Tooth Growth Analysis

About the data

Reviewing the help file “?ToothGrowth”, we can see the breakdown as follows:

A Study on the effect of two delivery methods Orange Juice or plain old Vitamin C for tooth growth.
Sample size of 60 guinea pigs.
Each animal recieving one of each three doses which are equally divided among 60 guinea pigs.
- 20 receiving .05 mg OJ/VC, 20 receiving 1 mg OJ/VC, and 20 receiving 2 mg OJ/VC
The reference link is a paywall but provides us with;
- a title name “The Growth of the Odontoblasts of the Incisor Tooth as a Criterion of the Vitamin C Intake of the Guinea Pig: Five Figures”
- Date Published: 01 May 1947.
- First page of the article summarises “Canadian Government undertaking study to provide good source of vitamin c for soldiers during war time.”

str(ToothGrowth) provided the technical information of the dataframe.

Environment

library(tidyverse)
library(ggpubr)
library(glue)
library(cowplot)
library(tinytex)
library(matrixStats)
downloadDate <- date()

glue("Simulation date is {downloadDate}")

## Simulation date is Thu May 21 00:08:26 2020

sessionInfo()

## R version 4.0.0 (2020-04-24)
## Platform: x86_64-apple-darwin17.0 (64-bit)
## Running under: macOS Catalina 10.15.5
## 
## Matrix products: default
## BLAS:   /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRblas.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRlapack.dylib
## 
## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
##  [1] matrixStats_0.56.0 tinytex_0.22       cowplot_1.0.0      glue_1.4.0        
##  [5] ggpubr_0.3.0       forcats_0.5.0      stringr_1.4.0      dplyr_0.8.5       
##  [9] purrr_0.3.4        readr_1.3.1        tidyr_1.0.2        tibble_3.0.1      
## [13] ggplot2_3.3.0      tidyverse_1.3.0   
## 
## loaded via a namespace (and not attached):
##  [1] Rcpp_1.0.4.6      lubridate_1.7.8   lattice_0.20-41   assertthat_0.2.1 
##  [5] digest_0.6.25     R6_2.4.1          cellranger_1.1.0  backports_1.1.6  
##  [9] reprex_0.3.0      evaluate_0.14     httr_1.4.1        pillar_1.4.3     
## [13] rlang_0.4.5       curl_4.3          readxl_1.3.1      rstudioapi_0.11  
## [17] data.table_1.12.8 car_3.0-7         rmarkdown_2.1     foreign_0.8-76   
## [21] munsell_0.5.0     broom_0.5.6       compiler_4.0.0    modelr_0.1.6     
## [25] xfun_0.13         pkgconfig_2.0.3   htmltools_0.4.0   tidyselect_1.0.0 
## [29] rio_0.5.16        fansi_0.4.1       crayon_1.3.4      dbplyr_1.4.3     
## [33] withr_2.2.0       grid_4.0.0        nlme_3.1-147      jsonlite_1.6.1   
## [37] gtable_0.3.0      lifecycle_0.2.0   DBI_1.1.0         magrittr_1.5     
## [41] scales_1.1.0      zip_2.0.4         cli_2.0.2         stringi_1.4.6    
## [45] carData_3.0-3     ggsignif_0.6.0    fs_1.4.1          xml2_1.3.1       
## [49] ellipsis_0.3.0    generics_0.0.2    vctrs_0.2.4       openxlsx_4.1.5   
## [53] tools_4.0.0       hms_0.5.3         abind_1.4-5       yaml_2.2.1       
## [57] colorspace_1.4-1  rstatix_0.5.0     rvest_0.3.5       knitr_1.28       
## [61] haven_2.2.0

Data Loading

attach(ToothGrowth)

Format

ToothGrowth shows 60 observations on 3 variables

len - tooth length
supp - Supplement type (VC or OJ)
dose - in milligrams/day

Using view(toothGrowth) we can see the following:

range 1:30 len have a supp valuing “VC”
range 31:60 len have a supp valuing “OJ”

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.   :2.000

Exploring the data

e<-ggpaired(ToothGrowth, x = "supp", y = "len",
         color = "supp", line.color = "red", line.size = 0.4,
         palette = "jco")


e1 <- ggpaired(ToothGrowth, x = "supp", y = "len",
          color = "supp", palette = "jco", 
          line.color = "red", line.size = 0.4,
          facet.by = "dose", short.panel.labs = FALSE)


plot_grid(e,e1)

The graph on the left tells us the following:

The median is higher in OJ vs VC
The range is more spread out in VC compared to OJ
The OJ minimum point is higher than VC
The VC max point is higher than OJ
Look at the quantiles below to see the spread of data across the boxplot
OJ quantiles between 50%-75% are in 22.70-25.725 vs. VC which are 16.5-23.1 the distance between them are also staggering 25.725-22.70 = 3.025 for OJ vs. 23.1-16.5 = 6.6. Mother nature is very effective at delivering vitamin C through OJ.

