Statistical evaluation of in vitro gas production kinetics

by

Lena Nathanaelsson Linda Sandström

Master thesis in Mathematical Statistics Umeå University, 2003

Supervisors: Peter Anton and Mårten Hetta

ABSTRACT At the Forage Research Centre, Swedish University of Agricultural Sciences in Umeå a technique has been developed to describe the degradation of feeds in ruminant animals. The development of this technique has been made in collaboration with Dr J.W. Cone, Nutrition and Food, Animal Sciences Group of Wageningen UR, Lelystad, the Netherlands. Experiments have been performed in laboratories where feed samples have been incubated with rumen fluid and the amount of gas produced during the digestion has been measured continuously. These feed samples were analysed at three separate occasions. The purpose of this thesis was to identify and describe statistical procedures for detecting differences between feeds analysed within the same laboratory as well as differences between the same feeds analysed in two different laboratories (in Sweden and the Netherlands). To determine the rate of digestion and to describe to what extent feeds are digested in the rumen a gas production model was fitted, using non linear regression. In order to test whether there are significant differences between the feeds, three methods were applied. For each method, the variances were estimated differently. In the first method, Hotelling’s T2 tests and two sample t tests were performed. From these tests, differences between the feeds that were analysed within the same laboratory were detected whereas no differences between the same feeds analysed in two different laboratories could be found. In the other two methods, tests were performed using an assumption of normality. These two methods detected a larger number of differences between the feeds than the first method, primarily due to extremely underestimated variances. The first method is to be preferred since the estimated variances in this method are unbiased. This causes the result to be more reliable. For future experiments it is recommended that the feed samples are analysed at considerably more than three occasions. This would lead to better estimations in the first method and consequently the result would be enhanced. REFERAT Vid Grovfodercentrum, Sverige lantbruksuniversitet i Umeå har en teknik utvecklats för att beskriva nedbrytningen av foder hos idisslare. Utvecklingen har genomförts genom ett samarbete med Dr J.W Cone, Nutrition and Food, Animal Sciences Group of Wageningen UR, Lelystad, the Netherlands. Försök har utförts i laboratorium där foderprover har inkuberats med vomvätska och mängden producerad gas under nedbrytningen har mätts kontinuerligt. Samma foderprover har analyserats vid tre olika tillfällen. Syftet med detta examensarbete var att identifiera och beskriva statistiska procedurer för att upptäcka skillnader mellan foder som har analyserats inom samma laboratorium samt upptäcka skillnader mellan samma foder analyserade vid två olika laboratorier (i Sverige och Nederländerna). För att bestämma nedbrytningshastigheten och för att beskriva i vilken utsträckning foder bryts ner i vommen har en gasproduktions modell anpassats med hjälp av icke linjär regression. För att testa om det finns signifikanta skillnader mellan fodren har tre metoder använts. I varje metod har variansen skattats på olika sätt. I den första metoden utfördes Hotelling’s T2 test samt t test. I dessa tester upptäcktes skillnader mellan foder som har analyserats inom samma laboratorium medan inga skillnader upptäcktes mellan samma foder analyserade vid två olika laboratorier. I de övriga två metoderna gjordes tester där antaganden om normalfördelning användes. Dessa två metoder upptäckte betydligt fler skillnader mellan fodren än den första metoden och detta beror främst på att variansen i de två metoderna är extremt underskattad. Den första metoden är att föredra eftersom den skattade variansen i denna metod är väntevärdesriktig vilket innebär att resultatet blir mer pålitligt. För framtida försök rekommenderas att foderproven analyseras vid betydligt fler än tre tillfällen. Då skulle skattningarna i den första metoden blivit bättre och resultatet skulle ha förbättrats.

PREFACE This Master’s thesis is the final part required for a Master of Science degree in Mathematical Statistics at Umeå University. The work has been carried out during the fall of 2003 under the supervision of Peter Anton and Mårten Hetta. The aspiration of this thesis was to identify and describe a statistical procedure for finding significant differences between feed samples analysed in Sweden and in the Netherlands. All tests and analysis based on the Swedish data have been performed by Lena Nathanaelsson whereas tests and analysis based on data from the Netherlands have been performed by Linda Sandström. Furthermore, the test and analysis of the same feeds evaluated in both Sweden and the Netherlands were performed together. ACKNOWLEDGEMENTS First, we would like to thank our supervisors Peter Anton and Mårten Hetta for their encouragement, guidance and assistance during the process of composing this thesis. Special thanks to the staff at the Department of Mathematical Statistics at Umeå University for their endless help and support during our time of study. Finally, we would also like to thank Dr Niclas Börlin at the Department of Computing Science, Umeå University for his computer expertise, shown interest and time set aside.

Lena Nathanaelsson Linda Sandström

Umeå, February 2004

TABLE OF CONTENTS

1 INTRODUCTION.............................................................................................................................................. 1

2 GAS PRODUCTION PROFILES .................................................................................................................... 2

3 MODEL FITTING............................................................................................................................................. 4

3.1 ANALYSIS OF RESIDUALS............................................................................................................................... 5 3.2 NON LINEAR REGRESSION ......................................................................................................... 7

3.1.1 Levenberg-Marquardt’s method ........................................................................................................... 7

4 ESTIMATION METHODS .............................................................................................................................. 9

4.1 METHOD (1) .................................................................................................................................................. 9 4.1.1 The Hotelling’s T2 test.......................................................................................................................... 9 4.1.2 The two sample t test .......................................................................................................................... 10

4.2 METHOD (2) ................................................................................................................................................ 11 4.3 METHOD (3) ................................................................................................................................................ 12

5 METHODS FOR GROUPING THE FEEDS................................................................................................ 13

5.1 APPROACH WHEN GROUPING THE FEEDS BASED ON RESULTS FROM METHOD (1) ........................................ 13 5.2 CLUSTER ANALYSIS..................................................................................................................................... 13

5.2.1 Non-hierarchical Clustering................................................................................................................ 13 5.2.2 Hierarchical Clustering....................................................................................................................... 14

6 RESULTS ......................................................................................................................................................... 15

6.1 METHOD (1) ................................................................................................................................................ 15 6.2 METHOD (2) ................................................................................................................................................ 16 6.3 METHOD (3) ................................................................................................................................................ 17 6. 4 RATIOS ....................................................................................................................................................... 18

6.4.1 Ratios of the estimated parameters ..................................................................................................... 18 6.4.2 Ratios of the estimated variances ....................................................................................................... 18

6.5 SAMPLE DESCRIPTION ................................................................................................................................. 19 6.6 GROUPING OF THE FEEDS............................................................................................................................. 19

7 DISCUSSION AND CONCLUSIONS ........................................................................................................... 25

7.1 METHOD (1) ................................................................................................................................................ 25 7.2 METHOD (2) ................................................................................................................................................ 26 7.3 METHOD (3) ................................................................................................................................................ 26 7.4 FINAL CONCLUSIONS ................................................................................................................................... 27

8 REFERENCES................................................................................................................................................. 28

8.1 BOOKS, ARTICLES AND INTERNET LINKS...................................................................................................... 28 8.2 COMPUTER SOFTWARE ................................................................................................................................ 28

APPENDIX A

A.1 METHOD (1) A.2 METHOD (2) A.3 METHOD (3) A.4 RATIOS A.5 CHEMICAL COMPOSITION OF THE FEEDS

APPENDIX B

B.1 METHOD (1) B.1.1 The Hotelling’s T2test B.1.2 The two sample t test

B.2 METHOD (2) B.3 METHOD (3)

1 INTRODUCTION Within the field of feed evaluation and nutrition more and more focus is brought on dynamic variables such as rate of degradation and passage. When investigating feed utilisation in ruminant animals, it is of interest to study the amount of undigested feed passing the rumen as well as the ruminal fermentation of feed. In vivo measurements on ruminant animals can be made to determine to what extent different feeds have been digested. The difference between the amount of eaten feed and the amount of produced waste represents the total quantity of feed utilized by the ruminant. Since these in vivo experiments are expensive and complicated, different in vitro techniques have been developed to describe ruminal digestion. One of the most interesting techniques is the in vitro gas production technique which can be used for feed evaluation, plant breeding and quality assurance. The in vitro gas experiments are carried out in a laboratory; small feed samples are incubated at 39°C in an anaerobic mixture of rumen fluid and a buffered solution. The amount of gas produced during the digestion is measured continuously. At the end of the incubation (72 h), the remains of the initially incubated substrate are measured. From the in vitro gas experiments, observations of the amount of gas produced during fermentation are obtained. These observations can be used to fit system models of the rumen function. The Forage Research Centre within the Swedish University of Agricultural Sciences (SLU) in Umeå is working with development of in vitro methods for feed evaluation for ruminants. In their research they are focusing on the microbial degradation in the rumen. One of their fields involves studying the degradation of feeds in the rumen and the investigation of whether there are differences between them, in terms of digestion. The gas production technique retrieved can be used for this purpose. This thesis aims to identify and describe a statistical procedure for detecting significant differences in gas production kinetics between feed samples analysed within the same in vitro laboratory. Furthermore, the thesis will also involve a statistical evaluation of possible significant differences between gas production data from the same feeds analysed in two different in vitro laboratories (in Umeå, Sweden and Lelystad, the Netherlands). In chapter 2, a background description of gas production profiles is presented. An explanation of how a non linear gas production model is fitted, the analysis of residuals as well as the numerical method used when fitting this type of model is found in chapter 3. Three statistical procedures are described in chapter 4 and a part of the cluster analysis theory along with a description of how to group the feeds are located in chapter 5. Finally, the results from the statistical procedures are given in chapter 6 and the final conclusions and discussion in chapter 7.

1

2 GAS PRODUCTION PROFILES To predict to what extent feeds are digested in the rumen and to determine the rate of digestion, system models of the rumen function have been developed (Pitt et al., 1999). In these models, the digestion of feed is described by a first-order rate constant. To obtain the rate constants, digestion experiments (in vitro) and measurements (in vivo) of intake, passage and amount of undigested carbohydrates in the rumen, have been conducted during the last decades. The high costs concerning previous techniques have led to the development of a new computerised in vitro gas production technique that can be used to determine digestion kinetics in batch fermentation. Gas production techniques generate a number of data points that can subsequently be related to the substrate digested. The data, consisting of observations of amount gas produced for each separate feed, can be described mathematically by the fit of non linear growth models (Pitt et al., 1999). The rate of digestion of feeds depends on passage rate from the rumen as well as the characteristics of the feed. For each model, a different gas production profile is obtained and thereby the results and conclusions will differ depending on which model that has been fitted. A gas production profile is a result of both the chemical composition of the feed and the chemical substances’ qualities. Even though the gas production technique gives a satisfactory description of the digestion of feeds there are many different models to consider. This results in difficulties when deciding which model that gives the best description of the data and consequently a “new” adaptable model is desirable (Groot et al., 1996). Groot et al., (1996) have presented a flexible multiphasic model for parameterisation of gas production profiles:

Y =∑=

+

n

iC

i

ii

tBA

11

. (1)

Y (ml/g OM) denotes the amount of gas produced per gram of organic matter, Ai (ml/g OM) denotes the asymptotic gas production (Ai > 0), Bi (h) denotes the time after incubation when half of the asymptotic amount of gas has been produced (Bi > 0) and Ci is a constant determining the sharpness of the changing characteristic of the profile (0 < Ci < ∞). The index i denotes the phases in the profile (i = 1,…, n). When a feed contains a large amount of degradable protein it leads to a “small” estimated value of parameter A. Similarly, if a feed contains a large amount of degradable carbohydrates it results in a “large” estimated value of parameter A. (Cone and Van Gelder, 1999). The chemical composition of the feeds can be viewed in Table A.5.1. For a given phase, the parameter C establishes the curvature and thus the position of the inflection point tI. No inflection point exists for C 1≤ but the profile becomes sigmoidal with an increasing slope when C increases. A step function with an initial slope of zero is obtained when C . ∞→

2

The inflection point is calculated as C

I CCB

/1

11

+−

=t and is equivalent to the t obtained to the

Y retrieved when solving 02

2

=dC

Yd .

When fitting a model that describes the gas production, the first step is to determine the number of phases and initial values for the parameters. In the second step, using the results obtained in the first step, a multiphasic model is fitted using a numerical method. When the number of phases is increased the fitted non linear model will become unstable and more sensitive to the choice of initial values. A better alternative for standard analysis is to fit the data to a monophasic model (i.e. equation (1) where n =1). This model is more solid since only three parameters have to be estimated (Cone et al., 2001).

3

3 MODEL FITTING

In this thesis the focus will be on the non linear monophasic model Y = ε+

+

C

tB

A

1 with

the assumption that ε . ),0( 2σN∈ For twenty different feeds, each feed sample has been analysed at three different occasions (dates) at in vitro laboratories in both Sweden and the Netherlands. The data set consists of 101 observations of the gas production measured during 72 hours of incubation at three different dates. In this thesis, this data set is referred to as the unstacked data. When the data from all three dates is merged together into one data set it consists of 303 observations and is referred to as the stacked data. The table below shows a sample description of the twenty feeds. Table 3.1: Description of the samples

Feed Sample Description Feed Sample Description 1 Palmkernel expeller A 11 Peas 2 Maizeglutenfeed A 12 Palmkernel expeller B 3 Coconut expeller 13 Soya beans, extracted A 4 Maize 14 Sunflower feed, extracted 5 Citruspulp 15 Coconut, extracted 6 Lupins, treated 16 Soya beans, extracted B 7 Lupins 17 Rapeseed, extracted 8 Maizeglutenfeed B 18 Soya bean hulls 9 Soya beans heat treated 19 Soya beans, extracted C 10 Sugarbeet pulp 20 Rapeseed, extracted, treated

In order to fit the monophasic model, non linear regression is performed to obtain least square estimates of the parameters A , B and . Initial values for the parameters were decided by calibrating the curve for each feed using the software Table curve (

Cwww.systat.com). For the

data from the analysis in Sweden and the Netherlands, the initial values for the parameters for each feed were set to; A = 200, B = 7 and C = 2. To achieve convergence for the unstacked data in the Netherlands, the initial values were changed into; A = 300, B = 5, C = 1 for feed 2, date 3, A = 300 for feed 10, date 3 and A = 400, B = 10 for feed 18, date 1. The lower graph in figure 3.1 shows the gas production profile for feed 1 for the stacked data in Sweden. The dotted lines in this graph are the observations from the three dates and the connected line is the fitted model Y. The upper graph in figure 3.1 shows the first derivative of Y with respect to t and can be interpreted as the gas production rate.

4

Feed 1Eqn 8001 totgas(a,b,c)

r^2=0.96325877 DF Adj r^2=0.96289013 FitStdErr=14.144666 Fstat=3932.6074a=191.81745 b=10.653212

c=3.0210774

0 20 40 60 80

Time

0

50

100

150

200

Y

0

50

100

150

200

Y

2.5

7.5

12.5

17.5

Y

2.5

7.5

12.5

17.5

Y

Figure 3.1: Gas production profile for feed 1(stacked data), analysed in Sweden

3.1 Analysis of residuals In addition to the estimated parameters A , B and , estimates of the variances are also obtained. The estimated variances are derived using asymptotic theory and are based on the assumption that all of the error terms are independent and – distributed. In many cases this assumption is not valid and hence the estimated variance of the parameter estimate is not trustworthy. To give an example of when this assumption is not valid, an analysis of the residuals for one of the feeds was performed using the software MINITAB (

C

),0( 2σN

www.minitab.com). Figure 3.1.1 shows a histogram of the residuals for feed 1 for the stacked Swedish data.

residuals

Freq

uenc

y

3020100-10-20-30

30

25

20

15

10

5

0

Mean 2,807StDev 13,81N 303

Normal Histogram of residuals

Figure 3.1.1: Histogram of the residuals

In the histogram, the residuals do not appear to be normally distributed.

5

Figure 3.1.2 shows a normal probability plot of the residuals for feed 1 for the stacked Swedish data.

residuals

Perc

ent

50403020100-10-20-30-40

99,9

99

9590

80706050403020

10

5

1

0,1

Mean

<0,005

2,807StDev 13,81N 303AD 1,331P-Value

Probability Plot of residualsNormal

Figure 3.1.2: Normal probability plot of the residuals

As in the histogram, the normal probability plot implies that the residuals are not normally distributed. Since the Anderson-Darling test gives a small p-value, the assumption that the residuals are normally distributed is not appropriate. Figure 3.1.3 shows a plot of the residuals versus the fitted values for feed 1 for the stacked Swedish data.

Fitted Value

Res

idua

l

7,55,02,50,0

30

20

10

0

-10

-20

-30

Residuals Versus the Fitted Values(response is residuals)

Figure 3.1.3: Plot of the residuals versus the fitted values

The plot of the residuals versus the fitted values indicates that the residuals can not be assumed to be independent and have the same variance. This residual analysis shows that the assumption that all of the error terms are independent and – distributed is not correct. The same goes for residuals of other feeds analysed in Sweden as well as the Netherlands for both stacked and unstacked data.

),0( 2σN

6

3.2 Non linear regression Nonlinear regression aims to describe the relationship between a response variable Y and one or more explanatory variables X in a non linear way. The parameters in a non linear model on the form Y = f(X1,X2,…,Xp| θ1,θ2,…,θq) + ε where the function f is known except for the q unknown parameters θ1,θ2,…,θq, can be estimated using the least square method where the residual sum of squares are minimized.

To estimate the unknown parameters A, B and C in the non linear model Y = ε+

+

C

tB

A

1

the numerical method Levenberg-Marquardt is used (Nash and Sofer, 1996). This method is available in the software SPSS (www.spss.com) and Table Curve. 3.2.1 Levenberg-Marquardt’s method The Levenberg-Marquardt algorithm can be used to solve non linear least square problems (Nash and Sofer, 1996). It combines steepest descent with Gauss-Newton and is a trust region method. The trust region methods refer explicitly to a model of the objective function (Nash and Sofer, 1996). The model is derived from the Taylor series evolved around the point xk:

pxfppxfxfp kTT

kkk )(21)()()( 2∇+∇+=ψ

The method will only “trust” this model within a limited area around xk defined by the condition|| . This will limit the length of the step from xkp ∆≤|| k to xk+1. The value of k∆ is adjusted depending on how well the model )( pkψ and the object function f (xk + p) matches. If they match well, the model can be trusted and the value of k∆ is increased, otherwise the value of is decreased. At iteration k of a trust region method the following sub problem is solved:

k∆

pxfppxfxfp kTT

kkkp)(

21)()()(min 2∇+∇+=ψ

s.t.

kp ∆≤||||

The direction of the step is obtained by solving the linear equation system:

)())(( 2kkk xfpIxf −∇=+∇ λ

where 0≥λ is a scalar and the matrix is positive definite and))(( 2 Ixf k λ+∇ 0||)||( =−∆ kk pλ

7

The trust region algorithm can be performed in the following way; Specify some initial guess of the solution x0. Select the initial trust region bound > 0. Specify the constants

0∆10 <<< ηµ .

