A study of lactation curves in dairy cattle using the optimal design of experiments methodology

Figures & data

Figure 1. Standard shape of the lactation curve for dairy cattle. The plot shows the milk production as function of the weeks after the parturation.

Figure 2. Plot for some lactation models: QM (dashed line), HM (dotted line), IM (dot-dashed line) and WEM (solid line).

TABLE 1. Efficiencies for WM D-optimal designs.

	WM D-optimal design
WM design	True values $k_1$
Miss values $k_2$	$0.050$	$0.065$	$0.100$	$0.610$
$0.050$	$1$	$0.995$	$0.970$	$0.897$
$0.065$	$0.994$	$1$	$0.988$	$0.917$
$0.100$	$0.956$	$0.984$	$1$	$0.945$
$0.610$	$0.544$	$0.629$	$0.743$	$1$
${\xi }_{305}$	$0.428$	$0.397$	$0.350$	$0.221$

Efficiency is the behaviour of D-optimal designs for Wilmink models (WM) with $k_2$ when used to estimate WM with true value $k_1$.

TABLE 2. Support points of WEM locally D-optimal designs in $[1,305]$.

$k$	$t_1$	$t_2$	$t_3$	$t_4$
$0.050$	$1$	$19.85$	$92.32$	$305$
$0.065$	$1$	$15.75$	$77.05$	$305$
$0.100$	$1$	$10.75$	$56.28$	$305$
$0.610$	$1$	$2.64$	$13.70$	$305$

D-optimal designs are equally supported at $t_1$, $t_2$, $t_3$, and $t_4$. WEM: Wilmink extended model.

TABLE 3. Efficiencies for WEM D-optimal designs.

	WEM D-optimal design
WEM design	True value $k_1$
Miss value $k_2$	$0.050$	$0.065$	$0.100$	$0.610$
$0.050$	$1$	$0.975$	$0.834$	$0.015$
$0.065$	$0.976$	$1$	$0.934$	$0.049$
$0.100$	$0.851$	$0.938$	$1$	$0.190$
$0.610$	$0.184$	$0.245$	$0.384$	$1$
${\xi }_{305}$	$0.333$	$0.291$	$0.221$	$0.092$

Efficiency is the behaviour of the D-optimal designs for WEM with $k_2$ when used to estimate WEM with true value $k_1$. WEM: Wilmink extended model.

TABLE 4. Efficiencies of locally D-optimal designs for WEM with respect to those for WM, for different $k$ values.

	WM D-optimal design
WEM design	Value $k$
Value $k$	$0.050$	$0.065$	$0.100$	$0.610$
$0.050$	$0.860$	$0.861$	$0.874$	$0.864$
$0.065$	$0.865$	$0.861$	$0.868$	$0.880$
$0.100$	$0.871$	$0.867$	$0.863$	$0.902$
$0.610$	$0.616$	$0.673$	$0.762$	$0.865$

WEM: Wilmink extended model; WM: Wilmink model.

TABLE 5. Support points for D-optimal designs for some lactation models.

Model	Support points
LM	$1$			14
EM^(1)	$1$			$14$
HM	$1$	$3.74$		$14$
IM^(2)	1	$3.41$		$14$
WDM^(3)	$1$	$4.56$		$14$
EMC^(1)	$1$	$7.04$		$14$
QM	$1$	$7.50$		$14$
WM	$1$	$7.15$		$14$
WEM^(4)	$1$	$4.32$	$10.12$	$14$

Lactation models namely, linear (LM), exponential (EM), hyperbolic (HM), inverse quadratic (IM), Wood (WDM), exponential with constant term (EMC), quadratic (QM), Wilmink (WM), and Wilmink extended (WEM), in $[1,14]$. With best guesses from Landete-Castillejos and Gallego (2000): ⁽¹⁾${\beta }_2^0=0.03307$; ⁽²⁾${\beta }_0^0=0.0002865$; ${\beta }_1^0=0.0002117$; ${\beta }_2^0=0.0000301$; ⁽³⁾${\beta }_1^0=0.33269$; ${\beta }_2^0=0.09319;$ and ⁽⁴⁾$k_0=0.05$.

TABLE 6. Efficiency study for optimal design for lactation models.

		D-optimal design for lactation models
	LM	EM	HM	IM	WDM	EMC	QM	WM
LM	$1$	$1$
EM	$1$	$1$
HM	$0.861$	$0.845$	$1$	$0.994$	$0.980$	$0.803$	$0.762$	$0.794$
IM	$0.869$	$0.853$	$0.997$	$1$	$0.958$	$0.756$	$0.714$	$0.747$
WDM	$0.844$	$0.832$	$0.987$	$0.967$	$1$	$0.894$	$0.859$	$0.886$
EMC	$0.817$	$0.826$	$0.858$	$0.702$	$0.889$	$1$	$0.997$	$1$
QM	$0.817$	$0.829$	$0.825$	$0.653$	$0.851$	$0.997$	$1$	$0.998$
WM	$0.817$	$0.827$	$0.850$	$0.690$	$0.879$	$1$	$0.998$	$1$
WEM	$0.775$	$0.864$	$0.864$	$0.876$	$0.898$	$0.865$	$0.866$	$0.865$
${\xi }_{14}$	$0.620$	$0.629$	$0.676$	$0.676$	$0.681$	$0.671$	$0.671$	$0.671$

LM: linear; EM: exponential; HM: hyperbolic; IM: inverse quadratic; WDM: Wood; EMC: exponential with constant term; QM: quadratic; WM: Wilmink; WEM: Wilmink extended; ${\xi }_{14}$: monthly design on space $[1,14]$.

Landete-Castillejos T, Gallego L. 2000. Technical note: the ability of mathematical models to describe the shape of lactation curves. J Animal Sci. 78:3010–3013.

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