NCSU Statistics Department Consulting Project
Effect of Heat Stress on Lactating Sows
Client : Santa Mendoza Benavides, Department of Animal Science
Consulting Team: Sihan Wu, Bo Ning
Faculty Advisor: Dr.Bloomfield
April 2014
1 Introduction
The consulting project studies the effect of high ambient temperature and humidity on
feed intake of lactating sows based on data collected from a commercial farm. In particular, the goal is to find the thresholds of temperature and humidity above which sows will
be most stressed, and to study the behavior above such thresholds.
1
2 Background
Seasonal variation in temperature has significant negative influences on reproductive performance of sows and subsequently profitability. Sows are sensitive to heat stress during
the summer.They use reduction in feed intake as a mechanism to control body temperature (Williams, 1998). Past studies have shown that feed intake of sows reduce by 25%
when housed at 30 ◦ C (Mullan, 1992). Overall, heat stress generates economic losses of
$300 million per year (St-Pierre et al, 2003). This project studies the effect of heat stress
on feed intake of lactating sows in hope of innovating farm management to offset the
negative effect caused by high ambient temperature.
3 Data Description
The data was collected from a commercial farm in Oklahoma during summer and early
fall of 2013. It was originally used for another project to study the diet effect of fatty acid
on lactating sows. In the study, 480 sows were randomly assigned to 10 diets during 21
days of lactation. The diets consists of 9 experimental diets with two fatty acid additives
each at 3 levels, and a control diet without fatty acid additives. The daily feed intake
of each sow was measured during lactation. Sows were divided into 21 groups based on
their start dates of location. Those with similar start dates are arranged into the same
group. Ambient temperature and humidity were recorded every 5 minutes for each group.
Weight of each sow was recorded 1-2 days before farrow, at the begining and the end of
the lactation. The sows in the study were on the first, third, fourth, or fifth parity. Parity
stage is the number of litters sows delivered. Since sows with high parity in general have
different physical characteristic from those with low parity, the second parity sows were
purposedly excluded from the study to separate the low parity group from high parity
2
groups. To eliminate the effect of litter size, each sow was assigned 12 piglets.
Before we fit the models, exploratory data analysis was performed to evaluate data.
Some data with values out of physical bounds, such as humidity measurements exceeding
100%, are removed. From a summary plot of the maximum and minimum of daily
temperature and humidity over the experiment period, we observe significant drops of
both temoerature and humidity after day 330 of the calender year. This conincides with
the time that farm adjusting room control facility in early fall. Thus we deleted the small
portion of data after day 330.
4 Linear Mix Effect Model Using Summary
Statistics
Initially, it is hard to come up an idea to incorporate the 5 minutes observations of
temperature and humidity into the model. To have a good understanding of the effects
of temperature and humidity on feed intake, we fit linear mixed effect models using the
summary statistics. The summary statistics being explored are daily minimum, daily
average, and daily maximum of temperature and humidity.
In the first model, only summary statistics of temperature are included. The farm where
the data was collected is regulated by thermostats. When barn is hot during the day,
the farm uses sprinkler and fans to cool down the environment and to pull the air out.
The humidity decreases subsequently. This is an affordable strategy to comfort the sows
during the summer. Figure 4.1 shows that ambient temperature and humidity are negatively correlated during the day, contrary to their positive correlation outdoor. Thus it is
reasonable to explore temperature effect before studying its combined effect with humidity. Using AIC for stepwise selection of fixed effects, the best model including maximum
3
temperature, average temperature, and their interaction as fixed effects is
y i j = β0 + β1 AT j + β2 HT j + β3 AT j HT j
(4.1)
+ β4 parityi j + β5Wi + ns(lacday) + αi j + γ(α)i j + ²i j
where y i j is the response of feed intake for i t h sow on j t h day, AT j is the average
temperature for j t h day, HT j is the maximum temperature for j t h day, Wi is the weight
of sow after farrow and before lactation.1 For i t h sow, ns(lacday) is the nature spline
function2 for lactation days with 8 degree of freedom, αi j is the random effect of group
for i t h sow on j t h day, γ(α)i j is the random effect of sow ID nested in group for i t h sow
on j t h day, ²i j ∼ N (0, σ2 ) is the random error for i t h sow and j t h day and the correlation
of ² follows an AR(1) structure, where
cor r (²i j , ²i 0 j 0 ) =




ρ| j − j |,



0,
0
if i = i 0 and j 6= j 0
if i 6= i 0
The summary table of the model is in Table 4.1.
