Week 4-Principle Component Analysis Flashcards
What does PCA allow us to do?
■ A PCA does allows us to take many items and reduce the
dimensionality of a construct.
■ Eg Allison et al 2014 found that “Scripted responses”, “talking about unrelated topics” and “revealing well known information” were all highly associated with each other, therefore can be combined into a single measure
verbal CITS
■ Ergo we have reduced the number of dimensions to analyse three have gone into one.
■ Reduces likelihood of false positives because we test the one construct not all three separate measures.
What does PCA look at?
Essentially a PCA looks at which items correlate (R-Matrix) with each other and calls them the
component/factor.
What does PCA tell us?
■ The PCA essentially finds underlying constructs from a
larger data set.
■ It will also tell us how important each component is (i.e. what % of variance in the data set it accounts for).
■ This statistic is called an Eigenvalue (note it is one word).
■ Any PCA will find a very large number of components and
you use the Eigenvalues to judge whether the components are worth keeping.
■ Kaiser’s rule: Eigenvalues greater than 1 mean the component is valid.
■ Joliffe’s rule (1972, 1986) Eigenvalues > .7 are valid in a PCA (no strict rule!).
What is an issue with PCA particularly given that it is used on Likert scales?
-Large sample sizes are needed in the hundreds
-You’re using a parametric test on a non-parametric measure
What do the Scree plots simply graph?
-Scree plots simply graph Eigen values and are another way to judge the number of factors/components.
-When the graph flattens there are no more relevant factors (i.e., the inflection point which you use to determine the number of components you have)
What do Eigenvalues tell us?
Tells us how many components or factors there are.
What are Component loadings?
■ The PCA will give us a number of components, but as you can see from the Eigenvalue table and the scree plot, this doesn’t tell us which of our individual measures make up each component.
■ Component loading tell us this, they tell you what the association is between each item and each component.
■ Essentially they are a Pearson’s correlation between the item and the factor/component.
■ The component matrix
*Component loadings can be negative (simply it negatively correlates with that component).
*A component can load onto more than one component- this is very common, a measure may simply tap into two underlying constructs.
What is classed as a strong enough loading?
■ Generally people take a component loading of .4 to be a
strong enough loading.
■ Sometimes people will actually calculate if it is significant as a loading by treating it as a correlation coefficient.
■ You can therefore work out the p value if you know the sample size.
■ Often PCA/FA are done on large samples.
What is Rotation?
■ To simplify data and the analysis we do not interpret the component
matrix; we apply a rotation first.
Yaremko, et al. (1986), defines rotation as:
■ “In factor or principal-component analysis, rotation of the factor axes(dimensions) identified in the initial extraction of factors, in order to obtain simple and interpretable factors.” (essentially we need to adjust the axes on the graph)
■ This is because the components (i.e. the groups of variables that make them) are just mathematical outputs and directly linked to
psychological phenomena in their raw form. This is due the axes being indiscriminate
What are the 2 forms of Rotation?
- Orthogonal rotation methods assume that the factors in the analysis are uncorrelated (i.e., no association) (Gorsuch 1983), most commonly used is the varimax rotation.
- Oblique rotation methods assume that the factors are correlated, most commonly used is oblimin rotation (we try to avoid rotating it into right angles).
■ This is theoretically driven; you choose the rotation according to whether evidence suggests your components/factors will be correlated or not.
What table would you look at with Oblimin rotation?
At the point in the annotated output where we look at the rotated component matrix (for orthagonal rotations) you would look at the table entitled “Pattern Matrix” if an Oblimin rotation was used (and interpret it in the same way).
What are the assumptions of PCA?
■ Variables should not be nominal (this requires a CATPCA), given what I said about
factor loadings being Pearson’s correlations this should be somewhat obvious!
Sampling adequacy (when we have enough data to conduct PCA):
– Kaiser-Meyer-Olkin measure of sampling adequacy should be above .5 to be
acceptable (if less collect more data).
– 0.5 - 0.7 = acceptable
– 0.7- 0.8 = good
– 0.8- 0.9 = great
– 0.9+ = Superb
– (Hutcheson and Sofroniou 1999)
■ Sufficient correlations between individual variables to run a PCA (We make sure that the R matrix is not an identity matrix).
■ Bartlett’s test of sphericity: This tests the null hypothesis that the correlations represent an identity matrix, so we want this to be significant (p<.05).
What is meant by an identity matrix?
Nothing is correlated with each other apart from the question with itself e.g., “what is your favourite colour?” x “what is your favourite food?” would have a correlation of 0 WHEREAS “what is your favourite colour?” x “what is your favourite colour?” would have a correlation of 1.
