Does tooth length increase with dosage?

Y = Tooth length: continuous normally distributed

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 ...
hist(ToothGrowth$len)

X = Dosage: keeping as numerical

Test to use: ANOVA

There is a statistically significant difference in tooth length with dose (F(1, 58) = 105.1, p < 0.001), and this is a quite strong linear relationship (R²=0.64).

len_model <- lm(len ~ dose, data = ToothGrowth)
summary(len_model)
## 
## Call:
## lm(formula = len ~ dose, data = ToothGrowth)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -8.4496 -2.7406 -0.7452  2.8344 10.1139 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   7.4225     1.2601    5.89 2.06e-07 ***
## dose          9.7636     0.9525   10.25 1.23e-14 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.601 on 58 degrees of freedom
## Multiple R-squared:  0.6443, Adjusted R-squared:  0.6382 
## F-statistic: 105.1 on 1 and 58 DF,  p-value: 1.233e-14
#plotting the results
r2 <- summary(len_model)$r.squared

ggplot(ToothGrowth, aes(x = dose, y = len, color = supp)) +
  geom_point(position = position_jitter(width = 0.05), size = 2, alpha = 0.7) + geom_smooth(method = "lm", se = FALSE, color = "red")+
  labs( x = "Dose (mg/day)", y = "Tooth Length (cm)",
       color = "Supplement",
        caption = paste0(
      "R² = ", round(r2, 2)
    )) +
  theme_minimal()

Which supplement type (VC or OJ) is more effective? What is the relationship between dose and supplement?

Y = Tooth length

X1 = dose (continuous)

X2 = supplement (2 levels)

Test: 2-way ANOVA

model_lm <- lm(len ~ supp * dose, data = ToothGrowth)
anova(model_lm)
## Analysis of Variance Table
## 
## Response: len
##           Df  Sum Sq Mean Sq  F value    Pr(>F)    
## supp       1  205.35  205.35  12.3170 0.0008936 ***
## dose       1 2224.30 2224.30 133.4151 < 2.2e-16 ***
## supp:dose  1   88.92   88.92   5.3335 0.0246314 *  
## Residuals 56  933.63   16.67                       
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Dose: extremely statistically significant (F(1,56) =133.415, p < 0.001) as tooth length differs significantly among the three dose levels

Supplement: There is an overall difference in tooth length between the two supplements (F(1,56) = 12.317, p < 0.001)

Supp:dose - The effect of supplement depends on the dose level is statistically significant with a (F(1,56) = 5.334, p = 0.025)

interaction.plot(x.factor = ToothGrowth$dose, 
                 trace.factor = ToothGrowth$supp, 
                 response = ToothGrowth$len,
                 type = "b", col = c("red","darkturquoise"),
                 xlab = "Dose", ylab = "Mean Tooth Length",
                 trace.label = "Supplement")