Case study: Lactation curves for camels
Data provided by Fred Aloo (Institut 480, Uni Hohenheim, Prof. Valle-Zarate, June 2003)
Short description of objective of study and research question
Rendille pastoralists in northern Kenya group their camels (all belonging to the Rendille
camel breed) into different performance and adaptation types. There is
- the Dabach type, said to have the highest milk yield during rainy season but loses
body condition rapidly during dry season, with consequent drop in milk yield,
- the Godan type with has a medium but stable milk yield during both rainy and dry
season and
- the Coitte type with a low milk yield but good body condition throughout.
- The Aitimaso type is said to be in-between the Dabach and the Godan type having a
higher milk yield then the Godan but a better drought tolerance then the Dabach type.
The study aims at estimating the milk yield of these different Rendille camel types in order to
assess whether the indigenous classification can be trusted and hence used as part of on-farm
performance estimation.
Performance estimation of livestock kept under their natural production conditions is
specifically difficult in remote areas under adverse conditions, such as the marginal dryland
areas. Performance and adaptation of such livestock species and breeds can also hardly be
determined on research stations, because of strong genotype-environment interactions.
Therefore the development and use of methods for on-farm performance testing are necessary
when aiming at assessing and improving livestock production systems in marginal areas.
These methods seem to be more practical when the livestock keepers are involved in the
performance recording. The results of the study will therefore contribute to the assessment of
coherence and complementarities of indigenous and scientific knowledge and further be used
for the fine-tuning of on-farm performance testing methods suitable for the area.
 Cross-sectional study, i.e., survey was conducted at one point in time (2,5 months data
collection period from March to May 2003, this was towards the end of the dry season)
 Camels are classified into four different types: Dabach, Aitimaso, Godan and Coitte
 For each type, camels were surveyed which were in lactation months 1, 2, ..., 12. For each
lactation month and camel type, there are replicate camels. Each camel was surveyed only
once, i.e., there are no repeated measurements.
 Lactation curves were to be fitted to each type
 Questions: What model to fit to the lactation curve? Are regression curves different among
types and if so, in what respect?
 The following model has been shown to fit well to lactation curves of camels:
 x  exp x 
where x is the number of lactation months. This model is known as the incomplete gamma
model due to its relation to the incomplete gamma function and as Wood's function (Wood,
1967, 1972).
Suggestion for analysis
The model can be linearized by taking logarithms:
log( )  log    log( x) x   z1 z2
1
where
   log  
z1  log( x)
z2  x
This can be fitted by multiple linear regression of y' = log(y) on z1 and z2.
To compare lactation curves, it is useful to fit curves simultaneously for all four types. The
joint model can be written
   i   i z1t  i z 2 t  eijt
yijt
where
 = logged milk yield of the j-th camel of the i-th type at the t-th month
y ijt
Since there are replicate camels per month and type, the lack-of-fit can be tested by adding a
lack-of-fit effect:
  i   i z1t  i z 2 t  it  eijt
yijt
(1)
where
it = lack-of-fit effect for the i-th type and t-th month
The ANOVA for model (1), using sequential SS (Type I SS in SAS), yields
Source
DF
Type I SS
Mean Square
F Value
Pr > F
Type
Lactmonth*Type
LOG_Lactmonth*Type
Type*lackfit
3
4
4
33
5.21121050
14.11557956
4.19206207
10.05184539
1.73707017
3.52889489
1.04801552
0.30460138
8.86
17.99
5.34
1.55
<.0001
<.0001
0.0004
0.0314
The lack-of-fit is significant, so one may consider modifications of model (1). Specifically,
we tried the following three models:
  i   i z1t  i z 2 t  i z3t  it  eijt
yijt
(2)
where
(a) z 3t  xt2 ,
(b) z3t  log( xt ) and
(c) z3t  xt log( xt ) .
