Advanced
Research
Methods
Quantitative - Book: Multivariate Data Analysis – Hair – 21-12-2020
Radboud University – Master Business Administration
v
,Multivariate Data analysis
Multivariate analysis = All statistical techniques that simultaneously analyse multiple measurements
on individuals or objects under investigations: more than two variables
A structured approach to multivariate model building
To aid the researcher, a six-step approach to multivariate analysis is presented. This is should not be
regarded as a set of procedures, but more like a guideline that emphasizes a model-building
approach.
1. Define the research problem, objectives and multivariate technique to be used
The researcher must first view the problem in conceptual terms by defining the concepts and
identifying the relationships to be investigated. With the objective and conceptual model
specified, the researcher must choose the appropriate technique, based on measurement
characteristics of the dependent and independent variables.
2. Develop the analysis plan
Attention turns to the implementation issues. General considerations like minimum or
desired sample sizes and allowable or required types of variables and estimation methods.
3. Evaluate the assumptions underlying the multivariate technique
The statistical and conceptual underlying assumptions of the multivariate model affect their
ability to represent relationships. For example: normality, linearity, independence of the
error terms and equality of variances. Each technique has a series of assumptions that need
to be met.
4. Estimate the multivariate model and assess overall model fit
When assumptions are met, the actual estimation of the model follows. After that, the
overall model fit is evaluated to see whether it achieves acceptable levels on statistical
criteria, identifies proposed relationships and achieves practical significance.
5. Interpret the variate(s)
The actual interpretation of effects. The objective is to identify empirical evidence of
multivariate relationships in the sample data that can be generalized to the total population.
6. Validate the multivariate model
Before accepting the results, the model must be validated as insurance that the results are
the most descriptive of the data, yet generalizable.
,Factor analyse
The purpose of factor analysis is to estimate a model which explains variance/covariance
between a set of observed variables (in a population) by a set of (fewer) unobserved factors
& weightings
Factor analysis a interdependence technique
(you look into how the different items
interrelate with each other)
You use it for two reasons; to summarize
data and look into underlying dimensions or
to reduce data
Factor analysis is anything where you would
like to assess higher-order dimensions
Two types of factor analysis
o Exploratory factor analysis -> useful
in searching for structure among a set of variables or as data reduction. The analysis
is led by the data.
o Confirmatory factor analysis -> useful when the researcher has preconceived
thoughts on the actual structure based on theoretical support or prior research.
Two types of extraction methods:
o Principal component analysis – Factors are estimated on the whole variance ->
factor are called principal components
o Common factor analysis – Factors are estimated based on only the common
variance -> Communalities are
inserted in correlation matrix
Common factor analysis equation;
Factor loadings are significant around 0.5,
but are desirable above 0.7.
All components or factors need to have an
eigenvalue above 1
The cumulative variance must be above
60%, but less important than eigenvalue.
We do factor rotation because we want to achieve that each factor has some variables that
load on it, but usually not all, since we then can not reduce or summarize the data. Decision
for the kind of rotation should be based on theory.
o Orthogonal - Assumes that factors are not correlated (Varimax)
o Oblique – Assumes that factors are correlated (Oblimin)
Data reduction methods;
To look at the internal consistency of the factors that are produced you can use the reliability
analysis. This uses the Cronbach’s alpha. Ideal is around .80, under .60 is not accepted.
,Interpreting Exploratory Factor analysis
Stage 1 Objectives of factor analysis
The purpose of factor analysis is to summarize data or reduce the data. There are four issues;
1. Specifying the unit of analysis – can be variables or respondents which you want to
summarize or reduce.
a. R factor analysis -> analyses set of variables into factors
b. Q factor analysis -> analyses respondents into smaller groups
2. Achieving data summarization and/or data reduction
a. Data summarization -> deriving underlying dimensions that describe the data in a
smaller number of concepts.
b. Data reduction -> simplify the following analysis by reducing the items for a factor.
