What is dimensionality and what is factor analysis? - Chapter 4

When we measure a physical or psychological trait of an object or person, we only measure one trait of object or person. However, you can investigate multiple questions/items that ultimately lead to a certain dimension/trait. These are called composite scores.

In this chapter, the concept of dimensionality wil be discussed. This is done on the basis of three fundamental questions, and associated relevant information from an exploratory (explanatory) factor analysis (EFA):

1. How many dimensions does a test have?
   
   1. One-dimensional
   2. Two (+) dimensional
   3. Relevant information from EFA: eigenvalues, scree plot, factor loads etc.
2. Are the dimensions correlated?
   
   1. Yes: type of scale = multidimensional with beaded dimensions
   2. No: type of scale = multidimensional with uncorrelated dimensions
   3. Relevant information from EFA: rotation method, interfactor correlations
3. What is the psychological service of the dimensions?
   
   1. Factor analysis
   2. Relevant information from EFA: factor loadings.

What is the dimensionality of a test?

Unidimensional

When a psychological test contains items that reflect a single trait of a person and the reactions are not influenced by other traits of that person, this means that the test is unidimensional. The concept of conceptual homogeneity means that all responses to the items / questions are influenced by one and the same psychological trait.

If a psychological test contains items that reflect more than one trait of a person, the test can be subdivided into dimensions (multidimensional). These dimensions are multidimensional with correlating dimensions or multidimensional without correlating dimensions.

Multi-dimensional with correlating dimensions

A test that is multi-dimensional with correlating dimensions is also called test with higher-order factors. This means that there is one higher (general) factor that merges all subtests. Subtests are groups of questions that identify different psychological characteristics. These subtests correlate with each other to a larger whole.

Subtests are specific factors that are in themselves one-dimensional and the questions within the subtest are conceptually homogeneous.

Full scale score is a combination of subscores to a general trait, this is called the higher-order factor.

• Unidimensional: a single score of a single psychological characteristic.
• Multidimensional: added subscores.

Multi-dimensional without correlating dimensions

With this type of test, the sub-tests do not correlate with each other and therefore the sub-scores cannot be added up and combined into a larger whole (higher-order factor).

What is factor analysis?

Factor analysis is the most commonly used statistical procedure to measure and test dimensionality. There are two types of factor analysis: explorative factor analysis (EFA) and confirmatory factor analysis (CFA). EFA is the type that is used most often.

Exploratory factor analysis

Suppose you have a test with six items and you want to know how many dimensions are being measured. To measure this, you take the test with, for example, a hundred people, this data is entered into a computer program and then correlations are calculated. This helps to identify and interpret the number of underlying dimensions. Each set of items that correlate relatively high with each other represents a psychological dimension, also called a factor.

If all items of a test correlate to each other to about the same degree, there is only one set (factor) and then the scale is one-dimensional. If there are two or more sets (factors), the scale is multidimensional.

If the items in set one correlate with the items in set two, we can speak of correlated factors and therefore of a multidimensional test with correlated dimensions. If the items from one set do not correlate with those from the other set, the factors are not correlated and we speak of a multidimensional test without correlated dimensions.

Viewing and reviewing all this data is almost impossible if a test contains many items, so EFA is usually used.

Implementing and interpreting an EFA

Step 1: First choose the statistical technique that you will use. The most commonly used techniques are principal axis factoring (PAF) and principal components analysis (PCA).

Step 2: Identify the number of factors (dimensions). There is no simple rule for this, you must use guidelines and subjective assessment. This often uses eigenvalues ​​(eigenvalues) that you can view in three ways. Based on examples, I provide a clear picture of what the terms mean:

• Eigenvalues: there is a big difference between eigenvalue two and three. This term says that there are two dimensions within this test. Just like if there were a big difference between eigenvalue four and five, this would mean that there are four dimensions within this test.
   
• Eigenvalue greater than one rule: the rule applies that the number of eigenvalues ​​that are greater than one determines the number of dimensions. In other words, there are three eigenvalues ​​that have values ​​above one. This means that there are three dimensions within the test.
   
• Screeplot: this term is a graphical representation of the eigenvalues ​​within the test. You can see from the graph that the line flattens from eigenvalue three. A clear smoothing point suggests that the number of factors is one less than the factor number of the smoothing point. From this you can therefore also conclude that there are two dimensions.

Step 3: if the evidence indicates that a scale is multidimensional then we use factor rotation to see if there is a correlation between the dimensions. There are two types of rotations:

• Orthogonal rotation: produces dimensions that are not related to each other.
• Skew (oblique) rotation: produces dimensions that may be interrelated.

The idea of ​​a factor rotation is sometimes viewed with restraint or skepticism. How can it be legitimate to change the results so that it "clarifies the psychological significance of factors", while preferring a simple structure? However, this restraint is not necessary. In short, a factor rotation does not change the relative location of the items. The purpose of factor analysis is to find a perspective that summarizes the items and describes their mutual relationship. For any given set of correlations between items countless perspectives are possible that are legitimate and statistically valid. Factor rotation is a tool to find a perspective that is clear and simple.

Step 4: After the cohesion between dimensions has been established by means of factor rotation, the meaning of the dimensions can be determined. This is done by means of factor loads. Factor loads are a link between items and factors (dimensions). Which test items are most strongly linked to a dimension is the question that is answered. The stronger they are linked, the clearer the meaning of the dimension is. It is of course better if test items are strongly linked to only one dimension and not to multiple dimensions, because this makes the meaning complicated. In addition, there can be a positive or negative charge. A positive charge indicates that people who score high on the item also score high on the underlying factor. A negative charge indicates that people who score high on the item score low on the underlying factor.

Simple structure: if items are strongly linked with only one factor.

When you use an oblique rotation you also have to look at the correlations between the factors.

Confirmative factor analysis

EFA is used in situations where little is known about the dimensionality of a test. CFA is used when there are already clear ideas about the dimensionality of a test. For example if you have a test with fourteen items that is designed so that seven questions belong to one dimension and seven questions belong to a second dimension. Then you can use CFA to test whether this is also true.

Chapter 12 deals with CFA in more detail.

When we measure a physical or psychological trait of an object or person, we only measure one trait of object or person. However, you can investigate multiple questions/items that ultimately lead to a certain dimension/trait. These are called composite scores.

In this chapter, the concept of dimensionality wil be discussed. This is done on the basis of three fundamental questions, and associated relevant information from an exploratory (explanatory) factor analysis (EFA):

1. How many dimensions does a test have?
   
   1. One-dimensional
   2. Two (+) dimensional
   3. Relevant information from EFA: eigenvalues, scree plot, factor loads etc.
2. Are the dimensions correlated?
   
   1. Yes: type of scale = multidimensional with beaded dimensions
   2. No: type of scale = multidimensional with uncorrelated dimensions
   3. Relevant information from EFA: rotation method, interfactor correlations
3. What is the psychological service of the dimensions?
   
   1. Factor analysis
   2. Relevant information from EFA: factor loadings.