Overview and Summary of Data

Data Description

The Effect of Vitamin C on Tooth Growth in Guinea Pigs: The response 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).

Data Format

A data frame with 60 observations on 3 variables.

  • len numeric Tooth length
  • supp factor Supplement type (VC or OJ).
  • dose numeric Dose in milligrams/day
data("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

Data Exploration

boxplot(len ~ as.factor(dose)+supp, ToothGrowth, col = c("lightgreen", "brown", "yellow", "lightblue", "gold", "orange"), main = "Tooth Length by Dose and Supplement", ylab = "Tooth Length (mm)", xlab = "Dose (mg) & Supplement Type")

Analysis

As can be seen in the plots above, there is a clear and positive relationship between the tooth length (len) and the dose levels of Vitamin C (dose) for both delivery methods (supp).

The effect of the dose can also be identified using regression analysis.

fit <- lm(len ~ dose + supp, data=ToothGrowth)
summary(fit)
## 
## Call:
## lm(formula = len ~ dose + supp, data = ToothGrowth)
## 
## 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 ***
## dose          9.7636     0.8768  11.135 6.31e-16 ***
## suppVC       -3.7000     1.0936  -3.383   0.0013 ** 
## ---
## 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

Conclusions

The model explains 69.3% of the variance in the data. The intercept is 9.27, meaning that with no supplement of Vitamin C, the average tooth length is 9.27 units. The coefficient of dose is 9.76. It can be interpreted as increasing the delievered dose 1 mg, all else equal, would increase the tooth length 9.76 units. The computed coefficient is for suppVC and the value is -3.7 meaning that delivering a given dose as ascorbic acid, without changing the dose, would result in 3.7 units of decrease in the tooth length. Since there are only two categories, we can also conclude that on average, delivering the dosage as orange juice would increase the tooth length by 3.7 units.

95% confidence intervals for two variables and the intercept are as follows.

confint(fit)
##                 2.5 %    97.5 %
## (Intercept)  6.704608 11.840392
## dose         8.007741 11.519402
## suppVC      -5.889905 -1.510095

The confidence intervals mean that if we collect a different set of data and estimate parameters of the linear model many times, 95% of the time, the coefficient estimations will be in these ranges. For each coefficient (i.e. intercept, dose and suppVC), the null hypothesis is that the coefficients are zero, meaning that no tooth length variation is explained by that variable. All p-values are less than 0.05, rejecting the null hypothesis and suggesting that each variable explains a significant portion of variability in tooth length, assuming the significance level is 5%.

References

  • McNeil, D. R. (1977) Interactive Data Analysis. New York: Wiley.

  • Crampton, E. W. (1947) The growth of the odontoblast of the incisor teeth as a criterion of vitamin C intake of the guinea pig. The Journal of Nutrition 33(5): 491–504. http://jn.nutrition.org/content/33/5/491.full.pdf