3. Variable selection – Researcher should always think about the conceptual theory when
selecting or deleting items in the factor analysis.
4. Using factor analysis results with other multivariate techniques – Factor analysis makes clear
for the researcher which and how much variables can be expected to have an impact in other
analysis. Forms the basis for creating new variables
Stage 2 Designing and exploratory factor analysis
There are three basic decisions in the design of a factor analysis:
1. Calculation of the input data to meet the specified objectives of grouping variables or
respondents.
2. Design the study in terms of number of variables, measurement properties of variables and
the types of allowable variables
3. The sample size necessary, both in absolute terms and as a function of the number of
variables in the analysis.
Factor analysis is performed only with metric variables. If there are categoric variables they must be
dummified, but only a small set of dummies can be a part of a factor analysis.
Step 1 Mention the number of respondents who participate in the study or the number of variables.
Step 2 name all the variables and their measurement levels -> they must be metric
Step 3 Mention the univariate descriptive statistics like the mode, mean or standard deviation
Step 4 Look if the data is normal distributed, if this is not the case transform the variables
a. You can measure this by looking at the kurtosis and the skewness. Skewness =
scheefheid & Kurtosis = steilheid
b. De kurtosis and skewness lay inside 3 times the standard error.
c. You can calculate this by dividing the skewness by the standard error of the
skewness. This is the same for the kurtosis. Example -.652/212= 3.08
d. If you have multiple variables in which the 3 is exceeded, than you look at the worst
variable.
e. You could then transform by; Inverse, logarithmic, square and square root. If this
does not make it better you can continue with the old table.
,Stage 3 Assumptions in exploratory factor analysis
The most important assumptions for factor analysis is the normal distribution which we looked into
in stage 2 and the KMO and Bartlett test.
Every time you have to do a new factor analysis, because you have deleted an item or are gonna
delete an item is to look if the factor analysis is still suitable to be done. This can be looked into with
the KMO-test and the Bartlett’s test of sphericity.
- KMO-test -> the value must be bigger than .50 to be significant
- Bartlett’s test -> significance (p) must be lower than alpha (.05)
Stage 4 Deriving factors and assessing overall fit
You have to chose between two methods for extraction;
- Common Factor analysis; This method considers only the common or shared variance,
assuming that both the unique and error variance are not of interest in defining the structure
of the variables. This one uses communalities -> Most appropriate when the objective is to
identify underlying dimensions.
- Principal component analysis; This method considers the total variance and derives factors
that contain small portions of unique variance and error variance. This one uses unities ->
Most appropriate when data reduction is the goal.
We look into how much factors we want to extract in a few ways; the eigenvalue, Screeplot and the
variance.
a. Eigenvalue must be above 1 -> most important to have
b. Screeplot -> everything before the kink are the factors that need to be extracted
c. Variance -> Preferably above 60%
d. You can also chose for a determined set of factors based on previous research
, Stage 5 Interpreting the factors
Factor rotation
Factor rotation is used to achieve simpler and theoretically more meaningful factor solutions. It
improves the interpretation by reducing ambiguities. There are two methods of rotation;
Orthogonal rotation -> The axes are maintained at 90 degrees. This one is mostly used and
you will use it when the research goal is data reduction to either a smaller number of
variables or a set of uncorrelated measures for subsequent use in other multivariate
techniques.
Oblique rotation -> Axes can vary, wants to cluster the groups more obviously. This one is
best suited to the goal of obtaining several theoretically meaningful factors or constructs.
Orthogonal Oblique
You have a few rules, besides your own choice of a certain rotation
You can rotate Oblique (coherent factors) if one of the correlations in the correlation matrix
is bigger than |.30| -> You look at the pattern matrix
You can rotate Orthogonal (independent factors) if all correlations are smaller than |.30|->
You look at the Rotated factor matrix
Communalities
Communalities are representing the amount of variance accounted for by the factor solution for each
variable. Communalities should be above .50, but under .20 is unacceptable. You always look at the
extraction column.
This one is lower
than .20
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