Exploratory Data Analysis Utilizing ToothGrowth Data Set in R
Data Analysis Series: Statistical Inference
Courtney D. Shelley
August 22, 2014
Purpose
The purpose of this assignment is to demonstrate an exploratory data analysis suitable for presentation. Data utilized is the ToothGrowth dataset in the R datasets package.
library(datasets)
data = ToothGrowth
observations <- nrow(data)
variables <- ncol(data)
names <- names(data)Description
Effects of Vitamin C on tooth growth in guinea pigs was studied using 10 guinea pigs each receiving three dose levels of Vitamin C: 0.5, 1, and 2 mg, with each of two delivery methods: orange juice (OJ) and ascorbic acid (VC). The resulting dataset consists of 60 observations on 3 variables: tooth length, supplement type (VC or OJ), and dose in milligrams.
Exploratory Analysis
low<-subset(data, data$dose==0.5)
med<-subset(data, data$dose==1.0)
high<-subset(data, data$dose==2.0)
VC<-subset(data, data$supp=="VC")
OJ<-subset(data, data$supp=="OJ")
c(min(low$len), max(low$len), mean(low$len)) # Low dosage min, max, mean## [1] 4.20 21.50 10.61c(min(med$len), max(med$len), mean(med$len)) # Med dosage min, max, mean## [1] 13.60 27.30 19.73c(min(high$len), max(high$len), mean(high$len)) # High dosage min, max, mean## [1] 18.5 33.9 26.1c(min(VC$len), max(VC$len), mean(VC$len)) # Acid dosage min, max, mean## [1] 4.20 33.90 16.96c(min(OJ$len), max(OJ$len), mean(OJ$len)) # Juice dosage min, max, mean## [1] 8.20 30.90 20.66t.test(VC$len,OJ$len, paired=FALSE)$conf## [1] -7.571 0.171
## attr(,"conf.level")
## [1] 0.95par(mfrow=c(1,2))
plot(density(low$len), col="blue", xlim=c(0,40), ylim=c(0,0.15), lwd=2, main="Dosage Amount",
xlab="Tooth Length(mm)")
lines(density(med$len), col="red", lwd=2)
lines(density(high$len), col="green", lwd=2)
legend("topright", pch=16, col=c("blue", "red", "green"),
legend=c("Low (0.5ml)", "Med (1.0ml)", "High (2.0ml"))
plot(density(OJ$len), col="blue", lwd=2, main="Delivery Method",
xlab="Tooth Length (mm)")
lines(density(VC$len), col="red", lwd=2)
legend("topright", pch = 16, col = c("blue", "red"),
legend= c("Orange Juice", "Ascorbic Acid"))
A difference in tooth growth is observed between delivery methods of Vitamin C. Guinea pigs receiving orange juice supplements had tooth growth ranging from 8.20mm to 30.90mm with a mean growth length of 16.96mm. Guinea pigs receiving ascorbic acid supplements had tooth growth ranging from 4.20mm to 33.90mm, with a mean of 16.96. A Gosset’s t-Test does not support a true difference in means between the two groups, as demonstrated by the confidence interval spanning 0. Thus, we must dig deeper into differences caused by dosage level intervals.
