ToothGrowth: T-test Simulation

Synopsis

Data

In this page, I’m going to analyze the ToothGrowth data in the R datasets package.

ToothGrowth data set contains the result from an experiment studying the effect of vitamin C on tooth growth in 60 Guinea pigs. Each animal received one of three dose levels of vitamin C (0.5, 1, and 2 mg/day) by one of two delivery methods, (orange juice or ascorbic acid (a form of vitamin C and coded as VC).

This data set contain three columns.

• len : Tooth length

• supp : Supplement type (it has 2 type, VC for Vitamin C and OJ for Orange Juice)

• dose : Dose of supplement given per mg in 1 day

Processing Steps

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

2. Provide a basic summary of the data.

3. Use confidence intervals and/or hypothesis tests to compare tooth growth by supp and dose.
4. State conclusions and the assumptions.

Loading and Preprocessing Data

Loading the data set

data <- data.frame(ToothGrowth)

Showing some observations and the structure of the data set

# showing the first 5 observations
    head(data)

# showing the last 5 observations
    tail(data)

# showing data set structure
    str(data)

'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 ...

From the result above, we have 60 Observations and 3 variables.

len and dose are numerical value. len is the dependent variable of the experiment, . Otherwise, dose is the one of the control variable of the experiment. Let’s see how many unique value in dose

unique(data$dose)

0.5

1

2

There are 3 different treatment about dosage of the supplement are given, which are 0.5, 1.0, and 2.0. Also, there 2 different treatment on the type of supplement are given.

Showing plot of the experiment result.

# import necessary library
    library(ggplot2)

# create a box plot
    ggplot(data, aes(x=supp, y=len, fill=supp)) + 
        geom_boxplot() +
        facet_wrap(~dose) +
        labs(title = "Tooth Growth Experiment", x = "Supplement Type by Dosage", y = "Tooth Length") +
             scale_fill_discrete(name = "Supplement", breaks = c("OJ", "VC"), 
                                 labels = c("Orange Juice", "Vitamin C"))

Data Analysis

The data analysis examine by using statistical t-test. We will run 4 t-test analysis, one by supp factor variable and 3 by dose treatment. We will see the p-value and Confidence Interval.

t-test by Supplement

with(data, 
     t.test(len~supp, paired = F, var.equal = F))

	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 

We see that p-value is 0.06 (greater that 0.05) and the confidence interval is contain 0. So, we can reject the null hypotheses. There is no significant statistical difference in supplement treatment.

t-test by Dose

with(data, 
     t.test(len[dose == 1.0],
            len[dose == 0.5],
            paired = F, var.equal = F))

	Welch Two Sample t-test

data:  len[dose == 1] and len[dose == 0.5]
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:
  6.276219 11.983781
sample estimates:
mean of x mean of y 
   19.735    10.605 

with(data, 
     t.test(len[dose == 2.0],
            len[dose == 1.0],
            paired = F, var.equal = F))

	Welch Two Sample t-test

data:  len[dose == 2] and len[dose == 1]
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:
 3.733519 8.996481
sample estimates:
mean of x mean of y 
   26.100    19.735 

with(data, 
     t.test(len[dose == 2.0],
            len[dose == 0.5],
            paired = F, var.equal = F))

	Welch Two Sample t-test

data:  len[dose == 2] and len[dose == 0.5]
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:
 12.83383 18.15617
sample estimates:
mean of x mean of y 
   26.100    10.605 

We see that all combinations of dosage level analysis has very small p-value and lower that 0.05. So, we have not enough evidence to reject the null hypotheses. There are statistically significant difference to all combinations of dosage level that give a positive effect to tooth growth.

Conclusions

From the result of Data Analysis section, we conclude that :

1. It seem the type of supplement doesn’t give any effect to subjects tooth growth.
2. The level of supplement dosages give effect to subjects tooth growth. Its mean that the increase of thee dosage level can stimulate subject’s tooth to grow more.