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)

attraction

hear

t bea

t x10

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 x 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. (Russ on The Bachelorette)

attraction

hear

t bea

t

attractionta

lkin

gattraction

phon

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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.

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 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

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. Fraley

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.

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?

Simple Example

• Let’s consider a relatively simple example: Let’s try to measure crying.

• Before we can do so, we need to decide “what counts” as crying behavior.

• What examples come to mind?

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.

Complex Example

• Now let’s 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.

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.

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).

Measuring Latent Variables

• Let’s 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 defined—can be considered observable indicators of the unobserved state of “finding something humorous.”

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.

Mapping

• Mapping—specifying the relationship between the latent and manifest variable—tends 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?

LINEAR RELATIONSHIP

LOVE

TIM

E S

PE

NT

TA

LK

ING

-4 -2 0 2 4

68

10

12

14

NONLINEAR RELATIONSHIP

LOVE

TIM

E S

PE

NT

TA

LK

ING

-4 -2 0 2 4

05

10

15

20

> 1One

Lin

ear

Non

line

ar

Multiple non-linear indicators

(Very Complex)

Single non-linear

relationship (Complex)

Multiple linear indicators

(Simple)

Equivalence relation

(Simplest)

How many indicators?

Mat

hem

atic

al M

appi

ng

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.

Humor

Lau

ghin

g

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.

Equivalence Relationship

• Advantages: – Explicit and straight-forward

– Doesn’t require complicated mathematics

– Other researchers can easily determine what you did

• Disadvantages: – Behaviors are influenced by many things. Thus, part of what

you’re 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.