DATA DESCRIPTION

The response is the length of odontoblasts (teeth) in each of 10 guinea pigs at each of three dose levels of Vitamin C (0.5, 1, and 2 mg) with each of two delivery methods (orange juice or ascorbic acid). We’re going to analyze the ToothGrowth data in the R datasets package by performing the following task: 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 supplement and dose by using Only the techniques from class, (even if there’s other approaches worth considering) 4. State your conclusions and the assumptions needed for your conclusions.

We start the program by loading neccesary library and the ToothGrowth Data to investigate its structure

library(knitr)
library(ggplot2)
library(dplyr)
library(datasets)
library(gridExtra)
data(ToothGrowth)
attach(ToothGrowth)

PRE-PROCESSING AND EXPLORATORY DATA ANALYSIS

The following code will give us a brief on the nature of data we are dealing with.

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 ...
head(ToothGrowth)
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.   :2.000
unique(ToothGrowth$len)
##  [1]  4.2 11.5  7.3  5.8  6.4 10.0 11.2  5.2  7.0 16.5 15.2 17.3 22.5 13.6 14.5
## [16] 18.8 15.5 23.6 18.5 33.9 25.5 26.4 32.5 26.7 21.5 23.3 29.5 17.6  9.7  8.2
## [31]  9.4 19.7 20.0 25.2 25.8 21.2 27.3 22.4 24.5 24.8 30.9 29.4 23.0
unique(ToothGrowth$supp)
## [1] VC OJ
## Levels: OJ VC
unique(ToothGrowth$dose)
## [1] 0.5 1.0 2.0

From the above, our data is a data frame of 60 observations and 3 varaibles vis: len(length), supp (supplements) and dose(dose level administered). We also notice that len and dose are number class whereas supp is a factor variable with two levels: OJ (Orange juice) and VC (vitamin C or ascorbic acid). The summary statistics indicates that the minimum and maximum tooth length is 4.20 and 33.90 respectively with an average of 18.81, minimum and maximum dose level is 0.5 and 2.0 respectively with an average of 1.167, suppliment have a minimum level of 30 for both OJ and VC. We will plot a scater plot to visualize our data.

plot(ToothGrowth, main = "The toothgrowth data", col = c("blue","red", "green"))

It seems that variable dose is rather a factor then a numeric value as seen in its unique value entries. Therefore, it will be converted into a factor variable.

ToothGrowth$dose <- as.factor(ToothGrowth$dose)

We will produce a scatter box plot to depict tooth length by dose and type of adminstration

set.seed(123)
boxcat <- ggplot(ToothGrowth, aes(dose, len)) + 
  geom_boxplot(aes(fill = supp)) + 
  geom_jitter(alpha = I(3/4), aes(color = supp)) + 
  scale_color_manual(values = c("orange","darkgreen")) + 
  theme(legend.position = c(1,0.3), legend.justification = c(1,1)) + 
  labs(title = "Scatter Plot of Tooth Length and Dose Levels",x = "Dose Levels", y = "Tooth Length")

boxcat

The box plots seem to show, increasing the dosage increases the tooth growth. Orange juice is more effective than ascorbic acid for tooth growth when the dosage is 0.5 to 1.0 milligrams per day. Both types of supplements are equally as effective when the dosage is 2.0 milligrams per day. To get a clearer picture a density plot for comparison between Tooth Lengths with respect to Dose Levels and supplement is performed in the following codes below.

# Plotting Marginal Density of `Tooth Lengths`. 

plt <- ggplot(ToothGrowth,aes(len,fill = supp)) + 
  geom_density(alpha = .5) + 
  coord_flip() + 
  scale_fill_manual(values = c("orange","purple")) + 
  theme(legend.position = "none") + 
  labs(title = "Density Plot", y = "Density", x = "Tooth Length")

plt

grid.arrange(boxcat, plt, ncol = 2, nrow = 1, widths = c(4, 2))

FURTHER ANALYSIS:

We will compute the mean and varaince by application method (supp). from our results OJ have a mean of 20.66 and a varaince of 43.63. Ascorbic acid have a mean of 16.96 and a varinace of 68.32.

appmthd <- split(ToothGrowth$len, ToothGrowth$supp)
sapply(appmthd, mean)
##       OJ       VC 
## 20.66333 16.96333
# Varaince of supplement
sapply(appmthd, var)
##       OJ       VC 
## 43.63344 68.32723

