Transcript Answering Descriptive Questions in Multivariate Research • When we are studying more than one variable, we are typically asking one (or more) of the.

Answering Descriptive Questions in
Multivariate Research
• When we are studying more than one
variable, we are typically asking one (or
more) of the following two questions:
– How does a person’s score on the first variable compare to
his or her score on a second variable?
– How do scores on one variable vary as a function of scores
on a second variable?
Some example multivariate
descriptive questions
• How does marital satisfaction vary as a
function of relationship length?
• What is the relationship between coffee
consumption and happiness?
• What is the relationship between violent TV
viewing and aggressive behavior in children?
• How does one’s mood vary as a function of
the quality of one’s sleep?
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- 50
- 050
0 0.8
0.8
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-10
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0 1.0
-30
20
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- 30
- 20
1- 2* x ^ 2
-20
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11+- 22** xx^^22
1.0 30
11++20**xx^ 2
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- 5 0.9
11+- 02** xx
- 10
-40
10
1.1 40
1.
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0
0
50
1.250
1.10
2
Some possible functional
relationships between two variables
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6
4
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0
• Many of the
relationships we’ll focus
on in this course are of
the linear variety.
• The relationship
between two variables
can be represented as
a line.
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Graphic presentation
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2
4
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violent television viewing
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0
0
2
2
4
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6
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8
8
10
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• Linear relationships can be negative or
positive.
0
2
4
6
8
violent television viewing
10
0
2
4
6
8
violent television viewing
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• How do we determine whether there is a
positive or negative relationship between two
variables?
Scatter plots
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One way of determining the
form of the relationship
between two variables is to
create a scatter plot or a
scatter graph.
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The form of the relationship
(i.e., whether it is positive or
negative) can often be seen
by simply inspecting the
graph.
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violent television viewing
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y
-2 -1 0 1 2
How to create a scatter plot
A
D
B
Use one variable as the xaxis (the horizontal axis)
and the other as the y-axis
(the vertical axis).
Plot each person in this two
dimensional space as a set
of (x, y) coordinates.
F
E
C
- 2
- 1
0
x
1
2
Person
A
B
C
D
E
F
Zx
1.55
0.15
-0.75
0.48
-1.34
0.08
Zy
1.39
0.28
-1.44
0.64
-0.69
-0.19
negative relationship
no relationship
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-1
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18
0
10
1
20
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2
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3
positive relationship
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Quantifying the relationship
• How can we quantify the linear relationship
between two variables?
• One way to do so is with a commonly used
statistic called the correlation coefficient
(often denoted as r).
Some useful properties of the
correlation coefficient
(1) Correlation coefficients range between –1
and + 1.
Note: In this respect, r is useful in the same way
that z-scores are useful: they both use a
standardized metric.
Some useful properties of the
correlation coefficient
(2) The value of the correlation conveys
information about the form of the relationship
between the two variables.
– When r > 0, the relationship between the two variables is
positive.
– When r < 0, the relationship between the two variables is an
inverse relationship. (Higher scores on x correspond to
lower scores on y.)
– When r = 0, there is no relationship between the two
variables.
r = -.80
r=0
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-1
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r = .80
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Some useful properties of the
correlation coefficient
(3) The correlation coefficient can be interpreted
as the slope of the line that maps the
relationship between the two variables.
slope as rise over run
takes you
up .5 on y
y
1
2
3
r = .50
0
rise
-1
run
-2
moving from
0 to 1 on x
-2
-1
0
x
1
2
How do you compute a correlation
coefficient?
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N
z
x
zy  r
• First, transform each variable to standardized
scores.
• Multiply each person’s z-scores together.
• Finally, average these products.
Example
Person
The on-line correlation calculator is available
at: http://P034.psch.uic.edu/correlations.htm
Violent TV
viewing (zscores): Zx
Aggressive
behavior (zscores): Zy
1
N
z
Mike
1
1
1
Claudia
1
1
1
Len
-1
-1
1
Kramer
-1
-1
1
Average
0
0
1=
1
N
x
zy  r
z
x
zy  r
Some example correlations from our
class
Variable
Correlation with “attentive
mood”
Within the last few days, I have had a fight or
disagreement with someone close to me.
-.25
I slept well last night.
.17
I ate breakfast this morning.
.26
I have or had an overwhelming amount of work to
do today.
-.09
link
Magnitude of correlations
• When is a correlation “big” versus “small?”
• There is no real cut-off, but, on average,
correlations between variables in the “real
world” rarely get larger than .30.
• Why is this the case?
– Any one variable can be influenced by a hundred other
variables. To the degree to which a variable is multidetermined, the correlation between it and any one variable
must be small.