The Effect of Vitamin C on Tooth Growth in Guinea Pigs
Synopsis
In this report we explore and analyze the ToothGrowth data in the R datasets package and infer statistically by performing hypothesis tests to compare tooth growth by supp and dose.
Load Data and Perform Basic Exploratory Data Analysis
data(ToothGrowth)
str(ToothGrowth)## '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 ...unique(ToothGrowth$dose)## [1] 0.5 1.0 2.0From the basic exploratory data analyses above and references provided in the appendix section below, we see that there are:
* 60 observations with each observation collected from one of the 60 guinea pigs.
* 2 types of supplements: OJ (orange juice) and VC (ascorbic acid) with 30 observations each.
* 3 dose levels: 0.5, 1.0, and 2.0, unit in milligrams per day.
For more basic exploratory data analyses with plots, please refer to the appendix section below.
Confidence Intervals and Hypothesis Tests
Test by Supplement Type - OJ vs. VC
# Test: OJ vs. VC
t <- t.test(len ~ supp, data = ToothGrowth, paired = FALSE, var.equal = FALSE)
t$p.value## [1] 0.06063451t$conf.int[1:2]## [1] -0.1710156 7.5710156Result: With P-value of 6.063% and 95% confidence interval of (-0.171, 7.571) for mean(OJ)-mean(VC), we can’t reject the null hypothesis that there is no significant difference in tooth length between the two supplement types.
Test by Dose Level - 0.5 vs. 2.0, 1.0 vs. 2.0, and 0.5 vs. 1.0, respectively
# Test: Dose level comparison - 0.5 vs. 2.0
doseL1L3 <- subset(ToothGrowth, dose %in% c(0.5, 2.0))
t <- t.test(len ~ dose, data = doseL1L3, paired = FALSE, var.equal = FALSE)
t$p.value## [1] 4.397525e-14t$conf.int[1:2]## [1] -18.15617 -12.83383Result: With P-value of < 5% and 95% confidence interval of (-18.156 -12.834) for mean(0.5)-mean(2.0), we reject the null hypothesis that there is no significant difference in tooth length between the two dose levels. Dose level at 2.0 mg results in significantly longer tooth length compared to that at 0.5 mg.
# Test: Dose level comparison - 1.0 vs. 2.0
doseL2L3 <- subset(ToothGrowth, dose %in% c(1.0, 2.0))
t <- t.test(len ~ dose, data = doseL2L3, paired = FALSE, var.equal = FALSE)
t$p.value## [1] 1.90643e-05t$conf.int[1:2]## [1] -8.996481 -3.733519Result: With P-value of < 5% and 95% confidence interval of (-8.996 -3.734) for mean(1.0)-mean(2.0), we reject the null hypothesis that there is no significant difference in tooth length between the two dose levels. Dose level at 2.0 mg results in significantly longer tooth length compared to that at 1.0 mg.
# Test: Dose level comparison - 0.5 vs. 1.0
doseL1L2 <- subset(ToothGrowth, dose %in% c(0.5, 1.0))
t <- t.test(len ~ dose, data = doseL1L2, paired = FALSE, var.equal = FALSE)
t$p.value## [1] 1.268301e-07t$conf.int[1:2]## [1] -11.983781 -6.276219Result: With P-value of < 5% and 95% confidence interval of (-11.984 -6.276) for mean(0.5)-mean(1.0), we reject the null hypothesis that there is no significant difference in tooth length between the two dose levels. Dose level at 1.0 mg results in significantly longer tooth length compared to that at 0.5 mg.
Test by Supplement Type and Dose Level
# Test: OJ vs. VC on dose at 0.5 mg
doseL1 <- subset(ToothGrowth, dose == 0.5)
t <- t.test(len ~ supp, data = doseL1, paired = FALSE, var.equal = FALSE)
t$p.value## [1] 0.006358607t$conf.int[1:2]## [1] 1.719057 8.780943Result: With P-value of < 5% and 95% confidence interval of (1.719 8.781) for mean(OJ)-mean(VC), we reject the null hypothesis that there is no significant difference in tooth length between the two supplement types. Using OJ with dose level at 0.5 mg results in significantly longer tooth length compared to using VC with the same dose level.
# Test: OJ vs. VC on dose at 1.0 mg
doseL2 <- subset(ToothGrowth, dose == 1.0)
t <- t.test(len ~ supp, data = doseL2, paired = FALSE, var.equal = FALSE)
t$p.value## [1] 0.001038376t$conf.int[1:2]## [1] 2.802148 9.057852Result: With P-value of < 5% and 95% confidence interval of (2.802 9.058) for mean(OJ)-mean(VC), we reject the null hypothesis that there is no significant difference in tooth length between the two supplement types. Using OJ with dose level at 1.0 mg results in significantly longer tooth length compared to using VC with the same dose level.
# Test: OJ vs. VC on dose at 2.0 mg
doseL3 <- subset(ToothGrowth, dose == 2.0)
t <- t.test(len ~ supp, data = doseL3, paired = FALSE, var.equal = FALSE)
t$p.value## [1] 0.9638516t$conf.int[1:2]## [1] -3.79807 3.63807Result: With P-value of 96.385% and 95% confidence interval of (-3.798 3.638) for mean(OJ)-mean(VC), we can’t reject the null hypothesis that there is no significant difference in tooth length between the two supplement types. With dose level at 2.0 mg, there is in no significant difference in tooth length between OJ and VC.
Conclusions
- There is no significant difference in teeth length between the two supplement types when dose level is not considered.
- Higher dose level of either supplement type results in longer tooth length.
- At dose levels of 0.5 mg and 1.0 mg, OJ results in longer teeth than VC at the same respective levels. However, at dose level of 2.0 mg, there is no significant difference in teeth length between the two supplement types.
Assumptions
- The populations are independent. In fact, they should be independent as each guinea pig should received only one type of supplement at one dose level everyday during the whole period of experiment. None of them should be “reused” to receive different supplement type or dose level. Hence, “paired = FALSE” was used for all the t-tests.
- The guinea pigs are independent and identically distributed. All have the similar age, similar physical and health condition, and were raised under the same environment.
- The variances between the groups being tested are different. i.e. Unequal variances. Hence, “var.equal = FALSE” was used for all the t-tests.
Appendix
References
R documentation of ToothGrowth data.
Description improvement of ToothGrowth data.
Plots for Basic Exploratory Data Analyses
# Tooth length by supp type
ggplot(ToothGrowth, aes(x = supp, y = len, fill = supp)) +
geom_boxplot() +
labs(title = "Guinea Pigs Tooth Growth Analysis",
x = "Supp Type - Orange Juice (OJ) vs. Ascorbic Acid (VC)",
y = "Length of Tooth") +
scale_fill_discrete(name = "Supp Type")
# Tooth length by dose
ggplot(ToothGrowth, aes(x = factor(dose), y = len, fill = factor(dose))) +
geom_boxplot()+
labs(title = "Guinea Pigs Tooth Growth Analysis",
x = "Dose (mgs/day)", y = "Length of Tooth") +
scale_fill_discrete(name = "Dose")
# Tooth length by supp type and dose
ggplot(data = ToothGrowth, aes(x = supp, y = len, fill = supp)) +
geom_boxplot() + facet_grid(. ~ dose) +
labs(title = "Guinea Pigs Tooth Growth Analysis\nGrouped by Dose\n(mgs/day)",
x = "Supp Type - Orange Juice (OJ) vs. Ascorbic Acid (VC)",
y = "Length of Tooth") +
scale_fill_discrete(name = "Supp Type")