Page 255 - Statistics II for Dummies

Chapter 13: Forming Associations with Two-Way Tables 239

Step back and think about this scenario for a minute. Table 13-5 shows that
females won a higher percentage of the video games they played overall. But
Table 13-6 shows that males won more of the level-one games and more of the
level-two games. What’s going on? No need to check your math. No mistakes
were made — no tricks were pulled. This inconsistency in results happens in
real life from time to time in situations where an important third variable is
left out of a study, a situation aptly named Simpson’s Paradox. (See why it’s
called a paradox?)

Figuring out why Simpson’s
Paradox occurs

Lurking variables are the underlying cause of Simpson’s Paradox. A lurking
variable is a third variable that’s related to each of the other two variables and
can affect the results if not accounted for.

In the video game example, when you look at the video game outcomes (won
or lost) broken down by gender only (Table 13-5), females won a higher
percentage of their overall games than males (70 percent overall winning
percentage for females compared to 55 percent overall winning for males).
Yet, when you split up the results by the level of the video game (level one or
level two; see Table 13-6), the results reverse themselves, and you see that
males did better than females on the level-one games (90 percent compared
to 80 percent), and males also did better on the level-two games (50 percent
compared to 40 percent).
To see why this seemingly impossible result happens, take a look at the mar-
ginal row probabilities versus the marginal row totals for the level-one games
in Table 13-6. The percentage of times a male won when he played an easy
video game was 90 percent. However, males chose level-one video games
only 10 times out of 80 total level-one games played by men. That’s only
12.5 percent.

To break this idea down further, the males’ nonstellar performance on the
challenging video games (50 percent — but still better than the females)
coupled with the fact that the males chose challenging video games 87.5 per-
cent of the time (that’s 70 out of 80 times) really brought down their overall
winning percentage (55 percent). And even though the men did really well on
the level-one video games, they didn’t play many of them (compared to the
females), so their high winning percentage on level-one video games (90 per-
cent) didn’t count much toward their overall winning percentage.
Meanwhile, in Table 13-6, you see that females chose level-one video games 90
times (out of 120). Even though the females only won 72 out of the 90 games
(80 percent, a lower percentage than the males, who won 9 out of 10 of their
games), they chose to play many more level-one games, therefore boosting
their overall winning percentage.

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