the profile likelihood confidence intervals when c-hat is not equal to 1, the user has to open up the full output file for a model.\u00a0 Profile likelihood confidence intervals are displayed in the real parameters window only when c-hat = 1.\u00a0 Because the user can change the value of c-hat, and profile likelihood confidence intervals are only computed when the log-likelihood for the model is optimized, the profile likelihood values reported in the full output are defined by the c-hat in use at the time.\u00a0 If the user changes the value of c-hat, the profile intervals in the output will not change.\u00a0 Thus, to avoid potential mistakes, the value of c-hat used to compute the interval is reported beside the interval.\u00a0\u00a0<\/span><\/p>\n

Warning:<\/b> The approach described above does not always work correctly because of numerical problems, notably for parameters estimated on the boundary.\u00a0 I have found that often the optimization routine is not always able to move the parameter estimate away from the boundary.\u00a0 As an example, consider the 3rd real parameter in the {phi(g*t) p(g*t)} model of the AFS Monograph example distributed with Program MARK and described by Burnham et al. (1987)<\/a>.\u00a0 The following results are for the first version of the profile likelihood confidence interval code.\u00a0 Since that version, I have modified the code to provide better results, but I am not confident that the code will always produce correct results.\u00a0 Thus, if you should check the profile likelihood confidence interval estimates to be sure that proper optimization is achieved.\u00a0\u00a0<\/span><\/p>\n

With the identity link, the following estimate and default confidence interval is obtained:<\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Real Function Parameters of {Phi(g*t) p(g*t) PIM coding}<\/span>
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 95% Confidence Interval<\/span>
\nParameter\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Estimate\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Standard Error\u00a0\u00a0\u00a0\u00a0\u00a0 Lower\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Upper<\/span>
\n————————-\u00a0 ————–\u00a0 ————–\u00a0 ————–\u00a0 ————–<\/span>
\n\u00a0\u00a0\u00a0 3:Phi\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 1.0734483\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.2038269\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.6739477\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 1.4729490\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n

Note that the confidence interval is computed as the estimate +\/- 1,96 times the standard error in the above output, because the real parameter estimate is outside the 0-1 interval.\u00a0 Had the estimate been in the 0-1 interval, the confidence interval would have been computed on the logit scale, and then back-transformed.\u00a0 When a profile likelihood confidence interval is requested with the identity link, the following estimates are obtained:<\/span><\/p>\n

\u00a0\u00a0\u00a0 3:Phi\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 1.0734483\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.2038269\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.7269042\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 1.5390461\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n

and are not particularly different from the default values shown above.\u00a0 Next, consider the estimate for parameter 3 when the logit link is used to constrain the estimate to the 0-1 interval, and the following default output is obtained.<\/span><\/p>\n

\u00a0\u00a0\u00a0 3:Phi\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.9999728\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0032660\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.5107488E-06\u00a0\u00a0 1.0000000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n

The confidence interval is nonsensical because the standard error is estimated as zero, and so the confidence interval extends from 0 to 1.\u00a0 Now, when a profile likelihood interval is requested, the following results are obtained:<\/span><\/p>\n

\u00a0\u00a0\u00a0 3:Phi\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.9999728\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.0032660\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.9999728\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.9999728\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n

Note that the optimization procedure was not able to move the parameter estimate away from the boundary, so the profile interval is not correct.\u00a0 Similar results are obtained if the sin link is used:<\/span><\/p>\n

\u00a0\u00a0\u00a0 3:Phi\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 1.0000000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.1856222E-03\u00a0\u00a0 1.0000000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 1.0000000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n

Interestingly enough, when the confounded parameters phi5 and phi10 are fixed at one, a correct profile likelihood interval is obtained with the sin link:<\/span><\/p>\n

\u00a0\u00a0\u00a0 3:Phi\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.9999997\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.6026331E-03\u00a0\u00a0 0.7725536\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 0.9999997\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Profile c-hat=1.0000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n

Evidently fixing these confounded parameters makes the optimization more stable, and proper convergence is obtained.<\/span><\/p>\n

In this particular case, because the parameter at the boundary is a function of a single beta (i.e., the parameter is not a function of multiple beta values because an identity link is being used), you can compute the profile likelihood parameter manually by fixing the parameter to your guess of the profile likelihood interval endpoint.\u00a0 Thus, run the {phi(g*t} p(g*t)} model with phi3 fixed<\/a> to 0.7725536, and check that the deviance of this new model is approximately 3.84 units larger than the deviance of the model with none of the unconfounded parameters fixed.<\/span><\/p>\n

As noted above, the code has been modified to provide correct estimates for this example.\u00a0 However, be suspicious!<\/span><\/p>\n

In summary, care should be taken in using the profile likelihood capability because of the numerical problems caused with parameters at the boundary.\u00a0 Unfortunately, parameters estimated at the boundary is the primary reason that the prof<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

Contents – Index Profile Likelihood Confidence Intervals The Run Window includes a check box to request profile likelihood confidence intervals be computed for the real parameter estimates.\u00a0 The default confidence intervals for real parameter estimates in the 0-1 interval are …<\/p>\n

Profile Likelihood Confidence Intervals<\/span> Read More »<\/a><\/p>\n","protected":false},"author":117,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-361","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/pages\/361","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/users\/117"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/comments?post=361"}],"version-history":[{"count":1,"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/pages\/361\/revisions"}],"predecessor-version":[{"id":362,"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/pages\/361\/revisions\/362"}],"wp:attachment":[{"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/media?parent=361"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}