Load Data and Exploration

R Gives the Following description of the data

“The response is the length of odontoblasts (teeth) in each of 10 guinea pigs at each of three dose levels of Vitamin C (0.5, 1, and 2 mg) with each of two delivery methods (orange juice or ascorbic acid).”

library(ggplot2)

Tooth <- datasets::ToothGrowth
Tooth
##     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
## 7  11.2   VC  0.5
## 8  11.2   VC  0.5
## 9   5.2   VC  0.5
## 10  7.0   VC  0.5
## 11 16.5   VC  1.0
## 12 16.5   VC  1.0
## 13 15.2   VC  1.0
## 14 17.3   VC  1.0
## 15 22.5   VC  1.0
## 16 17.3   VC  1.0
## 17 13.6   VC  1.0
## 18 14.5   VC  1.0
## 19 18.8   VC  1.0
## 20 15.5   VC  1.0
## 21 23.6   VC  2.0
## 22 18.5   VC  2.0
## 23 33.9   VC  2.0
## 24 25.5   VC  2.0
## 25 26.4   VC  2.0
## 26 32.5   VC  2.0
## 27 26.7   VC  2.0
## 28 21.5   VC  2.0
## 29 23.3   VC  2.0
## 30 29.5   VC  2.0
## 31 15.2   OJ  0.5
## 32 21.5   OJ  0.5
## 33 17.6   OJ  0.5
## 34  9.7   OJ  0.5
## 35 14.5   OJ  0.5
## 36 10.0   OJ  0.5
## 37  8.2   OJ  0.5
## 38  9.4   OJ  0.5
## 39 16.5   OJ  0.5
## 40  9.7   OJ  0.5
## 41 19.7   OJ  1.0
## 42 23.3   OJ  1.0
## 43 23.6   OJ  1.0
## 44 26.4   OJ  1.0
## 45 20.0   OJ  1.0
## 46 25.2   OJ  1.0
## 47 25.8   OJ  1.0
## 48 21.2   OJ  1.0
## 49 14.5   OJ  1.0
## 50 27.3   OJ  1.0
## 51 25.5   OJ  2.0
## 52 26.4   OJ  2.0
## 53 22.4   OJ  2.0
## 54 24.5   OJ  2.0
## 55 24.8   OJ  2.0
## 56 30.9   OJ  2.0
## 57 26.4   OJ  2.0
## 58 27.3   OJ  2.0
## 59 29.4   OJ  2.0
## 60 23.0   OJ  2.0
str(Tooth)
## '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 ...
summary(Tooth)
##       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

Analysis

First I make dose a factor variable so it can be used by ggplot. Then the analysis starts by comparing the resulting lengths in relationship to suppplement type. The Violin plots are not necessary but I didn’t want to remove them ( show same thing as scatters).

Tooth$dose <- as.factor(Tooth$dose)

ggplot(Tooth, aes(x = supp, y = len)) +
        geom_point(aes(col= supp)) + 
        xlab("Supplement Type") +
        ylab("Tooth Length") +
        ggtitle("OJ vs Asorbic Acid")

ggplot(Tooth, aes(x = supp, y = len)) +
        geom_violin(aes(fill=supp))+
        geom_boxplot(width= 0.1 ,aes(fill= supp)) +
        xlab("Supplement Type") +
        ylab("Tooth Length") +
        ggtitle("OJ vs Asorbic Acid")

Next to see if dose seems to have an effect. There definitly seems to be a relationship.

ggplot(Tooth, aes(x = dose, y = len)) +
        geom_point(aes(col= dose)) + 
        xlab("Dose (mg)") +
        ylab("Tooth Length") +
        ggtitle("OJ vs Asorbic Acid")

ggplot(Tooth, aes(x = dose, y = len))+
        geom_violin(aes(fill= dose)) +
        geom_boxplot(width= 0.1,aes(fill= dose)) +
        xlab("Dose (mg)") +
        ylab("Tooth Length") +
        ggtitle("Effect of Supplement Size")

