1. Mixed model on visit length at the feeder

1.1. Models with set 1: next visit 74415 records

1.1.1. Mixed model: Location + Hours + median weight + Eartag

\[y=X\beta + Zu + e\] Where, $y$ is a $n x 1$ vector of visit length at the feeder (minutes), $X\beta$ are the fixed effects, $location-trial \ (14)$, $hour \ entry \ at \ the \ feeder \ (23)$ and $animal \ median \ weigth$ as covariate; $Z$ is a $n$ x $q$ desing matrix ($q$ is the number of pigs) relates records in $y$ to the random vector of additive genetic effects $u$ ($q$ x $1$); $e$ ($n$ x $1$) is the random residuals vector.

1.1.2. Mixed model: Location + Hours + median weight + Eartag + Follower

\[y=X\beta + Zu + Z_fa_f + e\] Where, $y$ is a $n x 1$ vector of visit length at the feeder (minutes), $X\beta$ are the fixed effects, $location-trial \ (12)$, $hour \ entry \ at \ the \ feeder \ (23)$ and $animal \ median \ weigth$ as covariate; $Z$ is a $n$ x $q$ desing matrix ($q$ is the number of pigs) relates records in $y$ to the random vector of additive genetic effects $u$ ($q$ x $1$); $Z_f$ is the design matrix of the next individual that visited the feeder, named $followers$, relating to $y$ with the random vector effects $a_f$ ($q$ x $1$) and $e$ ($n$ x $1$) is the random residuals vector.

1.1.3. Plot Blup Eartag vs Blup Follower

ggplot(BlupEF, aes(x=EarBLUP, y=FollBLUP))+
  geom_point(aes(color = EarBLUP, size = FollBLUP)) +
  geom_smooth(method = "lm",se = FALSE, color = "red")+
  scale_color_continuous(low = "black", high = "blue") +
  guides(color = FALSE, size = FALSE) +
  geom_point(aes(y = FollBLUP), shape = 1) +
  labs(title = "Direct BV vs Follower BV",
       subtitle = paste("r =",round(cor(BlupEF$EarBLUP,BlupEF$FollBLUP), 5), ";","p-val =",
                        round(cor.test(BlupEF$EarBLUP,BlupEF$FollBLUP)$p.val,5)))+
  theme(plot.subtitle = element_text(size = 8,face = "bold"))+
  xlab("Direct Breeding value")+ylab("Follower Breeding value")

2. Average Daily Gain (ADG) by Animal corected by the group mean

The Average daily gain $ADG$ for each individual was calculated as:

\[ADG_i=\frac{\ median(Wt_{initial}) \ - median(Wt_{end})}{total\ trial \ days}\] Where:

$ADG_i$ is the Average daily gain in kilograms of an individual $i$ in any trial, with $i=1,..,135$

$median(Wt_{initial})$ is the median weight in the start day of any trial.

$median(Wt_{end})$ is the median weight in the end day of any trial.

$total\ trial \ days$ are the number of total days of each trial.

ggplot(ADG, aes(x=Loc_trial, y=ADG))+geom_boxplot()+
  ggtitle("Average Daily Gain (kg) by Trial") + 
  xlab("Location by Trial")+ ylab("Average Daily Gain (Kg)")+
  theme(axis.text.x = element_text(angle = 45, hjust = 1),legend.position="none")

2.1. Linear model on Average daily Gain (ADG)

\[y=X\beta + e\] Where:

$y$ is a $n x 1$ vector of Average daily gain ($ADG_i$ kg) of an individual

$X\beta$ are the fixed effects, $location-trial \ (14)$

$e$ ($n$ x $1$) is the random residuals vector.

and $y_i-x_i\hat{\beta}=\hat{e_i}$ is the ADG of the animal $i$

$Average\ daily\ gain\ as\ deviation\ of\ group$

3. Regression model for quantile 0.5 of the Weight by individual

The regression model for quantile level $\tau$ of the Weight by individual is:

\[Q_{\tau}(y_i)= \beta_{0}(\tau) + \beta_1(\tau)x_{i1}\] $\tau=0.5$ median quantile

$(y_i)$ is the $i-th$ observation of the Weight of an animal

$\beta_{0}$ is the intercept parameter

$\beta_1$ is the slope

$x_{i1}$ is the trial day

The goodness of fit for quantile regression by individual, which is analog to $R^2$, was calculated following to Koenker and Machado (1999), and then

\[R^1(\tau)=1-\frac{\hat{V}(\tau)}{\tilde{V}(\tau)}\] Where:

$R^1(\tau)$ is a local measure of goodness of fit for a particular quantile

$\hat{V}(\tau)$ is a model with the intercept parameter only

$\tilde{V}(\tau)$ is the unrestricted model

The Weight gain ($weight.gain.qr$)

4. Random regression model on median weight

The random regression model on median weight in each trial in matrix notation is:

\[Y_{j}= X_j\beta + Z_ju_{j} + \ e_{j}\]

Where:

$Y_{ij}$ is a matrix $n_j$x$1$ with all observation of median weight for each day, in $jth$ trial

$X_j$ is the design matrix $n_j$x$2$ of fixed effects for each group

$\beta$ is the fixed effects vector, with $\beta=(\beta_{0}, \beta_{1})'$

$Z_j$ is the design matrix $n_j$x$2$ of random effects for each group

$u_j$ is the random effects vector, with $u=(u_{0j}, u_{1j})'$

$e_{ij}$ is the error vector

and the the $ith$ row ($i = 1,...n_j$) of the response vector is:

\[y_{ij}= \beta_0 + \beta_1x_{ij} + u_0 + u_{1j}x_{ij} + e_{ij}\]

4.1. Plot individual Weight gain estimated random regression and ADG_i model

$weight.gain.rr_{pig}= group slope + trial.day.rr = (1 + u{1j})x_{ij} $

ggplot(blps.gr, aes(x=ADG, y=Weight.gain.rr))+geom_point() + geom_abline()+
  labs(title = bquote( "Weight gain estimated by random regression and" ~ "ADG"[i] ~ "models" ) )+ xlab(bquote("ADG"[i]))+ ylab("Weight gain random regression")

5. Daily Feed Intake by Animal in trial day

The animal average daily feed intake was calculated as:

\[AFI_{i} = \frac{\sum_{j=1}^n X_j}{total\ days\ trial} \]

Where $X_{j}$ is the $jth$ consumed of an individual ($j=1,..n$)

Feed.intake<-trials.data%>%group_by(Loc_trial,Eartag_trial)%>%
                            summarize(animal.intake=sum(Consumed),
                               total.days=(max(trial.day)-min(trial.day)),
                               Average.daily.intake=animal.intake/total.days)%>%
          ungroup()%>%as.data.frame()

rownames(Feed.intake)<-Feed.intake$Eartag_trial
#----------
ggplot(Feed.intake, aes(x=Loc_trial, y=Average.daily.intake))+geom_boxplot()+
  ggtitle("Average Daily Feed Intake (kg) by Trial") + 
  xlab("Location by Trial")+ ylab("Average Daily Feed Intake (Kg)")+
  theme(axis.text.x = element_text(angle = 45, hjust = 1),legend.position="none")

#----------

5.1. Linear model on Average daily feed intake

\[y=X\beta + e\] Where:

$y$ is a $n x 1$ vector of Average feed intake ($AFI_i$ kg) of an individual

$X\beta$ are the fixed effects, $location-trial \ (14)$

$e$ ($n$ x $1$) is the random residuals vector.

and $y_i-x_i\hat{\beta}=\hat{e_i}$ is the AFI of the animal $i$

6. Correlation between blups and Weight gain, feed intake

$EarBLUP =\ animal\ ramdom\ effect\ of\ visit\ length\ at\ the\ feeder\\ less\ than\ or\ equal\ to\ 60 s$

$FollBLUP = animal\ ramdom\ effect\ as\ next\ individual\ that\ visited\ the\ feeder\\ less\ than\ or\ equal\ to\ 60 s$

$ADG= ADG_i$

$ADG.locmean=ADG_i-\hat{\ loc:trial}$

$weight.gain.qr= estimated\ weight\ gain\ by\ median\ regression\ model$

$trial.day.rr = trial\ day\ random\ effect\ of\ the\ ith\ animal\ estimated\ by\ random\ regression$

$Weight.gain.rr = group\ slope + trial.day.rr,\ estimated\ by\ random\ regression$

$Average.daily.intake= AFI_i$

$feed.intake.fitloc = AFI_i- \hat{\ loc:trial}$

ADGs.FI.BLUPs%>%select(EarBLUP,FollBLUP,ADG,ADG.locmean,
                       weight.gain.qr, trial.day.rr, Weight.gain.rr,
                       Average.daily.intake, feed.intake.fitloc)%>%cor()%>%
  kable()%>%kable_styling(bootstrap_options = c("striped", "hover", "condensed"),full_width = F,
                          position = "left",font_size = 12)

