Does Orange Juice or Ascorbic Acid and Dosage Levels Effect How Fast Teeth Grow?

Author: Russ Robbins

Code Repository (right click, open new window or tab)

Special Note: Despite my use of this odd data set, this work shows a firm understanding of comparing two approaches and determining whether each had an effect that couldn’t happen just by chance. Thanks for bearing with me.




Overview

In this data, with several caveats, it is found that increasing a 0.5mm dose to a 1mm does have a greater effect on teeth growing than a dose change from 1mm to 2mm.

In real life, a comparable finding could be that we discovered that increasing the number of ads from 500 to 1000, in a period of time, perhaps to a particular customers, has a greater effect on customers purchasing a streaming service than increasing the number of ads from 1000 to 2000.

Subjects

There were six different groups of ten guinea pigs each.

Measurements

The before and after length of teeth in millimeters.

Approaches

Question 1: Compare two delivery mechanisms. Vitamin C provided using two differing delivery mechanisms (orange juice and absorbic acid).

Question 2: Compare three different dosage levels (0.5, 1, and 2 milligrams).

Answers

Question 1: Did guinea pigs that received vitamin C via orange juice grow teeth faster than those who received vitamin C in ascorbic acid?

Answer: It is possible that guinnea pig populations that receive Vitamin C via orange juice or via ascorbic acid vary in their tooth growth rates. However, the Vitamin C delivery mechanism data in this data set do not show a statistically significant difference in the growth rates.

Question 2: Did guinea pigs that received different dosage levels (0.5, 1, 2 milligrams) grow teeth at different rates?

Answer: It is entirely possible that guinnea pig populations that are provided Vitamin C at 2 mm per dose have faster growing teeth then those given 1 mm per dose. This relationship also holds between guinea pigs with 1 mm vs. 005 mm dose amounts.


At this point, for simplicity, I will stop showing how what I did with guinea pigs can also be done with customers.


Caveats

Note that these results are tentative because of the following reason. The sample sizes, ten in any group, is very small. Larger sample sizes would have provided more information that could be relied upon to make stronger claims. While the possible error related to accepting the null hypotheses incorrectly (that there wasn’t a difference between any two groups) was considered, the possible error related to accepting the alternative hypothesis incorrectly, was not considered. It was not considered, because the standard method for addressing this risk, is increasing sample size. In this situation, the data was provided to I the analyst, and I did not have access to the experiment which occurred before 1952, which was fourteen years before I was born. Nonetheless, in an ideal world, I would have rerun the experiment with many more guinnea pigs.

Technical Discussion

This section discusses the results after testing six hypotheses. The results are summarized in the table below.

# Null Alt t-score t-crit p-value alpha conf-int Reject Null?
1 OJ = VC OJ not = VC 1.92 2.00 0.06 .05 -0.16, 7.56 no
2 OJ <= VC OJ > VC 1.92 1.67 .03 .05 0.4708, Inf yes
3 2mm = 1mm 2mm not = 1mm 4.90 2.00 <.00001 .05 3.74, 8.99 yes
4 2mm <= 1mm 2mm > 1mm 4.90 1.67 <.00001 .05 4.17, Inf yes
5 1mm = 0.5mm 1mm not = 0.5mm 6.48 2.00 <.00001 .05 6.28, 11.98 yes
6 1mm <= 0.5mm 1mm > 0.5mm 6.48 1.67 <.000001 .05 6.75, Inf yes

Table 1: Hypotheses Tests Results

Hypothesis 1: Did guinea pigs that received vitamin C via orange juice grow teeth faster than those who received vitamin C in ascorbic acid?

No. When the two groups’ averages were compared, they were not statistically different, since t-score is not greater than t-critical, the p-value is greater than alpha, and mean difference of zero is within the confidence interval.

Hypothesis 2: Was the mean for the subjects receiving orange juice greater than the mean of those subjects receiving ascorbic acid?

A small effect was seen, as evidenced by the t-score that is somewhat greater than the t-critical, the p-value which is somewhat less than alpha, and the fact that a mean difference of zero was not in the confidence interval. However, the confidence interval of prospective mean differences did reach close to zero. Since the confidence interval almost included zero, we should be careful about indicating that there is an effect, expecially since Hypothesis 1 was found faulty.

Hypothesis 3: Did the groups receiving 2mm and 1mm doses have a difference in mean tooth growth?

Yes. The two groups (those receiving 2mm and those receiving 1mm doses) did not have a difference of means that could be explained only by chance. The t-score value was greater than the t-critical value, the p-value was less than the alpha value, and the estimated difference in means ranged from 3.74 mm to 8.99 mm.

Hypothesis 4: Did the groups receiving 2mm have greater mean tooth growth than subjects that received 1mm doses?

Yes. The t-score value was greater than the t-critical value, the p-value was less than the alpha value, and the estimated difference in means was greater than 4.17 mm.

Hypothesis 5: Did subjects receiving 2mm and those receiving 1mm doses have a statistically significant difference of means?

Yes. The t-score value was greater than the t-critical value, the p-value was less than the alpha value, and the estimated difference in means ranged from 6.28 mm to 11.98 mm.

Hypothesis 6: Did the groups receiving 1mm have greater mean tooth growth than subjects that received 0.5mm doses?