In this model, LacDay is modeled as a natural cubic spline function with 8 degree of
freedom. A natural cubic spline function is a piecewise cubic function passes through
given points called knots, and with constraints that the first and second derivative are
continuous at the knots. Since it is hard to incorporate all the features of lactation days
using a single cubic function, we decide to break down lactation days into 8 stages, and
model lacDay as 8 segments of piecewise cubic function with 7 interior knots, which
is a natural cubic spline function with 8 degree of freedom. From Table 4.1, parity is
significant, which suggests low parity group is difference from high parity group. Weight
1 Among
the three weights measured for sows, the weight before lactation is chosen for the model
according to commercial practice.
reason to apply nature spline function here is because lactation can not be modeled using a
2 The
polynomial model to feedintake.
4
(Intercept)
average Temp
high Temp
average Temp * high Temp
I (parity > 1)
weight
ns(LacDay, 8)1
ns(LacDay, 8)2
ns(LacDay, 8)3
ns(LacDay, 8)4
ns(LacDay, 8)5
ns(LacDay, 8)6
ns(LacDay, 8)7
ns(LacDay, 8)8
Value Std.Error t-value p-value
29.195
14.529
2.010
0.045
-1.082
0.561 -1.927
0.054
-0.414
0.511 -0.811
0.418
0.022
0.020
1.096
0.273
3.254
0.266 12.220
0.000
-0.016
0.004 -4.334
0.000
5.117
0.277 18.458
0.000
6.415
0.279 22.970
0.000
9.697
0.272 35.639
0.000
9.532
0.277 34.447
0.000
9.191
0.271 33.931
0.000
8.476
0.243 34.942
0.000
15.762
0.384 41.066
0.000
5.179
0.199 25.977
0.000
Table 4.1: Model 1 summary of parameters
is also significant, and has a negative relationship with feed intake. The high temperature
is not significant and average temperature is at the edge of 95% significant level.
In the next step, humidity is added into the model. Among different candidate models,
the quadratic model with maximum temperature and average humidity gives the best
explanation of variability of daily feed intake. This is consistent with a former study to
evaluate the effects of climatic variables on feed intake of lactating sows (Bergsma and
Hermesch, 2012). The model is,
y i j = β0 + β1 HT j + β2 HT2j + β3 AH j + β4 AH2j
(4.2)
+ β5 parityi j + β5Wi + ns(lacday) + αi j + γ(α)i j + ²i j
where, HT j is the high temperature for j t h day. Table 4.2 summarized the model that
includes humidity.
From the result, parity and weight both have similar interpretation as in Table 4.1. After
humidity is added, high temperature and average humidity both are statistical significant.
5
Figure 4.1: Plot of average temperature and humidity over 24 hour period
But it is interesting to find that temperature has negative effect on the feed intake in
model 4.1 while it becomes positive in the model 4.2. It is hard to find a reasonable
explanation between temperature and humidity. Actually, from Figure 4.1, by plotting
the temperature and humidity during 24 hours, we find the curves are nonlinear. In order
to find out the critical values of temperature and humidity that corresponds to maximum
sow feed intake, we need to investigate a nonlinear estimation approach.
6
(Intercept)
poly(high Temp, 3)2
poly(high Temp, 3)3
poly(average Hum, 3)2
poly(average Hum, 3)3
I (parity > 1)
weight
ns(LacDay, 8)1
ns(LacDay, 8)2
ns(LacDay, 8)3
ns(LacDay, 8)4
ns(LacDay, 8)5
ns(LacDay, 8)6
ns(LacDay, 8)7
ns(LacDay, 8)8
Value Std.Error t-value p-value
5.260
0.728
7.227
0.000
5.215
3.278
1.591
0.112
14.110
3.155
4.472
0.000
13.401
3.748
3.576
0.000
5.865
3.603
1.628
0.104
3.226
0.267 12.094
0.000
-0.016
0.004 -4.191
0.000
4.933
0.279 17.670
0.000
6.269
0.281 22.271
0.000
9.529
0.274 34.748
0.000
9.304
0.279 33.388
0.000
9.092
0.273 33.276
0.000
8.350
0.247 33.850
0.000
15.611
0.387 40.377
0.000
5.106
0.203 25.155
0.000
Table 4.2: Model 2 summary of parameters
5 Nonlinear Model using High Frequency Data
5.1 Incorporating High Frequency data
Now that we have some basic understanding of how heat stress affect feed intake from the
mixed effect model using summary statistics, we decide to incorporate the high frequency
observations of temperature and humidity into the model. We hope to use an exploratory
model to find the thresholds of temperature and humidity above which sows will be most
stressed.
