Table 1.

Hierarchical models describing 11 624 laying dates of 2262 common gull females over 37 years as a function of the annual spring temperature. (Qualitatively, the same results were obtained when the females that bred only once were excluded (see §3). The estimated variance with standard error in parentheses is given for terms (as specified in equations (2.1) and (2.2)) that were included in the model, with ‘—’ indicating terms that were not included. Models are nested with increasingly higher-order terms, where the significance of each higher-order term is based on the increase in log likelihood (log L), starting with the model with environment-specific residuals only (model 1) and ending with model 6 which includes all terms in equation (2.2). Terms are year (y) or a function of either the phenotypic variance across individuals (ind) or its components, the additive genetic effect (a) and permanent environmental effect (pe) as a polynomial function with the order (0 or 1) indicated by subscript. For all the models, the number of estimated (co)variance terms are provided with the likelihood ratio test (Χ² statistic with associated d.f.) between the given model and the model that is one hierarchical step higher (i.e. 2 versus 1, 3 versus 2, etc.). The statistics of the most parsimonious model is indicated in italics.)

	random regression model terms							test
model	I (ind0)	I×E (ind1)	G (a0)	G×E (a1)	PE (pe0)	PE×E (pe1)	y	log L	Χ2	d.f.	p
model 1	—	—	—	—	—	—	—	−24351.64
model 2	—	—	—	—	—	—	6.08 (1.48)	−23201.36	2300.6	1	≪0.001
model 3	7.32 (0.341)	—	—	—	—	—	5.84 (1.41)	−22259.64	116.6	1	≪0.001
model 4	7.38 (0.347)	1.25 (0.313)	—	—	—	—	5.80 (1.40)	−22220.55	78.2	2	<0.001
model 5	—	1.24 (0.313)	3.00 (1.20)	—	4.36 (1.20)	—	5.85 (1.42)	−22217.47	6.2	1	0.013
model 6	—	—	3.98 (1.24)	0.987 (1.21)	3.98 (1.24)	0.235 (1.22)	5.86 (1.42)	−22216.47	2.0	2	0.37