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Principal Components An Introduction

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Principal ComponentsAn Introduction

exploratory factoring

meaning & application of “principal components”

Basic steps in a PC analysis

PC extraction process

# PCs determination

Statistical approaches

Mathematical approaches

“Nontrivial factors” approaches

Exploratory Factoring – when we do not have RH: about . . .

- the number of factors
- what variables load on which factors
- we will “explore” the factor structure of the variables, consider multiple alternative solutions, and arrive at a post hoc solution
Weak Confirmatory Factoring – when we have RH: about the # factors and factor memberships

- we will “test” the proposed weak a priori factor structure
Strong Confirmatory Factoring – when we have RH: about relative strength of contribution to factors by variables

- we will “test” the proposed strong a priori factor structure

“Component” analyses are those that are based on the “full” correlation matrix

- 1.00s in the diagonal
- yep, there’s other kinds, more later
“Principal” analyses are those for which each successive factor...

- yep, there’s other kinds, more later
- accounts for maximum available variance
- is orthogonal(uncorrelated, independent) with all prior factors
- full solution (as many factors as variables) accounts for all the variance

Components analysis is a kind of “data reduction”

- start with an inter-related set of “measured variables”
- identify a smaller set of “composite variables” that can be constructed from the “measured variables” and that carry as much of their information as possible
A “Full components solution” ...

- has as many PCs as variables
- accounts for 100% of the variables’ variance
- each variable has a final communality of 1.00 – all of its variance is accounted for by the full set of PCs
A “Truncated components solution” …

- has fewer PCs than variables
- accounts for <100% of the variables’ variance
- each variable has a communality < 1.00 -- not all of its variance is accounted for by the PCs

- Compute the correlation matrix
- Extract a full components solution
- Determine the number of components to “keep”
- total variance accounted for
- variable communalities

- “Rotate” the components and “interpret” (name) them
- Structure weights > |.3|-|.4| define which variables “load”

- Compute “component scores”
- “Apply” components solution
- theoretically -- understand meaning of the data reduction
- statistically -- use the component scores in other analyses

- interpretability
- replicability

- Extraction is the process of forming PCs as linear combinations of the measured variables
PC1 = b11X1 + b21X2 + … + bk1Xk

PC2 = b12X1 + b22X2 + … + bk2Xk

PCf = b1fX1 + b2fX2 + … + bkfXk

- Here’s the thing to remember…
- We usually perform factor analyses to “find out how many groups of related variables there are” … however …
- The mathematical goal of extraction is to “reproduce the variables’ variance, efficiently”

X1 X2X3 X4

X1 1.0

X2 .7 1.0

X3.3 .31.0

X4.3 .3.5 1.0

- Consider R on the right
- Obviously there are 2 kinds of information among these 4 variables
- X1 & X2 X3 & X4

- Looks like the PCs should be formed as,

PC1 = b11X1 + b21X2 -- capturing the information in X1 & X2

PC2 = b32X3 + b42X4 -- capturing the information in X3 & X4

- But remember, PC extraction isn’t trying to “group variables” it is trying to “reproduce variance”
- notice that there are “cross correlations” between the “groups” of variables !!

- So, because of the cross correlations, in order to maximize the variance reproduced, PC1 will be formed more like ...
PC1 = .5X1 + .5X2 + .4X3 + .4X4

- Notice that all the variables contribute to defining PC1
- Notice the slightly higher loadings for X1 & X2

- Because PC1 didn’t focus on the X1 & X2 variable group orX3 & X4 variable group, there will still be variance to account for in both, and PC2 will be formed, probably something like …
PC2 = .3X1 + .3X2 - .4X3 - .4X4

- Notice that all the variables contribute to defining PC2
- Notice the slightly higher loadings for X3 & X4

- While this set of PCs will account for lots of the variables’ variance -- it doesn’t provide a very satisfactory interpretation
- PC1 has all 4 variables loading on it
- PC2 has all 4 variables loading on it and 2 of then have negative weights, even though all the variables are positively correlated with each other

- The goal here was point out what extraction does (maximize variance accounted for) and what it doesn’t do (find groups of variables)

Determining the Number of PCs

- Determining the number of PCs is arguably the most important decision in the analysis …
- rotation, interpretation and use of the PCs are all influenced by the how may PCs are “kept” for those processes
- there are many different procedures available – none are guaranteed to work !!
- probably the best approach to determining the # of PCS…
- remember that this is an exploratory factoring -- that meansyou don’t have decent RH: about the number of factors
- So … Explore …
- consider different “reasonable” # PCs and “try them out”
- rotate, interpret &/or tryout resulting factor scores from each and then decide

To get started we’ll use the SPSS “standard” of λ > 1.00

- PC analyses are extracted from a correlation matrix
- PCs should only be extracted if there is “systematic covariation” in the correlation matrix
- This is know as the “sphericity question”
- Note: the test asks if there the next PC should be extracted

- There are two different sphericity tests
- Whether there is any systematic covariation in the original R
- Whether there is any systematic covariation left in the partial R, after a given number of factors has been extracted
- Both tests are called “Bartlett’s Sphericity Test”

- Applying Bartlett’s Sphericity Tests
- Retaining H0: means “don’t extract another factor”
- Rejecting H0: means “extract the next factor”

- Significance tests provide a p-value, and so a known probability that the next factor is “1 too many” (a type I error)
- Like all significance tests, these are influenced by “N”
- larger N = more power = more likely to reject H0: = more likely to “keep the next factor” (& make a Type I error)