	EarBLUP	FollBLUP	ADG	ADG.locmean	weight.gain.qr	trial.day.rr	Weight.gain.rr	Average.daily.intake	feed.intake.fitloc
EarBLUP	1.0000000	0.1231659	0.0657784	0.1200577	0.0245560	0.0778275	0.0472014	0.0586624	0.0854993
FollBLUP	0.1231659	1.0000000	-0.0012882	-0.0023511	0.0006074	-0.0058356	-0.0035392	0.0019494	0.0028412
ADG	0.0657784	-0.0012882	1.0000000	0.5478900	0.9250629	0.4983372	0.9567826	0.7109197	0.3885109
ADG.locmean	0.1200577	-0.0023511	0.5478900	1.0000000	0.5723908	0.9095570	0.5516344	0.4865273	0.7091039
weight.gain.qr	0.0245560	0.0006074	0.9250629	0.5723908	1.0000000	0.6489043	0.9870484	0.7318058	0.4860840
trial.day.rr	0.0778275	-0.0058356	0.4983372	0.9095570	0.6489043	1.0000000	0.6064868	0.5149118	0.7504736
Weight.gain.rr	0.0472014	-0.0035392	0.9567826	0.5516344	0.9870484	0.6064868	1.0000000	0.7302546	0.4551523
Average.daily.intake	0.0586624	0.0019494	0.7109197	0.4865273	0.7318058	0.5149118	0.7302546	1.0000000	0.6861158
feed.intake.fitloc	0.0854993	0.0028412	0.3885109	0.7091039	0.4860840	0.7504736	0.4551523	0.6861158	1.0000000

7. Visit length by animal

7.1. linear model of Visit length mean of each individual fitted by group mean

The sum of visit length total at the feeder for each individual was calculated as:

\[Vl_i= \sum_{j=1}^{n_i}X_j\]

Where:

$Vl_i$ is the sum of the visit length total time at the feeder from the $ith$ animal

$X_j$ is the $jth$ observation of visit length time at the feeder from the $ith$ animal

$n_i$ total number of observations from the $ith$ animal

Visit length total time as deviation of estimated mean of location-trial group

The linear model fitted was: \[y=X\beta + e\] Where:

$y$ is a $n x 1$ vector of the sum of the visit length total $Vl_i$ at the feeder from the $ith$ animal

$X\beta$ are the fixed effects, $location-trial \ (12)$

$e$ ($n$ x $1$) is the random residuals vector.

and $y_i-x_i\hat{\beta}=\hat{e_i}$ is the visit length total as deviation of estimated mean of location-trial group of the animal $i$

7.2. Random regression model on Visit lenght

The random regression model on visit length in each trial in matrix notation is:

\[Y_{j}= X_j\beta + Z_ju_{j} + \ e_{j}\]

Where:

$Y_{ij}$ is a matrix $n_j$x$1$ with all observation of visit length total for each day, in $jth$ trial

$X_j$ is the design matrix $n_j$x$2$ of fixed effects for each group (location-trial, trial day)

$\beta$ is the fixed effects vector, with $\beta=(\beta_{0}, \beta_{1})'$

$Z_j$ is the design matrix $n_j$x$2$ of random effects for each group (Animal, trial day)

$u_j$ is the random effects vector, with $u=(u_{0j}, u_{1j})'$

$e_{ij}$ is the error vector

and the the $ith$ row ($i = 1,...n_j$) of the response vector is:

\[y_{ij}= \beta_0 + \beta_1x_{ij} + u_0 + u_{1j}x_{ij} + e_{ij}\]

The visit length by animal is

$Visit.length_{pig}= group\ slope + trial.day.rr = (\beta_1 + u_{1j})x_{ij}$

7.3. Average Feeder Occupation

The animal Average Feeder Occupation time was calculated as:

\[AFO_{i} = \frac{\sum_{j=1}^n X_j}{total\ days\ trial} \]

Where $X_{j}$ is the $jth$ time of occupation event of an individual in minutes ($j=1,..n$)

AFO<-trials.data%>%group_by(Loc_trial,Eartag_trial)%>%
                            summarize(animal.AFO=sum(visit.length),
                               total.days=(max(trial.day)-min(trial.day)),
                               AFO=animal.AFO/total.days)%>%
          ungroup()%>%as.data.frame()

rownames(AFO)<-AFO$Eartag_trial
#----------
ggplot(AFO, aes(x=Loc_trial, y=AFO))+geom_boxplot()+
  ggtitle("Average Daily Feeder Occupation Time (min) by Trial") + 
  xlab("Location by Trial")+ ylab("Average Daily Feeder Occupation Time (min)")+
  theme(axis.text.x = element_text(angle = 45, hjust = 1),legend.position="none")