For k = 0, 1, …

1. If xk is optimal, stop.

2. For the trial step pk, solve

pxfppxfxfp kTT

kkkp)(

21)()()(min 2∇+∇+=ψ

subject to || kp ∆≤||

3. Compute

reduction predictedreduction actual

)()()()(

=−

+−=

kk

kkkk pxf

pxfxfψ

ρ

4. If µρ ≤k

then xk+1 = xk (unsuccessful step) else xk+1 = xk + pk (successful step)

5. Update : k∆

µρ ≤k kk ∆=∆⇒ + 21

1

ηµρ <<k kk ∆=∆⇒ +1

ηρ ≥k . kk ∆=∆⇒ + 21

The value of ρk indicates how well the model predicts the reduction in the function value. If ρk

is small (that is, ρk µ≤( kk p

), then the actual reduction in the function value is much smaller than that predicted by )ψ , indicating that the model cannot be trusted for a bound as large as

; in this case the step pk∆ k will be rejected and k∆ will be reduced. If ρk is large (that is, ρk

η≥ ), then the model is adequately predicting the reduction in the function value, suggesting that the model can be trusted over an even wider region; in this case the bound ∆ will be increased.

k

8

4 ESTIMATION METHODS The observations of gas production for each feed are fitted to a monophasic model. To discover differences between the feeds, a comparison of these models can be made. This is accomplished by comparing the parameters A, B and C. Three methods have been applied in order to test whether there are any significant differences between the feeds. 4.1 Method (1) In method (1) the tests are performed using the unstacked data. When using non linear regression, three estimates of A, B and C are respectively obtained for each feed. The mean values of these estimates are used when comparing the feeds. The variance of the estimated

parameters is calculated as ∑=

−−

=i

i

n

kiik

ix xx

ns

1

22 )(1

1 where ni denotes the number of estimates

of parameter x for feed i. xik is the estimated parameter where index i denotes the feed and index k the date. To determine if there are differences between two feeds the three parameters A, B and C for each pair of feeds are initially tested simultaneously using the Hotelling’s T2 test. If this test indicates differences between two feeds the next step is to determine which parameter/parameters that differ. This is accomplished by performing a two sample t test. 4.1.1 The Hotelling’s T2 test

Hotelling’s T2 test is used when testing for differences between two populations, where each population is multivariate (Johnson, 1998) and (Chatfield & Collins, 1980). This test is a multivariate generalization of the two sample t test. One sample xi consists of ni observations and p parameters. When using the Hotelling’s T2 test the observations are assumed to be random samples from multivariate normal distributions Np(µi, Σi), where both µi and Σi are unknown (i = 1,2). We assume that the covariance matrices of the populations are the same, i.e. Σ1 = Σ2.

Consider two populations where each population has a distribution which contains three parameters. To simultaneously test whether the two populations differ, their mean values are compared and the following hypothesis is stated:

H0:

13

12

11

µµµ

=

23

22

21

µµµ

H1: µ1i ≠ µ2i for some i

where µ1i and µ2i are the population means (i = 1, 2, 3). The Hotelling’s T2 test statistic for comparing the two populations’ means is:

T2 = )2(1

21

21

−+−−+

nnppnn * τ where τ = )()( 21

121

21

21 xxSxx −−+

−T

nnnn

.

The pooled within-groups estimate of Σ is given by2

)1()1(

21

2211

−+−+−

=nn

nn SSS .

9

1

_x and denote the sample means and , the sample sizes. 2

_x 1n 2n

If the null hypothesis is true and when Σ is unknown then

+−− ΣµµXX

212121

11,~nn

N p and )2(1

21

21

−+−−+

nnppnn τ )1,(~ 21 −−+ pnnpFα .

The null hypothesis is rejected at the 100(1- α ) % significance level if T2 )1,( 21 −−+> pnnpF . The assumption that the covariance matrices of the two populations are equal is a generalisation of the assumption of equal variances in the univariate two sample t test. It is possible to test the assumption using a likelihood-ratio test suggested by M. S. Bartlett and modified by Box. However, the T2-statistic is not sensitive to departures from the assumption when the sample sizes are approximately equal and is the only statistic in common use (Chatfield & Collins, 1980). 4.1.2 The two sample t test To determine if two populations are equal a two sample t test can be performed (Montgomery, 2000). This is accomplished by comparing the sample means of the two populations. The following assumptions must be satisfied when using the t test procedure; both samples must be drawn from independent populations that can be described by a normal distribution, the variances of both populations must be equal and the observations must be independent random variables. When testing if the means of two populations are equal the following hypothesis is stated: H0: µi = µj H1: µi ≠ µj

where µi and µj are the population means. When the two populations are assumed to have equal variances, i.e. , the test statistic for comparing the means of two populations is:

222 σσσ == ji

t0 =

jip

ji

nns

xx11

__

+

− where and denote the sample means and , the sample sizes. ix_

jx_

in jn

2ps is a pooled estimate of the common variance computed as = 2

ps2

)1()1( 22

−+−+−

ji

jjii

nnsnsn

where and are the two individual sample variances. 2is 2

jsIf the null hypothesis is true and when σ 2 is unknown then t0 ~ t (ni + nj - 2). The null hypothesis is rejected at the 100(1- α ) % significance level if |t0| > tα/2(ni + nj - 2).

10

4.2 Method (2) As in method (1), the test in method (2) is performed using the unstacked data and is based on respectively the same three estimates of A, B and C. The mean values of these estimates are used when comparing the feeds. The variance of the each estimated parameter, ix is now

calculated as ∑=

=n

kxx iki

sn 1

2ˆ

2ˆ

1s where n denotes the number of estimates of each parameter x for

feed i. is the estimated variance of the variable and is obtained when estimating x2ˆ ikxs ikx ik

using non linear regression in the statistical software SPSS. The index i denotes the feed and index k the date. If we view all the estimated parameters as normally distributed, the mean value for each parameter can also be considered as normally distributed since they are a linear combination of normally distributed parameters. Since each estimated parameter is based on 101 observations it seems reasonable to assume that they are normally distributed according to the Central limit theorem. To compare two feeds, the mean value for each parameter is tested separately to check if they are normally distributed with the same mean. When testing if the means of two populations are equal the following hypothesis is stated: H0: µi = µj H1: µi ≠ µj

where µi and µj are the population means. The test statistic for comparing the means of two populations is:

Z0 =

∑=

+

−n

kxx

ji

jkikss

n

xx

1

2ˆ

2ˆ

1

ˆˆ where ix = ∑

=

n

kikx

n 1

ˆ1 and jx = ∑=

n

kjkx

n 1

ˆ1 are the sample means for

population i and j. The sample sizes for each population is n.

ix and jx are independent for every i, j where i ≠ j. If the null hypothesis is true and when σ 2 is unknown then, according to asymptotic theory Z0 ~ N(0, 1), approximately. This variance is obtained when estimating x using non linear regression in the statistical software SPSS. The estimated variance is derived using asymptotic theory. (www.spss.com) The null hypothesis is rejected if | Z0| > Zα/2, where α is the significance level of the test.

11

4.3 Method (3) In method (3) the test is performed using the stacked data and one estimate of each parameter A, B and C is obtained for each feed using non linear regression. The variance of the estimated parameters, is obtained from SPSS for parameter x for feed i, i = 1,…, 20. 2

ˆixs It seems reasonable to assume that the three estimated parameters are approximately normally distributed according to the Central limit theorem since they are based on 303 observations. To compare two feeds, each parameter estimate is tested separately to check if the parameters are equal. If they are equal the corresponding test variable Z0 should be approximately N(0, 1)- distributed. When testing if the means of two populations are equal the following hypothesis is stated: H0: µi = µj H1: µi ≠ µj where µi and µj are the population means . The test statistic for comparing the means of two populations is:

Z0 = 2ˆ

2ˆ

ˆˆ

ji xx

ji

ss

xx

+

− where and are the estimated parameters obtained when using non

linear regression in the statistical software SPSS.

ix jx

The estimated variances of and , and , are obtained from SPSS. The estimated variances are derived using asymptotic theory (

ix jx 2ˆixs 2

ˆ jxswww.spss.com).

If the null hypothesis is true and when σ 2 is unknown then Z0 ~ N(0, 1), approximately. The null hypothesis is rejected if | Z0| > Zα/2, where α is the significance level of the test.

12

5 METHODS FOR GROUPING THE FEEDS To graphically illustrate what feeds that differ, a grouping example based on the results obtained in Method (1), for data from Sweden and the Netherlands, is presented. Cluster analysis has also been performed where two different clustering methods, one hierarchical and one non-hierarchical, have been used. The cluster analysis was performed in SPSS where the distance measure used was the Euclidean distance. 5.1 Approach when grouping the feeds based on results from Method (1) Based on the table A.1.6 for data from Sweden and the table A.1.13 for data from the Netherlands, grouping of the feeds that did not differ in any of the parameters was done in the following way; initially the feeds that differed from all other feeds where placed into single groups. Then, feeds that could only be placed with one other feed were grouped together. Finally the remaining feeds were grouped based on the tables. The grouping result was graphically displayed in 3D plots of the parameters A, B and C. This was done in order to investigate whether or not the groupings were reasonable. If feeds within the same group were located too far apart from each other the groupings were reconstructed. This procedure was repeated until the 3D plots and the groupings agreed. For data from Sweden, the feeds could be grouped in no less than ten partitions based on the table A.1.6. If fewer partitions had been chosen there would have been differences in some feeds in some of the parameters. To be able to compare the results from the Netherlands with the results from Sweden, the Dutch results were also grouped into ten partitions. For comparisons between the non-hierarchical method, the hierarchical method and the grouping result for Sweden, the number of clusters for both methods was set to ten. This was also done for Dutch results. 5.2 Cluster analysis Cluster analysis is used for solving classification problems (Johnson, 1998). The purpose is to sort cases into groups, or clusters, so that the objects are similar within the same cluster. The cluster analysis allows many choices when choosing a method for combining groups. Each method may result in a different combination of groups. There are two different types of clustering techniques, non-hierarchical and hierarchical. If the variables are in different units then standardization of the data is recommended. 5. 2.1 Non-hierarchical Clustering Initially, an arbitrary number of clusters is momentarily selected. The distance between the cluster seed and the object is calculated. The minimum of this distance determines to which cluster each object should be assigned. One of the non-hierarchical clustering methods is the K-mean method. The algorithm is described as follows:

13

K-means algorithm

1. Specify K number of clusters and initial center points (seeds) for each cluster. 2. Calculate the distance between each object and each cluster center. Assign every

object to the closest cluster. 3. Compute a new cluster center for every cluster. 4. Reassign every object to the closest cluster center. 5. Repeat step 3 and 4 as long as the old centers differ from the new.

5.2.2 Hierarchical Clustering Hierarchical clustering starts with each object in a separate cluster and combines clusters until only one remains. By specifying a distance measure between the objects the single clusters are connected by using a link method. This method determines how clusters are connected to form a new one, and what the new cluster distances will be. There are several linkage methods to choose from, for example between-groups linkage, within-groups linkage, nearest neighbor, furthest neighbour, centroid clustering, median clustering, and Ward’s method. When performing hierarchical clustering in SPSS, nearest neighbor was chosen as linkage method. It is done in the following way; Given a set of N objects to be clustered. 1. Assign each object to its own cluster. 2. Join together the clusters that are most similar, i.e. the two closest objects. 3. Compute distances (similarities) between the new cluster and each of the old clusters. 4. Repeat step 2 and 3 until all objects are clustered into a single cluster of size N. We have used the Euclidian distance, )()( k

Tkk XX µµ −−=d .

14

6 RESULTS 6.1 Method (1) Based on the unstacked data, three estimates of A, B and C are respectively obtained for each feed using non linear regression. The mean values of these estimates are shown in table A.1.1 for the Swedish results and table A.1.8 for the Dutch. The estimated variances are also displayed in these tables. Based on these values, 95% confidence intervals have been constructed and are shown in table A.1.2 for the Swedish and table A.1.9 for the Dutch. For the Swedish results, the estimates of parameter A varies from 137.21 for feed 9 to 382.95 for feed 18. For B, the estimates vary from 5.08 for both feed 5 and feed 9 to 12.55 for feed 12. For C, the estimates vary from 1.05 for feed 20 to 3.15 for feed 4. The estimated variance of parameter A varies from 9.55 for feed 13 to 899.66 for feed 18. For B, the estimated variance varies from 0.00014 for feed 5 to 0.95 for feed 12. For C, the estimated variance varies from 0.00098 for feed 18 to 0.12 for feed 4. For the Dutch results, the estimates of parameter A varies from 161.19 for feed 9 to 400.74 for feed 18. For B, the estimates vary from 4.81 for feed 5 to 17.71 for feed 20. For C, the estimates vary from 0.97 for feed 20 to 2.73 for feed 4. For feed 20, there exists no point of inflection since C . The estimated variance of parameter A varies from 36.27 for feed 3 to 1127.25 for feed 5. For B, the estimated variance varies from 0.019 for feed 8 to 15.83 for feed 20. For C, the estimated variance varies from 0.00063 for feed 20 to 0.17 for feed 4.

1≤

For the Hotelling’s T2 test and the two sample t test, the number of estimates per parameter, n, as well as the number of parameters, p, are both 3. Table A.1.3 and table A.1.10 contain p-values from the Hotelling’s T2 test for Swedish results and the Dutch respectively. When performing this test the null hypothesis is rejected if the p-value is less than 0.05. This is equivalent to rejecting the null hypothesis when the test statistic is greater than a percentile from the F distribution with 3 degrees of freedom in the numerator and 2 degrees of freedom in the denominator. On the 5% significance level this F-percentile is 19.1643. Table A.1.4, A.1.5 and A.1.6 for the Swedish data and table A.1.11, A.1.12 and A.1.13 for the Dutch contain p-values from the two sample t test of the parameters A, B and C respectively. When performing this test the null hypothesis is rejected if the p-value is less than 0.05. This is equivalent to rejecting the null hypothesis when the test statistic is greater than a percentile from the t distribution with 2 degrees of freedom. On the 5% significance level this t-percentile is 4.3027. The two-sample t test has identified differences between the feeds, primarily in parameter B for Swedish and parameter A for the Dutch results. The smallest number of differences is detected in the parameter C for both Swedish and the Dutch results. Which parameters that differ for each feed analysed in Sweden and in the Netherlands are presented in table A.1.7 and A.1.14 respectively. The parameters shown in the tables are significantly different both when using the Hotelling’s T2 test and the two sample t test. For Sweden, feed 5, feed11 and feed 20 differ from all other feeds whereas feed 18 differ from all feeds except for feed 12. Feed 9 differ from all feeds except feed 14. For the data from the Netherlands, feed 4 differ from all except feed 18 and vice versa. Feed 11 differ from all feeds except feed 10.

15

To identify differences between the same feeds analysed in two different in vitro laboratories, in Sweden and in the Netherlands, a Hotelling’s T2 test and a two sample t test were performed. The p-values from these tests are shown in Table A.1.15. The null hypothesis is rejected if the p-value is less than 0.05. From the Hotelling’s T2 test a difference in feed 17 between the two in vitro laboratories is discovered. However, the two sample t test reveals no significant differences between any of the parameters. In some cases there are significant differences between parameters when using the two sample t tests but no significant differences when using the Hotelling’s T2 test. Some books recommend that when the multivariate test, i.e. the Hotelling’s T2 test, reveals no significance differences analysis on individual parameters should not be performed (Johnson, 1998). However, Johnson, (1998) suggests that analysis on separate parameters can be made but on a much smaller significance level. 6.2 Method (2) In this method the parameter estimates are the same as in Method (1). The estimates are shown in table A.2.1 and A.2.6 for Sweden and the Netherlands along with the estimated variances received when performing non linear regression in SPSS. For the data from Sweden, the estimated variance of parameter A varies from 0.13 for feed 9 to 5.17 for feed 12. For B, the estimated variance varies from 0.00048 for feed 5 to 0.078 for feed 20. For C, the estimated variance varies from 0.00014 for feed 5 to 0.005 for feed 1. For the data from the Netherlands, the estimated variance of parameter A varies from 0.14 for feed 9 to 6.92 for feed 20. For B, the estimated variance varies from 0.00051 for feed 5 to 0.19 for feed 20. For C, the estimated variance varies from 0.00008 for feed 2 to 0.0022 for feed 1. Table A.2.2, A.2.3 and A.2.4 for the data from Sweden and table A.2.7, A.2.8 and A.2.9 for the data from the Netherlands contain p-values from the test in method (2) for the parameters A, B and C respectively. When performing this test the number of estimates per parameter, n, is 3. The null hypothesis is rejected if the p-value is less than 0.05. This is equivalent to rejecting the null hypothesis when the absolute value of the test statistic is larger than the 2.5% percentile of a normally distributed variable with mean 0 and variance 1, i.e. if the absolute value of the test statistic is larger than 1.96. This test has identified differences between the feeds, primarily in parameter A. The smallest amount of difference is detected in the parameter C. This conclusion can be made for both the Swedish and the Dutch results. Which parameters that differ for each feed analysed in Sweden and the Netherlands are respectively presented in table A.2.5 and table A.2.10. For Sweden, all the feeds differ significantly in almost every parameter, except for feed 16 and feed 19 where no differences could be detected. For the Netherlands, all the feeds also differ in almost every parameter except for feed 13 and feed 19 where no differences could be detected. To identify differences between the same feeds analysed in two different in vitro laboratories, in Sweden and in the Netherlands, the test in method (2) was performed once more. The p-values from this test are illustrated in Table A.2.11. The null hypothesis is rejected if the p-value is less than 0.05. The table shows that all feeds differ in parameter A. No differences could be found in parameter B for feed 6, feed 9 and feed 20. For parameter C, no differences could be found in feed 6 and feed 20.

16

6.3 Method (3) Based on the stacked data, one estimate of A, B and C is respectively obtained for each feed using non linear regression. The estimated parameters along with the estimated variances are displayed in table A.3.1 for the Swedish and table A.3.7 for the Dutch results. Based on these values, 95% confidence intervals have been constructed and are shown in table A.3.2 for Sweden and table A.3.8 for the Netherlands. For the Swedish results, the estimates of parameter A vary from 136.90 for feed 9 to 382.98 for feed 18. For B, the estimates vary from 5.06 for feed 9 to 12.54 for feed 12. For C, the estimates vary from 1.05 for feed 20 to 3.06 for feed 4. The estimated variance of parameter A varies from 0.62 for feed 16 to 42.31 for feed 20. For B, the estimated variance varies from 0.00091 for feed 5 to 0.66 for feed 20. For C, the estimated variance varies from 0.00025 for feed 16 to 0.013 for feed 1. For the Dutch results, the estimates of parameter A varies from 160.66 for feed 9 to 401.51 for feed 18. For B, the estimates vary from 4.66 for feed 5 to 17.70 for feed 20. For C, the estimates vary from 0.96 for feed 20 to 2.70 for feed 4. For feed 20, there exists no point of inflection since C . The estimated variance of parameter A varies from 1.04 for feed 9 to 24.16 for feed 20. For B, the estimated variance varies from 0.0031 for feed 10 to 0.62 for feed 20. For C, the estimated variance varies from 0.00034 for feed 17 to 0.004 for feed 1.

