Deriving a Scale (Principle Component Analysis)

Question 1

Define:

Principle Components Analysis

Answer 1

A tool that helps identify which questions form a cohesive group versus those that give unique information.

(i.e. a tool to help organise different items measuring the same construct into one scale).

This is done by plotting them in a 3D space with the axes representing certain ‘dimensions’ (i.e. the ‘Principle Components’).

The tool ‘rotates’ the data-set until the ‘best summary’ is found, and then groups the items into these overall ‘Principle Components’.

Question 2

List the FOUR main steps in developing a scale using PCA:

(PCA = Principle Components Analysis)

Answer 2

1. Create Questions/Items.
2. Apply Principle Components Analysis.
3. Check Assumptions.
4. Interpret & Identify Components.

Question 3

Give THREE tips for developing questions to be analysed using PCA:

(PCA = Principle Components Analysis)

Answer 3

• Generate a large number of (high-quality) questions.
• Create questions that are appropriate for the aim, sample, & operationalisation(s).
• Balance (reverse-)wording and valence when constructing questions.

Question 4

What is the name of the equivalent/similar tool to PCA that is specific to the field of Psychology?

(PCA = Principle Components Analysis)

Answer 4

‘Exploratory Factor Analysis’

Note: Although EFA is NOT the exact same as PCA, both are ‘variable reduction techniques’.

Question 5

What are the TWO main forms of ‘cheating’ when using PCA?

Note: ‘Cheating’ may be unintentional too!

Answer 5

Inputting…

1. many unrelated items.
2. related pairs of items that don’t connect to any others.

1. The PCA software will attempt to group regardless of ‘logical relatedness’ and will give results simply based on what was put in (“garbage in, garbage out”).
2. This is essentially putting in your own ‘predetermined groups’ in order to verify your own thoughts/beliefs.

Question 6

List:

The TWO key assumptions when applying PCA to your dataset, and the measures/tests for each to ensure they have been met.

Extra Knowledge Test: Define/describe each assumption and its test/measure.

Answer 6

1. Sphericity + Bartlett Test
2. Sampling Adequacy + Kaiser-Meyer-Olkin (KMO) Measure

1. Sphericity concerns whether there is equal enough variance (in your items/between groups etc.) to form multiple/different scales.

A significant value (p-value <.05) needs to be obtained in a Bartlett Test to indicate the above assumption has been met.

2. Sampling Adequacy and the KMO Measure relate to whether the items have unique variance (i.e. are not just pairs of concepts).

The value of the KMO Measure indicates the proportion of the variance in your variables/items that may be caused by underlying factors - the ‘cut-off’ value for this is >.5 , and the closer to 1.0 the better.

Question 7

What TWO strategies are used for interpretting components?

(In order to determine potential scales)

Answer 7

1. Analysing statistical indicators in a Scree plot.
2. Applying our theoretical knowledge to the components.

When applying theoretical knowledge, it is important to asses whether or not the grouping outcomes appear to retain construct and internal validity, as well as whether or not these groupings differ to the ones you theorised.

Question 8

Fill-in-the-Blanks:

A Scree plot helps determine the ____ of ____ that are statistically significant and best summarise the items inputted with PCA.

(Identify the missing labels for the axes and key)

Answer 8

A Scree plot helps determine the number of components that are statistically significant and best summarise the items inputted with PCA.

Question 9

Fill-in-the-Blanks:

In a Scree plot…

The ____ essentially represents the amount of variance/variation in the data that can be explained by each principle component.

This value is represented on the ____-axis.

(Identify the missing labels for the axes and key)

Answer 9

1. eigenvalue
2. y

The eigenvalue of the first few PCs (i.e. PC1, PC2, PC3, etc.) is usually highest, and thus we tend to only focus on the ones before the plateu.

Question 10

Fill-in-the-Blanks:

To determine the minimum number of components that best summarise the items, look for the point ____ the line drops to its plateu, and the final point ____ the simulations line.

(Identify the missing labels for the axes and key)

Answer 10

1. before
2. above

In the image example, both of these indicators suggest the first two principle components best summarise the items inputted to our PCA.

Question 11

How do you interpret item loadings and what do they tell you about the relationship between items and components?

(This is an example of a table representing item loadings in PCA).

Answer 11

Item loadings tell us how much of an item’s variation is explained by a particular component.

Therefore, the higher the item loading value for a certain item + component, the stronger the correlation between them.

Question 12

List:

The SIX steps for reporting a PCA

Answer 12

1. Examine the table of loadings.
2. If there are cross-loaded or non-loaded items, consider restarting.
3. Convert loadings table into APA format with item labels.
4. Identify common themes within components.
5. Average items together to form a scale.
6. Report the descriptive statistics/reliability of the newly formed scale.