Topic 10

When Components/Factors are Correlated

Example: mind-wandering – occurs when thoughts stray from the topic at hand to unrelated inner thoughts. Occurs in 2 ways:

  • Spontaneous: not under conscious control

  • Deliberate: under conscious control

We have 4 items on each scale. We will specify 2 factors in SPSS because this is the predicted structure.

Can see this is an FA because eigenvalues are lower when the model is extracted

  • Good sign the 2 extracted factors have eigenvalues over 1 and are far higher than any other eigenvalues – confirmed in the scree plot

Factor Matrix and Factor Plot:

All items load highly onto factor 1 – not unexpected because the initial solution will try to maximise the variance explained by factor 1.

The factor plot does indicate two distinct groups of items. Therefore, a rotation could be used to improve factor loadings.

Varimax Rotation:

This is better, however, some items have high cross-loadings – called a complex item. This suggests there is some correlation or overlap between the factors.

Complex Items

Complex items: have a factor loading > .35 on more than 1 item in the rotated solution.

  • Could be tapping into a general factor

  • Not explanatory for a data reduction technique.

What do we do about them?

  • Exclude them

  • May suggest constructs are not independent and therefore an oblique rotation would be more appropriate

    • These allow for correlations between constructs

    • Common in psychology (e.g., depression and anxiety as components of psychopathology)

Oblique Rotation

Oblique rotation: used when components aren’t independent, and are correlated

  • Allows the angle between components to be less than 90 degrees – the smaller the angle, the stronger the correlation.

  • Move each factor independently

Can see the distance between the groups of items gets larger with the oblique rotation – which is ultimately what we want

Oblique Rotation Matrices include:

  • A pattern matrix: provides the factor loadings (excluding any unique variance)

  • Get a structure matrix: we do not interpret this – )

    this includes shared variance (which we cannot count twice so we ignore it

Can see there are no more complex items now

  • Also note loadings are higher in the structure matrix, and there are many complex items – this is because shared variance is being counted twice

Conducting an oblique rotation will also produce a factor correlation matrix, which shows the correlation between factors – this will confirm or disconfirm whether the factors are correlated.

This must be reported in results.

Example Oblique Rotation (PCA):

Can see unlike in varimax rotation, there is no unique variance accounted for column.

  • Because an oblique rotation allows for components to be correlated and therefore we cannot cleanly extract their unique variance accounted for

While there are some cross-loadings in this example it looks pretty good overall (these may be items worth getting rid of)

Also good to check the correlations between components, and it does appear these are correlated

Example Oblique Rotation (FA):

Can tell this is from an FA because the initial column doesn’t include 1s.

  • Because all unique variance is excluded

  • Can see one item has low communality, may indicate this is a unique item

FA explains slightly less variance than PCA Can see factor total and extraction total are not the same – because extraction total excludes unique variance

Still have a few items that have cross-loadings but definitely looks a lot better

Can also see factor are more highly correlated than on PCA which is good.

  • We can safely remove the items with cross-loadings at this point

After problematic items are removed:

Can see there is more variance explained

This looks heaps better

  • No cross-loadings

  • All loadings above .40

  • Items load onto the factor they should

  • Have 5 items for each factor

PCA and FA are both about testing different things and just seeing what works.

  • Can force different numbers of factors

  • Consider that items may load onto different factors than expected

  • If the analysis is mostly exploratory – a PCA may be more appropriate

  • If the analysis is more about uncovering a predicted underlying factor, FA may be more appropriate

  • Rarely do PCA and FA produce entirely different solutions

  • Orthogonal versus oblique rotation is a more considered question

Assumptions of PCA and FA:

The assumptions of PCA and FA are Bartlett’s test of sphericity and KMOs measure of sampling adequacy

Bartletts Test of Sphericity

Bartletts: determines whether there are factors or components to be extracted in the correlation’s matrix

  • Must be significant

  • Nearly always sig when n is large (which it has to be for PCA/FA)

If nonsignificant, it means there are no factors present in the correlations matrix.

