Evaluating Tooth Growth in Guinea Pigs Under Varying Diets
Statistical Inference, Coursera
Class Project #2, August 2014

Using the ToothGrowth dataset, this paper explores tooth growth for six sets of ten guinea pigs. The six sets are differentiated by the type of supplement received (orange juice (OJ) or ascorbic acid (VC)) and the dose in milligrams received (0.5, 1, or 2) of the supplement.

# load data, change dose to factor with 3 levels
tg <- ToothGrowth
tg$dose <- as.factor(tg$dose) # convert dose to factor

The measurements of central tendency (mean and standard deviation) of tooth growth is calculated by supplement (supp), dosage (dose), and supp-dose pairs. The code is shown for the dose-supp pair, and the other code is hidden in order to save space.

plyr::ddply(tg, .(dose, supp), summarize,
      mean = round(mean(len),2), sd = round(sd(len),2))
supp mean sd
OJ 20.66 6.61
VC 16.96 8.27
dose mean sd
0.5 10.61 4.50
1 19.73 4.42
2 26.10 3.77
dose supp mean sd
0.5 OJ 13.23 4.46
0.5 VC 7.98 2.75
1 OJ 22.70 3.91
1 VC 16.77 2.52
2 OJ 26.06 2.66
2 VC 26.14 4.80

These boxplots present a visual representation of the spread of data suggesting
- Orange juice (OJ) promotes longer tooth growth than ascorbic acid (VC) overall.
- Increased dosage amounts of either supplement promotes longer tooth growth.
- Orange juice promotes longer tooth growth for doses at 0.5 and 1.0 mg.
- Orange juice and ascorbic acid have similar effects on tooth growth at doses of 2.0 mg.

However, independent samples t-tests on these three categories (dose only, supplement along, dose-supplement pair) will be run to determine if these differences are significant or if the difference is simply what would be expected to due chance. If the differences are significant, then the findings can be applied to the general population of guinea pigs.

For the supp t-test, the data is subsetted into 30 cases each of OJ or VC.
For the dose t-tests, the data is subsetted into 20 cases each of 0.5, 1.0, and 2.0 mg.
For the dose-supp t-tests, data is subsetted into 10 cases each of the six possible combinations (0.5OJ, 0.5VC, 1.0OJ, 1.0VC, 2.0OJ, 2.0VC).
Here is code for one of the subsets. All other code is suppressed to save space.

# oj vs. vc
supp.oj <- tg$len[tg$supp == "OJ"]; supp.vc <- tg$len[tg$supp == "VC"]

For each t-test, the null hypothesis is that the mean tooth length for each group pairing is equal. Using the supplement pair as an example, the null hypothesis is \(H_0\): mean tooth length of OJ = mean tooth length of VC and \(H_A\): the means are not equal. \(H_0\) is rejected at the 95% confidence interval when the p-value < 0.05 and the confidence interval range does not include zero as a value.

Prior to running the independent t-test, the assumption of homogeneity of variance was testing using Levene’s Test. Since the p-value for Levene’s Test was greater than 0.05, the independent t-test can proceed assuming equal variances among groups (i.e. var.equal = TRUE). The code for Levene’s Test follows this example.

car::leveneTest(len ~ supp, data = tg)

Again, using the supplement pair as an example, each two sample independent groups t-test is calculated following this code structure.

ttest.oj.vc <- t.test(supp.oj, supp.vc, paired = FALSE, var.equal = TRUE)

This table summarizes the results of the seven t-tests. Five of the seven tests returned significant p-values.

Category Pairing CI.Low CI.Upper t p-val df CI.Includes.Zero
Supplement supp.oj and supp.vc -0.17 7.57 1.92 0.060 58 TRUE
Doseage dose.0.5mg and dose.1mg -11.98 -6.28 -6.48 0.000 38 FALSE
Doseage dose.0.5mg and dose.2mg -18.15 -12.84 -11.80 0.000 38 FALSE
Doseage dose.1mg and dose.2mg -8.99 -3.74 -4.90 0.000 38 FALSE
Dose + Supp d.0.5mg.and.supp.oj and d.0.5mg.and.supp.vc 1.77 8.73 3.17 0.005 18 FALSE
Dose + Supp d.1mg.and.supp.oj and d.1mg.and.supp.vc 2.84 9.02 4.03 0.001 18 FALSE
Dose + Supp d.2mg.and.supp.oj and d.2mg.and.supp.vc -3.72 3.56 -0.05 0.964 18 TRUE

Summary of Findings

Detailed Fidings

Findings for Supplement Comparison
These data do not provide evidence that there is a significant difference in the tooth growth in guinea pigs receiving orange juice (mean = 20.66, sd = 6.61) from those receiving ascorbic acid (mean = 16.96, sd = 8.27), t(58) = 1.92, p = 0.06. Even though one might surmise from the boxplot that there is a noticeable difference in these two supplements, this difference can be accounted for what would occur by chance.

