Mark Hebblewhite, Evelyn Merrill, and Greg McDermid. 2008. A multi-scale test of the forage maturation hypothesis in a partially migratory ungulate population. Ecological Monographs 78:141–166.

Appendix F. Linear mixed-effects time-series models for forage (herbaceous and shrub) biomass exposure for migrant and resident elk GPS locations, 2002–2004.

We tested for differences in exposure biomass and DMD between migrant and resident elk telemetry locations using a random effects approach to account for variation at the individual elk level in sampling intensity (VHF, GPS), non-independence, and autocorrelation structure (Skrondal and Rabe-Hesketh 2004). Herbaceous biomass was ln-tranformed to satisfy normality assumptions; untransformed coefficients are reported for interpretation. Shrub biomass and DMD did not require transformation. We used a linear mixed-effects approach with a random effect for individual elk and autocorrelated error, and dummy covariates for migrant/resident status, open/closed habitats, MODIS-time interval (n = 2 to 11), and the interaction between migrant status and interval (which tests when strategies differed in forage biomass exposure) using XTREGAR in STATA 8.0 (Baltagi and Wu 1999; StataCorp 2003) following:

where Y_it is the dependent variable (biomass, DMD) predicted for elk i = 1 to n during time t, 's are the covariate coefficients (intervals, open/closed, and the interaction between migrant and interval, etc.),    is the random effect for individual elk i, and is the autocorrelated error structure where is the first-order autocorrelated error coefficient, and   is the independently and identically distributed (iid) portion of the error for individual elk i in time t. XTREGAR is robust to unbalanced observations in both i and t, unevenly spaced t, and estimates autocorrelation using the Durban-watson statistic (Baltagi and Wu 1999). Because our main interest was differences between migrants and residents, we used backwards-stepwise model selection to select a final model.

   Notes: Coefficients are given in untransformed values for ease of interpretation. The reference category is migrant elk in years 2002, 2003, in closed habitats during all not included forage intervals and interactions. Coefficient estimates for the 16-d forage intervals refer to interval specific differences, whereas strategy × interval coefficients reveal in which intervals migrants had lower forage biomass exposure than residents (indicated by a minus sign).

   Notes: The reference category is migrant elk in years 2002, 2003, in closed habitats during all not-included forage intervals and interactions. Coefficient estimates for the 16-d forage intervals refer to interval specific differences, whereas strategy × interval coefficients reveal in which intervals migrants had significantly different forage biomass exposure than residents.

   Notes: The reference category is migrant elk in years 2002, 2003, in closed habitats during all not included forage intervals and interactions. Coefficient estimates for the 16-d forage intervals refer to interval specific differences, whereas strategy × interval coefficients reveal in which intervals migrants had significantly different forage biomass exposure than residents.

   Notes: The reference category is migrant elk during all non-included forage intervals and interactions. Coefficient estimates for the 16-d forage intervals refer to interval specific differences, whereas strategy × interval coefficients reveal in which intervals migrants had significantly higher forage quality exposure than residents.

LITERATURE CITED

Baltagi, B. H., and P. X. Wu. 1999. Unequally spaced panel data regressions with Ar(1) disturbances. Econometric Theory 15:814–823.

Skrondal, A., and S. Rabe-Hesketh. 2004. Generalized latent variable modeling: multilevel, longitudinal, and structural equation models. Chapman and Hall, New York, New York, USA.

StataCorp. 2003. Stata Statistical Software: Release 8.0. Stata Corporation, College Station, Texas, USA.