  • Therefore there is really only one component that describes the items

  • Could also mean the construct is itself unidimensional

KMOs Measure of Sampling Adequacy

KMOs: describes the proportion of variance that may be described by the underlying factors for the overall solution and for each individual item

  • The larger the better (range from 0-1)


KMO Cutoffs - want to be above .70 at a minimum

  • < .5 = unacceptable

  • .5 - .59: very poor

  • .6-.69 = poor

  • .70-.70 = average

  • .8-.89 = good

  • .9-.99 = extremely good

A poor or unacceptable KMOs value would prompt investigation of whether the items are appropriate for the factors were trying to extract

The value provided in this table is for the overall scale Sometimes, if this is poor its because of a few dodgy items

These are the KMOs values for each item.

Scales Suitable for PCA/FA

Response items for scales should have at least 3 response options (but ideally 4-5)

  • Because we need a good range of responses for discrimination

  • Cannot do a PCA/FA with dichotomous items

Non-Discriminating Items

Individual item distributions should have a range of scores so that they are discriminatory.

  • Most scores are the same = non discriminatory

  • Means there will be low correlations with other items

    • If an item has inter-item correlations all < .30 it should be excluded

    • This is because we are looking for shared variance, and low correlations means little shared variance

  • Means there will be poor sampling adequacy

Extreme Scores

Extreme scores also cause problems for the same reasons as non-discriminatory items.

Example of good and poor items:

Strategy Analysis for a PCA/FA:

  1. Inspect item distributions

  2. Exclude items with inter-item correlations < .30

  3. Assess sampling adequacy

  4. Determine how many components to extract (try a few)

    1. Should have some theoretical idea of how many

  5. Remove items that should be discarded, and re-run the analysis

  6. Try a new solution if needed

  7. Check factor correlations and ensure the solution is conceptually meaningful

    1. Interpret correct rotation

    2. Check factor correlations

    3. Examine internal consistency

Reliability Analysis

Factor loadings tell us the correlations between each item and its corresponding factor

  • Doesn’t tell us the internal consistency or reliability of these factors

  • PCA/FA is not viable unless we get a measure of internal consistency

Internal consistency (Cronbach’s alpha): measures the extend to which different items on a scale are measuring a similar construct

  • For research: > .7

  • For clinical practice: > .9

Any measure should be:

  • Accurate

  • Repeatable

  • Able to minimise error

Measures of internal consistency measure all of these things

  • Cronbach’s alpha measures the internal consistency and reliability of each item on each subscale

  • Doesn’t measure test-retest reliability

Good reliability doesn’t equal good validity

Poor consistency can occur because of:

  • Small sample size

  • Few items on a scale

  • Items with too low correlations with other items

  • Items being coded in different directions

Cronbach’s alpha can be calculated directly for each factor (by including all items that load onto that factor)

Can also examine the item total statistics table:

  • Squared multiple correlation: like tolerance – the percentage of variance in the items that is accounted for by other items in the scale

    • The higher this is, the less independent the items are

  • Corrected item-total correlation: the correlation of each item to the total score, when added up

    • Like factor loading

    • Should be > .3

Using our Components/Factors in Other Analyses

It is possible to save components generated through PCA/FA into a datafile, and use this for further analyses – though this is not recommended

  • This is because the linear combination are based on factor loadings, which are sample-specific.

The descriptives produced for factor scores are not that meaningful either.

We can also produce item-total scores by adding up scores on the components to get a measure of the different dimensions

  • This is recommended

The descriptives here are much more meaningful

What to Report

Include the number of items in the final solution and how many were excluded.

State whether assumptions were met and their values

Mention the percentage of variance accounted for overall and how many factors were included

  • If orthogonal rotation was used: include percentage of variance for each component

Use the final solution – this should be based on conceptual meaningfulness of the outcome, not just the statistics.

Include a table with all items on each component and their loading

Describe each component and name it

Include alpha for each component