Operational definitions and latent variables

1
Operational definitions and latent variables

• As we discussed in our last class, many
  psychological variables of interest cannot be
  directly observed
• These latent variables can be quantified by (a)
  measuring the observable manifestations of these
  variables and (b) explicating the relationship
  between the latent and observable variable(s)

2
How many indicators?
gt 1
One
Multiple linear indicators (Simple)
Equivalence relation (Simplest)
Linear
Mathematical Mapping
Multiple non-linear indicators (Very Complex)
Single non-linear relationship (Complex)
Nonlinear
3
In our example, we were assuming 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 beat
talking
phone calls
attraction
4
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.
5
allow the lowest measured value to represent the
lowest value of the latent variable
Observed
allow the highest measured value to represent the
highest value of the latent variable
the line between these points maps the
relationship between them
Latent
6
observed 12
observed 10
observed 13
estimate 2
estimate 0
estimate 3
After the relationship has been specified between
the latent variable and each measured variable,
the latent score estimates for each measured
variable can be averaged to scale the person on
the latent variable.
7
Multiple linear indicators

• Advantages
• By using multiple indicators, the uniqueness of
  each one gets washed out by what is common to all
  of them
• 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 was
  done to obtain the measurements.

8
Multiple linear indicators Caution

• On that last note, I should mention an important
  problem.
• 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

9
Multiple linear indicators Caution

• Why may this be a problem?
• 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???

10
Multiple linear indicators Caution

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

11
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)

12
Mapping the relationship by placing anchors at
the highest and lowest values helps to minimize
this problem
Observed
Preview Standardization and z-scores
Latent
13
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?

14
Some possible indicators of stress

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

15
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.

16
Operationally defining the latent variable
6
4.2
2.4
Observed Hours of Lost Sleep
-.6
-1.2
-3
Latent Stress Level
17
Operationally defining the latent variable
15
12.6
10.2
Observed Things to do
7.8
5.4
3
Latent Stress Level
18
Estimating latent scores
19
Summary

• So, we find that Shamara has a higher stress
  level (9.5) than Marc (3).
• 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