2
The ANOVAs are:
(a) z 3t  xt2
2
Source
DF
Type I SS
Mean Square
F Value
Pr > F
Type
Lactmonth*Type
LOG_Lactmonth*Type
Lactmon*Lactmon*Type
Type*lackfit
3
4
4
4
30
5.21121050
14.11557956
4.19206207
4.30056889
5.75127650
1.73707017
3.52889489
1.04801552
1.07514222
0.19170922
8.82
17.93
5.32
5.46
0.97
<.0001
<.0001
0.0004
0.0003
0.5092
R2 = 0.31 (without lack-of-fit term)
(b) z3t  log( xt )
2
Source
DF
Type I SS
Mean Square
F Value
Pr > F
Type
Lactmonth*Type
LOG_Lactmonth*Type
LOG_Lac*LOG_Lac*Type
Type*lackfit
3
4
4
4
29
5.21121050
14.11557956
4.19206207
2.91910479
7.13274064
1.73707017
3.52889489
1.04801552
0.72977620
0.24595657
8.86
17.99
5.34
3.72
1.25
<.0001
<.0001
0.0004
0.0057
0.1788
R2 = 0.29 (without lack-of-fit term)
(c) z3t  xt log( xt )
Source
DF
Type I SS
Mean Square
F Value
Pr > F
Type
Lactmonth*Type
LOG_Lactmonth*Type
Lactmon*LOG_Lac*Type
Type*lackfit
3
4
4
4
30
5.21121050
14.11557956
4.19206207
3.64047308
6.41137231
1.73707017
3.52889489
1.04801552
0.91011827
0.21371241
8.82
17.93
5.32
4.62
1.09
<.0001
<.0001
0.0004
0.0012
0.3522
R2 = 0.30 (without lack-of-fit term)
While model (a) had the largest R2, it showed a slight upward twist for the highest lactation
month (not shown), which is not plausible biologically. This problem did not occur with
models (b) and (c), and among these, (c) fit slightly better. Thus, all subsequent analyses were
based on model (c).
We tested interaction of type-by-covariate based on the model
  i   i z1t  i z2 t  i z3t  eijt
yijt
(3)
where z3t  xt log( xt ) . Type-specific effects were partitioned into a main effect and an
interaction effect as follows:
 i   ai
 i    bi
 i    ci
 i   d i
The ANOVA is as follows:
Source
DF
()
()
(bi)
()
(ci)
Type I SS
Mean Square
3
F Value
Pr > F
Type
Lactmonth
Lactmonth*Type
LOG_Lactmonth
LOG_Lactmonth*Type
Lactmonth*LOG_Lactmo
Lactmon*LOG_Lac*Type
3
1
3
1
3
1
3
5.21121050
13.55550713
0.56007242
3.81456270
0.37749937
2.29964353
1.34082956
1.73707017
13.55550713
0.18669081
3.81456270
0.12583312
2.29964353
0.44694319
8.75
68.32
0.94
19.22
0.63
11.59
2.25
<.0001
<.0001
0.4211
<.0001
0.5935
0.0007
0.0822
All three interaction terms (bi, ci, di) were non-significant. The results suggest that curves are
identical for the four groups, except for the intercept (ai significant), i.e., the curves are
parallel. Thus, we fit the reduced model
  i  z1t z 2 t z3t  eijt
yijt
(4)
For illustration, the fitted model for Type 1 is shown in Figs. 1 (log-scale) and Fig. 2 (original
scale; obtained by back-transformation of model fitted on log-scale). A residual plot across all
types is inconspicuous and not suggestive of variance heterogeneity (Fig. 3).
log(yield)
Type=1
2
1
0
-1
1
2
3
4
5
6 7 8
Month
9
10 11 12
Fig. 1: Fitted model (4c) for Type=1 on log-scale, with 95% prediction limits.
4
Yield
Type=1
7
6
5
4
3
2
1
0
1
2
3
4
5
6
7 8
Month
9
10 11 12
Fig. 2: Model for Type 1 on original scale based on back-transformation of model (4c) fitted
to log-transformed data (y0.29).
Residual
3
2
1
0
-1
-2
-3
-4
-0.2
0.0
0.2
0.4
0.6
Predicted value
0.8
1.0
Fig. 3: Plot of studentized residual versus predicted value for fitted model (4c). All types.
Finally, we compute least squares means based on the reduced model (2). A letter display is
obtained by the method of Piepho (2003).
5
Tab. 1: Means for types based on model (4) using log-transformed data. Means in a column
followed by the same letter are not significantly different.
Scale
Type
log§
original$
0.601 a
0.476 b
0.309 c
0.486 abc
1
2
3
4
1.82 a
1.61 b
1.36 c
1.63 abc
§ least squares mean
$ back-transformed least squares mean; this is a median estimate
A final improvement
Statistical inference (tests of effects) is expected to be fairly robust to nonnormality.
