Measuring abstract concepts: Latent Variables and Factor Analysis

Correlation as the shape of an ellipse of plotted points

oo

oo

oo

oo

oo

oo

oo

o

oo o

ooo

oo o

ooo o

o

o

o o

oo

oo

oo

oo

oo

oo

oo

o

oo o

ooo

oo o

ooo o

o

oo

oo

oo

oo

oo

oo o

oo

o

oo

o

oo

oo

oo

oo

oo

oo

oo

oo

o

oo o

ooo

ooo

ooo o

o

o o

o

oo

oo

ooo

o o

o ooo o

oo

oo

ooo

o

o

o oo

High correlation(people’s arm &

leg lengths?)

Lower correlation(arm length & body weight?)

No correlation(arm length &

income?)

Correlation shows how accurately you can predict the score on a second variable if you are told the first. It suggests that

there may be some underlying connection: growth, for example.

Multiple dimensionsWe can show correlations among 3variables (e.g. lengthof your arm & leg, and headcircumference). If they are correlated, the diagram becomes an ellipsoid. It has a central axis runningthrough it, forming a singlesummary indicator of thelatent variable (size).

Mathematically, we can also summarizecorrelations betweenmore than 3 dimensions(but I can’t draw it)

Headcircum.

arm

leg

General measurement approach

ϕ ψ social

Health

Conceptualmodel

Selection ofindicators(sampling)

Scoringsystem e.g. ϕ x 2 + Ψ x 1.2 + soc.

Possible hierarchies

ϕ ψ social

ϕ ψ social

Health

Health

In a multi-level construct we need to specify how the different levels relate to each other.

This comes entirely from a conceptual approach:there is no empirical way to assert one or the other model.

? ?

Indicators

Latent trait

Indicators inter-correlatebecause they all reflect the

latent trait

Indicators correlate becausethey reflect a common (latent) trait

Modeling the link between manifest (measured) and latent (inferred) variables

Income

Expenditure

Indicator scores

Health (Probability model)

For variables likeincome & expenditure

we can give a relatively fixed model

For health thereis more variationbetween people,so a less precise

model.

Principal Components analysis

• Translates a complex system of correlations between many variables into fewer underlying dimensions (or ‘principal components’).

• Developed by Charles Spearman in 1904 to identify a simpler underlying structure in large matrices of correlations between measures of mental abilities.

• Later greatly misused in ‘defining’ intelligence.

Common variance(what we are trying

to measure)

Unique variance in thisitem (irrelevant biasin the measurement)

Spearman’s 1904 core idea: each item contains somecommon (shared) variance plus some specific variance.

The latter (red circles) sometimes raises and sometimes lowers the score, so they cancel out if you have enough items.

+

-

One principal component• Red lines show scores on 8

tests as vectors• Cosine of angles between

them represent correlations: if 2 vectors overlap the correlation is perfect (Cosine 0° = 1.0)

• Principal component 1resolves most of the variance in the 8 measures: it’s the bestfit, or grand average.

1

Dimensionality & Rotation. The principal component is that which accounts for the most variance;this depends on the conceptual shape of the latent trait being measured.For Chile, one dimension will account for most of the variance in distancebetween cities; for HK a more complex model is required. To find the dominant dimension with the maximal variation, axes need to be rotated.

Variance ‘explained’Here 2 vectors, B & C, are only partially correlated.

Resolving power of the principal component is shown by comparing length of the vector (B or C) and its projection onto the axis (Bʹ, C ʹ)

Here, axis 1 ‘explains’ more variance for B than for C (Bʹ > C ʹ)

A second (horizontal) component may be required for C: axis 2 resolves much of the variance in C, but very little for B.

Principalaxis

BBʹ

CCʹ

Secondaxis

Thurstone’s 1930 multi-factor idea: each item containssome common variance plus several types of unique variance.

The latter (colored circles) can compose an additional factor being measured, or just random ‘error’.

+

-

Common variance(what we are trying

to measure)

Unique variance in thisitem (irrelevant biasin the measurement)

Second theme inthe measurement

Factor Loadings and Validity

In the second example, the latent variable is more strongly reflected in the item; it has a higher loading on the variableand is a purer indicator of the underlying variable.

The blue rectangle represents the contribution of the latent variableto the indicator. The green segment represents the contribution of other latent variables; the red section shows all other sources of variance(error, etc).

Example of a two-factorsolution (here related toconcepts in the Health

Belief Model)

Source: K.S. Lewis, PhD thesis“An examination of the HealthBelief Model when applied to

Diabetes Mellitus”University of Sheffield, 1994.

Solution with rotated axes

1

Anxietyitems

Depressionitems

Using factor 1 alone = general mental health factor?

1

Using 2 factors clarifies different groups,but neither explains substantial variance

2

Anxietyfactor

Depressionfactor

To rotate or not to rotate?• Dimensions are traditionally shown perpendicular to each

other: independent & uncorrelated (measures of different things should not be confounded).

• Applied to example of anxiety & depression there are various options: 1. as they are both are both facets of mental distress, they could be

summarized along a single factor2. perhaps it is diagnostically useful to keep anxiety & depression

conceptually distinct: 2 orthogonal factors. If so, our indicators are not terrible good (low variance explained)

3. anxiety & depression often co-occur, so in reality are correlated; the axes could therefore be rotated obliquely to resolve the maximum variance (next slide)

Oblique rotation

Allow the axes to correlate• Resolves more variance• But does not create

conceptually independent entities

Do you like this approach?

Anxietyfactor

Depressionfactor

An example of turningprincipal componentsanalysis results into

linear modeling (LISREL),

The Health Belief Model.

Source: Cao Z-J, Chen Y, Wang S-M. BMC

Public Health 2014, 14:26

Cautions to ponder…• Correlations between measures do not prove that they record

anything concrete. • Test scores may or may not result from (or be caused by) the

underlying factor. • The principal component is a mathematical abstraction; it may

not represent anything real– (correlate your age for successive years with the population of Mexico,

the weight of your pet turtle, the price of cheese and the distance between any 2 galaxies: this will produce a strong principal component).

– Rotating the axes causes the principal component to disappear, so it has no reality

• We cannot declare that a factor represents an underlying reality (intelligence or health, etc.) unless we have clear evidence from other sources.

Questions to debate

• Would you use a 1- or a 2-factor solution for anxiety & depression questions?– What sort of rotation?

• What type of evidence could demonstrate that your presumed health measures really do measure health?

• Should we ever use oblique rotation?