Overview

The “ToothGrowth” dataset is one of the example datasets included in R. According to the R documentation:

“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).”

It is important to note that this dataset does not include a control/placebo group, we can only make comparisons between the two delivery methods (VC and OJ) and doseage. Another caveat worth noting is the “length” variable 1. does not include units and 2. does not state whether the number given is the difference between end of study and baseline measurements or just the end of study measurements.

Methods

To begin examining the dataset, we use str() to determine the overall structure of the dataset and how many datapoints the set contains. From this, we see the dataset contains 60 observations, presumably 30 each for the two delivery methods (VC and OJ) and 10 in each of those groups for the three doses (0.5, 1, and 2 mg/day).

str(ToothGrowth)
## 'data.frame':    60 obs. of  3 variables:
##  $ len : num  4.2 11.5 7.3 5.8 6.4 10 11.2 11.2 5.2 7 ...
##  $ supp: Factor w/ 2 levels "OJ","VC": 2 2 2 2 2 2 2 2 2 2 ...
##  $ dose: num  0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 ...

We can get an idea of the overall data by looking at the mean length for each group. The code below creates a grouped barplot in ggplot2 (shown in the Results section).

library(ggplot2)
p <- ggplot(ToothGrowth, aes(x=supp, y=len, fill=factor(dose)))

The dataset is broken down into separate groups for easier comparison and statistical testing. Comparisons can be made between 1. supplement type (OJ vs VC) and 2. dose (0.5 vs. 1.0 vs. 2.0 mg/day).

OJ<-ToothGrowth[ToothGrowth$supp=="OJ",c(1,3)]
VC<-ToothGrowth[ToothGrowth$supp=="VC",c(1,3)]
OJ.5<-OJ[OJ$dose==0.5,]; OJ1<-OJ[OJ$dose==1.0,]; OJ2<-OJ[OJ$dose==2.0,]
VC.5<-VC[VC$dose==0.5,]; VC1<-VC[VC$dose==1.0,]; VC2<-VC[VC$dose==2.0,]

Multiple t-tests are performed to determine whether the difference in group means were likely due to chance alone (\(\alpha\)=0.05). Because multiple comparisons are being made, a Bonferroni correction is used to adjust p-values.

p1<-t.test(OJ.5,OJ1)$p.value; p2<-t.test(OJ1,OJ2)$p.value; p3<-t.test(OJ.5,OJ2)$p.value;
p4<-t.test(VC.5,VC1)$p.value; p5<-t.test(VC1,VC2)$p.value; p6<-t.test(VC.5,VC2)$p.value;
p7<-t.test(OJ.5,VC.5)$p.value; p8<-t.test(OJ1,VC1)$p.value; p9<-t.test(OJ2,VC2)$p.value
pVals1<-c(p1,p2,p3,p4,p5,p6); pVals2<-c(p7,p8,p9)
pBonDose<-p.adjust(pVals1, method= "bonferroni", n=6) 
pBonSupp<-p.adjust(pVals2, method= "bonferroni", n=3)

Results

Preliminary analysis involved determining how the group means and variance compare for each delivery method and dose. In Figure 1, we can see that there appears to be a dose effect within both the VC and OJ groups, whereby length increases as dose increases. For the 0.5 and 1.0 mg/day dose groups, it appears that subjects supplemented using OJ had greater length than the VC group, but this is not true for the 2.0 mg/day groups.

p + geom_boxplot() +
    labs(x = "Supplement Type", y = "Length") +
    scale_fill_discrete(name = "Dose")
Mean tooth length by supplement type and dose. Box midline indicates median, box bottom and top indicate 1st and 3rd quartiles, respectively. Whiskers indicate min and max and outliers are indicated by black dots.

Mean tooth length by supplement type and dose. Box midline indicates median, box bottom and top indicate 1st and 3rd quartiles, respectively. Whiskers indicate min and max and outliers are indicated by black dots.

In evaluating this data, we pose the question: are these group differences meaningful? Or might they have occurred due to chance? T-tests were performed for each group comparison: 1. OJ vs. VC (for groups receiving the same dosage) and 2. dosage within each supplement type. Because multiple comparisons are being made, a Bonferroni correction is used to adjust the p-values. These p-values are shown (with and without correction) in Figure 2.

axisSupp<-c("0.5\nmg/day","1.0\nmg/day","2.0\nmg/day")
par(mar=c(4,4,2,2), mfrow=c(1,2))
plot(pVals2, xaxt="n", xlab="OJ vs VC Comparison", ylab="p-value", pch=1, ylim=c(0,1.0))
points(pBonSupp, pch=16, col="red"); abline(h=0.05, col="black")
mtext("A", side=3, adj=0, font=2, cex=2); axis(1, at=c(1,2,3),labels=axisSupp, tick=F)

axisDose<-c("0.5 v 1.0\n(OJ)", "1.0 v 2.0\n(OJ)", "0.5 v 2.0\n(OJ)", "0.5 v 1.0\n(VC)", 
            "1.0 v 2.0\n(VC)", "0.5 v 2.0\n(VC)")
plot(pVals1, xaxt="n", xlab="Dose Comparison", ylab="p-value", pch=2, ylim=c(0,1.0))
points(pBonDose, pch=17, col= "blue"); abline(h=0.05, col="black")
mtext("B", side=3, adj=0, font=2, cex=2); axis(1, at=c(1,2,3,4,5,6), labels=axisDose, tick=F)
P-values for all t-tests before (open symbols) and after (closed symbols) Bonferroni adjustment for supplement type (A) and doseage (B). Line indicates alpha=0.05

P-values for all t-tests before (open symbols) and after (closed symbols) Bonferroni adjustment for supplement type (A) and doseage (B). Line indicates alpha=0.05

For the supplement type comparisons (OJ vs. VC), we can see that none of the tests met the required \(\alpha\) level, even prior to correcting for multiple comparisons. This suggests that any differences in the OJ group vs. the VC group (for the same doseage) are likely due to chance. In comparing the doseage levels, we see that only the VC 0.5 vs. VC 2.0 comparison met the required \(\alpha\) level. This suggests that, while differences in group means exist when comparing the doseage levels, these differences are all likely due to chance (except the difference between the 0.5 and 2.0 mg/day doseages in the VC group.)

Conclusions

The dataset detailed in this study examing the tooth length in 60 guinea pigs treated with Vitamin C. The groups differed by supplement type (Ascorbic Acid vs. Orange Juice) and doseage level (0.5, 1.0, or 2.0 mg/day). Each subgroup contained a total of only n=10 subjects. Given the small gorup size, we know that statistical power will be low and any differences in means would have to be substantial in order to reach the required \(\alpha\) level (0.05). We assume all groupings are independent (no subjects overlapped between the groups). In comparing the two supplement types, none of the comparisons were statistically significant, meaning we cannot make a determination as to whether Ascorbic acid or orange juice is the better administration route. In comparing the doseage levels, 3/6 comparisons achieved \(\alpha\) < 0.05, but after correcting for multiple comparisons only one remained statistically significant. This suggests increasing the dose of Vitamin C can result in increased tooth growth, but only when comparing the lowest and highest doses for the VC group. Repeating this study with a larger sample size (and greater statistical power) might allow more definitive conclusions to be made. In addition, the inclusion of a control/placebo group would be prudent if attempting to demonstrate a dose effect.