Predictions of individual observations, however, are more sensitive. A Q-Q-plot of
studentized residuals based on model (4) with log-transformed data shows some departure
from normality (Fig. 4). Thus, we used the Box-Cox transformation, which yielded y0.29 as the
optimal normalising transformation. Normality of studentized residuals was satisfactory on
this scale (Fig. 5). Thus, we re-did all analyses on this new transformed scale. ANOVA results
were the same as on the log-scale (ANOVA tables not shown), including the comparison of
means (Tab. 2). Fitted curves and prediction intervals are also quite similar, though the
intervals have become a bit more narrow.
Residual
3
2
1
0
-1
-2
-3
-4
-2.9
2.9
Normal score
Fig. 4: Normal Q-Q-plot based on model (4c) using log-transformed data.
6
Residual
3
2
1
0
-1
-2
-3
-4
-2.9
2.9
Normal score
Fig. 5: Normal Q-Q-plot based on model (4c) using power-transformed data (y0.29).
Yield0.29
Type=1
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1.0
0.9
0.8
0.7
1
2
3
4
5
6 7 8
Month
9 10 11 12
Fig. 6: Fitted model (4c) for Type=1 on power-transformed-scale (y0.29), with 95% prediction
limits.
7
Yield
Type=1
6
5
4
3
2
1
0
1
2
3
4
5
6
7 8
Month
9
10 11 12
Fig. 7: Fitted model for Type=1 on original scale, with 95% prediction limits. Model on original scale based on back-transformation of model (4c) fitted to power-transformed data
(y0.29).
Tab. 2: Means for types based on model (4) using power-transformed data (y0.29). Means in a
column followed by the same letter are not significantly different.
Scale
Type
1
2
3
4
y0.29§
1.202 a
1.161 b
1.107 c
1.163 abc
original$
1.88 a
1.67 b
1.42 c
1.68 abc
§ least squares mean
$ back-transformed least squares mean; this is a median estimate
Other types of model
Inverse polynomials are a flexible class of models (McCullagh and Nelder, 1989, Generalized
linear models. Chapman and Hall, London) which may be used to fit lactation curves, e.g.,
  exp   x   / x
and
 1    x   / x
8
These are easily linearized and so can be used in the same way as the incomplete gamma
model. One may also consider using the power transformation to gain more flexibility. Yet
another model (Emmans and Fisher, 1986):
log(  )    x   exp(x)
Rerefences
Wood PDP 1967 Algebraic model of the lactation curve in cattle. Nature 216: 164-165.
Wood PDP 1972 A note on seasonal fluctuations in milk production. Anim. Prod. 15: 89-92.
SAS code
SAS macros for multiple comparisons and for Box-Cox-transformation are available at
www.uni-hohenheim.de/bioinformatik. Details on their use can be found in the lecture notes
"Gemischte Modelle in den Life Sciences" (see homepage under "Veranstaltungen").
/*lack of fit for regression model*/
proc glm data=s;
