Week 8 – Conditions for Inference & One-Way ANOVA

1 Textbook Reading

First, we are going to refresh on the conditions for linear regression. You might want to reference last week’s reading while completing the first reading guide: Model Conditions for Linear Regression

Reading Guide – Due Monday by the start of class

There are a few questions from this that are similar to the of last week’s reading guide. Feel free to copy and paste your previous answers to this reading guide!

Download the Word Document

Now, let’s read about how hypothesis testing extends into multiple linear regression. This allows us to connect to our new topic—ANOVA!

Required Reading: Hypothesis Tests for Multiple Linear Regression

Important

There is no reading guide for this chapter, just a concept quiz!

2 Concept Quizzes

We have two concept quizzes this week, one that is due by Monday (on model conditions) and one that is due by Wednesday (on ANOVA).

2.1 Due Monday by the start of class

1. What are the required conditions for linear regression? (select all that apply)

  • a random sample was taken
  • equal variance of residuals
  • a large sample was collected
  • linear relationship between x and y
  • independence of observations
  • independence of variables
  • normality of residuals
  • normality of observations

2. In the scatterplot of stem dry mass versus stem length, what condition appears to be violated? (select all that apply)

  • equal variance of residuals
  • linear relationship between x and y
  • independence of observations
  • normality of residuals

3. Which of the following would violate the condition of independence? (select all that apply)

  • collecting a non-random sample
  • observations related geographically (spatially)
  • observations that are related in time (temporally)
  • repeated observations on the same person
  • observations related biologically

4. When conducting a hypothesis test (in general), we assume the [observations / variables] are independent. When conducting a hypothesis test for a linear regression, the conditions require that the [observations / variables] are independent.

2.2 Due Wednesday by the start of class

term estimate std_error statistic p_value lower_ci upper_ci
intercept 27.113 1.632 16.609 0.000 23.902 30.324
body_mass_g 0.003 0.000 7.228 0.000 0.002 0.004
species: Chinstrap 5.061 3.310 1.529 0.127 -1.451 11.573
species: Gentoo -0.575 2.794 -0.206 0.837 -6.072 4.922
body_mass_g:speciesChinstrap 0.001 0.001 1.475 0.141 0.000 0.003
body_mass_g:speciesGentoo 0.001 0.001 1.558 0.120 0.000 0.002

  1. In the coefficient table, the body_mass_g:speciesChinstrap line is testing if the [slope / intercept] (of the relationship between body mass and bill length) is different for [Adelie / Chinstrap / Gentoo] penguins as compared to [Adelie / Chinstrap / Gentoo] penguins.

  2. In the coefficient table, the body_mass_g:speciesGentoo line is testing if the [slope / intercept] (of the relationship between body mass and bill length) is different for [Adelie / Chinstrap / Gentoo] penguins as compared to [Adelie / Chinstrap / Gentoo] penguins.

  3. True or false, you can use the regression table to test if the slope for Chinstrap and Gentoo penguins are different.

term df sumsq meansq statistic p.value
body_mass_g 1 3449.82317 3449.823171 591.920719 0.0000000
species 2 4553.33934 2276.669668 390.631021 0.0000000
body_mass_g:species 2 19.92386 9.961928 1.709268 0.1826114
Residuals 327 1905.81634 5.828185 NA NA
  1. Match the appropriate null and alternative hypothesis being tested in the body_mass_g:species line of the ANOVA table.

Null Hypothesis


Alternative Hypothesis

  • The relationship between body mass and bill length is the same for this sample of Adelie, Chinstrap, and Gentoo penguins

  • The relationship between body mass and bill length is the same for all Adelie, Chinstrap, and Gentoo penguins

  • The relationship between body mass and bill length is different for a least one species of penguins

  • The relationship between body mass and bill length is different for the Adelie penguins

  • The relationship between body mass and bill length is different for the Chinstrap penguins

  • The relationship between body mass and bill length is different for the Gentoo penguins

  1. The species line of the ANOVA table is testing if the [intercept / slope] is different for [every species / at least one species / at least two species].

  2. True or false, the body_mass_g line of the ANOVA table and the body_mass_g line of the coefficient table are performing the same hypothesis test.

3 R Tutorial – Due Monday by the start of class (note the new time!)

Required Tutorial: Evaluating the technical conditions in linear regression