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Restricted maximum likelihood

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Sample size determination

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Methods in Ecology and Evolution

Technological advances have steadily increased the detail of animal tracking datasets, yet fundamental data limitations exist for many species that cause substantial biases in home‐range estimation. Specifically, the effective sample size of a range estimate is proportional to the number of observed range crossings, not the number of sampled locations. Currently, the most accurate home‐range estimators condition on an autocorrelation model, for which the standard estimation frame‐works are based on likelihood functions, even though these methods are known to underestimate variance—and therefore ranging area—when effective sample sizes are small. Residual maximum likelihood (REML) is a widely used method for reducing bias in maximum‐likelihood (ML) variance estimation at small sample sizes. Unfortunately, we find that REML is too unstable for practical application to continuous‐time movement models. When the effective sample size N is decreased to N ≤ (10), which is common in tracking applications, REML undergoes a sudden divergence in variance estimation. To avoid this issue, while retaining REML’s first‐order bias correction, we derive a family of estimators that leverage REML to make a perturbative correction to ML. We also derive AIC values for REML and our estimators, including cases where model structures differ, which is not generally understood to be possible. Using both simulated data and GPS data from lowland tapir ( Tapirus terrestris ), we show how our perturbative estimators are more accurate than traditional ML and REML methods. Specifically, when (5) home‐range crossings are observed, REML is unreliable by orders of magnitude, ML home ranges are ~30% underestimated, and our perturbative estimators yield home ranges that are only ~10% underestimated. A parametric bootstrap can then reduce the ML and perturbative home‐range underestimation to ~10% and ~3%, respectively. Home‐range estimation is one of the primary reasons for collecting animal tracking data, and small effective sample sizes are a more common problem than is currently realized. The methods introduced here allow for more accurate movement‐model and home‐range estimation at small effective sample sizes, and thus fill an important role for animal movement analysis. Given REML’s widespread use, our methods may also be useful in other contexts where effective sample sizes are small.

Overcoming the challenge of small effective sample sizes in home‐range estimation