Ecological Archives M070-003 

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 SODIS2 was included in interactions with the climate covariates PE and TE. 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 SODIS2 and the interactions which included SODIS2. Second, models including interactions between SODIS2, PE, and TE 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 SODIS2, 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 SODIS2 and PE and TE were removed. Therefore, we did not use model {a2'+()*(LSOEDG+SODIS+SODIS2)+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+SODIS2} 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 SODIS2. Model {a2'+()*(LSOEDG+SODIS)+LSOCOR+SODIS2} was achieved by discarding the interactions between SODIS2, PE, and TE. This model did not have the problems encountered in other models which included interactions with SODIS2.


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 )AICci wi
a2'+()*(LSOEDG+SODIS+SODIS2)+LSOCOR 1119.28 18 0.00 0.516
a2'+()*(LSOEDG+SODIS)+LSOCOR +SODIS2 1121.46 16 2.18 0.174
a2'+()*(LSOCOR+LSOEDG+SODIS+SODIS2) 1122.51 20 3.23 0.103
a2'+()*(SODIS+SODIS2)+LSOCOR+LSOEDG 1123.16 16 3.88 0.074
a2'++LSOCOR+LSOEDG+SODIS+SODIS2 1124.23 12 4.95 0.043
a2'+()*(LSOCOR+SODIS+SODIS2)+LSOEDG 1124.96 18 5.68 0.030
a2'+()*(LSOEDG)+LSOCOR+SODIS+SODIS2 1125.42 14 6.14 0.024
a2'+()*(LSOCOR)+LSOEDG+SODIS+SODIS2 1125.48 14 6.20 0.023
a2'+()*(LSOCOR+LSOEDG)+SODIS+SODIS2 1126.87 16 7.59 0.012
a2'+LSOCOR+LSOEDG+SODIS+SODIS2 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)

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