1≤

Table A.3.3, table A.3.4 and table A.3.5 for the results from Sweden and table A.3.9, table A.3.10 and table A.3.11 for the results from the Netherlands contain p-values from test in method (3) for the parameters A, B and C respectively. When performing this test the number of observations, n, is 101. The null hypothesis is rejected if the p-value is less than 0.05. This is equivalent to rejecting the null hypothesis when the absolute value of the test statistic is greater than the 2.5% percentile of a normally distributed variable with mean 0 and variance 1, i.e. if the absolute value of the test statistic is larger than 1.96. This test has identified differences between the feeds, primarily in parameter A. The smallest number of difference is detected in the parameter C. This conclusion can be made for both Sweden and the Netherlands. The parameters that differ for each feed analysed in Sweden and in the Netherlands are respectively presented in table A.3.6 and A.3.12. This is the result from the test in method (3). For the Swedish results, all the feeds differ in almost every parameter except for feed 13 and feed 19 along with feed 16 and feed 19 where no differences could be detected. For the Dutch results, all the feeds also differ in almost every parameter except for feed 13 and feed 19 where no differences could be detected. To identify differences between the same feeds analysed in two different in vitro laboratories, in Sweden and in the Netherlands, the same test was performed. The p-values from the test are illustrated in Table A.3.13. The null hypothesis is rejected if the p-value is less than 0.05. The table shows that all feeds differ in parameter A. No differences could be found in parameter B for feed 2, feed 6, feed 7, feed 12, feed 14, feed 15 and feed18. For parameter C, no differences could be found in feed 12 and feed 20.

17

6. 4 Ratios 6. 4.1 Ratios of the estimated parameters To give a more clarified view of the differences between the estimated parameters for the Swedish and the Dutch results, ratios between the parameter estimates have been calculated. For the same feed, each parameter from the Swedish results was divided by the same parameter for the Dutch. The result is displayed in table A.4.1.1 for the unstacked data and table A.4.1.2 for the stacked data. For the unstacked data, the estimated parameter A from the Dutch results is larger than for the Swedish results for all feeds. The estimated parameter B is larger for the Swedish data than for the Dutch data for almost half of the feeds. Only for feed 12, the estimated parameter C for the Dutch results is larger than for the Swedish For the stacked data, the estimated parameter A for the Dutch data is larger than for the Swedish data for all feeds. The estimated parameter B is larger for the Swedish results than for the Dutch for almost half of the feeds. For all feeds, the estimated parameter C is larger for the Swedish results than for the Dutch. 6 4.2 Ratios of the estimated variances To illustrate the differences between the estimated variances for the three different methods, ratios of the variances have been calculated. For the same feed, each parameter’s variance from method (1) was divided by the same parameters’ variance from method (2) and method (3). Similarly, each parameter’s variance from method (2) was divided by the same parameters’ variance from method (3). This is displayed as ratio 1, ratio 2 and ratio 3 respectively in table A.4.2.1 for the Swedish results and table A.4.2.2 for the Dutch. Table A.4.2.1 implies that for the Swedish results, every estimated variance for parameter A is larger in method (1) than method (2). The same goes for parameter B except for feed 5 and feed 20 and for parameter C where the estimated variance for feed 18 is larger in method (2). When comparing method (1) and method (3) the same result is obtained, that is, almost every estimated variance is larger in method (1) than method (3). Finally, for all parameters the estimated variance is larger in method (3) than method (2). For the Dutch results table A.4.2.2 indicates that the estimated variance for all parameters is larger in method (1) compared to method (2) and method (3). A larger estimated variance is obtained in method (3) than method (2). Ratios have also been calculated for the three methods for the data from Sweden and from the Netherlands with the purpose of comparing the estimated variances of the parameters between the two in vitro laboratories. Table A.4.2.3 illustrates the ratios for each parameter where ratio 1 is the ratio when method (1) for Swedish data was divided by method (1) for the Dutch and similarly for ratio 2 and ratio 3. For the Swedish results the estimated variances of parameter A are larger for about half of the feeds for all three methods. Parameter B has an estimated variance that is larger for most of the feeds in method (1) for the results from Sweden, whereas in the other two methods the estimated variances are generally also smaller for the Swedish results. The same goes for parameter C.

18

6.5 Sample description Table A.5.1 shows the chemical composition of the twenty feeds. In this table feed 13, feed 16 and feed 19 contain similar amount of each chemical. Feed 17 and feed 20 as well as feed 6 and feed 7 along with feed 2 and feed 8 contain also similar amount of chemicals. Feed 11 differ from the rest of the feeds. 6.6 Grouping of the feeds Table 6.6.1 and table 6.6.2 illustrate the clusters from the two methods, non-hierarchical and hierarchical, for Sweden. Table 6.6.1: Non-hierarchical Clustering, Table 6.6.2: Hierarchical Clustering, for the data from Sweden for the data from Sweden

Cluster Feeds analysed in Sweden 1 1 2 2, 6, 7, 8, 13, 15, 16, 17, 19 3 3 4 4 5 5, 10 6 9, 14 7 11 8 12 9 18 10 20

Cluster Feeds analysed in Sweden 1 1 2 2, 6, 7, 8, 13, 14, 15, 16, 17, 19 3 3 4 4 5 5, 10 6 9 7 11 8 12 9 18 10 20

These tables show that there are many clusters that contain only one feed. The two clustering methods produce the same grouping of the feeds except for feed 14. Based on the Hotelling’s T2 test and the two sample t test from method (1), an example of grouping of the twenty feeds has been made. This is illustrated in table 6.6.3.

Table 6.6.3: An example of partitioning of the feeds analysed in Sweden Compared to the two clustering methods, there are fewer groups containing only one feed. As in the two clustering methods feed 11, feed 18 and feed 20 form three single groups. For the non-hierarchical method as well as the partitioning example feed 9 and feed 14 are in the same group. Feed 13, feed 16 and feed 19 are to be found in the same group for both of the clustering methods and in the partitioning example.

Partition Feeds analysed in Sweden 1 1, 6, 12 2 2, 3, 8, 15 3 4, 10 4 5 5 7, 17 6 9, 14 7 11 8 13, 16, 19 9 18 10 20

19

Figure 6.6.1 shows a 3D plot of the twenty feeds analysed in Sweden.

18400

41110

58300

152 7620 A 12

131619 1317200

14

9

14 3,512 3,0

10 2,52,08 1,56 1,0B C

Figure 6.6.1: 3D plot of the parameters A, B and C for the 20 feeds analysed in Sweden

In the 3D plot of the feeds, feed 18 and feed 20 appear to differ from the rest. Feed 13, feed 16 and feed 19 seem to be the most similar. Figure 6.6.2 is a graphical display of the table 6.6.3.

9400

373

42300

22 5110 A 1

888 125200

6

6

14 3,512 3,0

10 2,52,08 1,56 1,0B C

Figure 6.6.2: 3D plot of the parameters A, B and C for10 partitions from the Swedish result

The feeds in the ten partitions look like they are agreeable since the feeds within the same group are not that far apart.

20

Figure 6.6.3, figure 6.6.4 and figure 6.6.5 illustrate where the partitions differ in the parameters A, B and C.

400 A4009 9

7 7 3 4 3

43 3

300 2 300

Figure 6.6.3: Scatter plot of the parameters Figure 6.6.4: Scatter plot of the parameters A and B for 10 clusters for the Swedish data A and C for 10 clusters for the Swedish data

Figure 6.6.5: Scatter plot of the parameters

B and C for 10 clusters in the Swedish data

Based on these figures, partition 6 containing feed 9 and feed 14 as well as partition 8 containing feed 13, feed 16 and feed 19 seem to be satisfactory groupings since the feeds within each partition are very close. The distance between feed 1, feed 6 and feed 12 in partition 1 in addition to the distance between feed 4 and feed 10 in partition 3 is larger in every parameter compared to the feeds in the other partitions.

B

141210 8 6 4

A 8 8

200

100

108

5

22

6

1

6

5 252

21 1888AA 10

15

1 1 2 2200

6

6

1002,5 3,0 1,0 1,5 2,0 3,5

C

14

10 1

12 9

1

10 3B 21

8 222

888 75

6 5 6 3

4 6

1,4

0 1,5 2,0 2,5 3,0 3,5

C

21

Table 6.6.4 and table 6.6.5 illustrate the clusters from the two methods, non-hierarchical and hierarchical, for the Netherlands. Table 6.6.4: Non-hierarchical Clustering, Table 6.6.5: Hierarchical Clustering, for the data from the Netherlands for the data from the Netherlands

Cluster Feeds analysed in the Netherlands

1 1 2 2, 6, 8, 15 3 3, 13 16, 19 4 4 5 5, 7, 10 6 9, 14, 17 7 11 8 12 9 18 10 20

Cluster Feeds analysed in the Netherlands

1 1 2 2, 5, 6, 7, 10, 13, 15, 16, 19 3 3 4 4 5 8 6 9, 14, 17 7 11 8 12 9 18 10 20

The non-hierarchical method has four clusters that each contains three or four feeds. The rest of the clusters contain only one feed. In the hierarchical method, there are only two clusters containing three or more feeds. The feeds in cluster 2, cluster 3 and cluster 5 in the non-hierarchical method have in the hierarchical method been grouped into one cluster except for feed 3. Both clustering methods group feed 9, feed 14 and feed 17 together. Using the Hotelling’s T2 test and the two sample t test from method (1), an example of grouping of the twenty feeds has been made. This is illustrated in table 6.6.6.

Table 6.6.6: An example of partitioning of the feeds analysed in the Netherlands Partition Feeds analysed in the Netherlands

1 1, 12 2 2, 3, 13, 16, 19 3 4, 18 4 5, 10 5 6, 15 6 7 7 8 8 9, 14 9 11 10 17, 20

For the two clustering methods as well as for the partitioning example feed 9 and feed 14 are in the same group. As in the two clustering methods, feed 13, feed 16 and feed 19 are to be found in the same group. For both of the clustering methods and in the partitioning example, feed 6 and feed 15 as well as feed 5 and feed 10 are grouped together. Feed 11 is always placed in a single group.

22

Figure 6.6.6 shows a 3D plot of the twenty feeds analysed in the Netherlands.

500

184400

11

105820 A 300 7 1615

111212 6933

1720014

9

3,0 18 2016 14 2,5

Figure 6.6.6: 3D plot of the parameters A, B and C for the 20 feeds analysed in

the Netherlands In the 3D plot of the feeds, feed 20 appear to differ from the rest. Feed 2, feed 6, feed 13, feed 15, feed 16 and feed 19 seem to be the most similar. Figure 6.6.7 is a graphical display of the table 6.6.6.

Figure 6.6.7: 3D plot of the parameters A, B and C for10 partitions from the Dutch result

The feeds in the first nine partitions look like they are agreeable since the feeds within the same group are not too far apart. Feed 17 and feed 20 in partition ten seem to differ.

2,0

CB 12 1,510 8 1,06

500

33400

947 4

A 10 300 6 1522225

12

102008

8

3,0 18 2016 2,514 2,0

CB 12 1,510 8 1,06

23

Figure 6.6.8, figure 6.6.9 and figure 6.6.10 illustrate where the partitions differ in the parameters A, B and C.

Figure 6.6.8: Scatter plot of the parameters Figure 6.6.9: Scatter plot of the parameters A and B for 10 clusters for the Dutch data A and C for 10 clusters for the Dutch data

Figure 6.6.10: Scatter plot of the parameters

B and C for 10 clusters for the Dutch data

Based on these figures, partition 4 containing feed 5 and feed 10, partition 5 containing feed 6 and feed 15, as well as partition 8 containing feed 9 and feed 14 seem to be satisfactory groupings since the feeds within each partition are very close. The distance between feed 1 and feed 12 in partition 1 in addition to the distance between feed 4 and feed 18 in partition 3 is larger in every parameter compared to the feeds in the other partitions. The scatter plots imply that feed 17 and feed 20 in partition 10 should be in separate groups.

500 500

3 3 400 400

9 9 34 4

B

181614 12 10 8 6 4

A

300

200

100

1025 2 2 2

10

8

1

8

7 3

44A 7

6 56

5 300 2 5222101 1 2 2 1

10

200 88

1002,0 2,5 ,5 1,0 1,5 3,0

C

18 10

16

14 1

12 3

B 110 7 3

58 95222 22

10

C

3,02,52,01,51,0,5

6

4

8864

4

24

7 DISCUSSION AND CONCLUSIONS The purpose of this thesis was to identify and describe a statistical procedure for finding significant differences between feed samples analysed within the same in vitro laboratory as well as detecting differences between the same feeds analysed in two different in vitro laboratories (in Sweden and the Netherlands). To discover possible differences between the feeds, three methods have been applied. 7.1 Method (1) When performing the Hotelling’s T2 test, the covariance matrices of the different feeds are assumed to be equal. This assumption may not always be correct since the estimated variances differ substantially for some of the feeds. The estimated covariance matrices are unstable because each matrix is based on only three estimates of parameter A, parameter B and parameter C, respectively obtained when using non linear regression for the three dates. Since there are only three estimates for each parameter, the variance of the estimated variances is large. If the feed samples had been analysed at more than three occasions, i.e. dates, the number of estimates for each parameter would have increased. This would lead to a more stable estimated covariance matrix. The two sample t test has identified differences between the parameters for the results from the analyses from Sweden and the Netherlands. For the Swedish data, most of the differences were found in parameter B whereas for the Dutch data most differences were found in parameter A. This could possibly be a result of the differences in the estimated variances for Sweden and the Netherlands. The significant differences between results from Sweden and the Netherlands are displayed in table A.1.7 and table A.1.14. These tables also show in what parameters the differences lie. Based on the Hotelling’s T2 test and the two sample t test a partitioning of the feeds that were similar was performed. For Sweden the result was; Citruspulp (feed 5), Peas (feed 11), Soya beans hulls (feed 18) and Rapeseed, extracted, treated (feed 20) were all placed into single groups. Palmkernel expeller A (feed 1), Lupins, treated (feed 6) and Palmkernel expeller B (feed 12) were all grouped together. The same goes for Maizeglutenfeed A (feed 2), Coconut expeller (feed 3), and Maizeglutenfeed B (feed 8) and Coconut, extracted feed 15. Maize (feed 4) and Sugarbeet pulp (feed 10) were placed into the same group. Lupins (feed 7) and Rapeseed, extracted (feed 17) were grouped together. Likewise for Soya beans, heat treated (feed 9) and Sunflower feed, extracted (feed 14). Finally, Soya beans, extracted A (feed 13), Soya beans, extracted B (feed 16) and Soya beans, extracted C (feed 19) were grouped together. For the Netherlands the following partitions were made; Lupins (feed 7), Maizeglutenfeed B (feed 8) and Peas (feed 11) were all placed into single groups. Palmkernel expeller A (feed 1) and Palmkernel expeller B (feed 12) were all grouped together. The same goes for Maizeglutenfeed A (feed 2), Coconut expeller (feed 3), Soya beans, extracted A (feed 13), Soya beans, extracted B (feed 16) and Soya beans, extracted C (feed 19). Maize (feed 4) was grouped with Soya beans hulls (feed 18), Citruspulp (feed 5) was grouped with Sugarbeet pulp (feed 10), Lupins, treated (feed 6) was grouped with Coconut, extracted (feed 15), Soya beans, heat treated (feed 9) was grouped with Sunflower feed, extracted (feed 14) and finally Rape seed, extracted (feed 17) and Rape seed extracted treated (feed 20) were grouped together.

25

Based on the chemical composition of the feeds shown in table A.5.1 the partitions made for the results from Sweden and the Netherlands appear reasonable. The feeds that were placed into the same group have similar chemical qualities. The differences between the same feeds analysed in Sweden and the Netherlands are presented in table A.1.15. From the Hotelling’s T2 test a difference in Rapeseed, extracted (feed 17) between the two in vitro laboratories is discovered. However, the two sample t test shows no significant differences between any of the parameters. This result could be a consequence of the unstable covariance matrices. 7.2 Method (2) In this method, the variances are extremely underestimated and therefore not reliable. This is mainly due to the unrealistic assumption that the residuals,ε are N (0, σ2) discussed in the residual analysis in chapter 3.1. The significant differences between the results from Sweden and the Netherlands are displayed in table A.2.5 and table A.2.10. These tables also show in what parameters the differences lie. No differences were detected between Soya beans, extracted B (feed 16) and Soya beans, extracted C (feed 19) for the Swedish data. For the Dutch data, no differences were detected between Soya beans, extracted A (feed 13) and Soya beans, extracted C (feed 19). When comparing the results from Sweden and the Netherlands differences between all feeds in at least one parameter were detected. The main reason to why these tests result in a great number of differences between the feeds is that the variance is extremely underestimated. Consequently, the test shows differences that may not exist. 7.3 Method (3) The variances in this method are extremely underestimated and therefore not reliable. As in method (2), this is mainly due to the unrealistic assumption that the residuals,ε are N (0, σ2) discussed in the residual analysis in chapter 3.1. The significant differences between the results from Sweden and the Netherlands are displayed in table A.3.6 and table A.3.12. These tables also show in what parameters the differences lie. No differences were detected between Soya beans, extracted A (feed 13) and Soya beans, extracted C (feed 19) as well as between Soya beans, extracted B (feed 16) and Soya beans, extracted C (feed 19) for Sweden. For the Netherlands, no differences were detected between Soya beans, extracted A (feed 13) and Soya beans, extracted C (feed 19). As in method (2), when testing the results from Sweden and the Netherlands differences between all feeds in at least one parameter were detected. The tests show a great number of differences between the feeds. This is caused by the extremely underestimated variance. Consequently, this test also shows differences that may not exist.

26

7.4 Final conclusions The estimates of the parameters A, B and C do not differ considerably for the three methods which suggest that all methods are suitable. When detecting significant differences between the feeds, method (1) is to be preferred over the two other methods. Even though method (1) has large estimated variances and unstable covariance matrices, these estimates are approximately unbiased which leads to a more reliable result. Although the estimated variances in method (3) are extremely underestimated they are still larger than the estimated variances in method (2) for all feeds. The estimated variances in method (3) are therefore closer to the “true” estimated variances obtained in method (1) and make method (3) more accurate than method (2). The given data set consists of observations of the gas production measured during 72 hours of incubation at three different dates. Based on this data, fitting the model using non linear regression will cause very small variances once the asymptote has been reached. This could lead to a poorly fitted model. For most of the feeds the asymptote has been reached after approximately 48 hours since the feed samples at this time have been completely degraded. To avoid the small variances that are obtained after the asymptote has been reached, the in vitro experiments should suggestively be terminated after 48 hours instead of 72 hours. The results from the Hotelling’s T2 test and the two sample t test would then have been different. In addition, a shorter incubation would make the experiments more efficient and less expensive. Consequently, more experiments could be performed. If the feed samples were analysed at considerably more than three occasions, i.e. on additional dates, the estimates in method (1) would be better and the result would be even more reliable. For future research it would be of interest to consider these facts. Another consideration is the choice of model, which may not be the best for describing the data for all feeds. In this thesis, the model has not been evaluated. Furthermore, the time dependency between the observations should have been taken into account.

27

28

8 REFERENCES 8.1 Books, articles and internet links Chatfield C.& Collins A.J. 1980. Introduction to multivariate analysis. Cone, J.W. and van Gelder A.H. 1999. Influence of protein fermentation on gas production profiles. Animal Feed Science and Technology 79, 251-264. Cone J.W., van Gelder A.H., Bachmann H. and Hindle V.A. 2001. Comparison of organic matter degradation in several feedstuffs in the rumen as determined with the nylon bag and gas production techniques. Animal Feed Science and Technology 96, 55-67. Groot J.C.J., Cone J.W., Williams B.A., Debersaques F.M.A. and Lantinga E.A. 1996. Multiphasic analysis of gas production kinetics for in vitro fermentation of ruminant feeds. Animal Feed Science and Technology 64, 77-89. Johnson Dallas E. 1998. Applied Multivariate Methods for Data Analysts. Montgomery Douglas C. 2000. Design and Analysis of Experiments.

