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1.1 Factor AnalysisPurpose factor analysis – estimate a model which explains variance/covariance ⃰ between a set of
observed variable (in a population) be a set of (fewer) unobserved factors and weightings.
⃰ variantie – maat die iets zegt over de spreiding in een dataset
⃰ covariantie – maat voor spreiding van twee variabelen die weergeeft in welke mate twee
variabelen met elkaar samenhangen
First of all, you have observed variables, which means you collected data, you want to understand the
variance/covariance between the set of observed variables how do they interrelate with each othter?
Survey Example: first collect data by a sample from population.
e.g. 5 point Likert perceived fairness of grading and how satisfied students are.
Gra 1: The grading is fair. The collection of data (left figure) has been plugged in a data
Gra 2: I deserve the grades I received. set (see table below).
Gra 3: The grade is reasonable. In the rows respondent (ID).
Sat 1: I am content. In the columns all the observations from the different
Sat 2: I am satisfied. items, so for each respondent how he/she perceived a certain
Sat 3: I am happy. grading of satisfaction (Gra/Sat). This makes up the data set.
Now you are interested in how these six items relate to each
other. In this case, you would like to know how Gra1, Gra2 and Gra3 together form the perception of
fair grading. Sat1, Sat2 and Sat3 would then together form the perception of satisfaction.
fair grading satisfaction
ID Gra 1 Gra 2 Gra 3 Sat 1 Sat 2 Sat 3
243 3 4 5 5 4 5
244 2 3 2 3 2 3
245 4 3 4 3 3 4
You do by means of factor analysis.
Factor analysis interdependence technique – how do the different items interrelate with each
other? Not yet interested in prediction. Define structure among observed variables in the data set and
find out how they relate to each other. = interrelationships among large number of variables to
identify underlying dimension. These underlying dimensions are called factors.
Two purposes: data summarization and data reduction.
Measurement model
Construct denoted by the term Xi (ξ);
Underlying items X1, X2 and X3.
These together: e.g. construct Xi.
Besides that, interested in measurement error: are there any
systematic biases that really might influence how we
measured these items? this type of measurement error can
be assessed with factor analysis. This together is entire
measurement model.
Why do we do multi-item measurement at all?
Increase reliability and validity of measures
Allow measurement assessment
, o Measurement error
o Reliability
o Validity
Two forms of measurement models:
Formative (emerging) – there are more items and they together emerge as the construct.
Reflective (latent) – there is a construct and it the items really reflect this kind of construct.
we will see this one in most of the research we conduct.
Because they are nicely clustered together, which means they are reliable, but they
really are not on target, which means they are not valid.
The black points are nicely on target, more or less, but they really spread out, they
don’t cluster nicely together.
The black points spread out and are not on target.
What you want to achieve with your measures is to be both valid and reliable.
Reflective measurement models:
Are used direction of causality from the construct to the measure;
They are correlated indicators, meaning: items correlate with each other and these correlations
together are used in the factor analysis to explain the dimensions;
It also takes measurement error into account at the item level;
Validity of items is usually tested with factor analysis. We see the Xi, the construct, and then
we want to assess the factor loadings to each item, which is denoted by
lambda (λ) and we are also interested in the measurement error epsilon
(ε). And together we have this equation of X1 equals lambda1 (the
factorloading) times the Xi (the construct) + the measurement error. We
have that for X1, X2, X3 and so on.
Applications – assess the validity of construct measurements thesis! If you do
quantitative research you will have to do factor analysis when it comes to
validity of the measures. It is also widely used in practice, so market
segmentation / product research / price management.
… anything where you would like to assess higher-order dimensions.
1.2 Conducting a Factor Analysis
, Process factor analysis = problem formulation constructing correlation matrix selecting
extraction method determining number of factors rotating factors interpreting factors
using factors in other analyses determining model fit.
Problem formulation, important to:
Identify objectives: data summarization or data reduction?
Determine variables that are going to be measured. criteria:
o Based on past research, theory and judgement of the researcher;
o Measurement properties (ratio, interval) because factor analysis is a metric method.
o Sample size – 4-5 per number of respondents per variable. Too low = not enough
power, analysis won’t work.
Conducting a factor analysis distinguish between exploratory and confirmatory factor analysis.
Exploratory factor analysis – exploration of data. interested in finding an underlying structure of
higher-order dimensions. There are assumptions that factors cause correlations between variables, but
no insights on what these factors could be. It is mainly used to reveal interrelationships, which you do
not know yet. Main purpose = generation of hypotheses.
Confirmatory factor analysis – researchers have a priori ideas of underlying factors, derived from
theory. Therefore, the relationships between variables and factors are kind of assumed before
conducting the factor analysis. There are expectations. Main purpose = testing of hypotheses.
Example: research among consumers on their perceptions on toothpaste. respondent numbers and
six variables are visible. These six variables could be items/questions to ask to the consumers. How do
these items relate with each other and whether they provide higher-order dimensions.
Construct the correlation matrix. Why factor analysis is an
analytical process, based on the matrix of correlations between
Example
V1 It is important to buy a toothpaste that prevents cavities.
V2 I like toothpaste that gives shiny teeth.
V3 A toothpaste should strengthen your gums.
V4 I prefer a toothpaste that freshens breath.
V5 Prevention of tooth decay should be an important benefit
offered by a toothpaste.
V6 The most important consideration in buying a toothpaste is variables. See table: six variables on
attractive teeth. both the vertical as well as the
horizontal axes. It is visible how Correlation Matrix
the intercorrelation of each item Correlation V1
V1
1,000
V2
-,053
V3
,873
V4
-,086
V5
-,858
V6
,004
with each item is. V2
V3
-,053
,873
1,000
-,155
-,155
1,000
,572
-,248
,020
-,778
,640
-,018
V4 -,086 ,572 -,248 1,000 -,007 ,640
V5 -,858 ,020 -,778 -,007 1,000 -,136
Does this correlation matrix V6 ,004 ,640 -,018 ,640 -,136 1,000
helps in determining whether
factor analysis can be used at all?
Kaiser-Meyer-Olkin (KMO) measure of sampling adequacy – tells whether the sample
adequately represents the population. Should be at least above .5. the closer to 1, the better.
Bartlett’s test of sphericity – tests the null hypothesis that the variables are uncorrelated in
the population. If this null hypothesis needs to be accepted it means you have no correlations
in the population and then you wouldn’t not be able to do a factor analysis. So usually you
want to reject the null hypothesis and you
want to be sure that there are enough KMO and Bartlett's Test
Kaiser-Meyer-Olkin Measure of Sampling
correlations within the population. Sig. level Adequacy. ,660
should be smaller than .05 (typical value α). Bartlett's Test of Approx. Chi-Square 111,314
Sphericity df 15
Sig. ,000
1.3 Selecting an Extraction Method
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