Inferential Analysis of Supplement Efficacy with R

Inferential Data Analysis with ToothGrowth Dataset

Exploratory Analysis

summary(data)
## len supp dose
## Min. : 4.2 OJ:30 Min. :0.50
## 1st Qu.:13.1 VC:30 1st Qu.:0.50
## Median :19.2 Median :1.00
## Mean :18.8 Mean :1.17
## 3rd Qu.:25.3 3rd Qu.:2.00
## Max. :33.9 Max. :2.00

We can see that there are three columns in the ToothGrowth dataset: tooth length, supplement name, and dosage amount. It is clear to see only two supplements are being tested, abbreviated OJ and VC, and three different dosages with 10 data points each at values 0.5, 1.0, and 2.0.


We can see from this exploratory plot that there may be a correlation between dosage and tooth length, and
note that both treatments seem to perform similarly well, increasing as the dosage does.

 

Comparing Supplements and Dosages

Since there are only ten samples within each supplement/dose combination, we will use t-testing to analyze the significance of the data. Our null hypothesis H_0 is that the difference in means of the two supplements at a given dose is 0.

vc <- group_by(select(filter(data, data$supp == "VC"), len, dose), dose)
oj <- group_by(select(filter(data, data!$supp == "OJ"), len, dose), dose)
dose.small <- t.test(vc$len[vc$dose == 0.5], oj$len[oj$dose == 0.5])
dose.med <- t.test(vc$len[vc$dose == 1.0], oj$len[oj$dose == 1.0])
dose.large <- t.test(vc$len[vc$dose == 2.0], oj$len[oj$dose == 2.0])

The above tests are performed given unequal variances, unpaired data, and using a two-sided null hypothesis.

0.5 Dosage

In comparing VC vs OJ tooth length at a dosage of 0.5, we find a 95% confidence interval of [-8.7809, -1.7191] with a p-value of 0.0064. Thus at a dosage of 0.5 we reject the null hypothesis because there is a statistically significant difference in mean values between supplements, with OJ performing better.

1.0 Dosage

At a dosage of 1.0, we find a 95% confidence interval of [-9.0579, -2.8021] with a p-value of 0.001. Once again, we reject the null hypothesis, as OJ’s increase in tooth length is statistically significant given α = 0.05.

2.0 Dosage

At a dosage of 1.0, we find a 95% confidence interval of [-3.6381, 3.7981] with a p-value of 0.9639. This confidence interval includes 0, as well as has an enormous p-value, therefore we fail to reject the null hypothesis at this larger dosage.

Conclusions

Assuming a null hypothesis H 0 that the difference in means between the supplements at a given dose is 0, the OJ supplement increases tooth growth by a statistically significant margin compared to the VC supplement when the dosage concerned is 0.5 or 1.0. At a dosage of 2.0 our confidence interval includes our null hypothesis, so one cannot say that one is better than the other. It may be that the OJ supplement suffers diminishing returns, or that VC reaches a higher efficacy at this higher dosage.

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