Nash Stephen G. & Sofer Ariela 1996. Linear and Nonlinear Programming. Pitt R. E., Cross T.L., Pell A.N., Schofield P. and Duane P.H. 1999. Use of in vitro gas production models in ruminal kinetics. Mathematical biosciences 159, 145-163. 8.2 Computer software MATLAB Version 6.5, Release 13, 2002 MINITAB Version 14 for Windows, www.minitab.com SPSS Version 11.0 for Windows, www.spss.com Table Curve 2D Version 5.0 for Windows, www.systat.com

Appendix A A.1 Method (1)

Table A.1.1: Estimates of the parameters and the variances for the Swedish data

Estimates of the parameters Estimates of the variances Feed A B C VA VB VC

1 191.91 10.70 3.06 297.24 0.71 0.081 2 269.91 7.56 1.62 408.94 0.33 0.0045 3 203.47 7.76 2.13 241.22 0.43 0.017 4 303.07 8.97 3.15 139.33 0.75 0.12 5 319.09 5.08 1.65 55.03 0.00014 0.0026 6 254.29 8.49 1.73 114.24 0.046 0.0080 7 273.05 6.45 1.74 111.72 0.0078 0.0076 8 287.71 8.58 1.71 501.52 0.13 0.0094 9 137.21 5.08 1.72 219.91 0.31 0.023 10 327.53 5.80 1.75 484.29 0.012 0.0066 11 333.22 6.66 2.51 830.14 0.44 0.0048 12 217.30 12.55 2.04 237.38 0.95 0.0074 13 247.44 6.84 1.55 9.55 0.0039 0.0011 14 172.74 5.93 1.60 742.89 0.036 0.014 15 275.62 7.60 1.77 619.91 0.17 0.031 16 252.90 6.79 1.58 12.36 0.0046 0.0015 17 215.68 5.99 1.52 62.76 0.15 0.0035 18 382.95 11.63 2.10 899.66 0.060 0.00098 19 250.62 6.68 1.57 116.91 0.016 0.0053 20 233.44 12.26 1.05 788.55 0.045 0.0027

Table A.1.2: Confidence interval of the parameters for the Swedish data 95% Confidence Interval Feed

A B C 1 149.08 234.74 8.60 12.79 2.35 3.77 2 219.67 320.15 6.13 8.98 1.46 1.79 3 164.89 242.05 6.12 9.39 1.81 2.46 4 273.75 332.39 6.83 11.12 2.28 4.01 5 300.66 337.52 5.05 5.11 1.52 1.78 6 227.74 280.84 7.95 9.02 1.51 1.95 7 246.79 299.31 6.23 6.67 1.53 1.96 8 232.08 343.34 7.67 9.49 1.47 1.95 9 100.37 174.05 3.70 6.46 1.34 2.10 10 272.86 382.20 5.52 6.07 1.55 1.95 11 261.65 404.79 5.02 8.30 2.34 2.68 12 179.03 255.57 10.13 14.96 1.83 2.26 13 239.77 255.11 6.69 7.00 1.47 1.64 14 105.03 240.45 5.46 6.40 1.31 1.89 15 213.77 337.47 6.57 8.62 1.33 2.21 16 244.17 261.63 6.62 6.96 1.49 1.68 17 196.00 235.36 5.02 6.96 1.37 1.66 18 308.44 457.46 11.02 12.24 2.02 2.18 19 223.76 277.48 6.36 7.00 1.39 1.75 20 163.68 303.20 11.73 12.79 0.92 1.18

Table A.1.3: p-values from the Hotelling’s T2 test for the Swedish data Feed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

1 - - - - - - - - - - - - - - - - - - -2 0.0539 - - - - - - - - - - - - - - - - - -3 0.0471 0.1091 - - - - - - - - - - - - - - - - -4 0.0432 0.0230 0.0248 - - - - - - - - - - - - - - - -5 0.0262 0.0320 0.0052 0.0068 - - - - - - - - - - - - - - -6 0.0668 0.3898 0.0471 0.0243 0.0022 - - - - - - - - - - - - - -7 0.0493 0.1077 0.0164 0.0882 0.0006 0.0077 - - - - - - - - - - - - -8 0.0832 0.1210 0.1753 0.0182 0.0131 0.3417 0.0498 - - - - - - - - - - - -9 0.0034 0.0084 0.0017 0.0068 0.0009 0.0015 0.0186 0.0043 - - - - - - - - - - -

10 0.0386 0.0862 0.0086 0.0920 0.0157 0.0015 0.0277 0.0325 0.0245 - - - - - - - - - -11 0.0355 0.0039 0.0221 0.0182 0.0034 0.0124 0.0293 0.0034 0.0133 0.0369 - - - - - - - - -12 0.0627 0.0907 0.1031 0.0197 0.0040 0.0868 0.0403 0.0033 0.0153 0.0253 0.0053 - - - - - - - -13 0.0063 0.0847 0.0020 0.0077 0.0023 0.0011 0.1216 0.0181 0.0028 0.0039 0.0022 0.0478 - - - - - - -14 0.0221 0.0357 0.0116 0.0009 0.0006 0.0111 0.0564 0.0117 0.2638 0.0282 0.0001 0.0237 0.0607 - - - - - -15 0.0682 0.2147 0.0977 0.0168 0.0040 0.0741 0.0648 0.3633 0.0029 0.0160 0.0080 0.0215 0.0189 0.0181 - - - - -16 0.0047 0.1167 0.0025 0.0063 0.0023 0.0005 0.1653 0.0193 0.0009 0.0008 0.0017 0.0470 0.2425 0.0430 0.0123 - - - -17 0.0080 0.0246 0.0006 0.0173 0.0015 0.0076 0.0595 0.0038 0.0015 0.0455 0.0102 0.0243 0.0242 0.2543 0.0026 0.0041 - - -18 0.0457 0.0014 0.0104 0.0021 0.0014 0.0063 0.0021 0.0195 0.0021 0.0032 0.0097 0.0585 0.0024 0.0037 0.0077 0.0026 0.0028 - -19 0.0269 0.1190 0.0094 0.0291 0.0081 0.0036 0.2491 0.0295 0.0063 0.0272 0.0083 0.0487 0.0468 0.0647 0.0310 0.0787 0.0482 0.0026 - 20 0.0281 0.0053 0.0287 0.0006 0.0002 0.0019 0.0011 0.0061 0.0012 0.0003 0.0011 0.0107 0.0003 0.0007 0.0029 0.0000 0.0035 0.0060 0.0014

Table A.1.4: p-values from the two sample t test when comparing feeds, parameter A, for the Swedish data

Feed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 191 - - - - - - - - - - - - - - - - - - -2 0.0071 - - - - - - - - - - - - - - - - - -3 0.4367 0.0107 - - - - - - - - - - - - - - - - -4 0.0008 0.0702 0.0009 - - - - - - - - - - - - - - - -5 0.0003 0.0167 0.0003 0.1174 - - - - - - - - - - - - - - -6 0.0060 0.3025 0.0095 0.0061 0.0010 - - - - - - - - - - - - - -7 0.0023 0.8232 0.0030 0.0304 0.0035 0.0967 - - - - - - - - - - - - -8 0.0042 0.3647 0.0059 0.3524 0.0826 0.0801 0.3633 - - - - - - - - - - - -9 0.0141 0.0008 0.0059 0.0001 0.0000 0.0004 0.0002 0.0006 - - - - - - - - - - -

10 0.0011 0.0289 0.0013 0.1651 0.5635 0.0066 0.0181 0.0930 0.0002 - - - - - - - - - -11 0.0019 0.0357 0.0024 0.1688 0.4570 0.0113 0.0274 0.0969 0.0005 0.7991 - - - - - - - - -12 0.1300 0.0231 0.3353 0.0016 0.0005 0.0268 0.0067 0.0109 0.0029 0.0021 0.0036 - - - - - - - -13 0.0054 0.1299 0.0086 0.0014 0.0001 0.3462 0.0158 0.0368 0.0002 0.0034 0.0069 0.0293 - - - - - - -14 0.3615 0.0077 0.1650 0.0016 0.0009 0.0085 0.0040 0.0049 0.1183 0.0016 0.0022 0.0693 0.0092 - - - - - -15 0.0087 0.7733 0.0131 0.1595 0.0442 0.2445 0.8774 0.5657 0.0012 0.0538 0.0588 0.0260 0.1236 0.0085 - - - - -16 0.0039 0.2245 0.0058 0.0021 0.0002 0.8406 0.0351 0.0564 0.0002 0.0044 0.0087 0.0175 0.1135 0.0072 0.1927 - - - -17 0.0958 0.0124 0.2920 0.0004 0.0001 0.0073 0.0017 0.0063 0.0013 0.0012 0.0024 0.8795 0.0029 0.0588 0.0165 0.0017 - - -18 0.0007 0.0056 0.0008 0.0127 0.0232 0.0022 0.0039 0.0116 0.0002 0.0613 0.1071 0.0010 0.0015 0.0008 0.0088 0.0017 0.0007 - -19 0.0075 0.2187 0.0125 0.0048 0.0008 0.6966 0.0620 0.0611 0.0004 0.0056 0.0097 0.0374 0.6507 0.0100 0.1858 0.7452 0.0107 0.0020 - 20 0.0944 0.1419 0.1811 0.0167 0.0069 0.2955 0.0841 0.0590 0.0063 0.0103 0.0127 0.4320 0.4389 0.0549 0.1234 0.2994 0.3513 0.0032 0.3787

Table A.1.5: p-values from the two sample t test when comparing feeds, parameter B, for the Swedish data

Feed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 191 - - - - - - - - - - - - - - - - - - -2 0.0059 - - - - - - - - - - - - - - - - - -3 0.0089 0.7100 - - - - - - - - - - - - - - - - -4 0.0686 0.0769 0.1242 - - - - - - - - - - - - - - - -5 0.0003 0.0017 0.0021 0.0015 - - - - - - - - - - - - - - -6 0.0117 0.0577 0.1412 0.3980 0.0000 - - - - - - - - - - - - - -7 0.0010 0.0298 0.0269 0.0073 0.0000 0.0001 - - - - - - - - - - - - -8 0.0162 0.0594 0.1309 0.5081 0.0001 0.7242 0.0006 - - - - - - - - - - - -9 0.0006 0.0058 0.0058 0.0028 0.9998 0.0006 0.0137 0.0008 - - - - - - - - - - -

10 0.0006 0.0065 0.0070 0.0032 0.0004 0.0000 0.0013 0.0002 0.0942 - - - - - - - - - - 11 0.0028 0.1507 0.1112 0.0211 0.0145 0.0104 0.6135 0.0117 0.0342 0.0901 - - - - - - - - - 12 0.0677 0.0016 0.0021 0.0089 0.0002 0.0021 0.0004 0.0027 0.0003 0.0003 0.0010 - - - - - - - - 13 0.0014 0.0985 0.0743 0.0130 0.0000 0.0002 0.0033 0.0013 0.0055 0.0001 0.6596 0.0005 - - - - - - - 14 0.0007 0.0096 0.0098 0.0040 0.0015 0.0001 0.0127 0.0004 0.0671 0.3594 0.1399 0.0003 0.0014 - - - - - - 15 0.0046 0.9344 0.7308 0.0667 0.0005 0.0289 0.0094 0.0360 0.0033 0.0019 0.1071 0.0013 0.0359 0.0032 - - - - - 16 0.0013 0.0838 0.0648 0.0121 0.0000 0.0002 0.0059 0.0011 0.0062 0.0002 0.7466 0.0005 0.4079 0.0018 0.0297 - - - - 17 0.0009 0.0175 0.0162 0.0055 0.0157 0.0006 0.1205 0.0011 0.0811 0.4520 0.2076 0.0004 0.0206 0.8110 0.0082 0.0252 - - - 18 0.1390 0.0003 0.0007 0.0068 0.0000 0.0001 0.0000 0.0003 0.0000 0.0000 0.0003 0.1896 0.0000 0.0000 0.0001 0.0000 0.0000 - - 19 0.0012 0.0617 0.0499 0.0105 0.0000 0.0002 0.0597 0.0011 0.0083 0.0008 0.9553 0.0005 0.1252 0.0047 0.0219 0.2598 0.0440 0.0000 - 20 0.0358 0.0002 0.0004 0.0031 0.0000 0.0000 0.0000 0.0001 0.0000 0.0000 0.0002 0.6444 0.0000 0.0000 0.0001 0.0000 0.0000 0.0286 0.0000

Table A.1.6: p-values from the two sample t test when comparing feeds, parameter C, for the Swedish data

Feed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 191 - - - - - - - - - - - - - - - - - - -2 0.0010 - - - - - - - - - - - - - - - - - -3 0.0068 0.0038 - - - - - - - - - - - - - - - - -4 0.7612 0.0017 0.0091 - - - - - - - - - - - - - - - -5 0.0011 0.6057 0.0039 0.0018 - - - - - - - - - - - - - - -6 0.0015 0.1716 0.0115 0.0024 0.2484 - - - - - - - - - - - - - -7 0.0015 0.1445 0.0120 0.0024 0.2061 0.9170 - - - - - - - - - - - - -8 0.0015 0.2589 0.0109 0.0023 0.3799 0.8262 0.7489 - - - - - - - - - - - -9 0.0020 0.3622 0.0239 0.0029 0.4825 0.9418 0.8831 0.9294 - - - - - - - - - - -

10 0.0016 0.1034 0.0124 0.0025 0.1429 0.7834 0.8652 0.6272 0.7906 - - - - - - - - - -11 0.0311 0.0001 0.0114 0.0360 0.0001 0.0003 0.0003 0.0003 0.0012 0.0002 - - - - - - - - -12 0.0041 0.0026 0.3803 0.0060 0.0024 0.0117 0.0123 0.0112 0.0334 0.0124 0.0019 - - - - - - - -13 0.0008 0.1823 0.0017 0.0014 0.0506 0.0324 0.0267 0.0545 0.1355 0.0175 0.0000 0.0008 - - - - - - -14 0.0012 0.7826 0.0062 0.0019 0.5339 0.2008 0.1772 0.2698 0.3342 0.1412 0.0003 0.0061 0.5443 - - - - - -15 0.0026 0.2390 0.0469 0.0037 0.3087 0.7219 0.7698 0.6251 0.7218 0.8464 0.0025 0.0751 0.1009 0.2287 - - - - -16 0.0009 0.4435 0.0022 0.0015 0.1527 0.0612 0.0503 0.1019 0.2070 0.0332 0.0000 0.0011 0.3478 0.8431 0.1448 - - - -17 0.0008 0.1074 0.0017 0.0013 0.0408 0.0257 0.0219 0.0401 0.0956 0.0155 0.0000 0.0009 0.3966 0.3319 0.0748 0.1689 - - -18 0.0044 0.0004 0.7084 0.0066 0.0002 0.0024 0.0024 0.0027 0.0135 0.0022 0.0007 0.3355 0.0000 0.0020 0.0337 0.0001 0.0001 - -19 0.0009 0.4163 0.0028 0.0016 0.1983 0.0751 0.0639 0.1137 0.1980 0.0463 0.0001 0.0019 0.7188 0.7368 0.1402 0.7890 0.3675 0.0003 - 20 0.0003 0.0003 0.0002 0.0005 0.0001 0.0003 0.0003 0.0005 0.0020 0.0002 0.0000 0.0001 0.0001 0.0018 0.0024 0.0001 0.0005 0.0000 0.0005

Table A.1.7: Parameters that differ for each feed analysed in Sweden

Feed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 − BC A ABC ABC ABC ABC ABC ABC BC ABC BC AC ABC BC

2 − C AB AB AC AB AB ABC BC

3 BC − AC ABC AC ABC ABC ABC AC AC BC AC BC AB ABC BC

4 A C AC − BC AC C ABC BC ABC ABC ABC C ABC ABC ABC ABC ABC

5 ABC AB ABC BC − AB AB B A B BC ABC AB AB AB AB ABC ABC AB ABC 6 AC AC AB − B AB AB ABC BC AB B ABC ABC B BC

7 ABC ABC AB B − B AB AB AC ABC ABC BC

8 C B B − AB B BC ABC AB AB B ABC ABC B BC

9 ABC AB ABC ABC A AB AB AB − A ABC ABC AB AB AB A ABC AB ABC

10 ABC ABC B AB AB B A − C ABC ABC A B ABC AC BC ABC ABC

11 ABC AC AC BC BC ABC AC BC ABC C − ABC AC AC C AC AC BC AC ABC 12 ABC ABC ABC ABC ABC ABC ABC − ABC BC AB ABC BC ABC C

13 ABC AC ABC AB BC AB AB ABC AC ABC − B AB ABC BC

14 BC AB BC ABC AB AB AB A AC BC − AB AB ABC BC

15 C AB AB B C AB B AB − B AB ABC B BC

16 ABC AC ABC AB B B AB ABC AC ABC AB B − AB ABC BC

17 BC AB BC ABC ABC ABC ABC A AC AC BC AB AB AB − ABC AB BC 18 AC ABC AB ABC ABC ABC ABC ABC ABC BC BC ABC ABC ABC ABC ABC − ABC ABC

19 ABC ABC ABC AB B B AB ABC AC ABC B AB ABC − BC

20 BC BC BC ABC ABC BC BC BC ABC ABC ABC C BC BC BC BC BC ABC BC −

Table A.1.8: Estimates of the parameters and the variances for the Dutch data Estimates of the parameters Estimates of the variances Feed A B C VA VB VC

1 246.79 10.27 2.66 120.53 0.83 0.093 2 285.94 7.07 1.39 455.54 2.20 0.085 3 235.97 7.13 1.86 36.27 0.75 0.010 4 362.55 9.31 2.73 299.66 0.10 0.17 5 346.18 4.81 1.54 1127.25 0.23 0.15 6 278.75 8.33 1.55 108.79 0.75 0.026 7 304.78 6.33 1.55 122.76 0.11 0.020 8 327.98 9.50 1.47 387.97 0.019 0.027 9 161.19 5.71 1.51 94.15 0.20 0.039 10 348.13 5.47 1.59 185.97 0.22 0.056 11 356.13 7.81 2.28 120.26 0.12 0.11 12 233.89 12.43 2.07 260.14 1.81 0.080 13 272.63 7.19 1.46 292.81 0.43 0.014 14 179.35 6.27 1.39 115.64 0.74 0.039 15 290.34 7.49 1.57 65.59 0.41 0.015 16 267.51 7.23 1.50 94.01 0.38 0.011 17 223.67 6.35 1.34 134.47 0.40 0.022 18 400.74 11.78 1.97 199.83 1.29 0.027 19 271.50 7.18 1.47 54.13 0.51 0.010 20 269.43 17.71 0.97 795.58 15.83 0.00063

Table A.1.9: Confidence interval of the parameters for the Dutch data 95% Confidence Interval Feed

A B C 1 219.53 274.07 8.01 12.53 1.90 3.42 2 233.01 339.05 3.38 10.76 0.67 2.12 3 221.01 250.93 4.98 9.28 1.61 2.11 4 319.55 405.55 8.54 10.08 1.70 3.75 5 262.87 429.67 3.62 6.00 0.57 2.50 6 252.84 304.66 6.18 10.49 1.15 1.95 7 277.26 332.30 5.52 7.15 1.21 1.90 8 279.06 376.92 9.17 9.84 1.06 1.88 9 137.10 185.30 4.60 6.81 1.02 2.00 10 314.28 382.04 4.31 6.63 1.00 2.18 11 328.89 383.37 6.95 8.68 1.46 3.10 12 193.82 273.96 9.09 15.77 1.37 2.78 13 230.13 315.15 5.56 8.82 1.17 1.75 14 152.66 206.08 4.14 8.40 0.90 1.88 15 270.22 310.46 5.90 9.08 1.27 1.88 16 243.42 291.60 5.69 8.76 1.23 1.76 17 204.89 262.51 4.79 7.91 0.96 1.71 18 365.62 435.86 8.96 14.59 1.56 2.38 19 253.22 289.78 5.41 8.95 1.22 1.72 20 199.44 339.58 7.84 27.61 0.90 1.03

