Chapter 7, section 1: Paired designs
Dr. Paul Pearson, Hope College
Fall 2024
Chapter 7: Paired data: one quantitative variable
Section 7.1: Paired designs
So far in this unit the studies have compared whole groups of
individuals like mildly depressed patients randomly assigned to swim
with dolphins or without dolphins. These studies used an independent
groups design—there were no systematic connections relating individuals
in one group to individuals in another group. In this section you will
learn about a different study design based on pairs of units. For many
situations, these designs lead to more powerful tests for differences
and narrower confidence intervals.
Example 7.1: Can you study with music blaring
Many students like to study while listening to music. They seem to
think that it doesn’t hurt their ability to focus on their work. Is that
really true? What does the research say? The following excerpt is from
“Checking It Out: Does Music Interfere with Studying?” (Strauss
2009):
- According to Stanford University Professor Clifford Nass, the human
brain listens to song lyrics with the same part that does word
processing, which is the same part that supposedly is being employed for
studying.
- Instrumental music is another story. For the most part, it is
processed on the other side of the brain from the part that is
processing language. “So if you are reading and listening to
instrumental music you get virtually no interference”
Experimental designs
While students are listening to music with or without lyrics
(categorical explanatory variable), they will play a memorization game:
They will study a list of 25 common five-letter words for 90 seconds,
and then write down as many of the words as they can remember
(quantitative response variable).
Consider the following experimental designs:
- Experiment 1: Independent groups design with random
assignment
- Each student is randomly assigned to one of two groups (lyrics or no
lyrics) while playing the memorization game.
- Because each student plays the game once, we don’t know how each
person would do under the other condition (lyrics or no lyrics).
- A sensible statistic to use is the difference of averages \(\bar{x}_{no~lyrics} - \bar{x}_{lyrics}\),
which is the average number of words in the no lyrics group minus the
average number of words in the lyrics group.
- Experiment 2: Paired design using repeated measures with
random assignment
- Each student plays the memorization game under both conditions
(lyrics and no lyrics) and which condition is done first is randomly
assigned.
- We expect there to be less variability in the responses of the same
person than between different people.
- A sensible statistic is the average of the differences \(\bar{x}_d = \bar{x}_{differences}\) for
each individual by first computing the number of words under the no
lyrics condition minus the number of words under the lyrics condition
for each person, and then computing the average of these
differences.
- Experiment 3: Paired design using matching and random
assignment
- Before playing the memorization game, all the students are matched
into pairs by having similar “speed reading scores”. One student from
each pair is randomly assigned to play the memorization game under the
lyrics condition, while the other student from the pair plays the
memorization game under the no lyrics condition. Each student plays the
game once.
- We expect there to be less variability in the responses for the two
individuals within each pair than between pairs.
- A sensible statistic is the average of the differences \(\bar{x}_d\) for each pair by first
computing the number of words under the no lyrics condition minus the
number of words under the lyrics condition for each matched pair, and
then computing the average of these differences.
Why should we use paired designs?
Key idea: We expect less variability (i.e., smaller
standard deviations) in the paired designs than in the independent
groups design, and as a result paired designs can yield stronger
results.
When
would we use matching instead of repeated measures?
- When it’s easy or convenient to do matching using manifest
variables:
- Example: Suppose you wanted to compare football player 40 yard dash
times with or without pads and half of the players are already wearing
pads. You’d want to pair up wide receivers (one with pads and another
without) and record their sprint times. Similarly, you’d want to pair up
linebackers (one with pads and another without) and record their sprint
times. This naive pairing uses the manifest variable of body size (which
is related to position played) and the manifest variable of wearing
pads. We could do repeated measures here, but it would cost extra time
due to putting on or taking off pads.
- When it’s impossible to do repeated measures:
- Example: Suppose you were studying health outcomes for kidney donors
under a treatment (aspirin) and control (no aspirin) groups. Because
each person can only donate a kidney once, repeated measures (donating
twice under different conditions) is not possible. In this case,
creating pairs of kidney donors who are similar as possible on manifest
(age, sex, etc.) and other measurable variables (blood test results,
etc.) is the best paired design we can do.