Statistical Inference Course Project - Part 2: Basic Inferential Data Analysis
HH
March 25, 2018
Overview
This project analyzes the ToothGrowth data in the R datasets package. The response of the ToothGrowth data is the length of odontoblasts (cells responsible for 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).
Loading and Summarizing Data
library(datasets); data(ToothGrowth); library(reshape2)
head(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 ...summary(ToothGrowth)## 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. :2.000unique(ToothGrowth$dose)## [1] 0.5 1.0 2.0ToothGrowth$dose <- factor(ToothGrowth$dose)
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: Factor w/ 3 levels "0.5","1","2": 1 1 1 1 1 1 1 1 1 1 ...unique(ToothGrowth$dose)## [1] 0.5 1 2
## Levels: 0.5 1 2A data frame with 60 observations on 3 variables.
- len: Tooth length
- supp: Supplement type (VC or OJ).
- dose: Dose in milligrams/day (0.5, 1, 0r 2)
Assumptions
As Figure-1 in Appendix indicates, it appears that for both supplement types, the increase dose effects the teeth growth Also, it appears that for the dose 0.5 and 1, orange juice has a more siginificant effects than vitamin C, meanwhile for dose 2, supplement type is not correlated with tooth growth.
Hypothesis Tests
Refer the Hypothesis Tests section in Appendix.
Conclusions
According to the hypothesis test results from Test-1, Test-2, Test-3, and Test-4 in Appendix, p-values are less than 0 and all interval confidences are less than 0 (negative value). This indicates for both supplement type, the increase dose has sigificant effect on tooth growth.
According to the hypothesis test results from Test-5 and Test-6 in Appendix, p-values are less than 0.5 and all interval confidences are greater than 0 (positive value). This indicates for the dose 0.5 milligrams/day and 1 milligrams/day, orange juice has a more siginificant effect than vitamin C.
According to the hypothesis test results from Test-7 in Appendix, p-values are greater than 0.5 and Confidence Interval contains 0. This indicates for the dose 2 milligrams/day, supplement type is not correlated with tooth growth.
Appendix
Exploratory data analysis
library(ggplot2)
g <- ggplot(ToothGrowth, aes(x = dose, y = len,
colour = supp))
g <- g + geom_boxplot()
g <- g + stat_summary(aes(group = 1), geom = "line", fun.y = median, size = 1, col = "black")
g <- g + facet_grid(. ~ supp)+
labs(x="Dose in milligrams/day", y= "Tooth length",
title="Figure-1: The Effect of Vitamin C on Tooth Growth in Guinea Pigs")
g
Hypothesis Tests
Test-1
OJ <- subset(ToothGrowth, supp == "OJ")
OJ_d05_1 <- subset(OJ, dose %in% c(0.5,1))
OJ_d05_1_test.eq.true = t.test(len ~ dose, paired = FALSE, var.equal = TRUE, data = OJ_d05_1)
OJ_d05_1_test.eq.false = t.test(len ~ dose, paired = FALSE, var.equal = FALSE, data = OJ_d05_1)
rbind(
list(OJ_d05_1_test.eq.true$p.value,
OJ_d05_1_test.eq.true$conf.int[1], OJ_d05_1_test.eq.true$conf.int[2]),
list(OJ_d05_1_test.eq.false$p.value,
OJ_d05_1_test.eq.false$conf.int[1], OJ_d05_1_test.eq.false$conf.int[2])
)## [,1] [,2] [,3]
## [1,] 8.357559e-05 -13.41081 -5.529186
## [2,] 8.784919e-05 -13.41563 -5.524366Test-2
OJ_d1_2 <- subset(OJ, dose %in% c(1,2))
