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DAPR3 > Data Reduction: Exploratory Factor Analysis (EFA) > Flashcards

Data Reduction: Exploratory Factor Analysis (EFA) Flashcards

1
Q

What is the difference between PCA and EFA?

A

in PCA we don’t care why our variables are correlated, our only goal is to reduce the number of variables

in EFA we believe there are underlying causes as to why our variables are correlated and we have 2 goals:
1) reduce the number of variables
2) learn about and model the underlying (latent) causes of variables

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2
Q

What are latent variables?

A

latent variables are a theorised common cause of responses to a set of variables

  • they explain correlations between measured variables
  • they are held to be true
  • there is no direct test of this theory

PCA does not have latent variables

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3
Q

Practical steps

How do we move from data and correlations to EFA?

A

1) check the appropriateness of the data and decide the appropriate estimator

2) decide which methods to use to select a number of factors (same as PCA methods)

3) decide conceptually whether to apply rotation and how to do so

4) decide criteria to assess and modify your solution

5) run the analysisi

6) evaluate the solution (decided in step 4)

7) select a final solution and interpret the model, labelling the results

8) report your results

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4
Q

Interpreting EFA output

Factor loadings

A

M1 M2 (with numbers underneath)

the numbers show the relationship of each measured variable to each measured factor

we interpret our factor models by the pattern and size of these loadings

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5
Q

Interpreting EFA output

Factor loadings
What are primary loadings

A

= the factor for which a variable has its highest loading

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6
Q

Interpreting EFA output

Factor loadings
What are cross-loadings?

A

= refer to all the other factor loadings (not primary loading) for a given measured variable

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7
Q

Interpreting EFA output

h2

A

= explained item variance

the square of the factor loadings tells us how much item variance is explained

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8
Q

Interpreting EFA output

us

A

= uniqueness

unexplained item variance

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9
Q

Interpreting EFA output

com

A

= complexity

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10
Q

Interpreting EFA output

SSloadings

A

= give the strength of the relationship between the item and the component (same as PCA)

range from 1 to -1 → higher = stronger relationship

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11
Q

Differences in PCA vs EFA

Dependent variable

A

PCA = the component

EFA = observed measures

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12
Q

Differences in PCA vs EFA

Independent variable

A

PCA = observed measures (x1, x2 …)

EFA = the factor (is regressed on the item)

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13
Q

Differences in PCA vs EFA

Aim

A

PCA = explains as much variance in the measures (x1, x2, …) as possible

EFA = models the relationship (correlation) between the variables

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14
Q

Differences in PCA vs EFA

Components vs factors

A

PCA = components = are determinant (there is only one solution for the component)

EFA = factors = are indeterminant (there is an infinite number of factor solutions that could be extracted from a given dataset)

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15
Q

What does it mean to model the data in EFA

A

EFA tries to explain patterns of correlations

e.g. if there is a correlation between y1 and y2 die to some factor - if we remove the factor there should be no correlation

if the model (factors) is good, it will explain all the interrelationships

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16
Q

What is the modification index?

A

it is R (i think) saying “there should be a correlation here and you said there isn’t one”

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17
Q

Variance in EFA

total variance

A

total variance = common variance + specific variance + error variance

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18
Q

Variance in EFA

true variance

A

common variance + specific variance

common variance = variance common to one item and at least one other item

specific variance = variance specific to an item that is not shared with any other items

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19
Q

Variance in EFA

unique variance

A

specific variance + error variance

specific variance = variance specific to an item that is not shared with any other items

error variance = random noise

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20
Q

EFA assumptions

A

1) the residuals/error terms should be uncorrelated

2) the residuals/error terms should not correlate with factors

3) relationships between items and factors should be linear (there are models that can account for non-linear relationships)

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21
Q

Data suitability

How do we know our data is suitable for EFA?

A

this boils down to: “is the data correlated?”
so initially we check our correlations to check they are moderate (>0.2)

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22
Q

Data suitability

squared multiple correlations (SMC)

A

this is another way to check our data is suitable for EFA

SMC = tells us how much item variance in an item is explained by all other items

SMC are multiple correlations of each item regressed on all other (p-1) variables
- this tells us how much variation is shared between an item and all other items

this is one way to estimate communalities

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23
Q

Estimating EFA

What do we do?

A

for PCA we use eigen decomposition but this is not an estimation method, it is simply a calculation

As we have a model for the data in EFA we need to estimate the model parameters (primarily the factor loadings

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24
Q

Estimating EFA

what are communalities?