#----------

\[y=X\beta + e\] Where:

$y$ is a $n x 1$ vector of Average Daily feeder occupation time ($AFO_i$ min) of an individual

$X\beta$ are the fixed effects, $location-trial \ (14)$

$e$ ($n$ x $1$) is the random residuals vector.

and $\hat{e_i}=y_i-x_i\hat{\beta}$ is the ADFO of the animal $i$

7.4. Correlation between blups, visit length, weight gain, feed intake

$Estimates\ of\ Pearson,\ Kendall\ and\ Spearman\ coefficients\ correlations\ and\ their\ p-values$

$EarBLUP = animal\ ramdom \ effect\ of \ visit \ length \ at \ the \ feeder$

$FollBLUP = animal\ ramdom \ effect\ as \ next \ individual \ that \ visited \ the \ feeder$

$WG=trial.day.rr = u_{1i},\ trial\ day\ random\ effect\ of\ the\ ith\ animal\ on\ weight\ gain$

$AFI=feed.intake.fitloc = AFI_i- \hat{\ loc:trial}$

$FOT=visitl.fitloc = Vl_i-\hat{\ loc:trial}$

$FOD_1=Visit.length_{pig}= group\ slope + vl.day = (\beta_1 + u_{1j})x_{ij}$

$FOD_2=vl.day = trial\ day\ random\ effect\ of\ the\ ith\ animal\ on\ visit\ length\ at\ the\ feeder$

$ADFO=AFO.fitloc = AFO_i- \hat{\ loc:trial}$

kable(outcor, caption = " Estimates correlation between BLUP's of visit length and weight gain and feed intake, feeder occupation time") %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"),full_width = F, 
               position = "center",font_size = 14) %>%
  add_header_above(c(" Trait" = 1, "Direct BLUP" = 6, "Follower BLUP" = 6), italic = T,
                   bold = T)%>%
  column_spec(1, bold = T)%>%
  column_spec(c(2,3,8,9), background = "#AAEEAA")%>%
  column_spec(c(4,5,10,11), background = "#CCEAAA")%>%
  row_spec(c(3,6), bold = T)

Estimates correlation between BLUP’s of visit length and weight gain and feed intake, feeder occupation time
Trait	Direct BLUP						Follower BLUP
	Pearson	Pval	Spearman	Pval	Kendall	Pval	Pearson	Pval	Spearman	Pval	Kendall	Pval
WG	0.0778	0.3696	0.0826	0.3409	0.0543	0.3503	-0.0058	0.9464	0.0562	0.5174	0.0379	0.5141
AFI	0.0855	0.3241	0.1270	0.1422	0.0839	0.1488	0.0028	0.9739	0.0479	0.5810	0.0331	0.5695
FOT	0.4532	0.0000	0.4952	0.0000	0.3501	0.0000	-0.2150	0.0123	-0.1832	0.0334	-0.1257	0.0306
FOD_1	-0.0332	0.7025	-0.0677	0.4350	-0.0501	0.3889	-0.1464	0.0902	-0.1057	0.2225	-0.0700	0.2286
FOD_2	-0.0462	0.5950	-0.1054	0.2239	-0.0647	0.2658	-0.2037	0.0178	-0.1627	0.0593	-0.1076	0.0642
ADFO	0.5136	0.0000	0.5125	0.0000	0.3667	0.0000	-0.2601	0.0023	-0.2186	0.0109	-0.1472	0.0113

Analysis REML on visit length all records and Pearson, Spearman, Kendall correlations with productions traits

1. Mixed model on visit length at the feeder

1.1. Models with set 1: next visit 74415 records

1.1.1. Mixed model: Location + Hours + median weight + Eartag

1.1.2. Mixed model: Location + Hours + median weight + Eartag + Follower

1.1.3. Plot Blup Eartag vs Blup Follower

2. Average Daily Gain (ADG) by Animal corected by the group mean

2.1. Linear model on Average daily Gain (ADG)

3. Regression model for quantile 0.5 of the Weight by individual

4. Random regression model on median weight

4.1. Plot individual Weight gain estimated random regression and ADG_i model

5. Daily Feed Intake by Animal in trial day

5.1. Linear model on Average daily feed intake

6. Correlation between blups and Weight gain, feed intake

7. Visit length by animal

7.1. linear model of Visit length mean of each individual fitted by group mean

Visit length total time as deviation of estimated mean of location-trial group

7.2. Random regression model on Visit lenght

7.3. Average Feeder Occupation

7.4. Correlation between blups, visit length, weight gain, feed intake