Quiz

• Name one latent variable• Name 2 manifest variables that are indicators

for the latent variable.

Multiple linear indicators

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

• Example: Attraction

attraction

heart rate talking phone calls

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).

let’s assume an interval scale ranging from –4 (not at all attracted) to + 4 (highly attracted)

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 variables 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)

Mapping the relationship by placing anchors at the highest and lowest values helps to minimize this problem

Preview: Standardization and z-scores

Latent

Obs

erve

d

attraction

hear

t bea

t /10

attractionta

lkin

gattraction

phon

e ca

lls

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.

-4 0 4

020

4060

8010

0

Allow the lowest measured value to represent the lowest value of the latent variable

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

Obs

erve

d

attraction

hear

t bea

t / 1

0

attractionta

lkin

gattraction

phon

e ca

lls

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.

attraction

hear

t bea

t/10

attractionta

lkin

gattraction

phon

e ca

lls

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.

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, interval-based 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.

Latent: Stress Level

Obs

erve

d: H

ours

of

Los

t Sle

ep6

4.2

2.4

-.6

-1.2

-3

Operationally defining the latent variable

Latent: Stress Level

Obs

erve

d: T

hing

s to

do

15

12.6

10.2

7.8

5.4

3

Operationally defining the latent variable

Estimating latent scores

Person Indicator 1 (hours lost

sleep)

Latent score

estimate 1

Indicator 2

(to do list)

Latent score

estimate 2

Averaged latent score

Prof. Powell

4.2 10

b

c

d

e

Latent: Stress Level

Obs

erve

d: H

ours

of

Los

t Sle

ep6

4.2

2.4

-.6

-1.2

-3

Operationally defining the latent variable

Latent: Stress Level

Obs

erve

d: T

hing

s to

do

15

12.6

10.2

7.8

5.4

3

Operationally defining the latent variable

Estimating latent scores

Person Indicator 1 (hours lost

sleep)

Latent score

estimate 1

Indicator 2

(to do list)

Latent score

estimate 2

Averaged latent score

Prof. Powell

4.2 8 10 6

b

c

d

e

(8 + 6)/2 = 14/2 = 7

Average the latent score estimates

Estimating latent scores

Person Indicator 1 (hours lost

sleep)

Latent score

estimate 1

Indicator 2

(to do list)

Latent score

estimate 2

Averaged latent score

Prof. Powell

4.2 8 10 6 7

b

c

d

e

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.