Exploration of Tooth Growth among Guinea Pigs

Original study was conducted by EW Crampton at Macdonald College, McGill Univ(Quebec Canada). http://jn.nutrition.org/content/33/5/491.full.pdf During world war II, soldiers suffered sever Vitamin C deficiency and fell sick. This study was conducted to identify dosage levels and physical methods through which the problem to be addressed. 60 Subjects of Guinea Pigs are taken and divided into two groups. Each group was sujected to different ways through which Vitamin C is infused. Further the 2 groups were divided into 3 sub-groups for different dosage levels.

Following graph shows the variation in dosage levels(0.5, 1, 2) and length of odontoblasts. Above plot clearly indicates, as the dossage level increases, lenght of odontoblasts increase.

This is again proved with the following boxplot. However the difference in methods(Orange Jucie(OJ), Vitamin C ) is insignificant This is portrayed within the dosage levels also.

Confidence Interval & Hypothesis Tests

** Analysing based on supp**

vc <- tg[1:30,]$len
oj <- tg[31:60,]$len

test_by_supp <- sapply(data.frame(vc, oj), function(x){
    t.test(x)$conf.int
})

colnames(test_by_supp) <- c("VC", "OC")
rownames(test_by_supp) <- c("Lower", "Upper")
test_by_supp
##             VC       OC
## Lower 13.87675 18.19678
## Upper 20.04992 23.12989

Analysing individually by dosage and supplement method

We assume the data is IID normal and distributed roughly symmetric and gaussian. To get more confidence, we shal do a t-test on subsets of the data and observe the confidence interval

dt_M <- data.frame(tg[1:10,]$len, tg[11:20,]$len, tg[21:30,]$len, tg[31:40,]$len, tg[41:50,]$len, tg[51:60,]$len)

test_result <- sapply(dt_M, function(x){
    t.test(x)$conf.int
})

colnames(test_result) <- c("VC_0.5", "VC_1.0", "VC_2.0", "OJ_0.5", "OJ_1.0", "OJ_2.0")
rownames(test_result) <- c("Lower", "Upper")

test_result
##         VC_0.5   VC_1.0   VC_2.0   OJ_0.5   OJ_1.0   OJ_2.0
## Lower 6.015176 14.97066 22.70791 10.03972 19.90227 24.16069
## Upper 9.944824 18.56934 29.57209 16.42028 25.49773 27.95931

The above results are the independent group t confidence intervals. We have compared the mean mean length of odontoblasts with in the subgroups.

Two sample t-test

Since the groups are depenendet and of equal sample size we can do a paired t test to establish the hypothesis

tg_supp <- data.frame(tg[1:30,]$len, tg[31:60,]$len)
res_supp <- t.test(tg_supp[,1], tg_supp[,2], paired = FALSE, var.equal = F)

tg_05 <- data.frame(tg[1:10,]$len, tg[31:40,]$len)
tg_05_dose <- t.test(tg_05[,1], tg_05[,2], paired = FALSE, var.equal = F)

tg_10 <- data.frame(tg[11:20,]$len, tg[41:50,]$len)
tg_10_dose <- t.test(tg_10[,1], tg_10[,2], paired = FALSE, var.equal = F)

tg_20 <- data.frame(tg[21:30,]$len, tg[51:60,]$len)
tg_20_dose <- t.test(tg_20[,1], tg_20[,2], paired = FALSE, var.equal = F)

Conclusion

Through exploratory analysis we can very confidently conclude tooth growth is depend on dosage level. However It is quite impossible to figure out which method Orange Juice or Vitamin C supplement is good.

\(H_0\) \(H_a\) t-value Confidnece Int Conclusion
Tooth growth is affected by change in method(VC vs OJ) not-affected when there is change in method -1.9152683 -7.5710156, 0.1710156 Correctly accept \(H_0\)
Tooth growth affected by method within 0.5 dosage not-affected by method within 0.5 dosage -3.1697328 -8.7809427, -1.7190573 Correctly accept \(H_0\)
Tooth growth affected by method within 1.0 dosage not-affected by method within 1.0 dosage -4.0327696 -9.0578518, -2.8021482 Correctly accept \(H_0\)
Tooth growth affected by method within 2.0 dosage not-affected by method within 2.0 dosage 0.0461361 -3.6380705, 3.7980705 Correctly accept \(H_0\)