Table A.1.10: p-values from the Hotelling’s T2 test for the Dutch data

Feed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 191 - - - - - - - - - - - - - - - - - - -2 0.1509 - - - - - - - - - - - - - - - - - -3 0.0243 0.2674 - - - - - - - - - - - - - - - - -4 0.0015 0.0175 0.0011 - - - - - - - - - - - - - - - -5 0.0160 0.0185 0.0112 0.0001 - - - - - - - - - - - - - - -6 0.1227 0.5524 0.0774 0.0042 0.0072 - - - - - - - - - - - - - -7 0.0373 0.0155 0.0086 0.0061 0.0164 0.0006 - - - - - - - - - - - - -8 0.1255 0.0224 0.0184 0.0172 0.0017 0.0338 0.0145 - - - - - - - - - - - -9 0.0048 0.0252 0.0031 0.0004 0.0070 0.0078 0.0034 0.0035 - - - - - - - - - - -

10 0.0050 0.0160 0.0063 0.0206 0.4533 0.0029 0.0051 0.0055 0.0033 - - - - - - - - - - 11 0.0094 0.0398 0.0061 0.0111 0.0020 0.0164 0.0251 0.0261 0.0024 0.0680 - - - - - - - - - 12 0.0196 0.0254 0.0723 0.0042 0.0023 0.0080 0.0002 0.0060 0.0325 0.0007 0.0107 - - - - - - - - 13 0.1133 0.6906 0.2079 0.0103 0.0025 0.6855 0.0097 0.1424 0.0363 0.0018 0.0221 0.0176 - - - - - - - 14 0.0020 0.0144 0.0013 0.0008 0.0031 0.0017 0.0010 0.0055 0.2479 0.0011 0.0035 0.0228 0.0281 - - - - - - 15 0.0133 0.3217 0.0195 0.0022 0.0151 0.0450 0.0489 0.1074 0.0048 0.0032 0.0138 0.0073 0.2562 0.0007 - - - - - 16 0.1021 0.4680 0.1994 0.0019 0.0042 0.4592 0.0067 0.0550 0.0150 0.0002 0.0091 0.0061 0.8529 0.0075 0.0298 - - - - 17 0.0114 0.0358 0.0430 0.0010 0.0030 0.0133 0.0032 0.0145 0.0518 0.0004 0.0053 0.0994 0.0361 0.0330 0.0035 0.0002 - - - 18 0.0103 0.0081 0.0022 0.1534 0.0166 0.0045 0.0045 0.0026 0.0012 0.0203 0.0467 0.0090 0.0124 0.0013 0.0052 0.0063 0.0038 - - 19 0.0592 0.5464 0.0907 0.0023 0.0084 0.1061 0.0030 0.0150 0.0062 0.0030 0.0110 0.0169 0.9998 0.0004 0.0146 0.9028 0.0161 0.0023 - 20 0.0143 0.0166 0.0122 0.0039 0.0061 0.0136 0.0078 0.0081 0.1092 0.0049 0.0065 0.0336 0.0254 0.1279 0.0114 0.0186 0.0529 0.0032 0.0155

Table A.1.11: p-values from the two sample t test when comparing feeds, parameter A, for the Dutch dataFeed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

1 - - - - - - - - - - - - - - - - - - -2 0.0472 - - - - - - - - - - - - - - - - - -3 0.2086 0.0173 - - - - - - - - - - - - - - - - -4 0.0006 0.0084 0.0003 - - - - - - - - - - - - - - - -5 0.0081 0.0581 0.0049 0.4935 - - - - - - - - - - - - - - -6 0.0217 0.6268 0.0035 0.0020 0.0290 - - - - - - - - - - - - - -7 0.0030 0.2452 0.0007 0.0082 0.1117 0.0414 - - - - - - - - - - - - -8 0.0034 0.0658 0.0015 0.0843 0.4623 0.0186 0.1498 - - - - - - - - - - - -9 0.0005 0.0008 0.0003 0.0001 0.0008 0.0001 0.0001 0.0002 - - - - - - - - - - -

10 0.0006 0.0130 0.0002 0.3201 0.9297 0.0022 0.0128 0.2182 0.0000 - - - - - - - - - -11 0.0003 0.0071 0.0001 0.6164 0.6499 0.0009 0.0047 0.0964 0.0000 0.4722 - - - - - - - - -12 0.3158 0.0278 0.8445 0.0007 0.0064 0.0155 0.0033 0.0031 0.0026 0.0007 0.0004 - - - - - - - -13 0.0925 0.4458 0.0249 0.0031 0.0275 0.6247 0.0523 0.0213 0.0006 0.0039 0.0021 0.0462 - - - - - - -14 0.0016 0.0015 0.0013 0.0001 0.0012 0.0003 0.0001 0.0003 0.0951 0.0001 0.0000 0.0082 0.0013 - - - - - -15 0.0052 0.7543 0.0007 0.0028 0.0484 0.2029 0.1426 0.0375 0.0001 0.0032 0.0011 0.0056 0.1803 0.0001 - - - - -16 0.0705 0.2436 0.0087 0.0012 0.0174 0.2431 0.0118 0.0088 0.0002 0.0011 0.0005 0.0364 0.6752 0.0005 0.0351 - - - -17 0.0660 0.0112 0.1777 0.0003 0.0039 0.0036 0.0009 0.0014 0.0020 0.0003 0.0001 0.4229 0.0148 0.0082 0.0012 0.0073 - - -18 0.0001 0.0015 0.0000 0.0416 0.0599 0.0003 0.0008 0.0065 0.0000 0.0097 0.0125 0.0002 0.0006 0.0000 0.0003 0.0002 0.0001 - -19 0.0317 0.3284 0.0029 0.0011 0.0195 0.3803 0.0123 0.0096 0.0001 0.0010 0.0004 0.0213 0.9211 0.0003 0.0405 0.6004 0.0038 0.0001 - 20 0.2657 0.4643 0.1155 0.0082 0.0386 0.6207 0.1140 0.0422 0.0033 0.0122 0.0077 0.1316 0.8751 0.0067 0.2856 0.9165 0.0604 0.0020 0.9086

Table A.1.12: p-values from the two sample t test when comparing feeds, parameter B, for the Dutch data

Feed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 191 - - - - - - - - - - - - - - - - - - -

2 0.0335 - - - - - - - - - - - - - - - - - -

3 0.0123 0.9537 - - - - - - - - - - - - - - - - -

4 0.1589 0.0629 0.0147 - - - - - - - - - - - - - - - -

5 0.0008 0.0663 0.0153 0.0002 - - - - - - - - - - - - - - -

6 0.0559 0.2726 0.1645 0.1400 0.0035 - - - - - - - - - - - - - -

7 0.0021 0.4517 0.2113 0.0003 0.0104 0.0203 - - - - - - - - - - - - -

8 0.2234 0.0476 0.0093 0.3752 0.0001 0.0819 0.0001 - - - - - - - - - - - -

9 0.0015 0.2042 0.0649 0.0003 0.0756 0.0096 0.1209 0.0001 - - - - - - - - - - -

10 0.0012 0.1501 0.0429 0.0003 0.1624 0.0073 0.0576 0.0001 0.5520 - - - - - - - - - -

11 0.0120 0.4451 0.2724 0.0051 0.0009 0.3922 0.0058 0.0014 0.0030 0.0022 - - - - - - - - -

12 0.0821 0.0098 0.0046 0.0173 0.0008 0.0114 0.0016 0.0199 0.0012 0.0011 0.0045 - - - - - - - -

13 0.0090 0.8994 0.9226 0.0072 0.0071 0.1444 0.1122 0.0039 0.0316 0.0206 0.2217 0.0037 - - - - - - -

14 0.0052 0.4659 0.2890 0.0045 0.0617 0.0431 0.9069 0.0030 0.3729 0.2287 0.0448 0.0026 0.2126 - - - - - -

15 0.0124 0.6741 0.5914 0.0114 0.0044 0.2484 0.0495 0.0060 0.0167 0.0115 0.4855 0.0045 0.6043 0.1194 - - - - -

16 0.0087 0.8729 0.8819 0.0064 0.0059 0.1468 0.0919 0.0034 0.0261 0.0171 0.2247 0.0037 0.9535 0.1924 0.6333 - - - -

17 0.0036 0.4857 0.2769 0.0019 0.0277 0.0329 0.9687 0.0011 0.2216 0.1219 0.0244 0.0021 0.1839 0.8989 0.0928 0.1611 - - -

18 0.1472 0.0121 0.0049 0.0222 0.0006 0.0140 0.0013 0.0263 0.0010 0.0009 0.0044 0.5524 0.0038 0.0026 0.0047 0.0037 0.0019 - -

19 0.0098 0.9111 0.9401 0.0090 0.0087 0.1504 0.1347 0.0051 0.0385 0.0252 0.2386 0.0039 0.9824 0.2299 0.6044 0.9375 0.2052 0.0040 -

20 0.0345 0.0124 0.0109 0.0220 0.0051 0.0164 0.0079 0.0236 0.0066 0.0062 0.0128 0.0955 0.0108 0.0083 0.0119 0.0108 0.0082 0.0682 0.0108

Table A.1.13: p-values from the two sample t test when comparing feeds, parameter C, for the Dutch data

Feed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 191 - - - - - - - - - - - - - - - - - - -2 0.0066 - - - - - - - - - - - - - - - - - -3 0.0126 0.0592 - - - - - - - - - - - - - - - - -4 0.8200 0.0102 0.0238 - - - - - - - - - - - - - - - -5 0.0170 0.6307 0.2374 0.0217 - - - - - - - - - - - - - - -6 0.0052 0.4549 0.0490 0.0100 0.9582 - - - - - - - - - - - - - -7 0.0047 0.4375 0.0383 0.0095 0.9506 0.9868 - - - - - - - - - - - - -8 0.0041 0.7183 0.0255 0.0080 0.7867 0.5639 0.5313 - - - - - - - - - - - -9 0.0054 0.6043 0.0525 0.0099 0.9078 0.7780 0.7565 0.8053 - - - - - - - - - - -

10 0.0088 0.4091 0.1487 0.0144 0.8453 0.8163 0.8192 0.4967 0.6572 - - - - - - - - - -11 0.2206 0.0250 0.1013 0.2155 0.0644 0.0263 0.0247 0.0189 0.0254 0.0426 - - - - - - - - -12 0.0724 0.0438 0.2803 0.0861 0.1242 0.0496 0.0462 0.0329 0.0467 0.0864 0.4574 - - - - - - - -13 0.0032 0.7308 0.0112 0.0069 0.7538 0.4682 0.4236 0.9484 0.7395 0.4333 0.0154 0.0255 - - - - - - -14 0.0038 0.9965 0.0222 0.0072 0.5955 0.3429 0.3163 0.6450 0.5214 0.3267 0.0162 0.0269 0.6446 - - - - - -15 0.0047 0.3747 0.0380 0.0098 0.8813 0.8490 0.8514 0.4198 0.6402 0.9168 0.0257 0.0490 0.3060 0.2486 - - - - -16 0.0034 0.5896 0.0134 0.0075 0.8703 0.6544 0.6124 0.8057 0.9461 0.5622 0.0175 0.0300 0.6993 0.4672 0.4588 - - - -17 0.0025 0.7751 0.0074 0.0053 0.4440 0.1625 0.1386 0.3616 0.2964 0.1867 0.0107 0.0162 0.3191 0.7020 0.0990 0.2002 - - -18 0.0269 0.0399 0.3638 0.0420 0.1475 0.0339 0.0286 0.0202 0.0353 0.0846 0.2214 0.6180 0.0118 0.0176 0.0291 0.0139 0.0077 - -19 0.0030 0.7019 0.0090 0.0068 0.7703 0.4776 0.4286 0.9881 0.7648 0.4425 0.0150 0.0248 0.9471 0.6020 0.3016 0.7271 0.2764 0.0105 - 20 0.0007 0.0628 0.0001 0.0018 0.0613 0.0033 0.0020 0.0064 0.0091 0.0101 0.0023 0.0025 0.0020 0.0196 0.0011 0.0011 0.0130 0.0005 0.0011

Table A.1.14: Parameters that differ for each feed analysed in the Netherlands

Feed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 − BC A ABC ABC ABC ABC AB ABC ABC BC AC BC

2 − AC B A A AC ABC A A ABC B

3 BC − ABC AB AC ABC A AB A AC AC C AB BC

4 A AC ABC − BC AC ABC C ABC BC B AB ABC ABC ABC ABC ABC ABC ABC

5 ABC AB BC − AB B B A B AB AB A AB AB AB B AB AB 6 AC AB − AB A AB AB AC ABC AB AB ABC BC

7 ABC AC ABC B AB − B A A ABC ABC A B A A ABC A BC

8 B ABC C B A B − AB B BC ABC AB AB ABC AB ABC

9 ABC A A ABC A AB A AB − A ABC ABC AB AB AB ABC AB

10 ABC A AB BC AB A B A − AB AB A AB AB A AB AB ABC

11 AB AC A B B AC ABC BC ABC − AB AC ABC AC AC ABC AB AC ABC 12 ABC AB AB ABC ABC ABC ABC AB AB − ABC ABC ABC ABC A ABC C

13 ABC AB AB AB AC ABC − A A ABC BC

14 ABC A AC ABC A AB A AB A ABC ABC A − A A A ABC A

15 ABC AC ABC AB B AB AB AC ABC A − A A ABC A BC

16 ABC AB A AB AB AC ABC A A − A ABC BC

17 BC A C ABC AB AB A AB A ABC A A A A − ABC A 18 AC ABC AB B ABC ABC ABC ABC AB AB A ABC ABC ABC ABC ABC − ABC AC 19 ABC AB A AB AB AB AC ABC A A A ABC − BC

20 BC B BC ABC AB BC BC ABC ABC ABC C BC BC BC AC BC −

Table A.1.15: p-values from the Hotelling’s T2 test and the two sample t test when comparing feeds analysed in Sweden to the same feeds analysed in the Netherlands

2 sample t test Feed Hotelling’s T2 test parameter A parameter B parameter C

1 0.0750 0.0097 0.5811 0.1668 2 0.7381 0.3977 0.6242 0.2534 3 0.2993 0.0278 0.3739 0.0443 4 0.0703 0.0079 0.5591 0.2495 5 0.0587 0.2430 0.3745 0.6444 6 0.4399 0.0470 0.7784 0.1669 7 0.3328 0.0230 0.5920 0.1253 8 0.2156 0.0794 0.0147 0.0921 9 0.4746 0.0790 0.2026 0.2113 10 0.6402 0.2399 0.2981 0.3346 11 0.2899 0.2675 0.0555 0.3012 12 0.4284 0.2670 0.9107 0.8730 13 0.2464 0.0661 0.4066 0.2548 14 0.5238 0.7161 0.5404 0.1951 15 0.5974 0.3852 0.8312 0.1867 16 0.1462 0.0702 0.2933 0.2573 17 0.0428 0.3795 0.4484 0.1233 18 0.1014 0.4054 0.8422 0.2489 19 0.4361 0.0505 0.2986 0.2170 20 0.3594 0.1927 0.0773 0.0626

A.2 Method (2)

Table A.2.1: Estimates of the parameters and the variances for the Swedish data

Estimates of the parameters Estimates of the variances Feed A B C VA VB VC

1 191.91 10.70 3.06 1.32 0.0096 0.0050 2 269.91 7.56 1.62 0.82 0.0026 0.00027 3 203.47 7.76 2.13 1.68 0.0087 0.0022 4 303.07 8.97 3.15 0.25 0.00059 0.00064 5 319.09 5.08 1.65 0.36 0.00048 0.00014 6 254.29 8.49 1.73 0.72 0.0030 0.00027 7 273.05 6.45 1.74 0.44 0.0011 0.00022 8 287.71 8.58 1.71 0.83 0.0029 0.00028 9 137.21 5.08 1.72 0.13 0.00093 0.00033 10 327.53 5.80 1.75 1.08 0.0016 0.00043 11 333.22 6.66 2.51 1.33 0.0019 0.0015 12 217.30 12.55 2.04 5.17 0.047 0.0032 13 247.44 6.84 1.55 0.58 0.0020 0.00022 14 172.74 5.93 1.60 0.40 0.0022 0.00042 15 275.62 7.60 1.77 3.41 0.010 0.0014 16 252.90 6.79 1.58 0.58 0.0018 0.00023 17 215.68 5.99 1.52 0.50 0.0018 0.00028 18 382.95 11.63 2.10 4.20 0.011 0.0010 19 250.62 6.68 1.57 0.74 0.0023 0.00029 20 233.44 12.26 1.05 4.92 0.078 0.00024

Table A.2.2: p-values from the test in method (2) when comparing feeds, parameter A, for the Swedish data

Feed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 191 - - - - - - - - - - - - - - - - - - -2 0.0000 - - - - - - - - - - - - - - - - - -3 0.0000 0.0000 - - - - - - - - - - - - - - - - -4 0.0000 0.0000 0.0000 - - - - - - - - - - - - - - - -5 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - - - - - -6 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - - - - -7 0.0000 0.0051 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - - - -8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - - -9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - -

10 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - -11 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 - - - - - - - - -12 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - -13 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - -14 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - -15 0.0000 0.0055 0.0000 0.0000 0.0000 0.0000 0.1908 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - -16 0.0000 0.0000 0.0000 0.0000 0.0000 0.2212 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - -17 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.4971 0.0000 0.0000 0.0000 0.0000 - - -18 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - -19 0.0000 0.0000 0.0000 0.0000 0.0000 0.0023 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0059 0.0000 0.0000 0.0469 0.0000 0.0000 - 20 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Table A.2.3: p-values from the test in method (2) when comparing feeds, parameter B, for the Swedish data

Feed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 191 - - - - - - - - - - - - - - - - - - -2 0.0000 - - - - - - - - - - - - - - - - - -3 0.0000 0.0584 - - - - - - - - - - - - - - - - -4 0.0000 0.0000 0.0000 - - - - - - - - - - - - - - - -5 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - - - - - -6 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - - - - -7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - - - -8 0.0000 0.0000 0.0000 0.0000 0.0000 0.2305 0.0000 - - - - - - - - - - - -9 0.0000 0.0000 0.0000 0.0000 0.9984 0.0000 0.0000 0.0000 - - - - - - - - - - -

10 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - -11 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0000 0.0000 0.0000 - - - - - - - - -12 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - -13 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0036 0.0000 - - - - - - -14 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0343 0.0000 0.0000 0.0000 - - - - - -15 0.0000 0.7509 0.2273 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - -16 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0306 0.0000 0.4261 0.0000 0.0000 - - - -17 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0009 0.0000 0.0000 0.0000 0.3160 0.0000 0.0000 - - -18 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 - -19 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0000 0.0000 0.0000 0.7230 0.0000 0.0157 0.0000 0.0000 0.0902 0.0000 0.0000 - 20 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.4174 0.0000 0.0000 0.0000 0.0000 0.0000 0.0352 0.0000

