Factor Analysis- Path Analysis

(SEM Foundations)

Everything Up Until Now

• Does X Predict Y?• Does X predict Y after controlling for Z?

• Some list of Independent Variables and a list of dependent variables

Where We are Going

• Now, many of the hypothesis we are going to look at revolve around the idea of how well a theory can explain observed data.

Mediation and Moderation

• These are the ideas that gave birth to path analysis.

• Baron and Kenny (1986) Judd and Kenny (1981)

Mediation

• B mediates the effect of A on C if A causes B and B causes C

A

B

C

A C

A (probably) untrue example

• A fatty diet has been shown to cause heart disease

• But, lets say it turned out that the mechanism was weight gain

• So really, a fatty diet caused weight gain and weight gain caused heard disease.

• Weight gain, then mediates the relationship between fatty eating and heart disease

Fatty Foods

Weight Gain

Heart Disease

Full and Partial Mediation

• Full: If, after controlling for path BC, path AC is not different than zero, we say that B fully mediates the relationship between A and C

• Partial: If path AC is just reduced, we say this is partial mediation

A

B

C

A Real Example

• People have known for a while that Maternal Depression greatly increases a child’s likelihood of becoming depressed

• Mediation is a question of what is the mechanism by which the depression is transmitted? Is it genes? Enviromnent?

• Turns out it is several things, but a major one is explanatory style.

• So we say explanatory style partially mediates the relationship

Partial Mediation

Maternal Depression

Explanatory Style

Child Depression

Baron and Kenny Steps

• Step 1: Show A is Correlated with C

• Step 2: Show A is Correlated with B

• Step 3: Show B predicts C after controlling for A

• Step 4: (For full mediation) A does not significantly predict C after controlling for B

The Catch

• Turns out that one also needs to test the indirect effect of A on C through B.

• Kenny used to recommend the Sobel test. (Not worth learning anymore)– But this had all sorts of problems with the figuring

our confidence intervals• Kenny now recommends bootstrapping with

an SEM program

Moderation

• Some variable B affects the magnitude of the relationship between A and C

A C

B

β

Moderation Example

• Violent Attitudes are related Domestic Violence Behaviors in Adolescents

• But, turns out that the relationship is stronger for females than for males

• Gender is said to moderate the relationship between violent attitudes and DV behaviors

Testing

• In the pre-SEM world, people would often test moderation using an interaction term– In the previous example: +

• With SEM it will make much more sense (Coming Soon!)

Factor Analysis

• There are several uses of factor analysis, but the easiest way to think about it:– When you have many variables, but you think that

there are really only a couple of underlying “real” traits or constructs

• Essentially, factor analysis is the process of either uncovering those traits or confirming the existence of those traits

Thurstone’s Box Problem

• The problem (initially) with the adoption of factor analysis, is that psychometric concepts can be abstract.

• Louis Thurstone decided to take the most concrete examine he could think of.

• Imagine you had a ton of boxes of random sizes

• Then, imagine you create a ton of variables (maybe 30) that are measurements of the box– Perimeter around the base– Diagonal from top right to bottom left

What would be the essential elements from all of these different variables?

• Using factor analysis, the three variables are extracted:– Length– Width– Height

So…how is this useful?

• In psychometrics, the idea of a scale is a bunch of items that are measuring the same characteristic (with lots of error).

• If the items are really measuring the same thing, we should be able to extract this essential characteristic out

For Example:

• Depression:– What are some items that measure depression?– What is common to all of these items?

Factor Analysis: How does it work

• Each factor is a linear combination of each of the observed variables

• You are (in a way) drawing new axes through your data

• There are an infinite number of solutions to most problems.

How Many Factors?

Rotation

Types

• Most common: PCA- Principal Components Analysis

• (Other types that will only look at reliable variance. Don’t worry about these for now)

Rotation

• Un-rotated solution:– Gives the maximum variance to the first factor– (This is where G theories of intelligence came

from)• Orthogonal Rotation (Varimax)– Assumes that factors are uncorrelated– Simple Structure: maximize the variable loadings

of each factor

• Oblique Rotation– Allows factors to be correlated– Example: verbal, mathematical intelligence

Exploratory- Confirmatory

• Exploratory: How many factors make up this set of items?

• Confirmatory: Does my scale fit a one factor (or two factor) model?