Synopsis

To investigate the effect of vitamin C on tooth growth, Crampton et al. (1947) measured odontoblast length as a function of vitamin C dose in 60 guinea pigs. Two methods of vitamin C delivery were characterised: delivery by orange juice and delivery by ascorbic acid. Here I show that, regardless of delivery method, higher doses of vitamin C lead to increased tooth growth. At low doses of vitamin C, tooth growth is stronger when the vitamin C is delivered by orange juice. The reason for this effect is unclear, but one possible explanation is that absorption of vitamin C by the body is better if vitamin C is delivered by orange juice.

Data Processing and Exploratory Analysis

Load the ToothGrowth data:

library(datasets)
head(ToothGrowth)
##    len supp dose
## 1  4.2   VC  0.5
## 2 11.5   VC  0.5
## 3  7.3   VC  0.5
## 4  5.8   VC  0.5
## 5  6.4   VC  0.5
## 6 10.0   VC  0.5

The ToothGrowth dataset includes measurements of the length of odontoblasts (cells responsible for tooth growth) in 60 guinea pigs (column len). Each animal received one of three dose levels of vitamin C (0.5, 1, and 2 mg/day) by one of two delivery methods (column supp), (orange juice (encoded as OJ) or ascorbic acid (a form of vitamin C and encoded as VC).

So odontoblast length was measured as a function of vitamin C dose and delivery method. To explore the results, I therefore calculated the mean odontoblast length as a function of these two variables:

toothLength <- aggregate(len ~ ., data = ToothGrowth, mean)
print(toothLength[order(toothLength$supp),])
##   supp dose   len
## 1   OJ  0.5 13.23
## 3   OJ  1.0 22.70
## 5   OJ  2.0 26.06
## 2   VC  0.5  7.98
## 4   VC  1.0 16.77
## 6   VC  2.0 26.14

Here is a boxplot exploring the dependence of tooth growth on dose and delivery:

library(ggplot2)
g <- ggplot(data = ToothGrowth, aes(y = len, x = as.factor(dose), fill= as.factor(dose)))
g + geom_boxplot() + facet_grid(facets = .~supp) +
  labs(x = "Vitamin C dose", y = "Odontoblast length")

This preliminary analysis suggests that 1) odontoblast length increases with the dose of vitamin C for both methods of delivery, and 2) for low doses of vitamin C, the extent of tooth growth may also depend on the method of delivery.

Results

1. For either delivery method, higher doses of vitamin C result in more tooth growth.

To test this hypothesis, I performed a t test on the difference in odontoblast length for the smallest and largest dose of vitamin C delivered via orange juice or ascorbic acid. In both cases, the null hypothesis was that there is no difference in mean odontoblast length for the lowest vs the highest dose of vitamin C. The alternative hypothesis was that mean odontoblast length is greater for the higher dose of vitamin C.

# test for orange juice
# odontoblast length for supp = OJ and dose = 0.5
OJteeth05 <- ToothGrowth$len[ToothGrowth$supp == "OJ" & ToothGrowth$dose == 0.5]
# odontoblast length for supp = OJ and dose = 2
OJteeth20 <- ToothGrowth$len[ToothGrowth$supp == "OJ" & ToothGrowth$dose == 2]
# t test
testOJdose <- t.test(OJteeth20, OJteeth05, alternative = "greater", paired = FALSE, mu = 0)
# print p-value
cat(paste("Tooth growth is greater for high doses of Vitamin C
                 (Orange Juice delivery): p-value =", 
                testOJdose$p.value))
## Tooth growth is greater for high doses of Vitamin C
##                  (Orange Juice delivery): p-value = 6.61891938848617e-07
# test for ascorbic acid
# odontoblast length for supp = VC and dose = 0.5
VCteeth05 <- ToothGrowth$len[ToothGrowth$supp == "VC" & ToothGrowth$dose == 0.5]
# odontoblast length for supp = VC and dose = 2
VCteeth20 <- ToothGrowth$len[ToothGrowth$supp == "VC" & ToothGrowth$dose == 2]
# t test
testVCdose <- t.test(VCteeth20, VCteeth05, alternative = "greater", paired = FALSE, mu = 0)
# print p-value
cat(paste("Tooth growth is greater for high doses of Vitamin C 
                 (Ascorbic Acid delivery): p-value =", 
                testVCdose$p.value))
## Tooth growth is greater for high doses of Vitamin C 
##                  (Ascorbic Acid delivery): p-value = 2.34078870724605e-08

For both methods of delivery, odontoblast length is significantly larger for the higher vitamin C dose (both p-values <<< 0.05).

2. At low doses of vitamin C, delivery by orange juice results in more tooth growth than delivery by ascorbic acid.

To test this hypothesis, I performed a t-test on the change in tooth growth for the two delivery methods at the lowest dose of vitamin C.

# test for orange juice
OJteeth05 <- ToothGrowth$len[ToothGrowth$supp == "OJ" & ToothGrowth$dose == 0.5]
VCteeth05 <- ToothGrowth$len[ToothGrowth$supp == "VC" & ToothGrowth$dose == 0.5]
test05supp <- t.test(OJteeth05, VCteeth05, alternative = "greater", paired = FALSE, mu = 0)
cat(paste("Tooth growth for low doses of Vitamin C is greater when 
          Vitamin C is delivered by orange juice: p-value =", 
            test05supp$p.value))
## Tooth growth for low doses of Vitamin C is greater when 
##           Vitamin C is delivered by orange juice: p-value = 0.0031793033820484

The p-value = 0.0031793 indicates that at low doses of vitamin C, the effect on tooth growth is significantly larger if the vitamin C is delivered via orange juice.

Conclusion

I conclude that regardless of delivery method, higher doses of vitamin C result in increased tooth growth. At low doses of vitamin C, tooth growth is greater when the vitamin C is delivered by orange juice. A possible explanation for this effect is that absorption of vitamin C by the body is better if vitamin C is delivered by orange juice.

The analysis and conclusions are based on the assumption that the variables are iid, and that all other potential influences on tooth growth were controlled for in the original experiment.