Adding a Second Factor (Two-Way ANOVA)

Section 5.9 Adding a Second Factor (Two-Way ANOVA)

We have done a fantastic job of identifying the best studying technique out of the three in order to get the highest memory exam scores. In life, there are typically more than just two variables. What if there was another variable in this study? For instance, what if each person’s caffeine level while studying for the memory exam was also taken into consideration. Maybe caffeine level had an influence on how well they studied, and in turn impacts their memory score. And if that is taken into account, is that a better model for explaining differences in memory scores?

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To test this out, let’s first create the caffeine column using a similar technique to how we created our data in the beginning.

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# Let's say we also measured caffeine intake (low vs high)
set.seed(42)
memory2 <- memory |>
  mutate(
    caffeine = rep(c("Low", "High"), times = 30))

Now that we’ve established that study method affects memory, let’s ask a new question: does caffeine level also play a role? This moves us from a one-way ANOVA (one independent variable) to a two-way ANOVA (two independent variables, or factors)

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# To see the interaction between method and caffeine levels, we add a *
anova_2 <- aov(score ~ method * caffeine, data = memory2)

supernova(anova_2)

Analysis of Variance Table (Type III SS)
Model: score ~ method * caffeine

                                        SS df      MS      F   PRE     p
--------------- --------------- | -------- -- ------- ------ ----- -----
          Model (error reduced) | 2777.807  5 555.561 10.194 .4856 .0000
         method                 | 1746.636  2 873.318 16.025 .3725 .0000
       caffeine                 |   39.360  1  39.360  0.722 .0132 .3992
method:caffeine                 |  117.439  2  58.720  1.077 .0384 .3477
          Error (from model)    | 2942.885 54  54.498                   
--------------- --------------- | -------- -- ------- ------ ----- -----
          Total (empty model)   | 5720.691 59  96.961

Interestingly, when we go to the p-values, we see that neither caffeine nor the interaction (how caffeine influences method) are statistically significant. This is a huge indication that caffeine does not have any impact on memory scores whatsoever. We can visualize this

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ggplot(memory2, aes(x = method, y = score, color = caffeine, group = caffeine)) +
  stat_summary(fun = mean, geom = "point") +
  stat_summary(fun = mean, geom = "line") +
  labs(title = "Interaction of Study Method and Caffeine",
       x = "Study Method",
       y = "Mean Memory Score") +
  theme_minimal()

A line graph showing mean memory scores for High and Low caffeine groups across three study methods. The two lines are nearly parallel, indicating no significant interaction between caffeine level and study method. — Figure 5.9.1. Interaction plot between Study Method and Caffeine Level. The near-parallel lines indicate that caffeine does not significantly alter the effectiveness of different study methods.
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What the p-value is actually telling you:.

In the graph above, we see that the lines for both high and low are nearly parallel. This gives you an indication that caffeine does not have a significant impact on study method

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The results from the ANOVA are impactful, and since we now have two different ANOVA models, we can compare the models.

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