Table A.2.4: p-values from the test in method (2) when comparing feeds, parameter C, for the Swedish data

Feed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 191 - - - - - - - - - - - - - - - - - - -2 0.0000 - - - - - - - - - - - - - - - - - -3 0.0000 0.0000 - - - - - - - - - - - - - - - - -4 0.2615 0.0000 0.0000 - - - - - - - - - - - - - - - -5 0.0000 0.1803 0.0000 0.0000 - - - - - - - - - - - - - - -6 0.0000 0.0000 0.0000 0.0000 0.0001 - - - - - - - - - - - - - -7 0.0000 0.0000 0.0000 0.0000 0.0000 0.7169 - - - - - - - - - - - - -8 0.0000 0.0001 0.0000 0.0000 0.0025 0.4483 0.2465 - - - - - - - - - - - -9 0.0000 0.0001 0.0000 0.0000 0.0009 0.7457 0.4946 0.6898 - - - - - - - - - - -

10 0.0000 0.0000 0.0000 0.0000 0.0000 0.4399 0.6242 0.1510 0.3028 - - - - - - - - - -11 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - -12 0.0000 0.0000 0.2294 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - -13 0.0000 0.0016 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - -14 0.0000 0.3808 0.0000 0.0000 0.0343 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0653 - - - - - -15 0.0000 0.0002 0.0000 0.0000 0.0015 0.2825 0.3720 0.1312 0.2118 0.5855 0.0000 0.0001 0.0000 0.0000 - - - - -16 0.0000 0.0880 0.0000 0.0000 0.0007 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1364 0.5543 0.0000 - - - -17 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0970 0.0016 0.0000 0.0024 - - -18 0.0000 0.0000 0.5880 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.3763 0.0000 0.0000 0.0000 0.0000 0.0000 - -19 0.0000 0.0285 0.0000 0.0000 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.4278 0.2807 0.0000 0.5486 0.0213 0.0000 - 20 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Table A.2.5: Parameters that differ for each feed analysed in Sweden

Feed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 - ABC ABC AB ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC

2 ABC - AC ABC AB ABC ABC ABC ABC ABC ABC ABC ABC AB AC AB ABC ABC ABC ABC

3 ABC AC - ABC ABC ABC ABC ABC ABC ABC ABC AB ABC ABC AC ABC ABC AB ABC ABC

4 AB ABC ABC - ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC

5 ABC AB ABC ABC - ABC ABC ABC AC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC 6 ABC ABC ABC ABC ABC - AB A AB AB ABC ABC ABC ABC AB BC ABC ABC ABC ABC

7 ABC ABC ABC ABC ABC AB - AB AB AB ABC ABC ABC ABC B ABC ABC ABC ABC ABC

8 ABC ABC ABC ABC ABC A AB - AB AB ABC ABC ABC ABC AB ABC ABC ABC ABC ABC

9 ABC ABC ABC ABC AC AB AB AB - AB ABC ABC ABC ABC AB ABC ABC ABC ABC ABC

10 ABC ABC ABC ABC ABC AB AB AB AB - ABC ABC ABC ABC AB ABC ABC ABC ABC ABC

11 ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC - ABC ABC ABC ABC ABC ABC ABC AC ABC 12 ABC ABC AB ABC ABC ABC ABC ABC ABC ABC ABC - ABC ABC ABC ABC BC AB ABC AC

13 ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC - AB ABC A AB ABC AB ABC

14 ABC AB ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC AB - ABC AB AC ABC AB ABC

15 ABC AC AC ABC ABC AB B AB AB AB ABC ABC ABC ABC - ABC ABC ABC ABC ABC

16 ABC AB ABC ABC ABC BC ABC ABC ABC ABC ABC ABC A AB ABC - ABC ABC ABC

17 ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC BC AB AC ABC ABC - ABC ABC ABC 18 ABC ABC AB ABC ABC ABC ABC ABC ABC ABC ABC AB ABC ABC ABC ABC ABC - ABC ABC

19 ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC AC ABC AB AB ABC ABC ABC - ABC

20 ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC A ABC ABC ABC ABC ABC ABC ABC -

Table A.2.6: Estimates of the parameters and the variances for the Dutch data Estimates of the parameters Estimates of the variances Feed A B C VA VB VC

1 246.80 10.27 2.66 1.31 0.0061 0.0022 2 286.03 7.07 1.39 0.67 0.0016 0.00008 3 235.97 7.13 1.86 1.36 0.0051 0.0011 4 362.55 9.31 2.73 1.29 0.0024 0.0012 5 346.27 4.81 1.54 0.44 0.00051 0.00016 6 278.75 8.33 1.55 0.47 0.0018 0.00013 7 304.78 6.33 1.55 0.71 0.0015 0.00020 8 327.99 9.51 1.47 0.90 0.0031 0.00010 9 161.20 5.71 1.51 0.14 0.00089 0.00012 10 348.16 5.47 1.59 1.03 0.0013 0.00028 11 356.13 7.81 2.28 1.96 0.0033 0.0012 12 233.89 12.43 2.07 2.48 0.019 0.0016 13 272.64 7.19 1.46 0.79 0.0025 0.00019 14 179.37 6.27 1.39 0.30 0.0018 0.00020 15 290.34 7.49 1.58 1.49 0.0044 0.00040 16 267.51 7.23 1.50 0.75 0.0024 0.00021 17 223.70 6.35 1.34 0.37 0.0015 0.00012 18 400.74 11.77 1.97 3.97 0.0098 0.00070 19 271.50 7.18 1.47 0.83 0.0026 0.00021 20 269.51 17.73 0.96 6.92 0.19 0.00010

Table A.2.7: p-values from the test in method (2) when comparing feeds, parameter A, for the Dutch data Feed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

1 - - - - - - - - - - - - - - - - - - -2 0.0000 - - - - - - - - - - - - - - - - - -3 0.0000 0.0000 - - - - - - - - - - - - - - - - -4 0.0000 0.0000 0.0000 - - - - - - - - - - - - - - - -5 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - - - - - -6 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - - - - -7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - - - -8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - - -

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.00009 - - - - - - - - - - -10 0.0000 0.0000 0.0000 0.0000 0.1200 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - -11 0.0000 0.0000 0.0000 0.0004 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - -12 0.0000 0.0000 0.2888 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - -13 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - -14 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - -15 0.0000 0.0033 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - -16 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - -17 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - -18 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - -19 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.3716 0.0000 0.0000 0.0015 0.0000 0.0000 - 20 0.0000 0.0000 0.0000 0.0000 0.0000 0.0007 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.2600 0.0000 0.0000 0.4703 0.0000 0.0000 0.4745

Table A.2.8: p-values from the test in method (2) when comparing feeds, parameter B, for the Dutch data

Feed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 191 - - - - - - - - - - - - - - - - - - -2 0.0000 - - - - - - - - - - - - - - - - - -3 0.0000 0.4692 - - - - - - - - - - - - - - - - -4 0.0000 0.0000 0.0000 - - - - - - - - - - - - - - - -5 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - - - - - -6 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - - - - -7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - - - -8 0.0000 0.0000 0.0000 0.0084 0.0000 0.0000 0.0000 - - - - - - - - - - - -9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - -

10 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - -11 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - -12 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - -13 0.0000 0.0525 0.4542 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - -14 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.2543 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - -15 0.0000 0.0000 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0000 0.0003 0.0000 - - - - -16 0.0000 0.0140 0.2616 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.6460 0.0000 0.0013 - - - -17 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.7368 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1473 0.0000 0.0000 - - -18 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 - -19 0.0000 0.0880 0.5547 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.8524 0.0000 0.0002 0.5218 0.0000 0.0000 - 20 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Table A.2.9: p-values from the test in method (2) when comparing feeds, parameter C, for the Dutch data

Feed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 191 - - - - - - - - - - - - - - - - - - -2 0.0000 - - - - - - - - - - - - - - - - - -3 0.0000 0.0000 - - - - - - - - - - - - - - - - -4 0.2129 0.0000 0.0000 - - - - - - - - - - - - - - - -5 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - - - - - -6 0.0000 0.0000 0.0000 0.0000 0.4445 - - - - - - - - - - - - - -7 0.0000 0.0000 0.0000 0.0000 0.4336 0.9137 - - - - - - - - - - - - -8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - - -9 0.0000 0.0000 0.0000 0.0000 0.0589 0.0045 0.0096 0.0077 - - - - - - - - - - -

10 0.0000 0.0000 0.0000 0.0000 0.0111 0.0458 0.0813 0.0000 0.0000 - - - - - - - - - -11 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - -12 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 - - - - - - - -13 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.6409 0.0073 0.0000 0.0000 0.0000 - - - - - - -14 0.0000 0.9241 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0009 - - - - - -15 0.0000 0.0000 0.0000 0.0000 0.1194 0.2992 0.3739 0.0000 0.0026 0.5238 0.0000 0.0000 0.0000 0.0000 - - - - -16 0.0000 0.0000 0.0000 0.0000 0.0341 0.0034 0.0061 0.0855 0.6115 0.0000 0.0000 0.0000 0.0568 0.0000 0.0016 - - - -17 0.0000 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0012 0.0000 0.0000 - - -18 0.0000 0.0000 0.0069 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0328 0.0000 0.0000 0.0000 0.0000 0.0000 - -19 0.0000 0.0000 0.0000 0.0000 0.0002 0.0000 0.0000 0.9330 0.0257 0.0000 0.0000 0.0000 0.7488 0.0004 0.0000 0.1247 0.0000 0.0000 - 20 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Table A.2.10: Parameters that differ for each feed analysed in the Netherlands

Feed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 - ABC ABC AB ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC

2 ABC - AC ABC ABC ABC ABC ABC ABC ABC ABC ABC AC AB ABC ABC ABC ABC AC ABC

3 ABC AC - ABC ABC ABC ABC ABC ABC ABC ABC BC AC ABC ABC AC ABC ABC AC ABC

4 AB ABC ABC - ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC

5 ABC ABC ABC ABC - AB AB ABC AB BC ABC ABC ABC ABC AB ABC ABC ABC ABC ABC 6 ABC ABC ABC ABC AB - AB ABC ABC AB ABC ABC ABC ABC AB ABC ABC ABC ABC ABC

7 ABC ABC ABC ABC AB AB - ABC ABC AB ABC ABC ABC AC AB ABC AC ABC ABC ABC

8 ABC ABC ABC ABC ABC ABC ABC - ABC ABC ABC ABC AB ABC ABC AB ABC ABC AB ABC

9 ABC ABC ABC ABC AB ABC ABC ABC - ABC ABC ABC ABC ABC ABC AB ABC ABC ABC ABC

10 ABC ABC ABC ABC BC AB AB ABC ABC - ABC ABC ABC ABC AB ABC ABC ABC ABC ABC

11 ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC - ABC ABC ABC ABC ABC ABC ABC ABC ABC 12 ABC ABC BC ABC ABC ABC ABC ABC ABC ABC ABC - ABC ABC ABC ABC ABC ABC ABC ABC

13 ABC AC AC ABC ABC ABC ABC AB ABC ABC ABC ABC - ABC ABC A ABC ABC BC

14 ABC AB ABC ABC ABC ABC AC ABC ABC ABC ABC ABC ABC - ABC ABC AC ABC ABC ABC

15 ABC ABC ABC ABC AB AB AB ABC ABC AB ABC ABC ABC ABC - ABC ABC ABC ABC ABC

16 ABC ABC AC ABC ABC ABC ABC AB AB ABC ABC ABC A ABC ABC - ABC ABC A BC

17 ABC ABC ABC ABC ABC ABC AC ABC ABC ABC ABC ABC ABC AC ABC ABC - ABC ABC ABC 18 ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC - ABC ABC

19 ABC AC AC ABC ABC ABC ABC AB ABC ABC ABC ABC ABC ABC A ABC ABC - BC

20 ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC BC ABC ABC BC ABC ABC BC -

Table A.2.11: p-values from the test in method (2) when comparing feeds analysed in Sweden to the same feeds analysed in the Netherlands

Feed Parameter A Parameter B Parameter C

1 0.0000 0.0240 0.0000 2 0.0000 0.0000 0.0000 3 0.0000 0.0000 0.0000 4 0.0000 0.0000 0.0000 5 0.0000 0.0000 0.0000 6 0.0000 0.6565 0.6731 7 0.0000 0.0000 0.0000 8 0.0000 0.0000 0.0000 9 0.0000 0.4045 0.0000 10 0.0000 0.0000 0.0000 11 0.0000 0.0000 0.0000 12 0.0000 0.3225 0.0017 13 0.0000 0.0000 0.0000 14 0.0000 0.0000 0.0000 15 0.0000 0.0240 0.0000 16 0.0000 0.0000 0.0000 17 0.0000 0.0000 0.0000 18 0.0000 0.0000 0.0000 19 0.0000 0.0000 0.0000 20 0.0000 0.6565 0.6731

A.3 Method (3)

Table A.3.1: Estimates of the parameters and the variances for the Swedish data Estimates of the parameters Estimates of the variances Feed

A B C VA VB VC

1 191.82 10.65 3.02 3.11 0.024 0.013 2 269.96 7.53 1.62 7.28 0.023 0.0024 3 203.42 7.74 2.10 2.50 0.013 0.0038 4 303.81 9.01 3.06 1.70 0.0040 0.0036 5 319.07 5.08 1.65 0.68 0.00091 0.00027 6 254.19 8.48 1.73 1.77 0.0074 0.00073 7 272.94 6.44 1.74 1.63 0.0040 0.00084 8 287.82 8.57 1.70 5.79 0.019 0.0017 9 136.90 5.06 1.70 0.91 0.0066 0.0023 10 327.48 5.80 1.75 4.42 0.0066 0.0018 11 333.83 6.68 2.46 4.33 0.0066 0.0049 12 217.53 12.54 2.02 8.09 0.074 0.0049 13 247.42 6.84 1.55 0.69 0.0023 0.00026 14 172.55 5.93 1.59 6.07 0.034 0.0061 15 275.12 7.57 1.76 6.04 0.018 0.0026 16 252.85 6.79 1.58 0.62 0.0020 0.00025 17 215.49 5.98 1.51 0.70 0.0026 0.00038 18 382.98 11.62 2.10 13.49 0.035 0.0034 19 250.48 6.67 1.57 1.80 0.0057 0.00072 20 233.42 12.27 1.05 42.31 0.66 0.0020

Table A.3.2: Confidence interval of the parameters for the Swedish data 95% Confidence Interval Feed

A B C 1 188.35 195.29 10.35 10.95 2.80 3.24 2 264.65 275.26 7.23 7.83 1.52 1.71 3 200.30 206.53 7.52 7.97 1.99 2.23 4 301.24 306.37 8.89 9.14 2.95 3.18 5 317.46 320.69 5.02 5.14 1.62 1.68 6 251.57 256.81 8.31 8.65 1.68 1.78 7 270.43 275.46 6.32 6.57 1.68 1.80 8 283.08 292.55 8.30 8.85 1.62 1.78 9 135.02 138.78 4.90 5.22 1.61 1.80 10 323.34 331.61 5.64 5.96 1.66 1.83 11 329.74 337.92 6.52 6.84 2.32 2.59 12 211.93 223.13 12.01 13.07 1.89 2.16 13 245.79 249.06 6.74 6.93 1.52 1.58 14 167.70 177.40 5.56 6.29 1.44 1.74 15 270.28 279.95 7.31 7.84 1.66 1.86 16 251.30 254.40 6.70 6.88 1.55 1.62 17 213.83 217.14 5.88 6.08 1.48 1.55 18 375.75 390.21 11.26 11.99 1.99 2.22 19 247.84 253.13 6.52 6.82 1.52 1.62 20 220.62 246.22 10.68 13.86 0.96 1.13

Table A.3.3: p-values from the test in method (3) when comparing feeds, parameter A, for the Swedish data

Feed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 191 - - - - - - - - - - - - - - - - - - -2 0.0000 - - - - - - - - - - - - - - - - - -3 0.0000 0.0000 - - - - - - - - - - - - - - - - -4 0.0000 0.0000 0.0000 - - - - - - - - - - - - - - - -5 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - - - - - -6 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - - - - -7 0.0000 0.3171 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - - - -8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - - -9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - -

10 0.0000 0.0000 0.0000 0.0000 0.0002 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - -11 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0317 - - - - - - - - -12 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - -13 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - -14 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - -15 0.0000 0.1574 0.0000 0.0000 0.0000 0.0000 0.4327 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - -16 0.0000 0.0000 0.0000 0.0000 0.0000 0.3851 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - -17 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.4907 0.0000 0.0000 0.0000 0.0000 - - -18 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - -19 0.0000 0.0000 0.0000 0.0000 0.0000 0.0498 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0527 0.0000 0.0000 0.1286 0.0000 0.0000 - 20 0.0000 0.0000 0.0000 0.0000 0.0000 0.0018 0.0000 0.0000 0.0000 0.0000 0.0000 0.0252 0.0328 0.0000 0.0000 0.0030 0.0062 0.0000 0.0102

Table A.3.4: p-values from the test in method (3) when comparing feeds, parameter B, for the Swedish data

Feed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 191 - - - - - - - - - - - - - - - - - - -2 0.0000 - - - - - - - - - - - - - - - - - -3 0.0000 0.2588 - - - - - - - - - - - - - - - - -4 0.0000 0.0000 0.0000 - - - - - - - - - - - - - - - -5 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - - - - - -6 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - - - - -7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - - - -8 0.0000 0.0000 0.0000 0.0040 0.0000 0.5620 0.0000 - - - - - - - - - - - -9 0.0000 0.0000 0.0000 0.0000 0.8172 0.0000 0.0000 0.0000 - - - - - - - - - - -

10 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - -11 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0203 0.0000 0.0000 0.0000 - - - - - - - - -12 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - -13 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0921 0.0000 - - - - - - -14 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0085 0.0000 0.0000 0.5245 0.0002 0.0000 0.0000 - - - - - -15 0.0000 0.8264 0.3323 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - -16 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.2408 0.0000 0.4418 0.0000 0.0000 - - - -17 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0617 0.0000 0.0000 0.0000 0.7932 0.0000 0.0000 - - -18 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0054 0.0000 0.0000 0.0000 0.0000 0.0000 - -19 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0190 0.0000 0.0000 0.0000 0.9487 0.0000 0.0644 0.0002 0.0000 0.1880 0.0000 0.0000 - 20 0.0496 0.0000 0.0000 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.7529 0.0000 0.0000 0.0000 0.0000 0.0000 0.4356 0.0000

Table A.3.5: p-values from the test in method (3) when comparing feeds, parameter C, for the Swedish data

Feed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 191 - - - - - - - - - - - - - - - - - - -2 0.0000 - - - - - - - - - - - - - - - - - -3 0.0000 0.0000 - - - - - - - - - - - - - - - - -4 0.7331 0.0000 0.0000 - - - - - - - - - - - - - - - -5 0.0000 0.5384 0.0000 0.0000 - - - - - - - - - - - - - - -6 0.0000 0.0471 0.0000 0.0000 0.0123 - - - - - - - - - - - - - -7 0.0000 0.0327 0.0000 0.0000 0.0071 0.7914 - - - - - - - - - - - - -8 0.0000 0.1963 0.0000 0.0000 0.2520 0.5696 0.4452 - - - - - - - - - - - -9 0.0000 0.2099 0.0000 0.0000 0.2853 0.6412 0.5184 0.9664 - - - - - - - - - - -