Class type lackfit;
Model LOG_yield=type type*lactmonth
type*LOG_lactmonth
type*lactmonth*LOG_lactmonth
type*lackfit;
run;
quit;
/*fit model with interaction*/
proc glm data=s;
Class type;
Model LOG_yield=type lactmonth
log_lactmonth
lactmonth*log_lactmonth
type*lactmonth*log_lactmonth;
run;
type*lactmonth
type*LOG_lactmonth
/*interactions not significant, so reduce model -> curves run parallel on
log-scale/not on original scale*/
proc glm data=s;
Class type;
Model LOG_yield=type lactmonth log_lactmonth lactmonth*log_lactmonth;
output out=res student=student predicted=pred_log_scale LCL=LCL_log_scale
UCL=UCL_log_scale;
run;
proc sort data=res out=res;
by type lactmonth;
data res;
set res;
pred_original_scale=exp(pred_log_scale);
LCL_original_scale=exp(LCL_log_scale);
UCL_original_scale=exp(UCL_log_scale);
filename fileref 'd:\hpp\beratung\aloo\fig.cgm';
9
goptions reset=all dev=win target=cgmof97L gsfname=fileref keymap=winansi
ftext=hwcgm001 gsflen=8092
gsfmode=replace hsize=17 cm vsize=12 cm htext=1.1;
axis1 label=none offset=(0 cm, 0 cm) w=10 minor=none major=(w=10);
axis2 label=none offset=(0 cm, 0 cm) minor=none w=10 major=(w=10);
/* plot
symbol1
symbol2
symbol3
symbol4
fitted curves on log-scale*/
i=spline value=none color=black
i=none
value=dot color=black
i=spline value=none color=black
i=spline value=none color=black
w=2;
h=1.5;
w=2 line=3;
w=2 line=3;
proc gplot;
plot pred_log_scale*lactmonth log_yield*lactmonth
LCL_log_scale*lactmonth UCL_log_scale*lactmonth/overlay haxis=axis1
vaxis=axis2 nolegend;
by type;
run;
quit;
/* plot fitted curves on original scale*/
proc gplot;
plot pred_original_scale*lactmonth yield*lactmonth
LCL_original_scale*lactmonth UCL_original_scale*lactmonth/overlay
haxis=axis1 vaxis=axis2 nolegend;
by type;
run;
quit;
/*Residual plots*/
symbol i=none value=dot;
proc gplot;
plot student*pred_log_scale/overlay haxis=axis1 vaxis=axis2;
run;
quit;
proc univariate data=res;
qqplot student/normal nohlabel novlabel;
run;
/*pairwise comparisons*/
ods output diffs=diffs;
ods output lsmeans=lsmeans;
proc mixed data=s;
Class type;
Model LOG_yield=type lactmonth log_lactmonth lactmonth*log_lactmonth;
lsmeans type/pdiff;
%mult(trt=type);
run;
%boxcox(phimin=0.2,phimax=0.4, steps=20, model=type lactmonth log_lactmonth
lactmonth*log_lactmonth, class=type, data=s, response=yield);
Data
Type Lactmonth Yield
2
2
0.76
2
4
3.00
2
3
3.00
10
3
2
3
1
2
1
2
3
1
3
2
1
2
1
3
2
1
3
2
4
1
1
1
3
3
1
2
2
1
2
3
3
1
1
4
3
2
3
2
2
1
1
4
4
3
1
1
3
2
4
4
1
1
1
2
4
1
2
2
1
1
3
2
3
2
4
4
2
4
4
4
5
5
8
8
4
4
4
4
4
1
4
4
4
1
4
4
4
4
4
5
4
4
5
5
4
4
5
5
5
12
12
12
4
4
4
4
11
11
12
4
4
4
4
3
3
3
3
4
4
4
4
5
4
1
3
1.52
2.00
0.65
3.13
2.08
5.06
4.00
3.46
3.60
3.20
1.00
1.20
1.40
4.00
2.80
3.80
3.60
