This is a clear and comprehensive summary of factor analysis, which is half of the theory for the test of the WUR course 'Research Methods & Data Analysis' (YRM30806).
Good luck with studying!
Data reduction of a questionnaire through factor analysis
Steps guide
“What is an appropriate data reduction for this set of items/questions from a questionnaire?”
0. Assumptions before doing FA
o Correlations are sufficiently big (Bartlett’s test ‘p > 0.05’).
o Correlations are not too big, a.k.a. there is no multicollinearity (determinant / det >
0.00001 | correlations in correlation matrix < 0.8).
o Sample size is sufficient (KMO / overall MSA > 0.5).
When the assumptions are checked:
1. Decide how many factors should be maintained
a. Scree plot → maintain the number of factors on the left of the inflection point.
b. Kaisers criterium → maintain factors that have an eigenvalue above 1.0 (look in Scree
plot, or if not provided, look for an ev-table or SS loadings table to see eigenvalues).
c. Correlation cluster → look for groups/clusters of items that have high correlations
with each other, and consider maintaining those
2. Select a few factor solutions that you derived from the different decision criteria
3. Decide on what the most logical item clusters are
a. Set a criterium score (commonly 0.4) for factor loadings to be considered significant
(if an item has zero significant factor loadings, delete that item)
b. Assign each item to a factor based on for which factor the variable has the
highest factor loading.
c. If items a/d/g have highest factor loading for same factor (e.g. F5), they are a logical
cluster. You can also use the correlation matrix (if provided) to check for clusters.
4. Verify with the question content whether the items have a common theme
5. Decide which factor solution is best
a. Make clear what your criteria are.
i. The factor solution with the highest cumulative variance gets picked.
ii. If the factor solutions cumulative variances are (almost) equal (let’s say a
difference of 0.2 or less) → pick the solution with the least factors, because a
solution with fewer factors is simpler and might be easier to interpret.
iii. If the factor solutions have an even number of factors, pick the factor solution
in which the most items have a clear highest factor loading.
6. Label the clusters of items
a. Example: if you have a factor with high loadings on questions about job satisfaction,
motivation, and engagement, you can label that cluster ‘employee wellbeing’.
7. Examine the reliability through Cronbach’s alpha
a. “Are the decisions that respondents make in the survey consistent?”
b. Measures for reliability: raw alpha > 0.7 or 0.8 | r.drop > 0.3.
, Details and explanation
Introduction:
Questionnaires/surveys are a measurement instrument. Questionnaires work with scales.
• Scale = a set of observable variables / items (the survey questions) that collectively measure a
latent/unobservable variable / factor / component. It is not the same as a simple answer/response scale
which refers to responses provided for a single question in a survey.
Factor analysis / FA (a group of statistical methods) can be used for the construction and assessment of scales in
questionnaires. Purposes:
• Data reduction by trying to find whether items have common themes | identifying, constructing, and
assessing scales | solving multicollinearity issues.
Exploratory FA knows two models :
1. Principal Component Anlysis / PCA: aim =
identifying as many observed variables as possible to
explain the factor.
2. Common Factor Analysis: aim = identifying as few
latent variables (factors) as possible to explain the
common variance among the observable variables.
There is difference in the order of causality → (if factor = wellbeing) PCA says ‘the items predict your score for
the factor (wellbeing)’, whereas Common FA says ‘wellbeing predicts your score on the items’.
Terminology:
- Factor loading: correlation between variable/item V1 and factor F1 (each item has a
loading on all present factors).
- Communality: all squared factor loadings of an item summed together.
- Eigenvalue: all squared factor loadings of a factor summed together.
The basics:
Communality:
- Each variable has a variance (correlation with itself) of 1 (see correlation plot), of which a part is
common and a part is unique/random. The proportion of common variance is called communality.
- PS: the total variance is the total number of items, or ‘k’. If you have a questionnaire with 16 questions,
the total variance is 16. K or 16 is the maximum number of factors you can have, so max 1 factor per
variable. PS: if after data reduction your model has 8 variables, the tot. var. stays 16.
Data reduction:
- Data reduction means representing the data with a smaller number of variables that retain most of the
variability (so by merging them into factors or ‘principal components’).
- Dilemma → having a high number of factors (since that means a higher
communality), or maintaining the most important factors only by
performing data reduction. Decision criteria for this dilemma are the
following:
o Scree plot; decide on the point of inflection using your eyeball,
and retain/extract the number of factors to the left of the inflection
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