10 0.0000 0.0434 0.0000 0.0000 0.0296 0.6916 0.8531 0.4179 0.4761 - - - - - - - - - -11 0.0000 0.0000 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - -12 0.0000 0.0000 0.3719 0.0000 0.0000 0.0001 0.0002 0.0001 0.0001 0.0007 0.0000 - - - - - - - -13 0.0000 0.2146 0.0000 0.0000 0.0000 0.0000 0.0000 0.0010 0.0029 0.0000 0.0000 0.0000 - - - - - - -14 0.0000 0.7623 0.0000 0.0000 0.4554 0.0934 0.0734 0.2109 0.2153 0.0744 0.0000 0.0000 0.6518 - - - - - -15 0.0000 0.0439 0.0000 0.0000 0.0391 0.5841 0.7182 0.3639 0.4133 0.8606 0.0000 0.0022 0.0001 0.0681 - - - - -16 0.0000 0.5291 0.0000 0.0000 0.0048 0.0000 0.0000 0.0093 0.0184 0.0003 0.0000 0.0000 0.1598 0.9557 0.0011 - - - - 17 0.0000 0.0514 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0002 0.0000 0.0000 0.0000 0.1263 0.3545 0.0000 0.0049 - - - 18 0.0000 0.0000 0.9425 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.3968 0.0000 0.0000 0.0000 0.0000 0.0000 - - 19 0.0000 0.4006 0.0000 0.0000 0.0124 0.0000 0.0000 0.0085 0.0151 0.0004 0.0000 0.0000 0.5849 0.8185 0.0010 0.6390 0.0922 0.0000 - 20 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Table A.3.6: Parameters that differ for each feed analysed in Sweden

Feed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 - ABC ABC AB ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC AC

2 ABC - AC ABC AB ABC BC AB AB ABC ABC ABC AB AB C AB AB ABC AB ABC

3 ABC AC - ABC ABC ABC ABC ABC ABC ABC ABC AB ABC ABC AC ABC ABC AB ABC ABC

4 AB ABC ABC - ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC

5 ABC AB ABC ABC - ABC ABC AB A ABC ABC ABC ABC AB ABC ABC ABC ABC ABC ABC 6 ABC ABC ABC ABC ABC - AB A AB AB ABC ABC ABC AB AB BC ABC ABC BC ABC

7 ABC BC ABC ABC ABC AB - AB AB AB ABC ABC ABC AB B ABC ABC ABC ABC ABC

8 ABC AB ABC ABC AB A AB - AB AB ABC ABC ABC AB AB ABC ABC ABC ABC ABC

9 ABC AB ABC ABC A AB AB AB - AB ABC ABC ABC AB AB ABC ABC ABC ABC ABC

10 ABC ABC ABC ABC ABC AB AB AB AB - ABC ABC ABC A AB ABC AC ABC ABC ABC

11 ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC - ABC AC ABC ABC AC ABC ABC AC ABC 12 ABC ABC AB ABC ABC ABC ABC ABC ABC ABC ABC - ABC ABC ABC ABC BC AB ABC AC

13 ABC AB ABC ABC ABC ABC ABC ABC ABC ABC AC ABC - AB ABC A AB ABC ABC

14 ABC AB ABC ABC AB AB AB AB AB A ABC ABC AB - AB AB A ABC AB ABC

15 ABC C AC ABC ABC AB B AB AB AB ABC ABC ABC AB - ABC ABC ABC ABC ABC

16 ABC AB ABC ABC ABC BC ABC ABC ABC ABC AC ABC A AB ABC - ABC ABC ABC

17 ABC AB ABC ABC ABC ABC ABC ABC ABC AC ABC BC AB A ABC ABC - ABC AB ABC 18 ABC ABC AB ABC ABC ABC ABC ABC ABC ABC ABC AB ABC ABC ABC ABC ABC - ABC AC

19 ABC AB ABC ABC ABC BC ABC ABC ABC ABC AC ABC AB ABC AB ABC - ABC

20 AC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC AC ABC ABC ABC ABC ABC AC ABC -

Table A.3.7: Estimates of the parameters and the variances for the Dutch data

Estimates of the parameters Estimates of the variances Feed A B C VA VB VC

1 246.88 10.25 2.61 2.56 0.012 0.0040 2 286.40 7.09 1.35 13.32 0.039 0.0021 3 236.21 7.10 1.83 3.04 0.011 0.0023 4 361.76 9.28 2.70 1.92 0.0036 0.0018 5 342.08 4.66 1.50 3.54 0.0038 0.00097 6 279.03 8.33 1.53 2.92 0.011 0.00065 7 304.56 6.32 1.55 1.62 0.0032 0.00044 8 327.37 9.47 1.46 4.07 0.014 0.00046 9 160.66 5.65 1.50 1.04 0.0065 0.0010 10 347.54 5.44 1.57 2.47 0.0031 0.00065 11 355.21 7.77 2.26 2.36 0.0039 0.0015 12 234.70 12.47 2.01 6.45 0.050 0.0033 13 272.40 7.18 1.45 2.82 0.0089 0.00065 14 178.94 6.21 1.38 1.95 0.012 0.0010 15 290.23 7.46 1.56 2.05 0.0057 0.00052 16 267.29 7.21 1.49 1.39 0.0045 0.00037 17 223.19 6.31 1.33 1.19 0.0048 0.00034 18 401.51 11.78 1.94 12.53 0.032 0.0021 19 271.27 7.15 1.46 1.91 0.0060 0.00045 20 269.60 17.70 0.96 24.16 0.62 0.00038

Table A.3.8: Confidence interval of the parameters for the Dutch data

95% Confidence Interval Feed A B C

1 243.73 250.03 10.03 10.46 2.48 2.73

2 279.22 293.58 6.70 7.47 1.26 1.44

3 232.78 239.64 6.89 7.30 1.74 1.93

4 359.03 364.49 9.16 9.40 2.61 2.78

5 338.38 345.78 4.54 4.78 1.44 1.56

6 275.67 282.40 8.13 8.53 1.48 1.58

7 302.05 307.07 6.21 6.43 1.51 1.59

8 323.40 331.34 9.24 9.70 1.42 1.50

9 158.65 162.67 5.49 5.81 1.44 1.56

10 344.45 350.63 5.34 5.55 1.52 1.62

11 352.19 358.24 7.65 7.89 2.18 2.34

12 229.70 239.70 12.03 12.91 1.90 2.12

13 269.10 275.71 7.00 7.37 1.40 1.50

14 176.19 181.69 6.00 6.42 1.3 1.44

15 287.42 293.05 7.32 7.61 1.52 1.61

16 264.97 269.62 7.08 7.34 1.46 1.53

17 221.04 225.34 6.18 6.45 1.29 1.36

18 394.55 408.48 11.43 12.12 1.86 2.03

19 268.54 273.99 7.00 7.30 1.42 1.50

20 259.93 279.27 16.15 19.25 0.92 0.99

Table A.3.9: p-values from the test in method (3) when comparing feeds, parameter A, for the Dutch data

Feed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 - - - - - - - - - - - - - - - - - - - 2 0.0000 - - - - - - - - - - - - - - - - - - 3 0.0000 0.0000 - - - - - - - - - - - - - - - - - 4 0.0000 0.0000 0.0000 - - - - - - - - - - - - - - - - 5 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - - - - - - 6 0.0000 0.0676 0.0000 0.0000 0.0000 - - - - - - - - - - - - - - 7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - - - - 8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - - - 9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - -

10 0.0000 0.0000 0.0000 0.0000 0.0260 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - 11 0.0000 0.0000 0.0000 0.0016 0.0000 0.0000 0.0000 0.0000 0.0000 0.0005 - - - - - - - - - 12 0.0000 0.0000 0.6239 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - 13 0.0000 0.0005 0.0000 0.0000 0.0000 0.0057 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - 14 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - 15 0.0000 0.3276 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - 16 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0128 0.0000 0.0000 - - - - 17 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - 18 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - 19 0.0000 0.0001 0.0000 0.0000 0.0000 0.0004 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.6016 0.0000 0.0000 0.0289 0.0000 0.0000 - 20 0.0000 0.0061 0.0000 0.0000 0.0000 0.0700 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.5897 0.0000 0.0001 0.6479 0.0000 0.0000 0.7443

Table A.3.10: p-values from the test in method (3) when comparing feeds, parameter B, for the Dutch data

Feed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 - - - - - - - - - - - - - - - - - - - 2 0.0000 - - - - - - - - - - - - - - - - - - 3 0.0000 0.9629 - - - - - - - - - - - - - - - - - 4 0.0000 0.0000 0.0000 - - - - - - - - - - - - - - - - 5 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - - - - - - 6 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - - - - - 7 0.0000 0.0002 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - - - - 8 0.0000 0.0000 0.0000 0.1455 0.0000 0.0000 0.0000 - - - - - - - - - - - - 9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - - -

10 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0360 - - - - - - - - - - 11 0.0000 0.0009 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - 12 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - 13 0.0000 0.6418 0.5179 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - 14 0.0000 0.0001 0.0000 0.0000 0.0000 0.0000 0.3562 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - 15 0.0000 0.0726 0.0044 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0018 0.0000 0.0216 0.0000 - - - - - 16 0.0000 0.5498 0.3599 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.8429 0.0000 0.0118 - - - - 17 0.0000 0.0002 0.0000 0.0000 0.0000 0.0000 0.9168 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.4211 0.0000 0.0000 - - - 18 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0155 0.0000 0.0000 0.0000 0.0000 0.0000 - - 19 0.0000 0.7579 0.6740 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.7655 0.0000 0.0037 0.5627 0.0000 0.0000 - 20 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Table A.3.11: p-values from the test in method (3) when comparing feeds, parameter C, for the Dutch data

Feed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 - - - - - - - - - - - - - - - - - - - 2 0.0000 - - - - - - - - - - - - - - - - - - 3 0.0000 0.0000 - - - - - - - - - - - - - - - - - 4 0.2384 0.0000 0.0000 - - - - - - - - - - - - - - - - 5 0.0000 0.0080 0.0000 0.0000 - - - - - - - - - - - - - - - 6 0.0000 0.0007 0.0000 0.0000 0.4505 - - - - - - - - - - - - - - 7 0.0000 0.0001 0.0000 0.0000 0.2280 0.6509 - - - - - - - - - - - - - 8 0.0000 0.0384 0.0000 0.0000 0.2634 0.0290 0.0035 - - - - - - - - - - - - 9 0.0000 0.0096 0.0000 0.0000 0.9673 0.4328 0.2202 0.2956 - - - - - - - - - - -

10 0.0000 0.0000 0.0000 0.0000 0.0730 0.2453 0.4147 0.0006 0.0715 - - - - - - - - - - 11 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 - - - - - - - - - 12 0.0000 0.0000 0.0182 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0003 - - - - - - - - 13 0.0000 0.0593 0.0000 0.0000 0.2299 0.0289 0.0046 0.8569 0.2576 0.0008 0.0000 0.0000 - - - - - - - 14 0.0000 0.6919 0.0000 0.0000 0.0047 0.0001 0.0000 0.0294 0.0061 0.0000 0.0000 0.0000 0.0570 - - - - - - 15 0.0000 0.0000 0.0000 0.0000 0.1093 0.3569 0.5933 0.0009 0.1066 0.7611 0.0000 0.0000 0.0013 0.0000 - - - - - 16 0.0000 0.0055 0.0000 0.0000 0.8135 0.2210 0.0579 0.2415 0.8562 0.0114 0.0000 0.0000 0.2130 0.0016 0.0180 - - - - 17 0.0000 0.5886 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.1797 0.0000 0.0000 - - - 18 0.0000 0.0000 0.0906 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.3799 0.0000 0.0000 0.0000 0.0000 0.0000 - - 19 0.0000 0.0358 0.0000 0.0000 0.2785 0.0318 0.0040 0.9633 0.3116 0.0007 0.0000 0.0000 0.8238 0.0265 0.0010 0.2600 0.0000 0.0000 - 20 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Table A.3.12: Parameters that differ for each feed analysed in the Netherlands

Feed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 - ABC ABC AB ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC

2 ABC - AC ABC ABC BC ABC ABC ABC ABC ABC ABC A AB C AC AB ABC AC ABC

3 ABC AC - ABC ABC ABC ABC ABC ABC ABC ABC BC AC ABC ABC AC ABC AB AC ABC

4 AB ABC ABC - ABC ABC ABC AC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC

5 ABC ABC ABC ABC - AB AB AB AB AB ABC ABC AB ABC AB AB ABC ABC AB ABC 6 ABC BC ABC ABC AB - AB ABC AB AB ABC ABC ABC ABC AB AB ABC ABC ABC BC

7 ABC ABC ABC ABC AB AB - ABC AB AB ABC ABC ABC AC AB AB AC ABC ABC ABC

8 ABC ABC ABC AC AB ABC ABC - AB ABC ABC ABC AB ABC ABC AB ABC ABC AB ABC

9 ABC ABC ABC ABC AB AB AB AB - AB ABC ABC AB ABC AB AB ABC ABC AB ABC

10 ABC ABC ABC ABC AB AB AB ABC AB - ABC ABC ABC ABC AB ABC ABC ABC ABC ABC

11 ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC - ABC ABC ABC ABC ABC ABC ABC ABC ABC 12 ABC ABC BC ABC ABC ABC ABC ABC ABC ABC ABC - ABC ABC ABC ABC ABC AB ABC ABC

13 ABC A AC ABC AB ABC ABC AB AB ABC ABC ABC - AB ABC A ABC ABC BC

14 ABC AB ABC ABC ABC ABC AC ABC ABC ABC ABC ABC AB - ABC ABC A ABC ABC ABC

15 ABC C ABC ABC AB AB AB ABC AB AB ABC ABC ABC ABC - ABC ABC ABC ABC ABC

16 ABC AC AC ABC AB AB AB AB AB ABC ABC ABC A ABC ABC - ABC ABC A BC

17 ABC AB ABC ABC ABC ABC AC ABC ABC ABC ABC ABC ABC A ABC ABC - ABC ABC ABC 18 ABC ABC AB ABC ABC ABC ABC ABC ABC ABC ABC AB ABC ABC ABC ABC ABC - ABC ABC

19 ABC AC AC ABC AB ABC ABC AB AB ABC ABC ABC ABC ABC A ABC ABC - BC

20 ABC ABC ABC ABC ABC BC ABC ABC ABC ABC ABC ABC BC ABC ABC BC ABC ABC BC -

Table A.3.13: p-values from the test in method (3) when comparing feeds analysed in Sweden to the same feeds analysed in the Netherlands

Feed Parameter A Parameter B Parameter C 1 0.0000 0.0319 0.0015 2 0.0003 0.0761 0.0001 3 0.0000 0.0000 0.0004 4 0.0000 0.0022 0.0000 5 0.0000 0.0000 0.0000 6 0.0000 0.2673 0.0000 7 0.0000 0.1597 0.0000 8 0.0000 0.0000 0.0000 9 0.0000 0.0000 0.0004 10 0.0000 0.0003 0.0004 11 0.0000 0.0000 0.0134 12 0.0000 0.8388 0.8635 13 0.0000 0.0010 0.0009 14 0.0242 0.1877 0.0116 15 0.0000 0.4839 0.0004 16 0.0000 0.0000 0.0002 17 0.0000 0.0001 0.0000 18 0.0003 0.5545 0.0337 19 0.0000 0.0000 0.0014 20 0.0000 0.0000 0.0577

A.4 Ratios A.4.1 Ratios of the estimated parameters

In the ratios, each parameter for the Swedish data was divided by the same parameter for the Dutch data.

Table A.4.1.1: Ratios of the estimated parameters for the

unstacked data in Sweden and the Netherlands

Table A.4.1.2: Ratios of the estimated parameters for the stacked data in Sweden and the Netherlands

Ratios Ratios Feed A B C

Feed A B C

1 0.78 1.04 1.15 11 0.94 0.85 1.10 2 0.94 1.07 1.17 12 0.93 1.01 0.99 3 0.86 1.09 1.15 13 0.91 0.95 1.06 4 0.84 0.96 1.15 14 0.96 0.95 1.15 5 0.92 1.06 1.07 15 0.95 1.01 1.13 6 0.91 1.02 1.12 16 0.95 0.94 1.06 7 0.90 1.02 1.12 17 0.96 0.94 1.14 8 0.88 0.90 1.17 18 0.96 0.99 1.07 9 0.85 0.89 1.14 19 0.92 0.93 1.07 10 0.94 1.06 1.10 20 0.87 0.69 1.09

Ratios Ratios Feed A B C

Feed A B C

1 0.78 1.04 1.16 11 0.94 0.86 1.09 2 0.94 1.06 1.19 12 0.93 1.01 1.01 3 0.86 1.09 1.15 13 0.91 0.95 1.07 4 0.84 0.97 1.14 14 0.96 0.95 1.16 5 0.93 1.09 1.10 15 0.95 1.01 1.13 6 0.91 1.02 1.13 16 0.95 0.94 1.06 7 0.90 1.02 1.12 17 0.97 0.95 1.14 8 0.88 0.91 1.17 18 0.95 0.99 1.08 9 0.85 0.90 1.14 19 0.92 0.93 1.08 10 0.94 1.06 1.11 20 0.87 0.69 1.10

A.4.2 Ratios of the estimated variances In ratio 1, each parameter’s variance from method (1) was divided by the same parameters’ variance from method (2). In ratio 2, each parameter’s variance from method (1) was divided by the same parameters’ variance from method (3). In ratio 3, each parameter’s variance from method (2) was divided by the same parameters’ variance from method (3).