We will perform similar analysis for dose level.

dozmean <- split(ToothGrowth$len, ToothGrowth$dose)
sapply(dozmean, mean)
##    0.5      1      2 
## 10.605 19.735 26.100
# Variance for each dose
sapply(dozmean, var) 
##      0.5        1        2 
## 20.24787 19.49608 14.24421

We will perform a plot of the control varaible as a realtion to the target varaible to aid us in formulating our test of hyothesis for this analysis. The following plots produces plots for the measure of the following relationship:

  1. Plot of tooth length (len) against supplement (supp)
  2. box plot of len vs supp
  3. line plot of len vs dose
  4. box plot of len vs dose
  5. box plot of len vs dose and supp interaction effect.
par(mfrow = c(3,3), mar = c(4,4,2,0), oma = c(0,0,2,0))
plot(aggregate(len~supp,ToothGrowth,mean), ylab = "length mean", col = c("blue","red"))
boxplot(len~supp,ToothGrowth,xlab = "supp", ylab = "length", col = c("brown", "darkgreen"))
plot(aggregate(len~dose,ToothGrowth,mean), pch = 19, ylab = "mean length")
lines(aggregate(len~dose,ToothGrowth,mean), col = c("cyan4","red", "blue"))
boxplot(len~dose,ToothGrowth,xlab = "dose", ylab = "length", col = c("magenta","gold", "chocolate") )
boxplot(len~supp+dose,ToothGrowth,xlab = "supp+dose", ylab = "length", col = c("darkgoldenrod","coral4", "chocolate1"))
title(main = "Evaluation Of control varaible on target variable",outer = T)

HYPOTHESIS TESTING USING CONFIDENCE INTERVAL:

In this work we are going to evaluate the individual effect of control variables supp and dose on the target variable len, as well as their interaction effect. Assuming that a higher tooth length len value indicates a higher impact and a higher measure of dose indicates a higher dose, a first evaluation of the last plot above yields the following hypotheses : 1. Supp(OJ) has a higher impact on the target variable len. 2. Higher measure of control variable dose effect on target variable len. 3. The combined effect of the control variables supp and dose, shows that OJ has higher impact on target variable len for dose measures of 0.5 and 1. 4. For combined impact of control variables supp and dose, OJ and VC have same impact on target variable len for dose at measure 2mg.

Hypothesis #1:

Supp(OJ) has a higher impact on the target variable len.

HYP1 = t.test(len~supp, paired=F, var.equal=F,data=ToothGrowth)
print(HYP1)
## 
##  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
HYP1$conf.int
## [1] -0.1710156  7.5710156
## attr(,"conf.level")
## [1] 0.95
HYP1$p.value
## [1] 0.06063451

From our result, we notice that the p-Value = 0.0606 is greater than α=0.05 (α for confidence interval of 95%). The confidence interval =(-0.171, 7.571) for the difference of the means of each group spans 0, hence null hypothesis is Failed to Reject. Hence hypothesis one cannot be rejected implying that Orange juice has higher impact on tooth length.

Hypothesis #2:

Higher measure of control variable dose effect on target variable len.

For this hypothesis, we will consider it in three ways: A. That dose measure of 1.0mg has a higher impact on tooth length than dose measure of 0.5mg. B. That dose measure of 2.0mg has a higher impact on tooth length than dose measure of 1.0mg C. That dose measure of 2.0mg has a higher impact on tooth length than dose measure of 0.5mg

2A: dose measure of 1.0mg has a higher impact on tooth length than dose measure of 0.5mg———————————————————-

HYP2A <- t.test(len~dose,paired=F,var.equal=F,data=ToothGrowth[ToothGrowth$dose%in%c(0.5,1),])
print(HYP2A)
## 
##  Welch Two Sample t-test
## 
## data:  len by dose
## t = -6.4766, df = 37.986, p-value = 1.268e-07
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -11.983781  -6.276219
## sample estimates:
## mean in group 0.5   mean in group 1 
##            10.605            19.735
HYP2A$conf.int
## [1] -11.983781  -6.276219
## attr(,"conf.level")
## [1] 0.95
HYP2A$p.value
## [1] 1.268301e-07

From our result, we notice that the p-Value = 1.268301e-07 is less than α=0.05 (α for confidence interval of 95%). The confidence interval =(-11.983781 -6.276219) for the difference of the means of each measure level does not spans 0, hence null hypothesis is Rejected. Hence hypothesis 2A is failed to reject implying that dose level of 1.0mg does not really have higher impact on tooth length than dose level 0.5mg.