- Quandary?!?
- Samples large enough to have a stable R are likely to have “excessive power” and lead to “over factoring”
- Be sure to consider % variance, replication & interpretability

- The most commonly applied decision rule (and the default in most stats packages -- chicken & egg ?) is the > 1.00 rule … here’s the logic
Part 1

- Imagine a spherical R (of k variables)
- each variable is independent and carries unique information
- so, each variable has 1/kth of the information in R

- For a “normal” R (of k variables)
- each variable, on average, has 1/kth of the information in R

- Imagine a spherical R (of k variables)

Part 2

- The “trace” of a matrix is the sum of its diagonal
- So, the trace of R (with 1’s in the diag) = k (# vars)

- tells the amount of variance in R accounted for by each extracted PC
- for a full PC solution = k (accounts for all variance)
Part 3

- for a full PC solution = k (accounts for all variance)
- PC is about data reduction and parsimony
- “trading” fewer more-complex things (PCs - linear combinations of variables) for fewer more-simple things (original variables)

Putting it all together (hold on tight !)

- Any PC with > 1.00 accounts for more variance than the average variable in that R
- That PC “has parsimony” -- the more complex composite has more information than the average variable

- Any PC with < 1.00 accounts for less variance than the average variable in that R
- That PC “doesn’t have parsimony” -- the more complex composite has more no information than the average variable

There have been examinations the accuracy of this criterion

- The usual procedure is to generate a set of variables from a known number of factors (vk = b1k*PC1 + … +bfk*PCf, etc.) --- while varying N, # factors, # PCs & communalities
- Then factor those variables and see if > 1.00leads to the correct number of factors
Results -- the rule “works pretty well on the average”, which really means that it gets the # factors right some times, underestimates sometimes and overestimates sometimes

- No one has generated an accurate rule for assessing when which of these occurs
- But the rule is most accurate with k < 40, f between k/5 and k/3 and N > 300

These “common sense” approaches became increasing common as…

- the limitations of statistical and mathematical procedures became better known
- the distinction between exploratory and confirmatory factoring developed and the crucial role of “successful exploring” became better known
These procedures are more like “judgement calls” and require greater application of content knowledge and “persuasion”, but are often the basis of good factorings !!

Scree-- the “junk” that piles up at the foot of an glacier

a “diminishing returns” approach

- plot the for each factor and look for the “elbow”
- “Old rule” -- # factors = elbow (1966; 3 below)
- “New rule” -- # factors = elbow - 1 (1967; 2 below)

- Sometimes there isn’t a clear elbow -- try another “rule”
- This approach seems to work best when combined with attention to interpretability !!

4

2

0

# PC 1 2 3 4 5 6

A buddy in graduate school wanted to build a measure of “contemporary morality”. He started with the “10 Commandments” and the “7 Deadly Sins” and created a 56-item scale with 8 subscales. His scree plot looked like… How many factors?

λ

1? – big elbow at 2, so ’67 rule suggests a single factor, which clearly accounts for the biggest portion of variance

7? – smaller elbow at 8, so ’67 rule suggests 7

8? – smaller elbow at 8,’66 rule gives the 8 he was looking for – also 8th has λ > 1.0 and 9th had λ < 1.0

0 1 10 20

1 8 20 40 56

- Remember that these are subscales of a central construct, so..
- items will have substantial correlations both within and between subscales
- to maximize the variance accounted for, the first factor is likely to pull in all these inter-correlated variables, leading to a large λ for the first (general) factor and much smaller λs for subsequent factors
- This is a common scree configuration when factoring items from a multi-subscale scale!

Factor Rotations

- changing the “viewing angle” or “head tilt” of the factor space
- makes the groupings visible in the graph apparent in the structure matrix

Unrotated

Structure

PC1 PC2

V1 .7 .5

V2 .6 .6

V3 .6 -.5

V4 .7 -.6

PC1’

RotatedStructure

PC1 PC2

V1 .7 -.1

V2 .7 .1

V3 .1 .5

V4 .2 .6

PC2

V2

V1

PC1

V3

V4

PC2’

RotatedStructure

PC1 PC2

V1 .7 -.1

V2 .7 .1

V3 .1 .5

V4 .2 .6

Usually interpret factors using the rotated solutions using the rotated

- Factors are named for the variables correlated with them
- Usual “cutoffs” are +/- .3 - .4
- So … a variable that shares at least 9-16% of its variance with a factor is used to name that factor
- Variables may “load” on none, 1 or 2+ factors

This rotated structure is easy – PC1 is V1 & V2 PC2 is V3 & V4

It is seldom this easy !?!?!

- General Factor
- all or “almost all” variables load
- there is a dominant underlying theme among the set of variables which can be represented with a single composite variable

- Group Factor
- some subset of the variables load
- there is an identifiable sub-theme in the variables that must be represented with a specific subset of the variables
- “smaller” vs. “larger” group factors (# vars & % variance)

- Unique Factor
- single variable loads

- Univocal variable -- loads on a single factor
- Multivocal variable -- loads on 2+ factors
- Nonvocal variable -- doesn’t load on any factor

- You should notice a pattern here…
- a higher “cutoff” (e.g., .40) tends to produce …
- fewer variables loading on a given factor
- less likely to have a general factor
- fewer multivocal variables
- more nonvocal variables

- a lower “cutoff” (e.g., .30) tends to produce …
- more variables loading on a given factror
- more likely to have a general factor
- more multivocal variables
- fewer nonvocal variables