A

communalities = estimates of how much true variance any variable has

therefore they also indicate how much variance in an item is explained by other variables

if we consider that EFA is trying to explain true common variance, then communalities are more useful to us than total variance

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25
Q

Estimating EFA

Estimating communalities
Difficulties

A

Estimating communalities is hard as population communalities are unknown

  • they range from 0 (no shared variance) to 1 (all variance is shared)
  • Occasionally estimates will be >1 (Heywood cases)
  • methods of estimation are often iterative and ‘mechanical’
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26
Q

Estimating EFA

Methods of estimating communalities
Principal axis factoring (PAF)

A

this approach uses SMC to determine the stability of the values on the diagonal of our correlation matrix

1) compare initial communalities from SMC

2)Eigen decomposition = once we have these reasonable lower bounds, we substitute the 1s in the diagonal of our correlation matrix, with SMCs from step 1

3) obtain the factor loadings using eigen values and eigen vectors of the matrix obtained in step 2

some versions of PAF use an iterative process where they replace the diagonal with the communalities obtained in step 3, then do step 3 again, then replace the diagonal again etc.

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27
Q

Estimating EFA

Methods of estimating communalities
Method of minimum residuals (MINRES)

A

this is an iterative approach and the default of the FA procedure
it tries to minimise communalities on the diagonal

1) starts with some other solution e.g. PCA or principal axes, extracting a set number of factors

2) adjust the loadings of all factors on each variable so as to minimize the residual correlations for that variable

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28
Q

Estimating EFA

Methods of estimating communalities
Maximum likelihood estimation (MLE)

A

this is the best estimation method BUT it doesn’t always work (the other two will work no matter how bad your data is)

the procedure works to find values for these parameters that maximize the likelihood of obtaining the observed covariance matrix

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29
Q

Estimating EFA

Methods of estimating communalities
Maximum likelihood estimation (MLE)

Advantages

A
  • provides numerous ‘fit’ statistics that you can use to evaluate how good your model is compared to other data
  • MLE assumes a distribution for your data (e.g. normal distribution)
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30
Q

Estimating EFA

Methods of estimating communalities
Maximum likelihood estimation (MLE)

Disadvantages

A
  • it is sometimes not possible to find values for factor loadings that equal MLE estimates - this is referred to as non-convergence
  • MLE may produce impossible values of factor loadings (e.g. Heywood cases) or factor correlations (e.g. >1)

*MLE assumes data is continuous and this is not always the case

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31
Q

How to select a number of factors to keep in EFA?

A

We use the same methods mentioned in PCA
- variance explained …
- skree plots
- MAP
- parallel analysis

Use all of these to decide a plausible number of factors
- use MAP as the minimum
- use parallel analysis as the maximum

32
Q

Factor rotation

Why are factor solutions hard to interpret?

A
  • the pattern of factor loadings is not always clear
  • the difference between primary and cross loadings can be small
33
Q

Factor rotation

What is rotational interdeterminancy

A

= it means that there are an infinite number of pairs of factor loadings and factor score matrices which will fit the data equally well and are thus indistinguishable by any numeric criteria

in other words - there is no one unique solution to the factor problem
this is why the theoretical coherence of a model plays a bigger role in EFA than PCA

34
Q

Factor rotation

Analytic rotation

A

Rotation aims to maximise the relationship of a measured item with a factor
= make primary loadings big and cross loadings small

original correlations are very noisy and difficult to find patterns so we rotate to simplify

although we can’t tell the methods apart numerically, we can select the rotation with the most coherent solution

35
Q

Factor rotation

Simple structure

A

All factor rotations seek to optimize one or more aspects of simple structure:

1) each variable (row) should have at least one 0 loading

2) each factor (column) should have the same number of 0s as there are factors

3) every pair of factors (columns) should have several variables which load on one factor but not the other

4) when >4 factors are extracted, each pair of factors should have a large proportion of variables that do not load on either factor

5) every pair of factors should have a few variables that load on both factors

36
Q

Factor rotation

Orthogonal rotation

A

Correlations between factors are 0
Axes are at right angles

Includes varmax and quartimax rotations

37
Q

Factor rotation

Oblique rotation

A

This method is most recommended

Correlations between factors are NOT 0
this is useful as it is more like reality and as this whole thing is exploratory there is no need for this constraint
Axes are NOT at right angles

Includes promax and oblim rotations

38
Q

Factor rotation

Oblique rotation interpretation
Pattern matrix

A

pattern matrix = matrix of regression weights (loadings) from factors to variables