anova<-aov(data[,1]~as.factor(data[,2])*as.factor(data[,3]))
summary(anova)## Df Sum Sq Mean Sq F value
## as.factor(data[, 2]) 1 205 205 15.57
## as.factor(data[, 3]) 2 2426 1213 92.00
## as.factor(data[, 2]):as.factor(data[, 3]) 2 108 54 4.11
## Residuals 54 712 13
## Pr(>F)
## as.factor(data[, 2]) 0.00023 ***
## as.factor(data[, 3]) < 2e-16 ***
## as.factor(data[, 2]):as.factor(data[, 3]) 0.02186 *
## Residuals
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1par(mfrow=c(1,2))
lowVC <- subset(VC, VC[,3]==0.5)
medVC <- subset(VC, VC[,3]==1.0)
highVC <- subset(VC, VC[,3]==2.0)
plot(density(highVC$len), col="blue", lwd=2, xlim=c(0,40), ylim=c(0,0.25), main="Ascorbic Acid Dosage Amount",
xlab="Tooth Length (mm)")
lines(density(medVC$len), col="red", lwd=2)
lines(density(lowVC$len), col="green", lwd=2)
legend("topright", pch = 16, col = c("blue", "red", "green"),
legend= c("High (2.0ml)", "Med (1.0ml)", "Low (0.5ml)"))
lowOJ <- subset(OJ, OJ[,3]==0.5)
medOJ <- subset(OJ, OJ[,3]==1.0)
highOJ <- subset(OJ, OJ[,3]==2.0)
plot(density(highOJ$len), col="blue", lwd=2, xlim=c(0,40), ylim=c(0,0.25), main="Orange Juice Dosage",
xlab="Tooth Length (mm)")
lines(density(medOJ$len), col="red", lwd=2)
lines(density(lowOJ$len), col="green", lwd=2)
legend("topright", pch = 16, col = c("blue", "red", "green"),
legend= c("High (2.0ml)", "Med (1.0ml)", "Low (0.5ml)"))
Analysis of variance demonstrates that supplement type significantly predicts tooth growth outcome (p-value = 0+), as does dosage amount (p-value = 0+). Interactions between supplement type and dosage amount are weakly significant (p-value = 0.0219).
Null Hypothesis Significance Testing to Verify Analysis of Variance
To verify the above ANOVA conclusions, confidence intervals were fitted about factor level means.
VC.lm<-lm(VC$len~VC$dose)
summary(VC.lm)##
## Call:
## lm(formula = VC$len ~ VC$dose)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.226 -2.603 0.081 2.229 7.489
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.29 1.43 2.31 0.029 *
## VC$dose 11.72 1.08 10.86 1.5e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.68 on 28 degrees of freedom
## Multiple R-squared: 0.808, Adjusted R-squared: 0.801
## F-statistic: 118 on 1 and 28 DF, p-value: 1.51e-11t.test<-10.860
t.star<-qt(0.975,28)
if(t.test<t.star) print("Conclude H_0, no dosage level effect")
if(t.test>t.star) print("Conclude H_a: Significant dosage level effect")## [1] "Conclude H_a: Significant dosage level effect"p.value <- pt(-abs(t.test), 28, lower.tail=TRUE)
OJ.lm<-lm(OJ$len~OJ$dose)
summary(OJ.lm)##
## Call:
## lm(formula = OJ$len ~ OJ$dose)
##
## Residuals:
## Min 1Q Median 3Q Max
## -7.256 -3.798 -0.064 3.352 7.939
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 11.55 1.72 6.71 2.8e-07 ***
## OJ$dose 7.81 1.30 6.00 1.8e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.45 on 28 degrees of freedom
## Multiple R-squared: 0.563, Adjusted R-squared: 0.547
## F-statistic: 36 on 1 and 28 DF, p-value: 1.82e-06t.test<-6.001
t.star<-qt(0.975,28)
if(t.test<t.star) print("Conclude H_0, no dosage level effect")
if(t.test>t.star) print("Conclude H_a: Significant dosage level effect")## [1] "Conclude H_a: Significant dosage level effect"p.value <- pt(-abs(t.test), 28, lower.tail=TRUE)t.test(low$len, med$len, paired=FALSE)$conf## [1] -11.984 -6.276
## attr(,"conf.level")
## [1] 0.95t.test(low$len, high$len, paired=FALSE)$conf## [1] -18.16 -12.83
## attr(,"conf.level")
## [1] 0.95t.test(med$len, high$len, paired=FALSE)$conf## [1] -8.996 -3.734
## attr(,"conf.level")
## [1] 0.95Conclusions and Assumptions
While tooth growth attributable to differing treatment types (orange juice vs. ascorbic acid) appears to differ significantly, a simple Gosset’s t-Test does not support this conclusion. Separating data by dosage amount and retesting does show significant differences in tooth growth attributable to dosage amounts. Reanalysis favoring dosage amounts over treatment type is recommended.