## $OJ
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    8.20   15.53   22.70   20.66   25.73   30.90 
## 
## $VC
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    4.20   11.20   16.50   16.96   23.10   33.90

The graph on the right tell us the following:

The median increase as the dosage increase
VC delivery method is not an effective way of sourcing vitamin c unless it reaches the dose 2 mg per day.
- During wartime, carrying an orange juice vs. a tablet is probably a different story.
Orange juice is very effective delivery method for vitamin c.
The OJ range of the box plot on 2 mg decreases - sign of diminishing returns?
Their respective p-values
Check-out the quantiles below

## : OJ
## : 0.5
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    8.20    9.70   12.25   13.23   16.18   21.50 
## ------------------------------------------------------------ 
## : VC
## : 0.5
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    4.20    5.95    7.15    7.98   10.90   11.50 
## ------------------------------------------------------------ 
## : OJ
## : 1
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   14.50   20.30   23.45   22.70   25.65   27.30 
## ------------------------------------------------------------ 
## : VC
## : 1
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   13.60   15.28   16.50   16.77   17.30   22.50 
## ------------------------------------------------------------ 
## : OJ
## : 2
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   22.40   24.57   25.95   26.06   27.07   30.90 
## ------------------------------------------------------------ 
## : VC
## : 2
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   18.50   23.38   25.95   26.14   28.80   33.90

Hypothesis Testing

H0 = mean of both OJ and VC are equal Ha != H0 There is a difference in mean for VC and OJ

OJ <- ToothGrowth %>% filter(supp=="OJ")
VC <- ToothGrowth %>% filter(supp=="VC")

t.test(OJ$len,VC$len, paired = TRUE)

## 
##  Paired t-test
## 
## data:  OJ$len and VC$len
## t = 3.3026, df = 29, p-value = 0.00255
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  1.408659 5.991341
## sample estimates:
## mean of the differences 
##                     3.7

mu <- ToothGrowth %>% group_by(supp) %>% dplyr::summarise("means"=mean(len))

ggplot(data = ToothGrowth, aes(x=len, color = supp))+geom_density()+
  geom_vline(data=mu, aes(xintercept=means, color=supp),
             linetype="dashed")

x <- OJ$len
y <- VC$len

OJVC <-data.frame(x,y)
combOJVC <- OJVC %>% mutate("meanss" =rowMeans(OJVC))
mu1<-mean(combOJVC$meanss)

ggplot()+geom_density(data = combOJVC, aes(x=meanss), color="blue")+geom_vline(xintercept = mu1, linetype="dashed", color="blue")+labs(title = "OJ/VC Averages")

lmfit <- lm(ToothGrowth$len~ToothGrowth$supp+ToothGrowth$dose)
summary(lmfit)

## 
## Call:
## lm(formula = ToothGrowth$len ~ ToothGrowth$supp + ToothGrowth$dose)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -6.600 -3.700  0.373  2.116  8.800 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)          9.2725     1.2824   7.231 1.31e-09 ***
## ToothGrowth$suppVC  -3.7000     1.0936  -3.383   0.0013 ** 
## ToothGrowth$dose     9.7636     0.8768  11.135 6.31e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.236 on 57 degrees of freedom
## Multiple R-squared:  0.7038, Adjusted R-squared:  0.6934 
## F-statistic: 67.72 on 2 and 57 DF,  p-value: 8.716e-16

There is a statistically significant difference in means.

plot(lmfit)

Reject the null hypothesis because there are differences between the mean of OJ and VC.

OJdose_.5 <- ToothGrowth %>% filter(dose==0.5) %>%  filter(supp=="OJ")
VCdose_.5 <- ToothGrowth %>% filter(dose==0.5) %>%  filter(supp=="VC")

t.test(OJdose_.5$len, VCdose_.5$len, paired = F, var.equal = F)

## 
##  Welch Two Sample t-test
## 
## data:  OJdose_.5$len and VCdose_.5$len
## 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 of x mean of y 
##     13.23      7.98

OJdose_1 <- ToothGrowth %>% filter(dose==1.0) %>%  filter(supp=="OJ")
VCdose_1 <- ToothGrowth %>% filter(dose==1.0) %>%  filter(supp=="VC")

t.test(OJdose_1$len- VCdose_1$len)

## 
##  One Sample t-test
## 
## data:  OJdose_1$len - VCdose_1$len
## t = 3.3721, df = 9, p-value = 0.008229
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  1.951911 9.908089
## sample estimates:
## mean of x 
##      5.93

OJdose_2 <- ToothGrowth %>% filter(dose==2.0) %>%  filter(supp=="OJ")
VCdose_2 <- ToothGrowth %>% filter(dose==2.0) %>%  filter(supp=="VC")

t.test(OJdose_2$len- VCdose_2$len)

## 
##  One Sample t-test
## 
## data:  OJdose_2$len - VCdose_2$len
## t = -0.042592, df = 9, p-value = 0.967
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  -4.328976  4.168976
## sample estimates:
## mean of x 
##     -0.08

P-value is .967 not enough to reject the null hypothesis. The difference between OJ mean and VC Mean is -.08 is very close to 0 but still not equal to zero.

Conclusion

There is a significant difference between tooth length and dose levels across different delivery methods. A higher dosage led to longer teeth. Based on our evidence, the orange juice delivery method is more effective than the alternative method. At dosage 2 mg, the OJ still has a comparable impact vs. VC but the effectiveness is not the same as the lower dosage.