39
Q

Factor rotation

Oblique rotation interpretation
Structure matrix

A

structure matrix = matrix of correlations between factors and variables

Structure matrix = pattern matrix multiplied by factor correlations

(in orthogonal rotations, structure and pattern matrices are the same)

40
Q

Evaluating results

Checking the results

A

start by examining how much variance each factor accounts for and the total amount of variance

we evaluate factors based on the size and sign (+/-) of the loadings that you deem to be salient ( generally loadings >0.3)

41
Q

Evaluating results

Checking for trouble

A

REMEMBER - if you delete any items you must re-run FA starting from when you figure out how many factors to extract

*Heywood cases
= items with loadings >1 → this means that something is wrong and you should not trust these results

  • items with no salient loadings?
    = could be a signal of a problem which should be removed
    = could be a signal of an additional factor
  • items with multiple salient loadings? (cross-loadings)
    = indicated by item complexity values

*do any factors load on <3 items?
= 3 should be minimum
= may have over extracted
= might have too few items

42
Q

Evaluating results

EFA checklist

A

✅ all factors load on 3+ items at salient levels

✅ all items have at least one loading above the salient cut off

✅ No Heywood cases

✅ complex items are removed (in accordance with the research goals)

✅ solution accounts for an acceptable level of variance (given in the research goals)

✅ item content of factors is coherent and substantively meaningful

43
Q

Factor Congruence

Replicability

A

It is always good to test whether your study replicates well. This can be done by:
- collecting data on another sample
- splitting one large sample into two

then we can test these as exploratory vs confirmatory data
There are numerous methods for this

44
Q

Factor Congruence

Replicability
congruence coefficients

A

= correlations between vectors of factor loadings across samples

“how similar are the loadings for M1 across the two samples?”
“how similar are they for M2”

Calculating congruence:
1) run factor model on sample 1
2) run factor model on sample 2
* ensure the same items are included and same number of factors specified
3) Calculate congruence

45
Q

Factor Congruence

Replicability
(Tucker’s) congruence coefficients

A

= measures similarity independent of the mean size of the loadings

it is insensitive to a change in the sign of any pair of loadings

Basics:
<0.68 = terrible
>0.9 = good
>0.98 = excellent

46
Q

Factor Congruence

Replicability
Confirmatory FA (CFA)

A

This is the better solution

In EFA all factors load on all items - these loading are purely data driven

In CFA we specify a model and test how well it fits the data
- we explicitly state which items relate to which factor
- we can test if the loadings are the same in different samples / groups / across time etc

47
Q

Factor scores

What are factor scores?

A

they provide variables to represent what we measured in EFA so we can test our constructs

they use different pieces of information from the factor solution to compute a weighted score
- the scores are a combinations of observations, factor loadings and factor correlations (method dependant)

48
Q

Factor scoring

Types of scores
Sum scoring (unit weighting)

A

This is the simplest approach to factor scoring

= sum the raw scores on the observed variables which have primary loadings on each factor

  • which items to sum is a matter of defining what loadings are salient

These require strict properties to be present in the data (but these are rarely tested)

49
Q

Factor scoring

Types of scores
Ten Berge Scores

A

this is the preferred method
= focus on producing scores with correlations that match to the factor correlations

50
Q

Factor scoring

Types of scores
Structural equations modelling

A

= includes a measurement component (CFA) and a structural component (regression)

  • doesn’t require you to compute factor scores
  • requires good theory of measurement and structure

if your constructs don’t approximate simple structure you may have to turn to alternatives

51
Q

How do you determine sample size in FA?

A

in the past, rules were based around the participant to item (N:p) ratio

BUT the crucial determinant is the communalities and items to factors (p:m)

fewer participants needed if:
- communalities are high and wide
- p:m was high (e.g. 20:7 = interaction effect)

general rule of psychology = more is better

52
Q

GIGO

A

garbage in = garbage out

always check the quality of your data
PCA and FA can not make bad data → good data

53
Q

Reliability

Aim of measurement

A

= to develop and use measurements of constructs to test psychological theories

54
Q

Reliability

Measurement
Classical test theory

A

describes data from any measure as a combination of:
- the signal of the construct / the ‘true score’
- noise or ‘error’ the measure of other unintended things

observed score = true score + error

55
Q

Reliability

Measurement
True score theory

A

if we assume of our test that:

1) it measures some ability or trait
2) in the world, there is a ‘true’ value or score for this test for each individual

then the reliability of the test is a measure of how well it reflects the true score