Table A.4.2.1: Ratios of the estimated variances of the parameters for each method, for the Swedish data

Ratio 1 Ratio 2 Ratio 3 Feed A B C A B C A B C

1 224.67 73.97 16.16 95.63 30.24 6.32 0.43 0.41 0.39 2 497.28 126.24 16.73 56.19 14.06 1.87 0.11 0.11 0.11 3 143.92 49.76 7.55 96.32 33.24 4.52 0.67 0.67 0.60 4 552.02 1263.59 189.24 82.13 184.91 33.56 0.15 0.15 0.18 5 153.46 0.30 18.20 81.36 0.16 9.43 0.53 0.53 0.52 6 159.04 15.16 29.33 64.48 6.24 10.92 0.41 0.41 0.37 7 255.49 7.22 35.38 68.37 1.96 9.14 0.27 0.27 0.26 8 603.31 45.53 33.36 86.68 6.93 5.44 0.14 0.15 0.16 9 1711.53 333.25 71.14 241.28 46.69 10.38 0.14 0.14 0.15 10 446.95 7.59 15.26 109.66 1.86 3.63 0.25 0.25 0.24 11 624.68 226.62 3.32 191.86 66.57 0.98 0.31 0.29 0.30 12 45.94 20.19 2.30 29.35 12.83 1.51 0.64 0.64 0.66 13 16.38 1.97 5.04 13.87 1.67 4.21 0.85 0.85 0.84 14 1873.43 16.13 32.71 122.32 1.05 2.24 0.07 0.07 0.07 15 181.69 16.92 22.64 102.62 9.46 11.91 0.56 0.56 0.53 16 21.32 2.50 6.70 19.90 2.34 6.25 0.93 0.93 0.93 17 126.65 82.75 12.42 89.06 58.13 9.22 0.70 0.70 0.74 18 214.37 5.55 0.94 66.69 1.72 0.29 0.31 0.31 0.31 19 157.34 6.97 18.35 64.78 2.85 7.38 0.41 0.41 0.40 20 160.29 0.58 11.27 18.64 0.07 1.40 0.12 0.12 0.12

Table A.4.2.2: Ratios of the estimated variances of the parameters for each method, for the Dutch data

Ratio 1 Ratio 2 Ratio 3 Feed A B C A B C A B C

1 91.77 135.17 43.20 47.06 69.36 23.31 0.51 0.51 0.54 2 682.86 1354.12 1045.60 34.21 56.65 39.78 0.05 0.04 0.04 3 26.66 147.32 9.33 11.93 67.79 4.56 0.45 0.46 0.49 4 231.84 39.65 144.49 155.96 26.78 95.54 0.67 0.68 0.66 5 2541.40 447.33 931.37 318.86 60.27 155.96 0.13 0.13 0.17 6 233.21 430.82 199.31 37.21 70.82 39.62 0.16 0.16 0.20 7 171.95 73.09 96.75 75.73 32.78 44.56 0.44 0.45 0.46 8 432.88 6.02 277.93 95.33 1.34 59.81 0.22 0.22 0.22 9 681.93 222.40 333.37 90.27 30.51 37.83 0.13 0.14 0.11 10 181.41 167.68 199.30 75.43 70.59 85.77 0.42 0.42 0.43 11 61.50 36.73 88.56 50.87 30.93 73.05 0.83 0.84 0.82 12 104.79 93.44 49.90 40.32 35.82 24.46 0.38 0.38 0.49 13 372.06 173.57 70.51 103.78 48.59 21.18 0.28 0.28 0.30 14 384.48 413.88 192.54 59.25 63.04 38.74 0.15 0.15 0.20 15 43.89 94.19 38.76 32.00 71.40 29.51 0.73 0.76 0.76 16 125.03 157.34 54.60 67.41 85.13 31.22 0.54 0.54 0.57 17 365.68 262.56 180.20 112.71 81.83 66.56 0.31 0.31 0.37 18 50.36 130.88 38.97 15.95 40.78 13.07 0.32 0.31 0.34 19 65.25 194.39 47.29 28.30 84.66 22.24 0.43 0.44 0.47 20 114.99 83.13 6.43 32.93 25.54 1.65 0.29 0.31 0.26

In ratio 1 for method (1), each parameter’s variance in Sweden was divided by the same parameters’ variance in the Netherlands. In ratio 2 for method (2), each parameter’s variance in Sweden was divided by the same parameters’ variance in the Netherlands. In ratio 3 for method (3), each parameter’s variance in Sweden was divided by the same parameters’ variance in the Netherlands.

Table A.4.2.3: Ratios of the estimated variances of the parameters for each method,

for the data in Sweden and the Netherlands

Ratio 1 Ratio 2 Ratio 3 Feed A B C A B C A B C

1 2.47 0.86 0.87 1.01 1.57 2.33 1.21 1.97 3.22 2 0.90 0.15 0.05 1.23 1.60 3.30 0.55 0.60 1.12 3 6.65 0.58 1.65 1.23 1.71 2.04 0.82 1.18 1.67 4 0.46 7.78 0.71 0.20 0.24 0.54 0.88 1.13 2.03 5 0.05 0.00 0.02 0.81 0.94 0.87 0.19 0.24 0.28 6 1.05 0.06 0.31 1.54 1.73 2.11 0.61 0.69 1.13 7 0.91 0.07 0.39 0.61 0.74 1.07 1.01 1.23 1.90 8 1.29 7.21 0.34 0.93 0.95 2.84 1.42 1.40 3.75 9 2.34 1.56 0.60 0.93 1.04 2.81 0.87 1.02 2.19 10 2.60 0.06 0.12 1.06 1.24 1.53 1.79 2.13 2.77 11 6.90 3.60 0.04 0.68 0.58 1.18 1.83 1.67 3.29 12 0.91 0.52 0.09 2.08 2.43 2.00 1.25 1.46 1.49 13 0.03 0.01 0.08 0.74 0.80 1.12 0.24 0.26 0.40 14 6.42 0.05 0.35 1.32 1.25 2.08 3.11 2.93 6.09 15 9.45 0.42 2.03 2.28 2.32 3.48 2.95 3.14 5.03 16 0.13 0.01 0.13 0.77 0.76 1.10 0.45 0.44 0.67 17 0.47 0.39 0.15 1.35 1.23 2.25 0.59 0.54 1.12 18 4.50 0.05 0.04 1.06 1.10 1.50 1.08 1.10 1.62 19 2.16 0.03 0.53 0.90 0.90 1.35 0.94 0.96 1.58 20 0.99 0.00 4.36 0.71 0.41 2.49 1.75 1.06 5.13

A.5 Chemical composition of the feeds

Table A.5.1: Chemical composition (g kg -1 DM) of the feeds

a SE: solvent extracted b Data derived from Dutch Feeding Table (CVB, 2002)

Feeds No. Ash Crude Protein

Crude Fibre

Crude fat

Sugar

Palm kernel expeller A 1 41 156 182 131 20 Maize gluten feed A 2 70 260 77 45 26 Coconut expeller 3 68 232 141 131 66 Maize 4 16 96 20 53 15 Citrus pulp 5 72 68 128 22 243 Lupine (protected) 6 29 341 169 68 60 Lupine 7 29 342 162 69 63 Maize gluten feed B 8 70 210 77 39 49 Soybean meal (protected) 9 57 393 54 215 105 Beet pulp 10 81 93 149 8 225 Pea 11 28 217 56 13 58 Palm kernel expeller B 12 52 166 228 75 20b

Soybean meal SE A 13 63 505 76 29 104b Sunflower meal SE 14 73 291 293 20 104b Coconut meal SEa 15 75 227 133 39 84b

Soybean meal SE B 16 63 507 71 34 104b Rapeseed meal SE 17 74 388 136 32 103b

Soybean hulls 18 49 112 387 21 18b Soybean meal SE C 19 67 454 62 35 106b Rapeseed meal SE (protected) 20 78 371 150 41 104b

Appendix B The following Matlab code for the Swedish data has been used for obtaining the results from the tests used in the three methods. The same Matlab code was also used for obtaining the results for the Dutch data. B.1 Method (1) B.1.1 The Hotelling’s T2test % Hotelling’s T^2 test to compare feed i to feed j for unstacked data in Sweden. load param_estim_Sweden.txt % Reading the parameter estimates from a text file. load covar_estim_Sweden.txt % Reading the estimated covariance matrix from a text % file. Teststatistic = zeros(19,19); n1 = 3; % Number of estimates per parameter for feed i. n2 = 3; % Number of estimates per parameter for feed j. p = 3; % Number of parameters per feed. n = 1; m = 1; % Feed i is compared to feed j. (i = 1,...,19 and j = 2,...,20) for s = 1:3:55 A_feed_i = param_estim_Sweden(1:3,s); % The 3 estimates of A, % feed i, is defined as the values % of row 1-3, column s % in the textfile. mean_A_feed_i= mean(A_feed_i); % The mean value of A is calculated. B_feed_i = param_estim_Sweden(1:3,s+1); mean_B_feed_i= mean(B_feed_i); C_feed_i = param_estim_Sweden(1:3,s+2); mean_C_feed_i= mean(C_feed_i); feed_i = [mean_A_feed_i mean_B_feed_i mean_C_feed_i]; % A vector containing % the estimated mean % values for each % parameter. feed_i_S = covar_estim_Sweden(1:3,s:s+2); for k = 4 : 3 : 58 A_feed_j = param_estim_Sweden(1:3,k); mean_A_feed_j= mean(A_feed_j ); B_feed_j = param_estim_Sweden(1:3,k+1); mean_B_feed_j= mean(B_feed_j); C_feed_j = param_estim_Sweden(1:3,k+2); mean_C_feed_j= mean(C_feed_j); feed_j = [mean_A_feed_j mean_B_feed_j mean_C_feed_j]; feed_j_S = covar_estim_Sweden(1:3,k:k+2); Sigma = ((n1-1)*feed_i_S+(n2-1)*feed_j_S)/(n1+n2-2); % The pooled covariance matrix. Tao = ((n1*n2)/(n1+n2))*(feed_i-feed_j)*(Sigma)^(-1)*(feed_i-feed_j)';

Teststatistic(m,n) = ((n1+n2-p-1)/(p*(n1+n2-2)))*Tao; % The test statistic for comparing feed i and feed j. n = n + 1; end n = 1; m = m+1; end % The results are placed in an upper triangular matrix. Teststatistic = triu(Teststatistic) % Hotelling’s T^2-test to compare feed in Sweden to the same feed in % the Netherlands for the unstacked data. load param_estim_Sweden.txt % Reading the parameter estimates in Sweden % from a text file. load covar_estim_Sweden.txt % Reading the estimated covariance matrix in Sweden % from a text file. load param_estim_The Netherlands.txt load covar_estim_The Netherlands.txt Teststatistic = zeros(1,20); n1 = 3; % Number of estimates per parameter for feed in Sweden. n2 = 3; % Number of estimates per parameter for feed in the Netherlands. p = 3; % Number of parameters per feed. m = 1; % Feed from Sweden is compared to the same feed from The Netherlands. for s = 1:3:58 A_feed_Sweden = param_estim_Sweden(1:3,s); % The 3 estimates of A, % feed Sweden, is defined as the values % of row 1-3, column s, in the text file. mean_A_feed_Sweden= mean(A_feed_Sweden); % The mean value of A is calculated. B_feed_Sweden = param_estim_Sweden(1:3,s+1); mean_B_feed_Sweden= mean(B_feed_Sweden); C_feed_Sweden = param_estim_Sweden(1:3,s+2); mean_C_feed_Sweden= mean(C_feed_Sweden); feed_Sweden = [mean_A_feed_Sweden mean_B_feed_Sweden mean_C_feed_Sweden]; % A vector containing % the estimated mean % values for each % parameter in Sweden. feed_Sweden_S = covar_estim_Sweden(1:3,s:s+2); A_feed_The Netherlands = param_estim_The Netherlands(1:3,s); mean_A_feed_The Netherlands= mean(A_feed_The Netherlands ); B_feed_The Netherlands = param_estim_The Netherlands(1:3,s+1); mean_B_feed_The Netherlands= mean(B_feed_The Netherlands); C_feed_The Netherlands = param_estim_The Netherlands(1:3,s+2); mean_C_feed_The Netherlands= mean(C_feed_The Netherlands);

feed_The Netherlands = [mean_A_feed_The Netherlands mean_B_feed_The Netherlands mean_C_feed_The Netherlands]; feed_The Netherlands_S = covar_estim_The Netherlands(1:3,s:s+2); Sigma = ((n1-1)*feed_Sweden_S+(n2-1)*feed_The Netherlands_S)/(n1+n2-2); % The pooled covariance matrix. Tao = ((n1*n2)/(n1+n2))*(feed_Sweden-feed_The Netherlands)* (Sigma)^(-1)*(feed_Sweden-feed_The Netherlands)'; Teststatistic(m) = ((n1+n2-p-1)/(p*(n1+n2-2)))*Tao; % The test statistic for comparing feed in Sweden % and the Netherlands. m = m + 1; end % The results are placed in a vector. Teststatistic = Teststatistic' B.1.2 The two sample t test % Performs a two sample t test to determine whether two samples from a normal % distribution with unknown but equal variances could have the same mean, for % the Swedish data. load param_estim_Sweden.txt % Reading the parameter estimates in Sweden % from a text file. n = 1; m = 1; significance_A = zeros(19,19); significance_B = zeros(19,19); significance_C = zeros(19,19); alpha = 0.05; % desired significance level for s = 1:3:55 A_feed_i = param_estim_Sweden(1:3,s); B_feed_i = param_estim_Sweden(1:3,s+1); C_feed_i = param_estim_Sweden(1:3,s+2); % performs a t test for feed i and feed j for parameter A, B and C. for k = 4 : 3 : 58 A_feed_j = param_estim_Sweden(1:3,k); B_feed_j = param_estim_Sweden(1:3,k+1); C_feed_j = param_estim_Sweden(1:3,k+2); [h_A, significance_A(m,n)] = ttest2(A_feed_i,A_feed_j, alpha); % h_A =0 => "Do not reject null hypothesis at significance level of alpha." % h_A=1 => "Reject null hypothesis at significance level of alpha." % significance_A is the p-value, or the probability of observing the given % result by chance given that the null hypothesis is true. [h_B,significance_B(m,n)] = ttest2(B_feed_i,B_feed_j, alpha); [h_C,significance_C(m,n)] = ttest2(C_feed_i,C_feed_j, alpha); n = n + 1; end n = 1; m = m + 1; end % The results are placed in uppertriangular matrices. significance_A = triu(significance_A)’ significance_B = triu(significance_B)’ significance_C = triu(significance_C)’

% Performs a two sample t test when comparing the parameters for the Swedish and % the Dutch data. load param_estim_Sweden.txt load param_estim_The Netherlands.txt n = 1; m = 1; significance_A = zeros(1,20); significance_B = zeros(1,20); significance_C = zeros(1,20); for s = 1:3:58 A_feed_Sweden = param_estim_Sweden(1:3,s); B_feed_Sweden = param_estim_Sweden(1:3,s+1); C_feed_Sweden = param_estim_Sweden(1:3,s+2); A_feed_The Netherlands = param_estim_The Netherlands(1:3,s); B_feed_The Netherlands = param_estim_The Netherlands(1:3,s+1); C_feed_The Netherlands = param_estim_The Netherlands(1:3,s+2); % performs a t test for feed i, Sweden vs. feed i, the Netherlands, % for parameter A, B and C. [h_A, significance_A(n)] = ttest2(A_feed_Sweden,A_feed_The Netherlands,0.05); [h_B,significance_B(n)] = ttest2(B_feed_Sweden,B_feed_The Netherlands,0.05); [h_C,significance_C(n)] = ttest2(C_feed_Sweden,C_feed_The Netherlands,0.05); n = n + 1; end % The results are placed in upper triangular matrices. significance_A = significance_A' significance_B = significance_B' significance_C = significance_C'

B.2 Method (2) % The test in method (2) when comparing feed i to feed j for unstacked data, % Sweden. load unstacked_feed_Sweden.txt % Reading the unstacked data from a text file. teststatistic_A = zeros(19,19); teststatistic_B = zeros(19,19); teststatistic_C = zeros(19,19); s = 1; n = 1; m = 1; % Feed i is compared to feed j. (i = 1,...,19 and j = 2,...,20) for s = 1:3:55 A_feed_i = unstacked_feed_Sweden(1,s:s+2); % The 3 estimates of A, feed i, is % defined as the values of row 1, % column s until s+2 in the % text file. mean_A_feed_i = mean(A_feed_i); % The mean value of A is calculated. B_feed_i = unstacked_feed_Sweden(2,s:s+2); mean_B_feed_i = mean(B_feed_i); C_feed_i = unstacked_feed_Sweden(3,s:s+2); mean_C_feed_i = mean(C_feed_i); sA_feed_i = unstacked_feed_Sweden(4,s:s+2); % The standard deviations of the % estimated values of A, feed i, is % defined as the values of row 4, % column s until s+2 in the % text file. VA_feed_i = sA_feed_i.^2; % The variances of the estimated values of A, % feed i, is calculated. sB_feed_i = unstacked_feed_Sweden(5,s:s+2); VB_feed_i = sB_feed_i.^2; sC_feed_i = unstacked_feed_Sweden(6,s:s+2); VC_feed_i = sC_feed_i.^2; for k = 4 : 3 : 58 A_feed_j = unstacked_feed_Sweden(1,k:k+2); mean_A_feed_j = mean(A_feed_j ); B_feed_j = unstacked_feed_Sweden(2,k:k+2); mean_B_feed_j = mean(B_feed_j); C_feed_j = unstacked_feed_Sweden(3,k:k+2); mean_C_feed_j = mean(C_feed_j); sA_feed_j = unstacked_feed_Sweden(4,k:k+2); VA_feed_j = sA_feed_j .^2; sB_feed_j = unstacked_feed_Sweden(5,k:k+2); VB_feed_j = sB_feed_j.^2; sC_feed_j = unstacked_feed_Sweden(6,k:k+2); VC_feed_j = sC_feed_j.^2;

% The teststatistics for comparing feed i and feed j is calculated % for the parameters A, B and C.

teststatistic_A(m,n) = (mean_A_feed_i-mean_A_feed_j)/((1/3)*sqrt(sum(VA_feed_i+VA_feed_j)));

teststatistic_B(m,n) = (mean_B_feed_i-mean_B_feed_j)/((1/3)*sqrt(sum(VB_feed_i+VB_feed_j)));

teststatistic_C(m,n) = (mean_C_feed_i-mean_C_feed_j)/((1/3)*sqrt(sum(VC_feed_i+VC_feed_j)));

n = n + 1;

end n = 1; m = m + 1; end % The results are placed in an upper triangular matrix. teststatistic_A = triu(teststatistic_A) teststatistic_B = triu(teststatistic_B) teststatistic_C = triu(teststatistic_C)

B.3 Method (3) % The test in method (3) when comparing feed i to feed j for stacked data, Sweden. load stacked_feed_Sweden.txt % Reading the stacked data from a text file. teststatistic_A = zeros(19,19); teststatistic_B = zeros(19,19); teststatistic_C = zeros(19,19); s = 1; n = 1; m = 1; % Feed i is compared to feed j. (i = 1,...,19 and j = 2,...,20) for s = 1:19 A_feed_i = stacked_feed_Sweden(1,s); % The estimate of A, feed i, is defined % as the value of row 1, column s in % the text file. B_feed_i = stacked_feed_Sweden(2,s); C_feed_i = stacked_feed_Sweden(3,s); sA_feed_i = stacked_feed_Sweden(4,s);% The standard deviation of the estimated % value of A, feed i, is defined as the % value of row 4, column s in the textfile. VA_feed_i = sA_feed_i.^2; % The variance of the estimated value of A, % feed i,is calculated. sB_feed_i = stacked_feed_Sweden(5,s); VB_feed_i = sB_feed_i.^2; sC_feed_i = stacked_feed_Sweden(6,s); VC_feed_i = sC_feed_i.^2; for k = 2:20 A_feed_j = stacked_feed_Sweden(1,k); B_feed_j = stacked_feed_Sweden(2,k); C_feed_j = stacked_feed_Sweden(3,k); sA_feed_j = stacked_feed_Sweden(4,k); VA_feed_j = sA_feed_j .^2; sB_feed_j = stacked_feed_Sweden(5,k); VB_feed_j = sB_feed_j.^2; sC_feed_j = stacked_feed_Sweden(6,k); VC_feed_j = sC_feed_j.^2; % The test statistics for comparing feed i and feed j is calculated for the % parameters A, B and C. teststatistic_A(m,n) =(A_feed_i-A_feed_j)/(sqrt(sum(VA_feed_i+VA_feed_j))); teststatistic_B(m,n) =(B_feed_i-B_feed_j)/(sqrt(sum(VB_feed_i+VB_feed_j))); teststatistic_C(m,n) =(C_feed_i-C_feed_j)/(sqrt(sum(VC_feed_i+VC_feed_j))); n = n + 1; end n = 1; m = m + 1; end % The results are placed in an upper triangular matrix. teststatistic_A = triu(teststatistic_A) teststatistic_B = triu(teststatistic_B) teststatistic_C = triu(teststatistic_C)