A Guinea Pig

A Guinea Pig

Overview

Does increased dosages of Vitamin C (Vit-C) lead to longer tooth growth in Cavia Porcellus (aka the Guinea Pig)? Also, does the delivery system for Vit-C matter?

This is an analysis of a Tooth Growth Experiment conducted by C.I. Bliss (1952 - The Statistics of Bioassay. Academic Press.) on “The response is the length of odontoblasts (cells responsible for tooth growth) in 60 guinea pigs. Each animal received one of three dose levels of vitamin C (0.5, 1, and 2 mg/day) by one of two delivery methods, (orange juice or ascorbic acid (a form of vitamin C and coded as VC).”

Here is the detail of the original experiment :

A biological assay of the vitamin C activity of fresh orange juice may be used as a second example. The response of 10 guinea pigs on each of three doses of ascorbjc acid (8) and of fresh orange juice (U) during a six-week test period was measured by Crampton (26) from the average length of the odontoblasts in each animal (p. 257), ’with the results given in Table XXVIII. Five of the animals at each treatment combination were males and five were females but except for this sex restriction the guinea pigs were assigned to doses entirely at random. The assay could be considered, therefore, as forming a 2 X 3 X 2 factorial experiment and has been so analyzed. The complete analysis (not given here) showed that the effect of sex could be neglected, so that in the interests of simplicity the data will be treated here as a 2 X 3 assay with 10 responses at each dose.

Summary of Data

The dataset “ToothGrowth” is part of the standard R base system. Here’s a boxplot of the dataset using dose and suppliment (or supp in the dataset).

Note: All R code in this presentation can be found in the Appendix

The graph shows that the more Vit-C given to the test subjects, the longer their teeth grew. Also, that the use of Orange Juice (coded OJ) at lower doses (0.5 and 1) appear to promote more tooth growth than straight Vit-C (ascorbic acid), but we’ll need to explore further. To do this, we’ll factor the dataset into its 6 tests, and then find the mean and standard deviation (labeled mn and sig respectively):

##     test    mn      sig
## 1 OJ-0.5 13.23 4.459709
## 2 VC-0.5  7.98 2.746634
## 3   OJ-1 22.70 3.910953
## 4   VC-1 16.77 2.515309
## 5   OJ-2 26.06 2.655058
## 6   VC-2 26.14 4.797731

Assumptions

Since the experiment randomized the suppliment and dosage between we’ll treat these as unpaired results with a equal variance between them. Also, since the sample sizes are small (n = 10), we’ll use a T-distribution for all analysis. We’ll also assume a type I error, alpha = 5%.

Analysis

A exhaustive analysis of the data would require us to statistically compare every possible permuation of the experiment, but we can reduce this by computing the confidence intervals (labeled c.up and c.down) for each dose+supp and make judgements on which tests will be most important:

##     test    mn  sig  c.up c.down
## 1 OJ-0.5 13.23 4.46 16.42  10.04
## 2 VC-0.5  7.98 2.75  9.94   6.02
## 3   OJ-1 22.70 3.91 25.50  19.90
## 4   VC-1 16.77 2.52 18.57  14.97
## 5   OJ-2 26.06 2.66 27.96  24.16
## 6   VC-2 26.14 4.80 29.57  22.71

A cursory look at the confidence intervals shows many combinations have “clear daylight” between them. There are two cases that need more analysis:

  1. Is there a statistical difference between OJ supplement and a dosage of 1.0 and 2.0?
  2. Is there a statistical difference between a dosage of 2.0 and supplements OJ and VC?

T-tests

T-test Confidence Interval of Orange Juice, Dosage 2.0 v 1.0

## [1] 0.2194983 6.5005017
## attr(,"conf.level")
## [1] 0.95

The confidence interval entirely positive, meaning that the two dosages are different in their effect on tooth growth.

T-test Confidence Interval at Dosage of 2.0, Orange Juice v Ascorbic Acid

## [1] -3.562999  3.722999
## attr(,"conf.level")
## [1] 0.95

Here, the confidence interval spans a mean of 0, meaning that the the two supplements, at a dosage of 2.0, have the same (or similar) effect on tooth growth.

Conclusion

  1. The more vitamin C a guinea pig gets, the longer their teeth grow.
  2. At lower dosages (0.5 and 1), getting Vitamin C from orange juice results in higher tooth growth versus ascobic acid. However, at a higher dosage (2.0), tooth growth is similar.

Appendix

Code for presenting summary data

data("ToothGrowth")
library(ggplot2)
t <- ToothGrowth
t$dose <- as.factor(t$dose)
g <- ggplot(t, aes(group = dose, dose, len))
g + geom_boxplot() + facet_grid(.~ supp) + ggtitle("Vitamin C and Tooth Growth")

s <- aggregate(t$len, by=list(t$supp, t$dose), function(x) cbind(mean(x), sd(x)))
s$test <- paste(s$Group.1, s$Group.2, sep = "-")
s$mn <- s$x[1:6,1]
s$sig <- s$x[1:6,2]
s <- s[, 4:6]
s

Code for confidence intervals

library(dplyr)
s <- mutate(s, c.up = mn + qt(.975, 10-1)*sig/sqrt(10))
s <- mutate(s, c.down = mn - qt(.975, 10-1)*sig/sqrt(10))
s[,3:5] <- round(s[,3:5], 2)

s

Code for T-tests

oj1 <- subset(t, supp == "OJ" & dose == 1)
oj2 <- subset(t, supp == "OJ" & dose == 2)
t.test(oj2$len, oj1$len, var.equal = TRUE)$conf

vc2 <- subset(t, supp == "VC" & dose == 2)
t.test(vc2$len, oj2$len, var.equal = TRUE)$conf