#2B: dose measure of 2.0mg has a higher impact on tooth length than dose measure of 1.0mg ———————————————————-

HYP2B <- t.test(len~dose,paired=F,var.equal=F,data=ToothGrowth[ToothGrowth$dose%in%c(1,2),])

print(HYP2B)
## 
##  Welch Two Sample t-test
## 
## data:  len by dose
## 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:
##  -8.996481 -3.733519
## sample estimates:
## mean in group 1 mean in group 2 
##          19.735          26.100
HYP2B$p.value
## [1] 1.90643e-05
HYP2B$conf.int
## [1] -8.996481 -3.733519
## attr(,"conf.level")
## [1] 0.95

From our result, we notice that the p-Value = 1.90643e-05 is less than α=0.05 (α for confidence interval of 95%). The confidence interval =(-8.996481 -3.733519) for the difference of the means of each measure level does not spans 0, hence null hypothesis is Rejected. Hence hypothesis 2B is failed to reject implying that dose level of 2.0mg does not really have higher impact on tooth length than dose level 1.0mg.

2C: dose measure of 0.5mg has a higher impact on tooth length than dose measure of 2.0mg ———————————————————-

HYP2C <- t.test(len~dose,paired=F,var.equal=F,data=ToothGrowth[ToothGrowth$dose%in%c(0.5,2),])

print(HYP2C)
## 
##  Welch Two Sample t-test
## 
## data:  len by dose
## 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:
##  -18.15617 -12.83383
## sample estimates:
## mean in group 0.5   mean in group 2 
##            10.605            26.100
HYP2C$p.value
## [1] 4.397525e-14
HYP2C$conf.int
## [1] -18.15617 -12.83383
## attr(,"conf.level")
## [1] 0.95

From our result, we notice that the p-Value = 4.397525e-14 is less than α=0.05 (α for confidence interval of 95%). The confidence interval =(-18.15617 -12.83383) for the difference of the means of each measure level does not spans 0, hence null hypothesis is Rejected. Hence hypothesis 2C is failed to reject implying that dose level of 2.0mg does not really have higher impact on tooth length than dose level 1.0mg.

By way of conclusion from the above two analysis Hypothesis 2 is Failed to Reject.

Hypothesis #3:

In this hypothesis we are looking at: The combined effect of the control variables supp and dose, to show that OJ has higher impact on target variable len for dose measures of 0.5, 1 and 2. The following code will prepare the data for further analysis.

Dose0.5 <- subset(ToothGrowth, dose %in% c(0.5))
Dose1.0 <- subset(ToothGrowth, dose %in% c(1.0))
Dose2.0 <- subset(ToothGrowth, dose %in% c(2.0))

The Null hypothesis is: There is no correlation between the Delivery Method and Tooth Length for the given Dose Level. And we will consisder it in three different level

3A: “OJ” has higher impact for dose 0.5

HYP3a <- t.test(len~supp,paired = F,var.equal = F, data = Dose0.5)
print(HYP3a)
## 
##  Welch Two Sample t-test
## 
## data:  len by supp
## t = 3.1697, df = 14.969, p-value = 0.006359
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  1.719057 8.780943
## sample estimates:
## mean in group OJ mean in group VC 
##            13.23             7.98
HYP3a$p.value
## [1] 0.006358607
HYP3a$conf.int
## [1] 1.719057 8.780943
## attr(,"conf.level")
## [1] 0.95

From the result above p-Value =0.006358607 is less than α=0.05 (α for confidence interval of 95%). The confidence interval = (1.719057 8.780943) for the difference of the means of the supp and dose = 0.5 does not span 0, hence null hypothesis is Rejected, hence hypothesis 3a is Failed to Reject.

