Homework exercise

To be solved at home before the exercise session.

  1. Consider a data set with measurements of the variable y for three groups (x). Each group has sample size 15. Below are shown boxplots of the groups, along with outputs given by ANOVA and the Kruskal-Wallis test for the data.
    1. What are the conclusions of the two tests?
    2. Which test (if either) would you trust and why?
    3. How would you continue the analysis?
boxplot(y ~ x, data = my_data)

summary(aov(y ~ x, data = my_data))
##             Df Sum Sq Mean Sq F value Pr(>F)
## x            1   1.13   1.129   0.586  0.448
## Residuals   43  82.89   1.928
kruskal.test(y ~ x, data = my_data)
## 
##  Kruskal-Wallis rank sum test
## 
## data:  y by x
## Kruskal-Wallis chi-squared = 10.185, df = 2, p-value = 0.006142

Class exercise

To be solved at the exercise session.

  1. A botanist wants to test the hypothesis that the three iris species have equal expected value of Sepal.Width.
    1. Visualize the data.
    2. Conduct an analysis of variance.
    3. Are the assumptions of ANOVA satisfied?
    4. If the assumptions are fulfilled, conduct pairwise comparisons using the Bonferroni correction.
    5. State your conclusions.
  1. The data set mtcars has measurements for 32 cars. We investigate the relationship between mpg (miles/gallon, the response) and hp and am (horsepowers and transmission type, the explanatory variables) through an analysis of covariance.
    1. Find a suitable visualization for the data.
    2. Using the function lm, fit a regression model with the covariates hp, am and hp:am (the final one is an interaction effect, the product of the two covariates).
    3. Interpret the fitted model (homework problem 10.1.a might prove helpful).
  1. (Optional) Consider still the mtcars data set but replace the variable am with the variable gear (and make sure its type is factor). Fit the linear regression model mpg ~ hp + gear and find out how the function anova can be used to test whether all regression coefficients related to gear are equal to zero simultaneously. Note that the situation is different from problem 2 as gear has three classes (i.e., two coefficients) and thus the \(p\)-values from the model only relate to the hypotheses whether the two coefficients can be set to zero individually.