**A. B. Franklin, D. R. Anderson, R. J. Gutiérrez,
and K. P. Burnham. 2000. Climate, habitat quality, and fitness
in northern spotted owl populations in northwestern California.
Ecological Monographs 70: 539-590.**

Appendix B. Additional statistical considerations in estimating the sampling variances for territory-specific apparent survival and problems encountered in modeling apparent survival with interactions between climate and habitat covariates.

*Estimation of sampling variances for territory-specific
apparent survival*

We estimated and for each
territory using program MARK. Of the estimates for the 95 territories, 16 territories
had estimates of = 1 with
= 0 and four estimates of = 0 with = 0.
A problem arose with these 20 estimates in that their estimated standard errors
were zero, indicating that the uncertainty in these estimates was underestimated.
This estimated lack of uncertainty in the 20 estimates would inflate estimates
of process variance. To deal with this problem, we regressed those remaining
75 estimates of that had estimates of sampling variance,
against their corresponding sampling variances. The intercept of this regression
equation represented an estimate of the sampling variance for = 0 [= 0.1106]. However, the estimate
of var() was still zero based on this regression
equation. Therefore, we estimated var() by averaging the coefficients of sampling variation (CV)
for __>__ 0.90 as an estimate of
SE(). The rational behind this approach
was based on the fact that CV() = /1 and, hence, = CV.
This approach yielded = 0.0802 [= 0.00642]. Derived estimates of sampling
variance from these two methods yielded estimates that were within the ranges
of the 75 observed (non-zero) estimates (= 0.0033 - 0.1249).

*Problems in modeling N
with interactions between climate and habitat covariates*

The problems encountered in models of apparent survival, which included interactions
between climate and habitat covariates, can be summarized as follows. First,
models resulted in a near singular information matrix when SODIS^{2}
was included in interactions with the climate covariates P_{E} and T_{E}.
The near singular information matrix made parameter estimates questionable and
the estimated variance-covariance matrix unreliable. Often, this near singularity
resulted in standard errors of zero for SODIS^{2} and the interactions
which included SODIS^{2}. Second, models including interactions between
SODIS^{2}, P_{E}, and T_{E} had lower than expected
log-likelihoods based on neighboring models. This was probably due to difficulty
in estimating the maximum log-likelihood in a function with little or no peak.
Both problems were probably a result of confounding between climate covariates,
the quadratic term SODIS^{2}, and possibly one or two other habitat
covariates. We were unable to determine which covariates were involved but models
did not exhibit the above problems when interactions between SODIS^{2}
and P_{E} and T_{E} were removed. Therefore, we did not use
model {_{a2'+()*(LSOEDG+SODIS+SODIS}2_{)+LSOCOR}} ,which had the lowest
AICc (Table 1), as the best approximating model because of the above problems.
Instead, we used model {_{a2'+()*(LSOEDG+SODIS)+LSOCOR+SODIS}2} as the best approximating model. This
model was ranked second based on AICc (Table 1) and did not include interactions
between the climate covariates and SODIS^{2}. Model {_{a2'+()*(LSOEDG+SODIS)+LSOCOR+SODIS}2} was achieved by discarding the interactions between SODIS^{2},
P_{E}, and T_{E}. This model did not have the problems encountered
in other models which included interactions with SODIS^{2}.

Table 1. Comparison of climate,
habitat, and combined climate and habitat models for apparent survival
() in northern spotted owls in northwestern California.
All models are shown, including those with problems in estimation due to
singularities in the information matrix. |
||||

Model | AICc | K |
)AICc_{i} |
w_{i} |

†_{a2'+()*(LSOEDG+SODIS+SODIS}2_{)+LSOCOR} |
1119.28 | 18 | 0.00 | 0.516 |

_{a2'+()*(LSOEDG+SODIS)+LSOCOR} _{+SODIS}2 |
1121.46 | 16 | 2.18 | 0.174 |

†_{a2'+()*(LSOCOR+LSOEDG+SODIS+SODIS}2_{)} |
1122.51 | 20 | 3.23 | 0.103 |

†_{a2'+()*(SODIS+SODIS}2_{)+LSOCOR+LSOEDG} |
1123.16 | 16 | 3.88 | 0.074 |

_{a2'++LSOCOR+LSOEDG+SODIS+SODIS}2 |
1124.23 | 12 | 4.95 | 0.043 |

†_{a2'+()*(LSOCOR+SODIS+SODIS}2_{)+LSOEDG} |
1124.96 | 18 | 5.68 | 0.030 |

_{a2'+()*(LSOEDG)+LSOCOR+SODIS+SODIS}2 |
1125.42 | 14 | 6.14 | 0.024 |

_{a2'+()*(LSOCOR)+LSOEDG+SODIS+SODIS}2 |
1125.48 | 14 | 6.20 | 0.023 |

_{a2'+()*(LSOCOR+LSOEDG)+SODIS+SODIS}2 |
1126.87 | 16 | 7.59 | 0.012 |

_{a2'+LSOCOR+LSOEDG+SODIS+SODIS}2 |
1132.45 | 10 | 13.17 | 0.001 |

_{ } |
1136.01 | 7 | 16.73 | 0.000 |

†Models suspect because of singularities in information matrix (see text of Appendix) |