0.90
3.00
1.60
2.80
1.70
3.90
2.20
2.40
1.80
1.80
2.20
2.40
2.40
1.40
1.60
2.00
1.96
2.20
2.10
1.00
0.70
0.76
0.52
3.42
2.00
1.60
1.90
1.40
1.04
1.00
1.50
2.50
0.50
1.70
2.00
3.80
2.40
1.44
2.80
4.20
2.40
2.80
1.30
1.30
1.60
2.00
11
1
1
2
1
1
4
1
1
3
2
2
2
2
1
2
1
1
4
3
4
3
2
1
2
1
1
2
1
2
1
3
3
2
2
1
4
2
4
3
3
3
4
2
3
1
1
4
3
2
2
2
4
4
1
1
3
1
4
4
1
2
1
2
5
2
12
3
3
4
12
3
2
5
3
12
12
3
5
12
5
12
5
12
3
3
3
5
4
10
9
5
7
3
12
4
4
2
11
7
3
4
5
4
5
5
6
5
5
5
5
4
12
12
12
3
12
11
11
8
6
3
5
10
12
5
3
2.84
1.90
0.78
1.22
1.80
4.40
0.80
1.00
2.50
2.00
1.60
1.50
3.60
4.20
2.20
1.00
2.80
1.46
2.40
1.00
1.42
1.56
3.50
2.20
3.20
1.92
1.50
1.40
2.80
1.84
1.60
2.80
4.80
4.00
1.00
1.20
1.24
2.50
2.80
2.70
2.50
2.90
1.00
1.20
2.60
1.46
2.30
1.20
0.60
1.82
0.84
2.42
0.68
0.60
0.90
1.00
2.68
3.20
1.60
3.00
1.70
1.88
2.00
12
1
1
4
4
1
1
2
1
4
4
4
4
1
4
4
4
4
3
1
1
1
1
1
2
2
1
1
2
2
2
1
3
2
4
1
3
4
1
1
2
3
4
4
1
3
4
1
1
1
1
2
3
1
3
3
1
1
1
1
3
3
1
2
7
7
4
6
5
12
1
5
4
4
5
5
5
5
5
5
4
4
6
12
3
5
5
5
4
4
5
5
5
2
12
5
6
6
6
6
5
5
6
3
6
6
6
6
7
7
7
6
5
5
5
4
5
6
7
2
2
6
3
6
3
3
8
3.00
2.00
1.64
2.00
2.42
1.00
2.08
1.40
2.48
2.20
1.60
2.80
2.00
2.60
2.40
0.80
2.40
1.40
4.40
2.60
2.60
1.84
2.40
1.80
2.60
2.20
2.40
2.00
1.00
1.00
1.60
1.40
1.70
1.60
1.80
0.90
1.50
1.00
1.80
2.40
2.40
2.00
2.00
1.76
0.84
2.40
1.60
3.20
2.00
2.00
2.20
0.80
2.80
1.50
1.00
1.00
2.40
2.20
2.62
1.24
1.80
2.40
1.60
13
1
1
1
1
1
1
1
2
3
2
1
4
1
1
4
1
2
2
3
3
3
1
2
2
2
2
2
1
2
1
1
2
2
1
2
2
2
2
1
2
2
1
2
2
1
2
1
2
1
2
2
3
1
2
2
1
2
3
3
2
1
1
2
8
9
9
8
8
11
9
8
7
11
11
9
8
9
9
10
7
10
9
8
9
7
5
7
5
8
6
7
5
10
6
10
7
7
7
7
7
6
1
11
11
8
8
8
8
8
9
11
10
9
9
9
10
6
7
9
10
7
5
9
6
9
10
1.80
1.40
1.06
1.22
0.90
2.40
1.00
1.00
0.76
0.30
1.00
0.80
1.40
1.80
1.00
1.00
0.84
0.82
0.20
0.22
0.50
1.60
1.52
1.60
2.00
0.40
2.00
1.00
0.90
0.84
1.00
1.30
0.90
1.22
1.40
1.00
1.00
1.40
1.40
1.42
1.00
1.40
1.80
1.60
1.58
2.00
2.60
1.00
0.60
1.40
1.60
0.80
1.60
1.80
1.60
2.00
0.60
1.20
1.00
1.70
0.72
1.40
1.00
14
2
2
1
3
1
2
2
1
1
2
2
1
3
2
1
1
3
2
3
2
1
2
3
2
2
2
3
2
3
1
3
2
2
3
1
1
2
1
2
2
2
3
3
1
3
2
3
3
3
3
2
1
3
2
2
2
2
2
3
3
3
3
3
11
12
8
7
8
10
10
10
1
1
7
7
12
11
10
11
9
11
7
1
1
1
1
1
12
11
11
6
6
12
12
6
7
6
5
7
5
3
10
5
6
5
5
5
7
6
6
6
12
12
5
2
3
6
6
6
6
6
6
6
6
5
8
1.44
0.50
1.00
0.40
1.60
1.40
1.00
1.40
1.60
1.00
1.60
1.80
1.20
1.20
1.32
1.38
1.30
1.60
1.80
2.20
2.40
0.80
0.90
1.00
0.80
1.30
0.76
1.26
1.22
2.60
2.00
2.40
3.20
1.00
2.80
4.00
1.60
1.20
2.98
2.80
2.40
1.60
4.40
3.60
1.30
2.00
1.80
1.50
1.60
1.80
1.00
2.90
1.60
3.00
2.40
1.80
4.46
4.00
2.00
1.70
1.60
1.60
1.00
15
3
3
3
3
3
3
3
3
3
3
3
1
3
1
1
4
2
4
7
10
7
11
2
8
9
10
9
8
11
11
1
5
3
4
1
5
1.04
0.90
1.30
1.30
1.60
1.30
0.80
1.10
1.60
2.00
1.50
1.60
1.00
2.60
3.80
4.00
1.50
2.26
16