Martin Livingstone 2016-12-09

Synopsis

The ToothGrowth dataset from the R UsingR library has been analysed. Two-sample t-tests were used to compare the effect of vitamin C dosage and delivery method on tooth length.

1 Basic Summary of the Data

The ToothGrowth dataset contains data from a study on the effect of Vitamin C on tooth growth in 60 guinea pigs.

# Read in the ToothGrowth data
library(UsingR); data("ToothGrowth")

#Basic summary of data
str(ToothGrowth); table(ToothGrowth$supp,ToothGrowth$dose)

## '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 ...

##     
##      0.5  1  2
##   OJ  10 10 10
##   VC  10 10 10

There are 60 observations containing 6 different sample sets of n=10, one for each combination of dose (0.5, 1.0, 2.0 mg/day) and supplement (i.e. delivery method) (OJ - Orange Juice; VJ - Asorbic Acid).

2 Basic Exploratory Data Analysis

ggplot(data = ToothGrowth, aes(x = supp, y = len)) +
        geom_boxplot(aes(fill = supp)) + facet_wrap(~dose) + theme_bw()

The above box plot of the ToothGrowth data shows the effect of Vitamin C on tooth length by dose (0.5, 1.0, 2.0 mg/day) and delivery method (supp). The plot suggests the following:

For both delivery methods, tooth length increases as the dosage increases.
For doses of 0.5 and 1.0 mg/day, tooth length is greater when the supplement is OJ rather than VJ. However for a dose of 2.0 mg/day the difference is not significant.

We can use t-tests to see if the differences suggested above are statistically significant.

3 Hypothesis Testing

The null hypoothesis is that the dosage and supplement do not affect tooth length. The alternative hypothesis is that they do: \[H_0: \mu_1 = \mu_2\] \[H_1: \mu_1 \neq \mu_2\] \(H_0\) will be tested using t-tests for each combination of dose and for each supplement type. The level of significance we use is \(\alpha = 0.05\). We assume:

The samples contain independent and identically distibuted random variables.
Because the sample size is small, we assume that the populations are near normally distributed

3.1 Dosage as a Factor - Asorbic Acid (VC)

Is the mean tooth length affected by the dosage when supp is VC?

# Asorbic acid (VC) as the delivery method
vcA <- ToothGrowth %>% filter(dose %in% c(0.5,1.0), supp == "VC")
vcB <- ToothGrowth %>% filter(dose %in% c(1.0,2.0), supp == "VC")
vcC <- ToothGrowth %>% filter(dose %in% c(0.5,2.0), supp == "VC")
rbind(
        t.test(len~dose, paired=F, var.equal=F, data=vcA)$conf.int[1:2],
        t.test(len~dose, paired=F, var.equal=F, data=vcB)$conf.int[1:2],
        t.test(len~dose, paired=F, var.equal=F, data=vcC)$conf.int[1:2] )

##           [,1]       [,2]
## [1,] -11.26571  -6.314288
## [2,] -13.05427  -5.685733
## [3,] -21.90151 -14.418488

\(H_0\) should be rejected for all three dose combinations when supp is VC.

3.2 Dosage as a Factor - Orange Juice (OJ)

Is the mean tooth length affected by the dosage when supp is OJ?

# Orange juice (OJ) as the delivery method
ojA <- ToothGrowth %>% filter(dose %in% c(0.5,1.0), supp == "OJ")
ojB <- ToothGrowth %>% filter(dose %in% c(1.0,2.0), supp == "OJ")
ojC <- ToothGrowth %>% filter(dose %in% c(0.5,2.0), supp == "OJ")
rbind(
       t.test(len~dose, paired=F, var.equal=F, data=ojA)$conf.int[1:2],
       t.test(len~dose, paired=F, var.equal=F, data=ojB)$conf.int[1:2],
       t.test(len~dose, paired=F, var.equal=F, data=ojC)$conf.int[1:2] )

##            [,1]       [,2]
## [1,] -13.415634 -5.5243656
## [2,]  -6.531443 -0.1885575
## [3,] -16.335241 -9.3247594

\(H_0\) should be rejected for all three dose combinations when supp is OJ.

3.3 Delivery Method as a Factor

Is the mean tooth length affected by the delivery method (Asorbic Acid or Orange Juice)?

dose0.5 <- ToothGrowth %>% filter(dose == 0.5)
dose1.0 <- ToothGrowth %>% filter(dose == 1.0)
dose2.0 <- ToothGrowth %>% filter(dose == 2.0)
rbind(
       t.test(len~supp, paired=F, var.equal=F, data=dose0.5)$conf.int[1:2],
       t.test(len~supp, paired=F, var.equal=F, data=dose1.0)$conf.int[1:2],
       t.test(len~supp, paired=F, var.equal=F, data=dose2.0)$conf.int[1:2] )

##           [,1]     [,2]
## [1,]  1.719057 8.780943
## [2,]  2.802148 9.057852
## [3,] -3.798070 3.638070

\(H_0\) should be rejected for doses of 0.5 and 1.0 mg/day. However it should not be rejected for doses of 2.0 mg/day.