OJ_d1_2_test.eq.true = t.test(len ~ dose, paired = FALSE, var.equal = TRUE, data = OJ_d1_2)
OJ_d1_2_test.eq.false = t.test(len ~ dose, paired = FALSE, var.equal = FALSE, data = OJ_d1_2)
rbind(
list(OJ_d1_2_test.eq.true$p.value,
OJ_d1_2_test.eq.true$conf.int[1], OJ_d1_2_test.eq.true$conf.int[2]),
list(OJ_d1_2_test.eq.false$p.value,
OJ_d1_2_test.eq.false$conf.int[1], OJ_d1_2_test.eq.false$conf.int[2])
)## [,1] [,2] [,3]
## [1,] 0.0373628 -6.500502 -0.2194983
## [2,] 0.03919514 -6.531443 -0.1885575Test-3
VC <- subset(ToothGrowth, supp == "VC")
VC_d05_1 <- subset(VC, dose %in% c(0.5,1))
VC_d05_1_test.eq.true = t.test(len ~ dose, paired = FALSE, var.equal = TRUE, data = VC_d05_1)
VC_d05_1_test.eq.false = t.test(len ~ dose, paired = FALSE, var.equal = FALSE, data = VC_d05_1)
rbind(
list(VC_d05_1_test.eq.true$p.value,
VC_d05_1_test.eq.true$conf.int[1], VC_d05_1_test.eq.true$conf.int[2]),
list(VC_d05_1_test.eq.false$p.value,
VC_d05_1_test.eq.false$conf.int[1], VC_d05_1_test.eq.false$conf.int[2])
)## [,1] [,2] [,3]
## [1,] 6.492265e-07 -11.26435 -6.315654
## [2,] 6.811018e-07 -11.26571 -6.314288Test-4
VC_d1_2 <- subset(VC, dose %in% c(1,2))
VC_d1_2_test.eq.true = t.test(len ~ dose, paired = FALSE, var.equal = TRUE, data = VC_d1_2)
VC_d1_2_test.eq.false = t.test(len ~ dose, paired = FALSE, var.equal = FALSE, data = VC_d1_2)
rbind(
list(VC_d1_2_test.eq.true$p.value,
VC_d1_2_test.eq.true$conf.int[1], VC_d1_2_test.eq.true$conf.int[2]),
list(VC_d1_2_test.eq.false$p.value,
VC_d1_2_test.eq.false$conf.int[1], VC_d1_2_test.eq.false$conf.int[2])
)## [,1] [,2] [,3]
## [1,] 3.397578e-05 -12.96896 -5.77104
## [2,] 9.155603e-05 -13.05427 -5.685733Test-5
dose05 <- subset(ToothGrowth, dose %in% 0.5)
d05_test.eq.true = t.test(len ~ supp, paired = FALSE, var.equal = TRUE, data = dose05)
d05_test.eq.false = t.test(len ~ supp, paired = FALSE, var.equal = FALSE, data = dose05)
rbind(
list(d05_test.eq.true$p.value,
d05_test.eq.true$conf.int[1], d05_test.eq.true$conf.int[2]),
list(d05_test.eq.false$p.value,
d05_test.eq.false$conf.int[1], d05_test.eq.false$conf.int[2])
)## [,1] [,2] [,3]
## [1,] 0.005303661 1.770262 8.729738
## [2,] 0.006358607 1.719057 8.780943Test-6
dose1 <- subset(ToothGrowth, dose %in% 1)
d1_test.eq.true = t.test(len ~ supp, paired = FALSE, var.equal = TRUE, data = dose1)
d1_test.eq.false = t.test(len ~ supp, paired = FALSE, var.equal = FALSE, data = dose1)
rbind(
list(d1_test.eq.true$p.value,
d1_test.eq.true$conf.int[1], d1_test.eq.true$conf.int[2]),
list(d1_test.eq.false$p.value,
d1_test.eq.false$conf.int[1], d1_test.eq.false$conf.int[2])
)## [,1] [,2] [,3]
## [1,] 0.0007807262 2.840692 9.019308
## [2,] 0.001038376 2.802148 9.057852Test-7
dose2 <- subset(ToothGrowth, dose %in% 2)
d2_test.eq.true = t.test(len ~ supp, paired = FALSE, var.equal = TRUE, data = dose2)
d2_test.eq.false = t.test(len ~ supp, paired = FALSE, var.equal = FALSE, data = dose2)
rbind(
list(d2_test.eq.true$p.value,
d2_test.eq.true$conf.int[1], d2_test.eq.true$conf.int[2]),
list(d2_test.eq.false$p.value,
d2_test.eq.false$conf.int[1], d2_test.eq.false$conf.int[2])
)## [,1] [,2] [,3]
## [1,] 0.9637098 -3.722999 3.562999
## [2,] 0.9638516 -3.79807 3.63807