Pair-wise Comparisons for ANOVA

Recall the study done by the National Survey of Student Engagement, which found that, on average, college students spend 17 hours per week preparing and studying for their classes.^[1][2]

Previously, we conducted a one-way ANOVA to determine if there are any statistically significant differences between the means of the following majors:

• Arts and Humanities
• STEM
• Education
• Business

Using a 5% level of significance, we rejected the null hypothesis that the means are equal and accepted the alternative hypothesis that at least two means are different.

Question! Which means are different? To answer this question, we need to make multiple pair-wise comparisons.

We could begin by comparing business and education. This is an example of a pair-wise comparison. However, there are more!

Notice the following key concepts:

• “Did the original test reveal that at least one of the colors is linked to acne?”
  Show Answer

  No, [latex]P>0.05[/latex], so there was really no evidence to look at individual colors anyway.
• “How many tests did the stick researchers conduct?”
  Show Answer

  [latex]20[/latex]
• “What is the probability of committing a type I error (saying a color is significant when it is actually not) in each test?”
  Show Answer

  [latex]0.05[/latex]
• “How do we know the significance level?”
  Show Answer

  Each frame compares [latex]P[/latex] to [latex]0.05[/latex].
• “Is the conclusion adjusted for multiple comparisons?”
  Show Answer

  No, it reports a [latex]95\%[/latex] confidence level and a [latex]5\%[/latex] change for a type I error.