# BCB bowheads

Filed under:
Abundance estimation,
REML

Zeh & Punt (2005) combined data from two different counting methodologies in order to estimate the rate of increase of BCB bowhead whales. On a log-scale they fitted a linear mixed-effects models using REML estimates for the variance parameters. This example shows how random effects in ADMB can be used to obtain REML estimates.

Restricted maximum likelihood estimation (REML) is often used to estimate variance parameters in the linear mixed-model:

*y*=

*X*b*+

*Z*u*+

*e*

*X*and

*Z*are covariate matrices,

*b*are the fixed-effects,

*u*are the random effects and

*e*are the error terms. One way of defining the REML procedure is to:

- Assign a flat prior to
*b.* - Integrate the likelihood with respect to both
*b*and*u*.

*u*and

*e*).

To implement this in ADMB-RE we declare both

*b*and

*u*as random effects vectors. As a result, the likelihood is integrated with respect to both parameters. But, while there is a contribution to the objective function comming from the random effects distribution of

*u*, there should be no such contribution from

*b*(i.e. a flat prior).

Using this procedure you do not have to derive the "REML correction" to the log-likelihood function by hand. It is done automatically for you by ADMB-RE.

### References

Zeh, J.E., and A.E. Punt. "Updated 1978-2001 Abundance Estimates and Their Correlations for the Bering-Chukchi-Beaufort Seas Stock of Bowhead Whales." Journal of Cetacean

Research and Management 7, no. 2 (2005): 169-75.