Table 1.

Growth models^a of serial weight from birth to 15 months of age for 70 boys in the Born in Bradford birth cohort study.

		Model 1a conventional regression	Model 1b conventional quadratic polynomial regression	Model 1c mixed effects quadratic polynomial regression	Model 1d mixed effects fractional polynomial regression
	Notation	Beta (Standard Error) P-value	B (SE) P-value	B (SE) P-value	B (SE) P-value
Intercept	β0	3.884 (0.053) <0.001	3.355 (0.056) <0.001	3.328 (0.063) <0.001	21.662 (1.212) <0.001
Age (decimal years)	β1	6.641 (0.110) <0.001	11.367 (0.317) <0.001	11.303 (0.287) <0.001
Age2	β2		−4.821 (0.309) <0.001	−4.819 (0.262) <0.001
Age−2	β1				−2.938 (0.612) <0.001
Age−0.5	β2				−15.495 (1.782) <0.001
Variance (intercept)	${\mathrm{\sigma }}_{\mathrm{u}0}^2$			0.248 (0.047)	75.454 (17.880)
Variance (age)	${\mathrm{\sigma }}_{\mathrm{u}1}^2$			4.068 (0.974)
Variance (age2)	${\mathrm{\sigma }}_{\mathrm{u}2}^2$			2.657 (0.824)
Covariance (intercept, age)	σu01			−0.219 (0.159)
Covariance (intercept, age2)	σu02			0.145 (0.144)
Covariance (age, age2)	σu12			−2.680 (0.822)
Variance (age−2)	${\mathrm{\sigma }}_{\mathrm{u}1}^2$				16.993 (4.423)
Variance (age−0.5)	${\mathrm{\sigma }}_{\mathrm{u}2}^2$				156.281 (38.575)
Covariance (intercept, age−2)	σu01				32.236 (8.571)
Covariance (intercept, age−0.5)	σu02				−107.464 (26.159)
Covariance (age−2, age−0.5)	σu12				−49.011 (12.828)
Variance (residual)	${\mathrm{\sigma }}_{\mathrm{e}}^2$			0.059 (0.004)	0.047 (0.003)
Log Likelihood		−802	−699	−261	−214
−2 Log Likelihood (i.e., deviance)		1604	1398	522	428
Δ −2 Log Likelihood (P-value)b		--	−206 (<0.001)	−876 (<0.001)	−94 (not possiblec)
AICd		1608	1404	542	448
BICe		1617	1417	586	492

^aAll models were fit in Stata IC10 (College Station, Texas). Model formulae are shown in the methods section and the sample average curves are shown in Fig 1, with the exception of the curve of model 1b because if was nearly identical to that of model 1c.

^bΔ −2 Log Likelihood is the difference in the −2 Log Likelihood between the model and the model in the previous column. The p-value is a test of this difference using a likelihood ratio test (i.e., −2 log(likelihood for null model) + 2 log(likelihood for alternative model)).

^cModels 1c and 1d had the same number of estimated parameters (and thus the degrees of freedom did not differ), thereby making it impossible to calculate a p-value for the difference in −2 Log Likelihoods.

^dAkaike Information Criterion (AIC) = −2 Log Likelihood + 2(number of estimated parameters).

^eBayesian Information Criterion (BIC) = −2 Log Likelihood + number of estimated parameters*ln(number of observations).