Using the 24-hour observation, we can generate some relavent statistics. For example, a
statistics measuring the instensity of temperture over the 24-hour period.
s T 1t = T t
where T t is the 5 minutes observation of temperature during 24 hour period. The integral
7
of s 1t is the area under the temperature curve.
Since we do not have continuous data of the temperature, this can be approximated by
the 5 minutes observations. There are 288 such observations over a 24 hour period.
1
Z
0
Tt d t ≈
288
X
T t ÷ 288
t =1
Given the temperature threshold Tthresh , we can also generate a statistics to measure the
intensity of exposure above temperature threshold. Let
s T 2t = max(T t − Tthresh , 0)
Then the integral
1
Z
0
max(T t − Tthresh , 0)d t
is the intensity of exposure above temperature threshold. This is the area under the
temperature curve and above the threshold line. The intergral can also be approximated
by the 5 minutes observations.
Similarly, we can generate information from the high frequency data of humidity given a
humidity threshold.
s H 1t = H t
and
s H 2t = max(H t − Hthresh , 0)
8
5.2 The Conditional Model
To incorporate high frequency data, we consider a component S(T t , H t ) of temperature
and humidity using the tensor product of (1, s T 1t , s T 2t ) ⊗ (1, s H 3t , s H 4t ), such that
β1
Z
0
1
S(T t , H t )d t =
=
I
X
i =1
I
X
i =1
= β∗1
+ β∗5
β1 γi
β∗i
Z
1
0
Z 1
0
1
Z
Z
0
0
s i (T t , H t )d t
1
s i (T t , H t )d t
(5.1)
s T 1t d t + β∗2
1
Z
Z
1
Z
1
s T 2t d t + β∗3
s H 1t d t + β∗4
s H 2t d t
0
0
Z 1
Z 1
∗
∗
s T 1t s H 1t d t + β6
s T 1t s H 2t d t + β7
s T 2t s H 1t d t ,
0
0
0
given the thresholds for temperature and humidity, we can fit a linear mixed effect model
using integrated temperature and humidity stress as a predictor of feed intake, where
y i j | Tthresh , Hthresh = β0 + β1
1
Z
0
S(T t , H t )d t
+ β2 parityi j + ns(lacday) + I parityi j >1 ∗ ns(lacday)
(5.2)
+ β3Wi + αi j + γ(α)i j + ²i j
where Tthresh and Hthresh are thresholds for temperature and humidity.
In the model, β∗8
R1
0
s T 2t s H 2t d t was omitted based on AIC.
5.3 Optimal Threshold
For fixed thresholds, the model continues to be linear in parameters. We can obtain
the maximum likelihoods for the models given a range of temperature and humidity
thresholds. Among all the possible models, the model with the optimized threshold gives
the maximum profile likelihood. From
sup
Tthresh ∈A,Hthresh ∈B
sup L(β | Y , Tthresh , Hthresh )
β
9
A = [20.0, 32.5]
B = [47.3, 90.8]
where A and B are the ranges of observed temperature and humidity, we find the optimized
thresholds to be Tthresh = 28.06, Hthresh = 71.12.
5.4 The Final Model using Optimized Threshold
The final model using optimized threshold can be represented as an linear mixed effect
model
E(y i j | Tthresh = 28.06, Hthresh = 71.12) = β0 + β1
Z
0
1
S(T t , H t )d t
+ β2 I parityi j >1 + ns(lacday) + I parityi j >1 ∗ ns(lacday)
+ β3Wi + αi j + γ(α)i j
The summary of model parameters is listed in Table 5.1. All terms are significant except
for s T 1t . By using this model, we can construct a heat stress index based on expected
feed intake at different temperature and humidity, as shown in Figure 5.2. The contour
plot shows that when temperature is above 28 ◦ C, sows become more stressed as humidity
increases. They become most stressed when humidity is above 67%. The effect of humidity
is more obvious when temperature is high. When temperature reaches 32 ◦ C, sows eat
half as much when humidity is above 67% compared to when humidity is under 35%.
Note that the contour plot is less accurate around the bondaries since we have fewer
observations when temperature and humidity are extreme. Figure 5.3 presents the same
result in a 3D plot, where z is the expected feed intake given temperature and humidity.
