all.oxygen=c(11.10,11.22,11.29,11.49,11.32,11.40,11.71,11.60,11.78,12.05,10.61,10.88,11.12,11.24,11.43,10.92,11.20,11.30,11.62,11.70,11.70,11.79,11.91,12.15,11.33,11.41,11.62,12.15,12.30,11.32,11.65,11.96,12.15,11.54,11.89,12.04,10.93,11.01,11.08,11.12,11.28,11.37,11.35,11.43,11.50,11.57,11.92,11.95,12.01,12.25,12.30,12.39)

grps=c(1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,7,7,8,8,8,8,9,9,9,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,12)

source("...\\RSS.r")
#reduced model: all means are equal
RSS1=RSS(data=all.oxygen,group=grps,subset=1:12)
#full model: all means are different: add residual sums of squares for all groups
RSS2=RSS(data=all.oxygen,group=grps,subset=1)+RSS(data=all.oxygen,group=grps,subset=2)+RSS(data=all.oxygen,group=grps,subset=3)+RSS(data=all.oxygen,group=grps,subset=4)+RSS(data=all.oxygen,group=grps,subset=5)+RSS(data=all.oxygen,group=grps,subset=6)+RSS(data=all.oxygen,group=grps,subset=7)+RSS(data=all.oxygen,group=grps,subset=8)+RSS(data=all.oxygen,group=grps,subset=9)+RSS(data=all.oxygen,group=grps,subset=10)+RSS(data=all.oxygen,group=grps,subset=11)+RSS(data=all.oxygen,group=grps,subset=12)

#F1 for testing if all means are equal:
F1=((RSS1-RSS2)/((52-1)-(52-12)))/(RSS2/(52-12))
#p-value is:
1-pf(F1,11,40)#--->evidence that not all means are equal


### But note that we have two cases where the bones are coming from exactly the same part of the body. The temp should be equal here, is there a statistically significnat difference??

#One approach here is to consider the full model: all different means(12 parameters) and the reduced model: means for groups 2 and 3 are equal

#reduced model: group 2 and 3 are equal--->11 parameters
RSS3=RSS(data=all.oxygen,group=grps,subset=1)+RSS(data=all.oxygen,group=grps,subset=c(2,3))+RSS(data=all.oxygen,group=grps,subset=4)+RSS(data=all.oxygen,group=grps,subset=5)+RSS(data=all.oxygen,group=grps,subset=6)+RSS(data=all.oxygen,group=grps,subset=7)+RSS(data=all.oxygen,group=grps,subset=8)+RSS(data=all.oxygen,group=grps,subset=9)+RSS(data=all.oxygen,group=grps,subset=10)+RSS(data=all.oxygen,group=grps,subset=11)+RSS(data=all.oxygen,group=grps,subset=12)

#full model: all means are equal--->12 parameters; computed before
RSS2
#F2 is:
F2=((RSS3-RSS2)/((52-11)-(52-12)))/(RSS2/(52-12))
#p-value: 
1-pf(F2,1,40)#--->no evidence that the two gastralia means are different

#similarly with groups 4 and 5:
#reduced model: group 4 and 5 are equal--->11 parameters
RSS4=RSS(data=all.oxygen,group=grps,subset=1)+RSS(data=all.oxygen,group=grps,subset=2)+RSS(data=all.oxygen,group=grps,subset=3)+RSS(data=all.oxygen,group=grps,subset=c(4,5))+RSS(data=all.oxygen,group=grps,subset=6)+RSS(data=all.oxygen,group=grps,subset=7)+RSS(data=all.oxygen,group=grps,subset=8)+RSS(data=all.oxygen,group=grps,subset=9)+RSS(data=all.oxygen,group=grps,subset=10)+RSS(data=all.oxygen,group=grps,subset=11)+RSS(data=all.oxygen,group=grps,subset=12)

#full model: all means are equal--->12 parameters
RSS2
#F2 is:
F3=((RSS4-RSS2)/((52-11)-(52-12)))/(RSS2/(52-12))
#p-value: 
1-pf(F3,1,40)#--->suggestive but inconclusive evidence that there may be a difference in dorsal means

#Finally, if one wanted to test the claim that means are different for different parts of the body (refered to as "bone sites" in the problem), the reduced model could be that with 10 parameters, mu1, mu23, mu45, mu6, mu7, mu8, mu9, mu10, mu11, mu12. The full could be that all means are different (12 parameters)
RSS5=RSS(data=all.oxygen,group=grps,subset=1)+RSS(data=all.oxygen,group=grps,subset=c(2,3))+RSS(data=all.oxygen,group=grps,subset=c(4,5))+RSS(data=all.oxygen,group=grps,subset=6)+RSS(data=all.oxygen,group=grps,subset=7)+RSS(data=all.oxygen,group=grps,subset=8)+RSS(data=all.oxygen,group=grps,subset=9)+RSS(data=all.oxygen,group=grps,subset=10)+RSS(data=all.oxygen,group=grps,subset=11)+RSS(data=all.oxygen,group=grps,subset=12)

#full model: all means are equal--->12 parameters
RSS2
#F2 is:
F4=((RSS5-RSS2)/((52-10)-(52-12)))/(RSS2/(52-12))
#p-value: 
1-pf(F4,1,40)#--->no evidence that all means are different, in favor of means for different parts of the body are different