Findings for Doseage Comparison
0.5 mg vs. 1.0 mg: These data provide convincing evidence that tooth growth in guinea pigs receiving 0.5 mg of supplement (mean = 10.61, sd = 4.5) is shorter than guinea pigs receiving 1.0 mg of supplement (mean = 19.73, sd = 4.42), t(38) = -6.48, p < 0.005. The data suggests with 95% confidence that the tooth growth in guinea pigs receiving 0.5 mg of supplement will be between 6.28 and 11.98 units shorter than guinea pigs receiving 1.0 mg of supplement.

0.5 mg vs. 2.0 mg: These data provide convincing evidence that tooth growth in guinea pigs receiving 0.5 mg of supplement (mean = 10.61, sd = 4.5) is shorter than guinea pigs receiving 2.0 mg of supplement (mean = 26.1, sd = 3.77), t(38) = -11.8, p < 0.005. The data suggests with 95% confidence that the tooth growth in guinea pigs receiving 0.5 mg of supplement will be between 12.84 and 18.15 units shorter than guinea pigs receiving 2.0 mg of supplement.

1.0 mg vs. 2.0 mg: These data provide convincing evidence that tooth growth in guinea pigs receiving 1.0 mg of supplement (mean = 19.73, sd = 4.42) is shorter than guinea pigs receiving 2.0 mg of supplement (mean = 26.1, sd = 3.77), t(38) = -4.9, p < 0.005. The data suggests with 95% confidence that the tooth growth in guinea pigs receiving 1.0 mg of supplement will be between 3.74 and 8.99 units shorter than guinea pigs receiving 2.0 mg of supplement.

Finding for Doseage/Supplement Pairs
OJ vs. VC, at 0.5 mg: These data provide convincing evidence that tooth growth in guinea pigs receiving 0.5 mg of orange juice (mean = 13.23, sd = 4.46) is longer than guinea pigs receiving 0.5 mg of ascorbic acid (mean = 7.98, sd = 2.75), t(18) = 3.17, p = 0.005. The data suggests with 95% confidence that the tooth growth in guinea pigs receiving 0.5 mg of orange juice will be between 1.77 and 8.73 units longer than guinea pigs receiving 0.5 mg of ascorbic acid.

OJ vs. VC, at 1.0 mg: These data provide convincing evidence that tooth growth in guinea pigs receiving 1.0 mg of orange juice (mean = 22.7, sd = 3.91) is longer than guinea pigs receiving 1.0 mg of ascorbic acid (mean = 16.77, sd = 2.52), t(18) = 4.03, p < 0.005. The data suggests with 95% confidence that the tooth growth in guinea pigs receiving 1.0 mg of orange juice will be between 2.84 and 9.02 units longer than guinea pigs receiving 1.0 mg of ascorbic acid.

OJ vs. VC, at 2.0 mg: These data do not provide evidence that there is a significant difference in the tooth growth in guinea pigs receiving 2.0 mg of orange juice (mean = 26.06, sd = 2.66) is longer than guinea pigs receiving 2.0 mg of ascorbic acid (mean = 26.14, sd = 4.8), t(18) = -0.05, p = 0.964.

Session Info

## R version 3.2.3 (2015-12-10)
## Platform: x86_64-apple-darwin13.4.0 (64-bit)
## Running under: OS X 10.10.5 (Yosemite)
## 
## 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] grid      stats     graphics  grDevices utils     datasets  methods  
## [8] base     
## 
## other attached packages:
## [1] car_2.0-22      knitr_1.12.3    gridExtra_0.9.1 plyr_1.8.1     
## [5] ggplot2_1.0.1  
## 
## loaded via a namespace (and not attached):
##  [1] Rcpp_0.11.5      digest_0.6.4     MASS_7.3-45      gtable_0.1.2    
##  [5] formatR_1.3      evaluate_0.8.3   scales_0.2.4     highr_0.4       
##  [9] reshape2_1.4.1   rmarkdown_0.9.5  labeling_0.3     proto_0.3-10    
## [13] tools_3.2.3      stringr_0.6.2    munsell_0.4.2    colorspace_1.2-4
## [17] htmltools_0.2.6  nnet_7.3-11