hw7

1 a) the sires are modelled as random effects because each individual sire could have characteristics unique to itself. They are random smaple of all possible sires.

b)y_ij=u+a_i+e_ij where y_ij is the wiehgt gained from each sire, u is the mean for the popultaion, a_i is the random effect, and e_ij is the random error.

install.packages("lme4", repos= "https://CRAN.R-project.org/package=lme4" )

## Installing package into 'C:/Users/aleks/OneDrive/Documents/R/win-library/3.6'
## (as 'lib' is unspecified)

## Warning: unable to access index for repository https://CRAN.R-project.org/package=lme4/src/contrib:
##   cannot open URL 'https://CRAN.R-project.org/package=lme4/src/contrib/PACKAGES'

## Warning: package 'lme4' is not available (for R version 3.6.1)

## Warning: unable to access index for repository https://CRAN.R-project.org/package=lme4/bin/windows/contrib/3.6:
##   cannot open URL 'https://CRAN.R-project.org/package=lme4/bin/windows/contrib/3.6/PACKAGES'

library(lme4)

## Loading required package: Matrix

sires=c("A","B","C","D","A","B","C","D","A","B","C","D","A","B","C","D","A","B","C","D","A","B","C","D")
weight=c(1.46,1.17,0.98,0.95,1.23,1.08,1.06,1.10,1.12,1.20,1.15,1.07,1.23,1.08,1.11,1.11,1.02,1.01,0.83,0.89,1.15,0.86,0.86,1.12)

calves=data.frame(sires,weight)
calves

##    sires weight
## 1      A   1.46
## 2      B   1.17
## 3      C   0.98
## 4      D   0.95
## 5      A   1.23
## 6      B   1.08
## 7      C   1.06
## 8      D   1.10
## 9      A   1.12
## 10     B   1.20
## 11     C   1.15
## 12     D   1.07
## 13     A   1.23
## 14     B   1.08
## 15     C   1.11
## 16     D   1.11
## 17     A   1.02
## 18     B   1.01
## 19     C   0.83
## 20     D   0.89
## 21     A   1.15
## 22     B   0.86
## 23     C   0.86
## 24     D   1.12

2c)

modC<-lmer(weight~1+(1|sires), data=calves)

2d)

modD<-lm(weight~1, data=calves)

2e)

anova(modC, modD)

## refitting model(s) with ML (instead of REML)

## Data: calves
## Models:
## modD: weight ~ 1
## modC: weight ~ 1 + (1 | sires)
##      Df     AIC     BIC logLik deviance  Chisq Chi Df Pr(>Chisq)
## modD  2 -22.898 -20.542 13.449  -26.898                         
## modC  3 -22.095 -18.561 14.047  -28.095 1.1962      1     0.2741

It seems it was not worth making sires random effect as p vlaue is high at .2.

2f)

summary(modC)

## Linear mixed model fit by REML ['lmerMod']
## Formula: weight ~ 1 + (1 | sires)
##    Data: calves
## 
## REML criterion at convergence: -23.5
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -1.6639 -0.5601  0.1081  0.5645  2.3859 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  sires    (Intercept) 0.005078 0.07126 
##  Residual             0.015945 0.12627 
## Number of obs: 24, groups:  sires, 4
## 
## Fixed effects:
##             Estimate Std. Error t value
## (Intercept)  1.07667    0.04397   24.48

The variance is 0.005078 for sires, 0.015945 for residuals

2g)

confint(modC)

## Computing profile confidence intervals ...

##                  2.5 %    97.5 %
## .sig01      0.00000000 0.1795777
## .sigma      0.09534633 0.1783855
## (Intercept) 0.97994356 1.1733897