Next, a panel plot taking both dose and supplement type into account.

ggplot(Tooth, aes(x = supp, y = len))+
        geom_point(aes(col= dose)) +
        facet_wrap( ~ dose) +
        xlab("Dose (mg)") +
        ylab("Tooth Length") +
        ggtitle("Effect of Supplement Size")

ggplot(Tooth, aes(x = supp, y = len))+
        geom_violin(aes(fill= dose)) +
        geom_boxplot(width= 0.1,aes(fill= dose)) +
        facet_wrap( ~ dose) +
        xlab("Dose (mg)") +
        ylab("Tooth Length") +
        ggtitle("Effect of Supplement Size")

T comparison testing

First a t test to evaluate supplement type effect. Looking back to the first figures the relationship does not look confident.

t.test(len ~ supp, Tooth)
## 
##  Welch Two Sample t-test
## 
## data:  len by supp
## t = 1.9153, df = 55.309, p-value = 0.06063
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -0.1710156  7.5710156
## sample estimates:
## mean in group OJ mean in group VC 
##         20.66333         16.96333

The p value is small bu not < 0.05. This means we should not reject the null hypothesis but it is close.

Next to evaluate the dose amount. Again, by looking back on the figures there seems to be a relationship.

t.test( len ~ dose, Tooth[Tooth$dose != 2.0,])
## 
##  Welch Two Sample t-test
## 
## data:  len by dose
## t = -6.4766, df = 37.986, p-value = 1.268e-07
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -11.983781  -6.276219
## sample estimates:
## mean in group 0.5   mean in group 1 
##            10.605            19.735
t.test( len ~ dose, Tooth[Tooth$dose != 1.0,])
## 
##  Welch Two Sample t-test
## 
## data:  len by dose
## t = -11.799, df = 36.883, p-value = 4.398e-14
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -18.15617 -12.83383
## sample estimates:
## mean in group 0.5   mean in group 2 
##            10.605            26.100
t.test( len ~ dose, Tooth[Tooth$dose != 0.5,])
## 
##  Welch Two Sample t-test
## 
## data:  len by dose
## t = -4.9005, df = 37.101, p-value = 1.906e-05
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -8.996481 -3.733519
## sample estimates:
## mean in group 1 mean in group 2 
##          19.735          26.100

The t tests support the rejection of the null hypothesis that the true means are different.

Lastly, to evaluate the supplment types by each group of dosage amounts. These refer to the third set of figures.

t.test( len ~ supp, Tooth[Tooth$dose == 0.5,])
## 
##  Welch Two Sample t-test
## 
## data:  len by supp
## t = 3.1697, df = 14.969, p-value = 0.006359
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  1.719057 8.780943
## sample estimates:
## mean in group OJ mean in group VC 
##            13.23             7.98
t.test( len ~ supp, Tooth[Tooth$dose == 1.0,])
## 
##  Welch Two Sample t-test
## 
## data:  len by supp
## t = 4.0328, df = 15.358, p-value = 0.001038
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  2.802148 9.057852
## sample estimates:
## mean in group OJ mean in group VC 
##            22.70            16.77
t.test( len ~ supp, Tooth[Tooth$dose == 2.0,])
## 
##  Welch Two Sample t-test
## 
## data:  len by supp
## t = -0.0461, df = 14.04, p-value = 0.9639
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -3.79807  3.63807
## sample estimates:
## mean in group OJ mean in group VC 
##            26.06            26.14

Hmm, both the 0.5mg dosage and the 1mg are significantly different. This actually is shown in the figures but also explains why the t test for overall dosage type was almost significantly different.

Conclusion

To conclude there seems to be a relationship woth dosage to tooth growth. There also seems to be a relationship with supplement type but is more apparent at lower dosages. These tests were analzed to the 95% confidence, this means that this has a 1/20 false result rating.