Yes, and the test indicated that the difference between 0.5 mm and 1 mm doses may have a larger effect than the doses ranging from 1mm to 2mm.

Code

  1. Load the toothgrowth data
library(knitr); data(ToothGrowth); attach(ToothGrowth);
## Warning: package 'knitr' was built under R version 3.2.1
unique(supp); unique(dose)
## [1] VC OJ
## Levels: OJ VC
## [1] 0.5 1.0 2.0
  1. Provide a basic summary of the data.
summary(ToothGrowth)
##       len        supp         dose      
##  Min.   : 4.20   OJ:30   Min.   :0.500  
##  1st Qu.:13.07   VC:30   1st Qu.:0.500  
##  Median :19.25           Median :1.000  
##  Mean   :18.81           Mean   :1.167  
##  3rd Qu.:25.27           3rd Qu.:2.000  
##  Max.   :33.90           Max.   :2.000
  1. Use confidence intervals and/or hypothesis tests to compare tooth growth by supp and dose. (Only use the techniques from class, even if there’s other approaches worth considering).
VC<-subset(ToothGrowth,supp=="VC"); OJ<-subset(ToothGrowth,supp=="OJ")
Zero.5<-subset(ToothGrowth,dose==0.5); One<-subset(ToothGrowth,dose==1.0)
Two<-subset(ToothGrowth,dose==2.0)
meanVC<-mean(VC$len); meanOJ<-mean(OJ$len)
meanZero.5<-mean(Zero.5$len); meanOne<-mean(One$len); meanTwo<-mean(Two$len)

Hypothesis 1:   |  Null: meanOJ - meanVC = 0   |  Alternative: Mean-OJ - Mean_VC != 0

ojEqVc<-t.test(x=OJ$len,y=VC$len,var.equal=TRUE,conf.level=0.95,paired=FALSE,alternative=c("two.sided"))
ojEqVc
## 
##  Two Sample t-test
## 
## data:  OJ$len and VC$len
## t = 1.9153, df = 58, p-value = 0.06039
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -0.1670064  7.5670064
## sample estimates:
## mean of x mean of y 
##  20.66333  16.96333

Hypothesis 2:   |  Null: meanOJ - meanVC <= 0   |  Alternative: meanOJ - meanVC > 0

ojGtVc<-t.test(x=OJ$len,y=VC$len,var.equal=TRUE,conf.level=0.95,paired=FALSE,alternative=c("greater"))
ojGtVc
## 
##  Two Sample t-test
## 
## data:  OJ$len and VC$len
## t = 1.9153, df = 58, p-value = 0.0302
## alternative hypothesis: true difference in means is greater than 0
## 95 percent confidence interval:
##  0.4708204       Inf
## sample estimates:
## mean of x mean of y 
##  20.66333  16.96333

Hypothesis 3:   |  Null: mean2 - mean1 = 0   |  Alternative: Mean-2 - Mean_1 != 0

twoEqOne<-t.test(x=Two$len,y=One$len,var.equal=TRUE,conf.level=0.95,paired=FALSE,alternative=c("two.sided"))
twoEqOne
## 
##  Two Sample t-test
## 
## data:  Two$len and One$len
## t = 4.9005, df = 38, p-value = 1.811e-05
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  3.735613 8.994387
## sample estimates:
## mean of x mean of y 
##    26.100    19.735

Hypothesis 4:   |  Null: mean2 - mean1 <= 0   |  Alternative: Mean-2 - Mean_1 > 0

twoGtOne<-t.test(x=Two$len,y=One$len,var.equal=TRUE,conf.level=0.95,paired=FALSE,alternative=c("greater"))
twoGtOne
## 
##  Two Sample t-test
## 
## data:  Two$len and One$len
## t = 4.9005, df = 38, p-value = 9.054e-06
## alternative hypothesis: true difference in means is greater than 0
## 95 percent confidence interval:
##  4.175196      Inf
## sample estimates:
## mean of x mean of y 
##    26.100    19.735

Hypothesis 5:  |  Null: mean-1 - mean-Zero.5 = 0   |  Alternative: Mean-1 - Mean-Zero.5 != 0

oneEqZero.5<-t.test(x=One$len,y=Zero.5$len,var.equal=TRUE,conf.level=0.95,paired=FALSE,alternative=c("two.sided"))
oneEqZero.5
## 
##  Two Sample t-test
## 
## data:  One$len and Zero.5$len
## t = 6.4766, df = 38, p-value = 1.266e-07
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##   6.276252 11.983748
## sample estimates:
## mean of x mean of y 
##    19.735    10.605

Hypothesis 6:   |  Null: mean-1 - mean-Zero.5 <= 0   |  Alternative: Mean-1 - Mean-Zero.5 > 0

oneGtZero.5<-t.test(x=One$len,y=Zero.5$len,var.equal=TRUE,conf.level=0.95,paired=FALSE,alternative=c("greater"))
oneGtZero.5
## 
##  Two Sample t-test
## 
## data:  One$len and Zero.5$len
## t = 6.4766, df = 38, p-value = 6.331e-08
## alternative hypothesis: true difference in means is greater than 0
## 95 percent confidence interval:
##  6.753344      Inf
## sample estimates:
## mean of x mean of y 
##    19.735    10.605