Now in the second portion of the class, we’re going to analyze the ToothGrowth data in the R datasets package.

Load the ToothGrowth data and perform some basic exploratory data analyses Provide a basic summary of the data. 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) State your conclusions and the assumptions needed for your conclusions.

Some criteria that you will be evaluated on:

Question 1: Did you perform an exploratory data analysis of at least a single plot or table highlighting basic features of the data?

Question 2: Did the student perform some relevant confidence intervals and/or tests?

Question 3: Were the results of the tests and/or intervals interpreted in the context of the problem correctly?

Question 4: Did the student describe the assumptions needed for their conclusions?

Answer 1. Load the ToothGrowth data and perform some basic exploratory data analyses.

library(ggplot2)
## Warning: package 'ggplot2' was built under R version 3.2.2
library(datasets)
x <- ToothGrowth
# convert dose to factor
x$dose <- as.factor(x$dose)

str(x)
## '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: Factor w/ 3 levels "0.5","1","2": 1 1 1 1 1 1 1 1 1 1 ...
##      
##       OJ VC
##   0.5 10 10
##   1   10 10
##   2   10 10
ggplot(data=x, aes(x=dose, y=len, fill=supp)) + geom_bar(stat="identity",) + facet_grid(. ~ supp) + xlab("Dose in miligrams") + ylab("Tooth length") + guides(fill=guide_legend(title="Supplement type"))

ggplot(aes(x=dose, y=len), data=x) + geom_boxplot(aes(fill=dose)) + xlab("Dose in miligrams") + ylab("Tooth length") + guides(fill=guide_legend(title="Dose"))

ggplot(aes(x=supp, y=len), data=x) + geom_boxplot(aes(fill=supp)) + xlab("Supplement type") + ylab("Tooth length") + guides(fill=guide_legend(title="Supplement type"))

Answer 2. Provide a basic summary of the data.

summary(x)
##       len        supp     dose   
##  Min.   : 4.20   OJ:30   0.5:20  
##  1st Qu.:13.07   VC:30   1  :20  
##  Median :19.25           2  :20  
##  Mean   :18.81                   
##  3rd Qu.:25.27                   
##  Max.   :33.90

Answer 3. 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).

t.test(len ~ supp, data = x)
## 
##  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

Null hypothesis can not be rejected as confindence intervals contain zero and p-value is 0.06.

Supplement types seems to have no impact on Tooth growth.

# three groups as per dose level pairs
x.doses_0.5_1.0 <- subset (x, dose %in% c(0.5, 1.0)) 
x.doses_0.5_2.0 <- subset (x, dose %in% c(0.5, 2.0)) 
x.doses_1.0_2.0 <- subset (x, dose %in% c(1.0, 2.0)) 

# Check for dose levels (0.5, 1.0)
t.test(len ~ dose, data = x.doses_0.5_1.0)
## 
##  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
# Check for dose levels (0.5, 2.0)
t.test(len ~ dose, data = x.doses_0.5_2.0)
## 
##  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
# Check for dose levels (1.0, 2.0)
t.test(len ~ dose, data = x.doses_1.0_2.0)
## 
##  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 p-value is less than 0.05 and confindence intervals don’t contian 0. The average toot length increases with an inceasing dose. The null hypothesis can be rejected.

Answer 4. State your conclusions and the assumptions needed for your conclusions.

Conclusions

Supplement type seem to have no impact on tooth growth. Inreasing the dose level leads to increased tooth growth as well.