Multiple linear indicators

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Multiple linear indicators

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

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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
phone calls
attraction
lets assume an interval scale ranging from 4
(not at all attracted) to 4 (highly attracted)
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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.
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Allow the lowest measured value to represent the
lowest value of the latent variable
100
80
60
Observed
Allow the highest measured value to represent the
highest value of the latent variable
40
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The line between these points maps the
relationship between them
0
-4
0
4
Latent
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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. (Russ on The
Bachelorette)
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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)
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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.

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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

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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 doesnt make much sense.
• Person A 2 minutes talking 80 beats per minute
  
• 41 minutes talking/beats per minute???

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Multiple linear indicators Caution

• Second, the variables may have different ranges.
• If this is true, then some indicators will
  count more than others.

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Multiple linear indicators Caution

• Variables with a large range will influence the
  latent score more than variable with a small
  range
• Person Heart rate Time spent talking
  Average
• A 80 2 41
• B 80 3 42
• C 120 2 61
• D 120 3 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)

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Mapping the relationship by placing anchors at
the highest and lowest values helps to minimize
this problem
Observed
Preview Standardization and z-scores
Latent
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Some more examples

• Lets work through a detailed example in which we
  try to scale people on a latent psychological
  variable
• For fun, lets try measuring stress Some people
  feel more stressed than others
• Stress seems to be a continuous, interval-based
  variable
• What are some indicators of stress?

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Some possible indicators of stress

• Hours of sleep
• Number of things that have to be done by Friday

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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 Fridaythings that you cant
  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.

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Operationally defining the latent variable
6
4.2
2.4
Observed Hours of Lost Sleep
-.6
-1.2
-3
Latent Stress Level
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Operationally defining the latent variable
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12.6
10.2
Observed Things to do
7.8
5.4
3
Latent Stress Level
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Estimating latent scores
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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

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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

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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.

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Operational Definitions

• In our last class, we discussed (a) what it means
  to quantify psychological variables and (b) the
  different scales of measurement used for
  categorical and continuous variables.
• However, we deliberately side-stepped an
  important question How do we determine what
  matters when we try to measure a variable?

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Simple Example

• Lets consider a relatively simple example Lets
  try to measure crying.
• Before we can do so, we need to decide what
  counts as crying behavior.
• What examples come to mind?

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Definition of an Operational Definition

• It is critical that the set of rules, or
  operations, that we use to measure a behavior be
  explicit and as clear-cut as possible.
• These rules, or operations, constitute the
  operational definition of a variable.

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Complex Example

• Now lets consider a more complex variable the
  experience of humor.
• Whether or not someone finds something funny is a
  much more challenging (i.e., less tangible) thing
  to measure than crying.
• In-Class Example Two sets of operational
  definitions, and three students listening to
  jokes.

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Important Distinction

• Latent vs. Observed variables
• An observed variable, like crying, is behavioral
  and, therefore, directly observable.
• A latent variable or construct is not directly
  observable. Instead, it is inferred from
  variables that can be observed.

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Measuring Latent Variables

• Latent variables can be measured, but their
  measurement is much more complicated than that of
  observed variables.
• The first thing we need to do is identify,
  usually on an intuitive or theoretical basis, the
  scale of the latent variable. Is it categorical
  or continuous? If continuous, should we scale it
  on an interval metric or a ratio metric?
• Next, we need to identify the indicators of the
  latent variable (i.e., the observable
  consequences or manifestations of the latent
  variable).

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Measuring Latent Variables

• Lets answer the following question Someone who
  finds something funny should be likely to behave
  in the following ways __________.
• These things (e.g., laughing)which also need to
  be operationally definedcan be considered
  observable indicators of the unobserved state of
  finding something humorous.

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Measuring Latent Variables

• So, to operationally define a latent variable, we
  need to (a) specify the scale of the variable,
  (b) identify the observable manifestations of
  that latent variable, and (c) operationally
  define those observable manifestations.
• Next, we need to know how the operational
  definitions of the observable variables map onto
  the latent variable.

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Mapping

• Mappingspecifying the relationship between the
  latent and manifest variabletends to be handled
  differently by different researchers.
• Two considerations
• How many indicators to use?
• Can we assume a linear relationship between the
  measured variables and the latent variable?

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How many indicators?
1
One
Multiple linear indicators (Simple)
Equivalence relation (Simplest)
Linear
Mathematical Mapping
Multiple non-linear indicators (Very Complex)
Single non-linear relationship (Complex)
Nonlinear
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Equivalence Relationship

• Simplest case The equivalence relationship. In
  this case, we use one indicator and assume that
  the relation between the latent variable and the
  manifest variable is linear. The scale of the
  latent variable is identical to the scale chosen
  for the manifest variable.
• Example We may operationally define laughing,
  and then measure humor as if it is equal to
  laughing.

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For each extra laugh, we assume the person
thought the joke was one unit more funny Someone
who laughs 8 times would get a humor score of 8.
Laughing
Humor
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Equivalence Relationship

• Advantages
• Explicit and straight-forward
• Doesnt require complicated mathematics
• Other researchers can easily determine what you
  did
• Disadvantages
• Behaviors are influenced by many things. Thus,
  part of what youre measuring may be unrelated to
  the latent variable of interest.
• Latent variables manifest themselves in a variety
  of ways. By focusing on one variable, our
  measurements are not as rich or compelling.