Practice project: EFA interpretation

You conducted a survey with 10 items on job satisfaction. An exploratory factor analysis (EFA) was then conducted in R.

First (just to be on the safe side), a few definitions:

  • Uniqueness is the proportion of an item’s variance that is not explained by the extracted factors.
  • Loadings (factor loadings) indicate how strongly an item is related to a factor. They represent the correlation between the item and the factor. When interpreting them, both the loading itself is relevant (< 0.3 indicates a low loading, > 0.7 a very high loading) and the potential problem of multiple loadings (cross-loadings on several factors).
  • SS loadings (Sum of Squared Loadings): This is the sum of the squared loadings for each factor. It indicates how much of the total variance in the dataset a factor explains.
  • Proportion Var: Shows the proportion of the total variance explained by this one factor.
  • Cumulative Var: Cumulative explained variance up to and including this factor. It shows how much variance is explained by several factors together.

The following R output shows the most important results:

Call: factanal(x = datensatz, factors = 2, rotation = "varimax")

Uniquenesses:
Item1  0.18
Item2  0.20
Item3  0.72
Item4  0.23
Item5  0.21
Item6  0.70
Item7  0.25
Item8  0.28
Item9  0.80
Item10 0.27

Loadings:
        Factor1 Factor2
Item1     0.78   0.22
Item2     0.81   0.18
Item3     0.33   0.05
Item4     0.70   0.25
Item5     0.68   0.19
Item6     0.27   0.58
Item7     0.60   0.39
Item8     0.45   0.55
Item9     0.22   0.28
Item10    0.50   0.60

                 Factor1 Factor2
SS loadings        2.98    1.41
Proportion Var     0.30    0.14
Cumulative Var     0.30    0.44

Test of the hypothesis that 2 factors are sufficient.
The chi square statistic is 35.2 on 26 degrees of freedom.
The p-value is 0.10

✏️ Task:

  1. Name and describe the two extracted factors.
    • Focus on the items with high loadings.
    • What content dimensions could these factors represent?
  2. Which items should be excluded from the interpretation?
    • Justify your answer using the uniqueness values and/or weak or cross-loadings.
  3. How well do the two factors fit the dataset?
    • Address the variance explained and the model fit (chi-square test).
  4. What does the ‘varimax’ rotation mean, and how does it affect the interpretation?

Suggested solution

1. Name and describe the two extracted factors.

  • Factor 1: Items with high loadings on Factor 1 are Item1 (0.78), Item2 (0.81), Item4 (0.70), Item5 (0.68), Item7 (0.60).
    • → These items could, for example, relate to satisfaction with tasks and colleagues (depending on the content of the items).
  • Factor 2: High loadings on Factor 2 are found for Item6 (0.58), Item8 (0.55), Item10 (0.60).
    • → These items could relate to satisfaction with leadership/organization.
  • Note: Item7 and Item10 load on both factors (cross-loading) → potentially difficult to interpret.

2. Which items should be excluded from the interpretation?

  • Item3:
    • Low loadings on both factors (0.33 / 0.05).
    • High uniqueness value (0.72) → little of the item can be explained by the common factors.
    • → Recommendation: Exclude from the interpretation.
  • Item9:
    • Very low loadings (0.22 / 0.28), uniqueness = 0.80.
    • → Recommendation: Exclude.
  • Items with cross-loadings:
    • Item7 (0.60 / 0.39), Item10 (0.50 / 0.60): Both load on both factors → they cannot be clearly assigned.
    • → Recommendation: Discuss critically, possibly exclude, or assign deliberately.

3. How well do the two factors fit the dataset?

  • Variance explained:
    • Factor 1: 30%, Factor 2: 14%, 44% in total → moderate, but acceptable in the psychological context.
  • Chi-square test:
    • χ² = 35.2, df = 26, p = 0.10 → not significant
    • → The model fits the dataset adequately (no indication of poor model fit).

4. Meaning and impact of the rotation (‘varimax’)

Factors remain independent of one another (orthogonal).

Varimax rotation:

Orthogonal rotation that seeks to make the loadings on the factors as “simple” as possible.

Goal: Each variable should load as strongly as possible on one factor and weakly on the others.

It facilitates interpretation by producing clearer assignments.

Further:

  • SS loadings (Sum of Squared Loadings): Factor 1: SS loading = 2.98 → Factor 1 explains as much variance as 2.98 items. Since there are 10 items in total, this corresponds to 2.98 / 10 = 29.8 % of the total variance.
  • Proportion Var: Factor 1: 0.30 = 30 % of the variance; Factor 2: 0.14 = 14 % of the variance
  • Cumulative Var: After Factor 1: 30 %; after Factor 2: 44 % cumulative → (30 % + 14 %)