3B: “OJ” has higher impact for dose 1.0mg

HYP3b <- t.test(len~supp,paired = F,var.equal = F, data = Dose1.0)
print(HYP3a)
## 
##  Welch Two Sample t-test
## 
## data:  len by supp
## t = 3.1697, df = 14.969, p-value = 0.006359
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  1.719057 8.780943
## sample estimates:
## mean in group OJ mean in group VC 
##            13.23             7.98
HYP3b$p.value
## [1] 0.001038376
HYP3b$conf.int
## [1] 2.802148 9.057852
## attr(,"conf.level")
## [1] 0.95

From the result above p-Value =0.001038376 is less than α=0.05 (α for confidence interval of 95%), confidence interval = (2.802148 9.057852) for the difference of the means the supp and dose=1.0mg does not span 0, hence null hypothesis is Rejected, implying that hypothesis 3b is Failed to Reject.

3C: “OJ” has higher impact for dose 2.0mg

HYP3c <- t.test(len~supp, paired = F,var.equal = F, data = Dose2.0)
print(HYP3c)
## 
##  Welch Two Sample t-test
## 
## data:  len by supp
## t = -0.046136, df = 14.04, p-value = 0.9639
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -3.79807  3.63807
## sample estimates:
## mean in group OJ mean in group VC 
##            26.06            26.14
HYP3c$p.value
## [1] 0.9638516
HYP3c$conf.int
## [1] -3.79807  3.63807
## attr(,"conf.level")
## [1] 0.95

From the result above p-Value =0.9638516 is greater than α=0.05 (α for confidence interval of 95%), confidence interval = (-3.79807 3.63807) for the difference of the means the supp and dose=2.0mg does span 0, hence null hypothesis does not failed to Rejected, implying that hypothesis 3b is accepted.

By way of conclusion for Hypothesis 3 is Failed to Reject for lower doses of supp and does not fail to reject for higher dose.

Hypothesis #4:

In this hypothesis we are looking at: The combined effect of the control variables supp and dose, to show that OJ and VC has higher impact on target variable len for dose measures of 2.0mg. The following code will produce the result of our analysis.

HYP4 <- t.test(len~supp, paired = F,var.equal = F, data = ToothGrowth[ToothGrowth$dose == 2,])
print(HYP4)
## 
##  Welch Two Sample t-test
## 
## data:  len by supp
## t = -0.046136, df = 14.04, p-value = 0.9639
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -3.79807  3.63807
## sample estimates:
## mean in group OJ mean in group VC 
##            26.06            26.14
HYP4$p.value
## [1] 0.9638516
HYP4$conf.int
## [1] -3.79807  3.63807
## attr(,"conf.level")
## [1] 0.95

From the result above p-Value =0.9638516 is greater than α=0.05 (α for confidence interval of 95%), confidence interval = (-3.79807 3.63807) for the difference of the means the supp and dose=2.0mg does span 0, hence null hypothesis does not failed to Rejected, implying that hypothesis 4 is accepted.

It will be observed that the result for hypothesis four is similar to the result of hypothesis 3c. Hence we will say that supptypes and dose at higher levels does not differ in effect towards tooth length.

CONCLUSIONS:

Based on our analysis the following conclusions are arrived at:

  1. Increase in Supplement Dose Levels leads to overall increase in Tooth Length.
  2. Supplement types has no overall significant impact on Tooth Length, but for 0.5 and 1.0 Dose levels. Orange Juice increases Tooth Length more faster compared to Ascorbic Acid/ Vitamin, but for 2.0mg Dose Level there is no significant difference in the increase of Tooth Length by both Supplement.
  3. For combined impact of control variables, there is significant difference on target variable Tooth length for different values of supplement for dose 0.5 and 1. There is no significant difference for different values of supplement for dose 2.

ASSUMPTIONS NEEDED FOR THE CONCLUSIONS:

  1. Data provided is independently distributed. Members of the sample population, i.e. the 60 guinea pigs, are representative of the entire population of guinea pigs. This assumption allows us to generalize the results.
  2. The experiment was done with random assignment of guinea pigs to different Supplement Dose Level categories and Supplement Delivery Methods to take care of noise that might affect the outcome.
  3. For the t-tests, the variances are assumed to be different for the two groups being compared. This assumption is less stronger than the case in which the variances are assumed to be equal.
    4.Higher value of “length” indicates a higher impact of the supplement.
  4. Higher value of “dose” indicates increased dosages administration.
  5. Data follows T distribution as the observations are limited.