56
Q

Reliability

Parallel tests

A

under certain assumptions (parallelism, tau equivalence, congeneric tests) the correlations between two parallel tests of the same construct provided an estimate of reliability

Parallel tests can come from several sources

57
Q

Reliability

Sources of Parallel tests
Alternative forms of reliability

A

= correlation between two variants of a test

e.g. randomisation of stimuli, similar but not identical tests

Alternative tests (should) have equal means and variances
if the tests are perfectly reliable, then they should correlate perfectly
↳ since they won’t - this deviation provides the measure of reliability

alternatives can be expensive and time consuming - but they are becoming easier

58
Q

Reliability

Sources of Parallel tests
Split-half reliability

A

= indicates how internally consistent a measure is

1) split the items (randomly) into a pair of equal subsets on n items
2) score the two subsets
3) correlate the scores

With an increasing number of items, the number of random splits becomes increasingly large

59
Q

Reliability

Sources of Parallel tests
Split-half reliability
Cronbach’s alpha

A

the best known estimate for split half reliability is Cronbach’s alpha
- tells us “to what extent are observations consistent across items”

BUT it does not indicate whether items measure one unidimensional construct

cronbach’s alpha increases as you increase the number of items
this is represented by Spearman-Brown prophecy formula

60
Q

Reliability

Sources of Parallel tests
Split-half reliability
McDonald’s Omega

A

Any item may measure:
- a general factor that loads on all items
- a group or specific factor that loads on a subset of items

Given this, we can derive two internal consistency measures
1) Omega hierarchical (ωh) = the proportion of an item variance that is general
2) Omega total (ωt) = the total proportion of reliable item variance

These are much more robust to the structure of your data and how it will work in the real world

61
Q

Reliability

Sources of Parallel tests
Test-retest reliability

A

= correlation between tests taken at (at least) two different points in time

poses tricky questions:
- what is the appropriate time between measures?
- how stable should the construct be if we are to consider it a trait?

62
Q

Reliability

Sources of Parallel tests
Interrater reliability

A

= do all the raters involved have consistent measures

We can determine interrater reliability by means of intraclass correlation coefficients

63
Q

Reliability

Sources of Parallel tests
Interrater reliability
Intraclass correlation coefficients

A

this splits variance of ratings into multiple components:

  • variance between subjects (across targets)
  • variance within subjects (across raters, same targets)
  • variance due to raters (same rater, across targets)
64
Q

Uses of reliability?

A

it is useful to know how reliable our measure is for:
- implications of validity
- also allows us to correct for attenuation ( = estimates of effects are limited by reliability)

65
Q

Validity

What is validity?

A

there are debates over the definition but basically:

it determines whether a test really measures what it is supposed to measure

debates over the definition lead to debates over what counts as evidence for validity

66
Q

Evidence of validity …
… related to content

Content validity

A

= a test should contain only content relevant to the intended construct
= it should measure what it is intended to measure

67
Q

Evidence of validity …
… related to content

Face validity

A

= does the test ‘appear’ to measure what it was designed to measure?

68
Q

Evidence of validity …
… related to scale

Construct validity

A

= do the items measure a single intended construct

this is the most important
FA provides limited information towards this

69
Q

Evidence of validity …
… relationships with other concepts

Convergent validity

A

= measure should have high correlations with other measures of the same construct

70
Q

Evidence of validity …
… relationships with other concepts

Discriminate validity

A

= measure should have low correlations with measures of different constructs

71
Q

Evidence of validity …
… relationships with other concepts

Nomological Net validity

A

= measure should have expected pattern (+/-) correlations with different sets of constructs

also some measures should vary depending on manipulations

72
Q

Evidence of validity …
… relationships in terms of temporal sequence

Concurrent validity

A

= correlations with contemporaneous measures (tests done at the same time)

73
Q

Evidence of validity …
… relationships in terms of temporal sequence

Predictive validity

A

= related to expected future outcomes (longituinal)

74
Q

Evidence of validity …
… related to response processes

A

not commonly considered in validation studies
e.g. do tests of intelligence engage ‘problem solving’ behaviours

75
Q

Evidence of validity …
… related to consequences

A

= should potential consequences of the test be considered part of the evidence for the test’s validity

Important measures for the use of tests:
- is the measure systematically biased or is it fair for all groups of test takers?
- does bias have social ramifications?

76
Q

Reliability vs Validity

A

reliability = relation of true score to observed score

validity = correlations with other measures plays a key role

A score/measure can not correlate with anything more than it correlates with itself and therefore, reliability is the limit on validity

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