10
I(parity
I(parity
I(parity
I(parity
I(parity
I(parity
I(parity
I(parity
>
>
>
>
>
>
>
>
(Intercept)
BW2
I(parity > 1)TRUE
ns(LacDay, 8)1
ns(LacDay, 8)2
ns(LacDay, 8)3
ns(LacDay, 8)4
ns(LacDay, 8)5
ns(LacDay, 8)6
ns(LacDay, 8)7
ns(LacDay, 8)8
sT1
sT2
sH1
sH2
sT1sH1
sT1H2
sT2H1
1)TRUE:ns(LacDay, 8)1
1)TRUE:ns(LacDay, 8)2
1)TRUE:ns(LacDay, 8)3
1)TRUE:ns(LacDay, 8)4
1)TRUE:ns(LacDay, 8)5
1)TRUE:ns(LacDay, 8)6
1)TRUE:ns(LacDay, 8)7
1)TRUE:ns(LacDay, 8)8
Value Std.Error t-value p-value
-35.107
26.589 -1.320
0.187
-0.015
0.004 -3.958
0.000
0.895
0.418
2.140
0.033
3.563
0.406
8.781
0.000
4.744
0.446 10.649
0.000
8.170
0.427 19.144
0.000
7.408
0.429 17.283
0.000
7.314
0.458 15.983
0.000
7.847
0.392 20.036
0.000
13.625
0.606 22.487
0.000
4.346
0.326 13.347
0.000
1.419
1.018
1.394
0.163
21.000
4.389
4.785
0.000
0.876
0.396
2.211
0.027
-3.171
0.859 -3.691
0.000
-0.031
0.015 -2.058
0.040
0.123
0.033
3.701
0.000
-0.300
0.066 -4.527
0.000
4.426
0.568
7.793
0.000
1.551
0.622
2.492
0.013
1.616
0.596
2.711
0.007
4.013
0.599
6.700
0.000
2.852
0.638
4.467
0.000
0.927
0.547
1.696
0.090
3.510
0.842
4.169
0.000
1.597
0.454
3.517
0.000
Table 5.1: Model 3 summary of parameters
11
Figure 5.1: Coutour plot of expected feed intake under different temperature and
humidity
Figure 5.2: 3D plot of expected feed intake under different temperature and humidity
12
5.5 Summary
From the linear models, we found that weight has negative effect on feed intake, and
nature spline function fits well for lactation days. Also, we found there is a significant
difference between low partiy and high parity groups. The disadvantage of these models
is they cannot give the threshold values for temperature and humidity.
The nonlinear model gives critical values for temperature and humidity. By using 5
minutes recording data over 24 hour period, the maximum profile likelihoods method
yields critical values for temperature at 28.06 ◦ C and critical value of humidity at 71.12%.
6 Discussion
The two linear mixed effect models using summary statistics of temperature and humidity
gives an insight of how they affect feed intake. Although linear mixed effect models have
been conventionally used in other studies, they cannot fulfill the objectives of finding the
critial values. The model using high frequency data gives optimized thresholds for temperature and humidity, and reasonable explanation of heat stress impact using optimized
thresholds. However, the maximum profile likelihood method does not give confidence
intervals for the optimized thresholds. In addition, the exploratory model incoporates basis functions constructed based on intuitive understanding of the the project objectives.
It is diffucult to validate such model given the lack of literatures on analysis of similar
type of data. One possible but distinctive future approach we suggest is to use functional
data analysis for the project. Functional data analysis gives information from curves and
distributions. The plots of daily temperature and humidity, for instance, can be treated
as functional data. This approach could be explored in the future.
13
References
[1] Bergsma, R. and Hermesch, S., Exploring breeding opportunities for reduced thermal
sensitivity of feed intake in the lactating sow. J. Animal Science, 90:85-98, 2012
[2] Mullan, B.P., Brown, W. and Kerr, M., The response of the lactating sow to ambient
temperature. Proceedings of the Nutrition Society of Australia, 17, 1992
[3] Pluske, J.R., Williams, I.H., Zak, L.J., Clowes, E.J., Cegielski, A.C. and Aherne, F.X.,
Feeding lactating primiparous sows to establish three divergent metabolic states: III.
Milk production and pig growth. J. Animal Science, Vol. 76, No. 4 1165-1171, April,
1998
[4] St-Pierre, N.R., Gobanov, B and Schnitkey, G., Economic losses from heat stress by
US livestock industries. J. Dairy Sci., 86(Supplement): E52-E771, 2003
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