Transcript Multiple linear indicators • A better scenario, but one that is more challenging to use, is to work with multiple linear indicators. • Example:

Multiple linear indicators
• A better scenario, but one that is more
challenging to use, is to work with multiple
linear indicators.
• Example: Attraction
We assume that when someone is attracted to someone else
(a latent variable), that person is more likely to have an
increased heart rate, talk more, and make more phone calls
(all observable variables).
heart rate
talking
attraction
phone calls
let’s assume an interval scale
ranging from –4 (not at all
attracted) to + 4 (highly
attracted)
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talking
phone calls
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heart beat /10
-4-20
2
4
LO V
attraction
-4
-4
-20
-20
2
2
4
4
E
0 5 TIMESPNTALKIG 10 15 20
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-4
-20
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-20
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attraction
attraction
We assume that each observed variable has a
linear relationship with the latent variable.
Note, however, that each observed variable has
a different metric (one is heart beats per
minute, another is time spent talking). Thus,
we need a different metric for the latent
variable.
100
60
20
40
Allow the highest
measured value to
represent the highest value
of the latent variable
0
Observed
80
Allow the lowest
measured value to
represent the lowest value
of the latent variable
-4
0
Latent
4
The line between these
points maps the
relationship between them
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06
talking
phone calls
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heart beat / 10
-4-20
2
4
LO V
attraction
-4
-4
-20
-20
2
2
4
4
E
0 5 TIMESPNTALKIG 10 15 20
L
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A
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N E
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ON
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ON
A
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-4
-20
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-4
-20
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attraction
attraction
Now we can map the observed scores for each measured
variable onto the scale for the latent variable. For example,
the observed heart rate score of 120 maps onto an attraction
score of 2. Ten-minutes of talking maps onto an attraction
score of zero. Thirteen phone calls maps to a high
attraction score of 3.
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talking
phone calls
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6 8 TIMESPNTALKIG 10 12 14
heart beat/10
-4-20
2
4
LO V
attraction
-4
-4
-20
-20
2
2
4
4
E
0 5 TIMESPNTALKIG 10 15 20
L
I
N
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A
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N E
A
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ON
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ON
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ON
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-4
-20
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-4
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-20
2
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L
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attraction
attraction
This mapping process provides us with three estimates of
the latent score: 2, 0, and 3. Because we are trying to
estimate a single number for attraction, we can simply
average these three estimates to obtain our measurement of
attraction.
In this example: (2 + 0 + 3)/3 = 5/3 = 1.67 (somewhat
attracted)
Multiple linear indicators
• Advantages
– By using multiple indicators, the uniqueness of
each indicator gets washed out by what is
common to all of the indicators. (example: heart rate and
running up the stairs)
• Disadvantages
– More complex to use
– There is more than one way to scale the latent
variable, thus, unless a scientist is very explicit,
you might not know exactly what he or she did to
obtain the measurements.
Multiple linear indicators: Caution
• When using multiple indicators, researchers typically
sum or average the scores to scale people on the
construct
• Example:
(time spent talking + heart rate)/2 = attraction
Person A: (2 + 80)/2 = 82/2 = 41
Person B: (3 + 120)/2 = 123/2 = 62
Multiple linear indicators: Caution
• This can lead to several problems if each
manifest variable is measured on a different
scale.
• First, the resulting metric for the latent
variable doesn’t make much sense.
Person A: 2 minutes talking + 80 beats per minute
= 41 minutes talking/beats per minute???
Multiple linear indicators: Caution
• Second, the variables may have different
ranges.
• If this is true, then some indicators will “count”
more than others.
Multiple linear indicators: Caution
• Variables with a large range will influence the latent
score more than variable with a small range
Person
A
B
C
D
Heart rate
80
80
120
120
Time spent talking
2
3
2
3
Average
41
42
61
62
* Moving between lowest to highest scores matters more for one variable
than the other
* Heart rate has a greater range than time spent talking and, therefore,
influences the total score more (i.e., the score on the latent variable)
100
60
40
20
Preview: Standardization
and z-scores
0
Observed
80
Mapping the relationship
by placing anchors at the
highest and lowest values
helps to minimize this
problem
0
2
4
6
Latent
8
10
Some more examples
• Let’s work through a detailed example in
which we try to scale people on a latent
psychological variable
• For fun, let’s try measuring stress: Some
people feel more stressed than others
• Stress seems to be a continuous, intervalbased variable
• What are some indicators of stress?
Some possible indicators of stress
• Hours of sleep
• Number of things that have to be done by
Friday
Operationalizing our indicators
• We can operationally define these indicators as
responses to simple questions:
– “Compared to a good night, how many hours of sleep did
you lose last night?”
– “Please list all the things you have to accomplish before
Friday—things that you can’t really put off.”
• Note that each of these questions will give us a
quantitative answer. Each question is also explicit,
so we can easily convey to other researchers how we
measured these variables.
80
-1.2
20
-.6
60
2.4
40
4.2
-3
0
Observed: Hours of Lost Sleep
6
100
Operationally defining the latent
variable
0
2
4
6
8
Latent: Stress Level
10
80
5.4
20
7.8
60
10.2
40
12.6
3
0
Observed: Things to do
15
100
Operationally defining the latent
variable
0
2
4
6
8
Latent: Stress Level
10
Estimating latent scores
Person
Prof.
Fraley
b
c
d
e
Indicator 1
(hours lost
sleep)
4.2
Latent
score
estimate 1
8
Indicator 2
(to do list)
10
Latent
score
estimate 2
6
Averaged
latent
score
7
Summary
• Recap of what we did
– Determined the metric of the latent variable
– Identified two indicators of the latent variable
– Mapped the relationship between the latent
variable and each observed variable
– Using this mapping, estimated the latent scores
for each person with each observed variable
– Averaged the latent score estimates for each
person
Multiple linear indicators
• By mapping the measured variables explicitly
to the latent metric, we can avoid some of the
problems that emerge when variables are
assessed on very different metrics
Multiple linear indicators
• When the indicators are on the same metric
(e.g., questionnaire items that are rated on a
1 to 7 scale), the process of estimating the
latent score is easier, and researchers often
use the manifest metric as the latent metric
and average the observed scores to obtain a
score on the latent variable.