Page 256 - Statistics II for Dummies

240 Part IV: Building Strong Connections with Chi-Square Tests

Now the opposite situation happens when you look at the level-two video
games in Table 13-6. The males chose the harder video games 70 times (out
of 80), while the females only chose the harder ones only 30 times out of 120.
The males did better than the females on level-two video games (winning 50
percent of them versus 40 percent for the females). However, level-two video
games are harder to win than level-one video games. This factor means that
the males’ winning percentage on level-two video games, being only 50 per-
cent, doesn’t contribute much to their overall winning percentage. However,
the low winning percentage for females on level-two video games doesn’t
hurt them much, because they didn’t play many level-two video games.
The bottom line is that the occurrence or nonoccurrence of Simpson’s
Paradox is a matter of weights. In the overall totals from Table 13-5, the
males don’t look as good as the females. But when you add in the difficulty
of the games, you see that most of the males’ wins came from harder games
(which have a lower winning percentage). The females played many more of
the easier games on average, and easy games carry a higher chance of win-
ning no matter who plays them. So it all boils down to this: Which games
did the males choose to play, and which games did the females choose to
play? The males chose harder games, which contributed in a negative way
to their overall winning percentage and made the females look better than
they actually were.

Keeping one eye open

for Simpson’s Paradox

Simpson’s Paradox shows you the importance of including data about
possible lurking variables when attempting to look at relationships between
categorical variables.

Level of game wasn’t included in the original summary, Table 13-5, but it
should have been included because it’s a variable that affected the results.
Level of game, in this case, was the lurking variable. More men chose to play
the more difficult games, which are harder to win, thereby lowering their
overall success rate.

You can avoid Simpson’s Paradox by making sure that obvious lurking vari-
ables are included in a study; that way, when you look at the data you get the
relationships right the first time and there’s a lower chance of reversing the
results. And, as with all other statistical results, if it looks too good to be true
or too simple to be correct, it probably is! Beware of someone who tried to
oversimplify any result. While three-way tables are a little more difficult to
examine, they’re often worth using.

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